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Landau Lifshitz The Classical Theory of Fields Third Revised English Edition Course of Theoretical Physics Volume 2 CD CD o CO J.. fl* CD — E CO L. D. Landau (Deceased) and E. ML Lifshitz re Institute of Physical Problems USSR Academy of Sciences gamon Pergamon Press Course of Theoretical Physics Volume 2 THE CLASSICAL THEORY OF FIELDS Third Revised English Edition L, D. LANDAU (Deceased) and E, M. LIFSHITZ Institute of Physical Problems, USSR Academy of Sciences This third English edition of the book has been translated from the fifth revised and extended Russian edition published in 1967. Although much new material has been added, the subject matter is basically that of the second English translation, being a systematic presentation of electro- magnetic and gravitational fields for postgraduate courses. The largest additions are four new sections entitled "Gravitational Collapse", "Homogeneous Spaces", "Oscillating Regime of Approach to a Singular Point", and "Character of the Singularity in the General Cosmological Solution of the Gravitational Equations" These additions cover some of the main areas of research in general relativity. Mxcvn COURSE OF THEORETICAL PHYSICS Volume 2 THE CLASSICAL THEORY OF FIELDS OTHER TITLES IN THE SERIES Vol. 1. Mechanics Vol. 3. Quantum Mechanics — Non Relativistic Theory Vol. 4. Relativistic Quantum Theory Vol. 5. Statistical Physics Vol. 6. Fluid Mechanics Vol. 7. Theory of Elasticity Vol. 8. Electrodynamics of Continuous Media Vol. 9. Physical Kinetics THE CLASSICAL THEORY OF FIELDS Third Revised English Edition L. D. LANDAU AND E. M. LIFSHITZ Institute for Physical Problems, Academy of Sciences of the U.S.S.R. Translated from the Russian by MORTON HAMERMESH University of Minnesota PERGAMON PRESS OXFORD • NEW YORK • TORONTO SYDNEY ' BRAUNSCHWEIG Pergamon Press Ltd., Headington Hill Hall, Oxford Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford, New York 10523 Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto 1 Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright © 1971 Pergamon Press Ltd. All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Pergamon Press Ltd. First English edition 1951 Second English edition 1962 Third English edition 1971 Library of Congress Catalog Card No. 73-140427 Translated from the 5th revised edition of Teoriya Pola, Nauka, Moscow, 1967 Printed in Great Britain by THE WHITEFRIARS PRESS LTD., LONDON AND TONBRIDGE 08 016019 CONTENTS Preface to the Second English Edition ix Preface to the Third English Edition x Notation xi Chapter 1. The Principle of Relativity 1 1 Velocity of propagation of interaction 1 2 Intervals 3 3 Proper time 7 4 The Lorentz transformation 9 5 Transformation of velocities 12 6 Four-vectors 14 7 Four-dimensional velocity 21 Chapter 2. Relativistic Mechanics 24 8 The principle of least action 24 9 Energy and momentum 25 10 Transformation of distribution functions 29 1 1 Decay of particles 30 12 Invariant cross-section 34 13 Elastic collisions of particles 36 14 Angular momentum 40 Chapter 3. Charges in Electromagnetic Fields 43 15 Elementary particles in the theory of relativity 43 16 Four-potential of a field 44 17 Equations of motion of a charge in a field 46 18 Gauge invariance 49 19 Constant electromagnetic field 50 20 Motion in a constant uniform electric field 52 21 Motion in a constant uniform magnetic field 53 22 Motion of a charge in constant uniform electric and magnetic fields 55 23 The electromagnetic field tensor 60 24 Lorentz transformation of the field 62 25 Invariants of the field 63 Chapter 4. The Electromagnetic Field Equations 66 26 The first pair of Maxwell's equations • 66 27 The action function of the electromagnetic field 67 28 The four-dimensional current vector 69 29 The equation of continuity 71 30 The second pair of Maxwell equations 73 31 Energy density and energy flux 75 32 The energy-momentum tensor 77 33 Energy-momentum tensor of the electromagnetic field 80 34 The virial theorem 84 35 The nergy-momentum tensor for macroscopic bodies 85 VI CONTENTS Chapter 5. Constant Electromagnetic Fields 88 36 Coulomb's law 88 37 Electrostatic energy of charges 89 38 The field of a uniformly moving charge 91 39 Motion in the Coulomb field 93 40 The dipole moment 96 41 Multipole moments 97 42 System of charges in an external field 100 43 Constant magnetic field 101 44 Magnetic moments 103 45 Larmor's theorem 105 Chapter 6. Electromagnetic Waves 108 46 The wave equation 108 47 Plane waves 110 48 Monochromatic plane waves 114 49 Spectral resolution 118 50 Partially polarized light 119 51 The Fourier resolution of the electrostatic field 124 52 Characteristic vibrations of the field 125 Chapter 7. The Propagation of Light 129 53 Geometrical optics 129 54 Intensity 132 55 The angular eikonal 134 56 Narrow bundles of rays 136 57 Image formation with broad bundles of rays 141 58 The limits of geometrical optics 143 59 Diffraction 145 60 Fresnel diffraction 150 61 Fraunhofer diffraction 153 Chapter 8. The Field of Moving Charges 158 62 The retarded potentials 158 63 The Lienard-Wiechert potentials 160 64 Spectral resolution of the retarded potentials 163 65 The Lagrangian to terms of second order 165 Chapter 9. Radiation of Electromagnetic Waves 170 66 The field of a system of charges at large distances 170 67 Dipole radiation 173 68 Dipole radiation during collisions 177 69 Radiation of low frequency in collisions 179 70 Radiation in the case of Coulomb interaction 181 71 Quadrupole and magnetic dipole radiation 188 72 The field of the radiation at near distances 190 73 Radiation from a rapidly moving charge 193 74 Synchrotron radiation (magnetic bremsstrahlung) 197 75 Radiation damping 203 76 Radiation damping in the relativistic case 208 77 Spectral resolution of the radiation in the ultrarelativistic case 211 78 Scattering by free charges 215 79 Scattering of low-frequency waves 220 80 Scattering of high-frequency waves 221 CONTENTS Vii Chapter 10. Particle in a Gravitational Field 225 81 Gravitational fields in nonrelativistic mechanics 225 82 The gravitational field in relativistic mechanics 226 83 Curvilinear coordinates 229 84 Distances and time intervals 233 85 Covariant differentiation 236 86 The relation of the Christoffel symbols to the metric tensor 241 87 Motion of a particle in a gravitational field 243 88 The constant gravitational field 247 89 Rotation 253 90 The equations of electrodynamics in the presence of a gravitational field 254 Chapter 11. The Gravitational Field Equations 258 91 The curvature tensor 258 92 Properties of the curvature tensor 260 93 The action function for the gravitational field 266 94 The energy-momentum tensor 268 95 The gravitational field equations 272 96 Newton's law 278 97 The centrally symmetric gravitational field 282 98 Motion in a centrally symmetric gravitational field 287 99 The synchronous reference system 290 100 Gravitational collapse 296 101 The energy-momentum pseudotensor 304 1 02 Gravitational waves 311 103 Exact solutions of the gravitational field equations depending on one variable 314 104 Gravitational fields at large distances from bodies 318 105 Radiation of gravitational waves 323 106 The equations of motion of a system of bodies in the second approximation 325 Chapter 12. Cosmological Problems 333 107 Isotropic space 333 108 Space-time metric in the closed isotropic model 336 109 Space-time metric for the open isotropic model 340 110 The red shift 343 111 Gravitational stability of an isotropic universe 350 112 Homogeneous spaces 355 113 Oscillating regime of approach to a singular point 360 114 The character of the singularity in the general cosmological solution of the gravitational equations 367 Index 371 PREFACE TO THE SECOND ENGLISH EDITION This book is devoted to the presentation of the theory of the electromagnetic and gravitational fields. In accordance with the general plan of our "Course of Theoretical Physics", we exclude from this volume problems of the electrodynamics of continuous media, and restrict the exposition to "microscopic electrodynamics", the electrodynamics of the vacuum and of point charges. A complete, logically connected theory of the electromagnetic field includes the special theory of relativity, so the latter has been taken as the basis of the presentation. As the starting-point of the derivation of the fundamental equations we take the variational principles, which make possible the achievement of maximum generality, unity and simplicity of the presentation. The last three chapters are devoted to the presentation of the theory of gravitational fields, i.e. the general theory of relativity. The reader is not assumed to have any previous knowledge of tensor analysis, which is presented in parallel with the development of the theory. The present edition has been extensively revised from the first English edition, which appeared in 1951. We express our sincere gratitude to L. P. Gor'kov, I. E. Dzyaloshinskii and L. P. Pitaevskii for their assistance in checking formulas. Moscow, September 1961 L. D. Landau, E. M. Lifshitz PREFACE TO THE THIRD ENGLISH EDITION This third English edition of the book has been translated from the revised and extended Russian edition, published in 1967. The changes have, however, not affected the general plan or the style of presentation. An essential change is the shift to a different four-dimensional metric, which required the introduction right from the start of both contra- and covariant presentations of the four- vectors. We thus achieve uniformity of notation in the different parts of this book and also agreement with the system that is gaining at present in universal use in the physics literature. The advantages of this notation are particularly significant for further appli- cations in quantum theory. I should like here to express my sincere gratitude to all my colleagues who have made valuable comments about the text and especially to L. P. Pitaevskii, with whom I discussed many problems related to the revision of the book. For the new English edition, it was not possible to add additional material throughout the text. However, three new sections have been added at the end of the book, §§ 112-114. April, 1970 E. M. Lifshitz NOTATION Three-dimensional quantities Three-dimensional tensor indices are denoted by Greek letters Element of volume, area and length: dV, di, d\ Momentum and energy of a particle: p and $ Hamiltonian function: 2tf Scalar and vector potentials of the electromagnetic field: and A Electric and magnetic field intensities: E and H Charge and current density : p and j Electric dipole moment: d Magnetic dipole moment: m Four-dimensional quantities Four-dimensional tensor indices are denoted by Latin letters i, k, I, . . . and take on the values 0, 1, 2, 3 We use the metric with signature (H ) Rule for raising and lowering indices — see p. 14 Components of four-vectors are enumerated in the form A 1 = (A , A) Antisymmetric unit tensor of rank four is e iklm , where e 0123 = 1 (for the definition see P- 17) Radius four-vector: x* = (ct, r) Velocity four- vector: u l = dx l \ds Momentum four-vector: p = {Sic, p) Current four-vector : j* = (cp, pi) Four-potential of the electromagnetic field: A 1 = ($, A) Electromagnetic field four-tensor F ik = j± - — { (for the relation of the components of F ik to the components of E and H, see p. 77) Energy-momentum four-tensor T ik (for the definition of its components, see p. 78) CHAPTER 1 THE PRINCIPLE OF RELATIVITY § 1. Velocity of propagation of interaction For the description of processes taking place in nature, one must have a system of reference. By a system of reference we understand a system of coordinates serving to indicate the position of a particle in space, as well as clocks fixed in this system serving to indicate the time. There exist systems of reference in which a freely moving body, i.e. a moving body which is not acted upon by external forces, proceeds with constant velocity. Such reference systems are said to be inertial. If two reference systems move uniformly relative to each other, and if one of them is an inertial system, then clearly the other is also inertial (in this system too every free motion will be linear and uniform). In this way one can obtain arbitrarily many inertial systems of reference, moving uniformly relative to one another. Experiment shows that the so-called principle of relativity is valid. According to this principle all the laws of nature are identical in all inertial systems of reference. In other words, the equations expressing the laws of nature are invariant with respect to transforma- tions of coordinates and time from one inertial system to another. This means that the equation describing any law of nature, when written in terms of coordinates and time in different inertial reference systems, has one and the same form. The interaction of material particles is described in ordinary mechanics by means of a potential energy of interaction, which appears as a function of the coordinates of the inter- acting particles. It is easy to see that this manner of describing interactions contains the assumption of instantaneous propagation of interactions. For the forces exerted on each of the particles by the other particles at a particular instant of time depend, according to this description, only on the positions of the particles at this one instant. A change in the position of any of the interacting particles influences the other particles immediately. However, experiment shows that instantaneous interactions do not exist in nature. Thus a mechanics based on the assumption of instantaneous propagation of interactions contains within itself a certain inaccuracy. In actuality, if any change takes place in one of the inter- acting bodies, it will influence the other bodies only after the lapse of a certain interval of time. It is only after this time interval that processes caused by the initial change begin to take place in the second body. Dividing the distance between the two bodies by this time interval, we obtain the velocity of propagation of the interaction. We note that this velocity should, strictly speaking, be called the maximum velocity of propagation of interaction. It determines only that interval of time after which a change occurring in one body begins to manifest itself in another. It is clear that the existence of a 2 THE PRINCIPLE OF RELATIVITY § 1 maximum velocity of propagation of interactions implies, at the same time, that motions of bodies with greater velocity than this are in general impossible in nature. For if such a motion could occur, then by means of it one could realize an interaction with a velocity exceeding the maximum possible velocity of propagation of interactions. Interactions propagating from one particle to another are frequently called "signals", sent out from the first particle and "informing" the second particle of changes which the first has experienced. The velocity of propagation of interaction is then referred to as the signal velocity. From the principle of relativity it follows in particular that the velocity of propagation of interactions is the same in all inertial systems of reference. Thus the velocity of propaga- tion of interactions is a universal constant. This constant velocity (as we shall show later) is also the velocity of light in empty space. The velocity of light is usually designated by the letter c, and its numerical value is c = 2.99793 x 10 10 cm/sec. (1.1) The large value of this velocity explains the fact that in practice classical mechanics appears to be sufficiently accurate in most cases. The velocities with which we have occasion to deal are usually so small compared with the velocity of light that the assumption that the latter is infinite does not materially affect the accuracy of the results. The combination of the principle of relativity with the finiteness of the velocity of propaga- tion of interactions is called the principle of relativity of Einstein (it was formulated by Einstein in 1905) in contrast to the principle of relativity of Galileo, which was based on an infinite velocity of propagation of interactions. The mechanics based on the Einsteinian principle of relativity (we shall usually refer to it simply as the principle of relativity) is called relativistic. In the limiting case when the velocities of the moving bodies are small compared with the velocity of light we can neglect the effect on the motion of the finiteness of the velocity of propagation. Then relativistic mechanics goes over into the usual mechanics, based on the assumption of instantaneous propagation of interactions; this mechanics is called Newtonian or classical. The limiting transition from relativistic to classical mechanics can be produced formally by the transition to the limit c -*■ oo in the formulas of relativistic mechanics. In classical mechanics distance is already relative, i.e. the spatial relations between different events depend on the system of reference in which they are described. The state- ment that two nonsimultaneous events occur at one and the same point in space or, in general, at a definite distance from each other, acquires a meaning only when we indicate the system of reference which is used. On the other hand, time is absolute in classical mechanics ; in other words, the properties of time are assumed to be independent of the system of reference; there is one time for all reference frames. This means that if any two phenomena occur simultaneously for any one observer, then they occur simultaneously also for all others. In general, the interval of time between two given events must be identical for all systems of reference. It is easy to show, however, that the idea of an absolute time is in complete contradiction to the Einstein principle of relativity. For this it is sufficient to recall that in classical mechanics, based on the concept of an absolute time, a general law of combination of velocities is valid, according to which the velocity of a composite motion is simply equal to the (vector) sum of the velocities which constitute this motion. This law, being universal, should also be applicable to the propagation of interactions. From this it would follow § 2 VELOCITY OF PROPAGATION OF INTERACTION 3 that the velocity of propagation must be different in different inertial systems of reference, in contradiction to the principle of relativity. In this matter experiment completely confirms the principle of relativity. Measurements first performed by Michelson (1881) showed complete lack of dependence of the velocity of light on its direction of propagation; whereas according to classical mechanics the velocity of light should be smaller in the direction of the earth's motion than in the opposite direction. Thus the principle of relativity leads to the result that time is not absolute. Time elapses differently in different systems of reference. Consequently the statement that a definite time interval has elapsed between two given events acquires meaning only when the reference frame to which this statement applies is indicated. In particular, events which are simul- taneous in one reference frame will not be simultaneous in other frames. To clarify this, it is instructive to consider the following simple example. Let us look at two inertial reference systems K and K' with coordinate axes XYZ and X' Y'Z' respectively, where the system K' moves relative to K along the X(X') axis (Fig. 1). B— A— C -1 1 1 X' x Y Y' Fig. 1. Suppose signals start out from some point A on the X' axis in two opposite directions. Since the velocity of propagation of a signal in the K' system, as in all inertial systems, is equal (for both directions) to c, the signals will reach points B and C, equidistant from A, at one and the same time (in the K' system). But it is easy to see that the same two events (arrival of the signal at B and C) can by no means be simultaneous for an observer in the K system. In fact, the velocity of a signal relative to the A" system has, according to the principle of relativity, the same value c, and since the point B moves (relative to the K system) toward the source of its signal, while the point C moves in the direction away from the signal (sent from A to C), in the AT system the signal will reach point B earlier than point C. Thus the principle of relativity of Einstein introduces very drastic and fundamental changes in basic physical concepts. The notions of space and time derived by us from our daily experiences are only approximations linked to the fact that in daily life we happen to deal only with velocities which are very small compared with the velocity of light. § 2. Intervals In what follows we shall frequently use the concept of an event. An event is described by the place where it occurred and the time when it occurred. Thus an event occurring in a certain material particle is defined by the three coordinates of that particle and the time when the event occurs. It is frequently useful for reasons of presentation to use a fictitious four-dimensional space, on the axes of which are marked three space coordinates and the time. In this space 4 THE PRINCIPLE OF RELATIVITY § 2 events are represented by points, called world points. In this fictitious four-dimensional space there corresponds to each particle a certain line, called a world line. The points of this line determine the coordinates of the particle at all moments of time. It is easy to show that to a particle in uniform rectilinear motion there corresponds a straight world line. We now express the principle of the invariance of the velocity of light in mathematical form. For this purpose we consider two reference systems K and K' moving relative to each other with constant velocity. We choose the coordinate axes so that the axes X and X' coincide, while the Y and Z axes are parallel to Y' and Z'; we designate the time in the systems K and K' by t and t'. Let the first event consist of sending out a signal, propagating with light velocity, from a point having coordinates x t y ± z x in the K system, at time 1 1 in this system. We observe the propagation of this signal in the K system. Let the second event consist of the arrival of the signal at point x 2 y 2 z 2 at the moment of time t 2 . The signal propagates with velocity c; the distance covered by it is therefore c^ — 1 2 ). On the other hand, this same distance equals [(x 2 — x 1 ) 2 + (y 2 -y 1 ) 2 + (z 2 —z 1 ) 2 ] i . Thus we can write the following relation between the coordinates of the two events in the K system: (x 2 - Xl ) 2 + (y 2 - ytf + izi-tiY-fih-h) 2 = 0- (2-1) The same two events, i.e. the propagation of the signal, can be observed from the K' system: Let the coordinates of the first event in the K' system be xi y[ z[ t\, and of the second: x 2 y' 2 z' 2 t 2 . Since the velocity of light is the same in the K and K' systems, we have, similarly to (2.1): {A-AYHy'z-ytfHz'z-Af-c^-ttf = o. (2.2) If x x y x z t t ± and x 2 y 2 z 2 1 2 are the coordinates of any two events, then the quantity S12 = [c 2 (^-*i) 2 -(*2-*i) 2 -(y2-yi) 2 -(z2-Zi) 2 3* (2-3) is called the interval between these two events. Thus it follows from the principle of invariance of the velocity of light that if the interval between two events is zero in one coordinate system, then it is equal to zero in all other systems. If two events are infinitely close to each other, then the interval ds between them is ds 2 = c 2 dt 2 -dx 2 -dy 2 - dz 2 . (2.4) The form of expressions (2.3) and (2.4) permits us to regard the interval, from the formal point of view, as the distance between two points in a fictitious four-dimensional space (whose axes are labelled by x, y, z, and the product ct). But there is a basic difference between the rule for forming this quantity and the rule in ordinary geometry: in forming the square of the interval, the squares of the coordinate differences along the different axes are summed, not with the same sign, but rather with varying signs.f As already shown, if ds = in one inertial system, then ds' = in any other system. On the other hand, ds and ds' are infinitesimals of the same order. From these two conditions it follows that ds 2 and ds' 2 must be proportional to each other: ds 2 = ads' 2 where the coefficient a can depend only on the absolute value of the relative velocity of the t The four-dimensional geometry described by the quadratic form (2.4) was introduced by H. Minkowski, in connection with the theory of relativity. This geometry is called pseudo-euclidean, in contrast to ordinary euclidean geometry. § 2 INTERVALS 5 two inertial systems. It cannot depend on the coordinates or the time, since then different points in space and different moments in time would not be equivalent, which would be in contradiction to the homogeneity of space and time. Similarly, it cannot depend on the direction of the relative velocity, since that would contradict the isotropy of space. Let us consider three reference systems K, K X ,K 2 , and let V ± and V 2 be the velocities of systems K x and K 2 relative to K. We then have : ds 2 = a{Vi)ds\, ds 2 = a(V 2 )ds 2 2 . Similarly we can write ds\ = a(V x2 )ds\, where V 12 is the absolute value of the velocity of K 2 relative to K x . Comparing these relations with one another, we find that we must have -777\ = a(V 12 ). (2.5) But V 12 depends not only on the absolute values of the vectors V x and V 2 , but also on the angle between them. However, this angle does not appear on the left side of formula (2.5). It is therefore clear that this formula can be correct only if the function a(V) reduces to a constant, which is equal to unity according to this same formula. Thus, ds 2 = ds' 2 , and from the equality of the infinitesimal intervals there follows the equality of finite intervals: s = s'. Thus we arrive at a very important result: the interval between two events is the same in all inertial systems of reference, i.e. it is invariant under transformation from one inertial system to any other. This invariance is the mathematical expression of the constancy of the velocity of light. Again let x^y^Zxt^ and x 2 y 2 z 2 t 2 be the coordinates of two events in a certain reference system K. Does there exist a coordinate system K\ in which these two events occur at one and the same point in space ? We introduce the notation h-h = hi, (x 2 -x 1 ) 2 +(y 2 -y 1 ) 2 +(z 2 -z 1 ) 2 = \\ 2 . Then the interval between events in the K system is : 2 _ r 2,2 _;2 i 12 — C l 12 Ixi and in the K' system ~'2 _ _2,/2 j/2 s 12 — c '12 f 12' whereupon, because of the invariance of intervals, 2 f 2 _;2 _ 2./2 _//2 C Ii2 H2 — c l \2 l \2' We want the two events to occur at the same point in the K' system, that is, we require I' 12 = 0. Then ^12 = £ ^12 'l2 == C ^12 ^ ^* Consequently a system of reference with the required property exists if s\ 2 > 0, that is, if the interval between the two events is a real number. Real intervals are said to be timelike. Thus, if the interval between two events is timelike, then there exists a system of reference in which the two events occur at one and the same place. The time which elapses between THE PRINCIPLE OF RELATIVITY the two events in this system is §2 t'i2 = Uchl 2 -li 2 = S ^. (2.6) If two events occur in one and the same body, then the interval between them is always timelike, for the distance which the body moves between the two events cannot be greater than ct 12 , since the velocity of the body cannot exceed c. So we have always l 12 < ct 12 . Let us now ask whether or not we can find a system of reference in which the two events occur at one and the same time. As before, we have for the K and K' systems c t 12 -lj 2 = c 2 t'? 2 -l'? 2 . We want to have f 12 = 0, so that s 2 i2=-l'A<0. Consequently the required system can be found only for the case when the interval s 12 between the two events is an imaginary number. Imaginary intervals are said to be spacelike. Thus if the interval between two events is spacelike, there exists a reference system in which the two events occur simultaneously. The distance between the points where the events occur in this system is 'l2 = V/?2-C 2 *12 = «12- (2.7) The division of intervals into space- and timelike intervals is, because of their invariance, an absolute concept. This means that the timelike or spacelike character of an interval is independent of the reference system. Let us take some event O as our origin of time and space coordinates. In other words, in the four-dimensional system of coordinates, the axes of which are marked x, y, z, t, the world point of the event O is the origin of coordinates. Let us now consider what relation other events bear to the given event O. For visualization, we shall consider only one space dimension and the time, marking them on two axes (Fig. 2). Uniform rectilinear motion of a particle, passing through x = at t = 0, is represented by a straight line going through O and inclined to the t axis at an angle whose tangent is the velocity of the particle. Since the maximum possible velocity is c, there is a maximum angle which this line can subtend with the t axis. In Fig. 2 are shown the two lines representing the propagation of two signals Fig. 2 § 3 INTERVALS 7 (with the velocity of light) in opposite directions passing through the event O (i.e. going through x = at t = 0). All lines representing the motion of particles can lie only in the regions aOc and dOb. On the lines ab and cd, x = ±ct. First consider events whose world points lie within the region aOc. It is easy to show that for all the points of this region c 2 t 2 — x 2 > 0. In other words, the interval between any event in this region and the event O is timelike. In this region t > 0, i.e. all the events in this region occur "after" the event O. But two events which are separated by a timelike interval cannot occur simultaneously in any reference system. Consequently it is impossible to find a reference system in which any of the events in region aOc occurred "before" the event O, i.e. at time t < 0. Thus all the events in region aOc are future events relative to O in all reference systems. Therefore this region can be called the absolute future relative to O. In exactly the same way, all events in the region bOd are in the absolute past relative to O ; i.e. events in this region occur before the event O in all systems of reference. Next consider regions dOa and cOb. The interval between any event in this region and the event O is spacelike. These events occur at different points in space in every reference system. Therefore these regions can be said to be absolutely remote relative to O. However, the concepts "simultaneous", "earlier", and "later" are relative for these regions. For any event in these regions there exist systems of reference in which it occurs after the event O, systems in which it occurs earlier than O, and finally one reference system in which it occurs simultaneously with O. Note that if we consider all three space coordinates instead of just one, then instead of the two intersecting lines of Fig. 2 we would have a "cone" x 2 +y 2 +z 2 -c 2 t 2 = in the four-dimensional coordinate system x, y, z, t, the axis of the cone coinciding with the / axis. (This cone is called the light cone.) The regions of absolute future and absolute past are then represented by the two interior portions of this cone. Two events can be related causally to each other only if the interval between them is timelike; this follows immediately from the fact that no interaction can propagate with a velocity greater than the velocity of light. As we have just seen, it is precisely for these events that the concepts "earlier" and "later" have an absolute significance, which is a necessary condition for the concepts of cause and effect to have meaning. § 3. Proper time Suppose that in a certain inertial reference system we observe clocks which are moving relative to us in an arbitrary manner. At each different moment of time this motion can be considered as uniform. Thus at each moment of time we can introduce a coordinate system rigidly linked to the moving clocks, which with the clocks constitutes an inertial reference system. In the course of an infinitesi mal time interv al dt (as read by a clock in our rest frame) the moving clocks go a distance y/dx 2 + dy 2 +dz 2 . Let us ask what time interval dt' is indicated for this period by the moving clocks. In a system of coordinates linked to the moving clocks, the latter are at rest, i.e., dx' = dy' = dz' = 0. Because of the invariance of intervals ds 2 = c 2 dt 2 -dx 2 -dy 2 -dz 2 = c 2 dt' 2 t from which dt' = dtj\- dx 2 + dy 2 + dz 2 o THE PRINCIPLE OF RELATIVITY § 3 But dx 2 + dy 2 + dz 2 , — = v z , dt 2 where v is the velocity of the moving clocks; therefore ^ = - = ^^--2- (3.1) c Integrating this expression, we can obtain the time interval indicated by the moving clocks when the elapsed time according to a clock at rest is t 2 — t t : t-jfij t '2-fi = jdt^l-^. (3.2) tl The time read by a clock moving with a given object is called the proper time for this object. Formulas (3.1) and (3.2) express the proper time in terms of the time for a system of reference from which the motion is observed. As we see from (3.1) or (3.2), the proper time of a moving object is always less than the corresponding interval in the rest system. In other words, moving clocks go more slowly than those at rest. Suppose some clocks are moving in uniform rectilinear motion relative to an inertial system K. A reference frame K' linked to the latter is also inertial. Then from the point of view of an observer in the K system the clocks in the K' system fall behind. And con- versely, from the point of view of the K' system, the clocks in AT lag. To convince ourselves that there is no contradiction, let us note the following. In order to establish that the clocks in the K' system lag behind those in the K system, we must proceed in the following fashion. Suppose that at a certain moment the clock in K' passes by the clock in K, and at that moment the readings of the two clocks coincide. To compare the rates of the two clocks in A^and K' we must once more compare the readings of the same moving clock in K' with the clocks in K. But now we compare this clock with different clocks in K— with those past which the clock in K' goes at this new time. Then we find that the clock in K' lags behind the clocks in K with which it is being compared. We see that to compare the rates of clocks in two reference frames we require several clocks in one frame and one in the other, and that therefore this process is not symmetric with respect to the two systems. The clock that appears to lag is always the one which is being compared with different clocks in the other system. If we have two clocks, one of which describes a closed path returning to the starting point ■(the position of the clock which remained at rest), then clearly the moving clock appears to lag relative to the one at rest. The converse reasoning, in which the moving clock would be considered to be at rest (and vice versa) is now impossible, since the clock describing a closed trajectory does not carry out a uniform rectilinear motion, so that a coordinate system linked to it will not be inertial. Since the laws of nature are the same only for inertial reference frames, the frames linked to the clock at rest (inertial frame) and to the moving clock (non-inertial) have different properties, and the argument which leads to the result that the clock at rest must lag is not valid. § 4 THE LORENTZ TRANSFORMATION The time interval read by a clock is equal to the integral lh taken along the world line of the clock. If the clock is at rest then its world line is clearly a line parallel to the t axis; if the clock carries out a nonuniform motion in a closed path and returns to its starting point, then its world line will be a curve passing through the two points, on the straight world line of a clock at rest, corresponding to the beginning and end of the motion. On the other hand, we saw that the clock at rest always indicates a greater time interval than the moving one. Thus we arrive at the result that the integral b fds, a taken between a given pair of world points, has its maximum value if it is taken along the straight world line joining these two points.f § 4. The Lorentz transformation Our purpose is now to obtain the formula of transformation from one inertial reference system to another, that is, a formula by means of which, knowing the coordinates x, y, z, t, of a certain event in the K system, we can find the coordinates x', y', z', t' of the same event in another inertial system K'. In classical mechanics this question is resolved very simply. Because of the absolute nature of time we there have t = t'\ if, furthermore, the coordinate axes are chosen as usual (axes X, X' coincident, Y, Z axes parallel to Y', Z\ motion along X, X') then the co- ordinates v, z clearly are equal to y',z', while the coordinates x and x' differ by the distance traversed by one system relative to the other. If the time origin is chosen as the moment when the two coordinate systems coincide, and if the velocity of the K' system relative to K\s V, then this distance is Vt. Thus x = x'+Vt, y = y', z = z\ t = t'. (4.1) This formula is called the Galileo transformation. It is easy to verify that this transformation, as was to be expected, does not satisfy the requirements of the theory of relativity; it does not leave the interval between events invariant. We shall obtain the relativistic transformation precisely as a consequence of the require- ment that it leave the interval between events invariant. As we saw in § 2, the interval between events can be looked on as the distance between the corresponding pair of world points in a four-dimensional system of coordinates. Conse- quently we may say that the required transformation must leave unchanged all distances in the four-dimensional x, v, z, ct, space. But such transformations consist only of parallel displacements, and rotations of the coordinate system. Of these the displacement of the co- ordinate system parallel to itself is of no interest, since it leads only to a shift in the origin of the space coordinates and a change in the time reference point. Thus the required trans- t It is assumed, of course, that the points a and b and the curves joining them are such that all elements ds along the curves are timelike. This property of the integral is connected with the pseudo-euclidean character of the four-dimensional geometry. In euclidean space the integral would, of course, be a minimum along the straight line. 10 THE PRINCIPLE OF RELATIVITY § 4 formation must be expressible mathematically as a rotation of the four-dimensional x, y, z, ct, coordinate system. Every rotation in the four-dimensional space can be resolved into six rotations, in the planes xy, zy, xz, tx, ty, tz (just as every rotation in ordinary space can be resolved into three rotations in the planes xy, zy, and xz). The first three of these rotations transform only the space coordinates; they correspond to the usual space rotations. Let us consider a rotation in the tx plane; under this, the y and z coordinates do not change. In particular, this transformation must leave unchanged the difference (ct) 2 -x 2 , the square of the "distance" of the point (ct, x) from the origin. The relation between the old and the new coordinates is given in most general form by the formulas: x = x' cosh \\i + ct' sinh \J/, ct = x' sinh if/ + ct' cosh \J/, (4.2) where \j/ is the "angle of rotation"; a simple check shows that in fact c 2 t 2 -x 2 = c 2 t' 2 -x' 2 . Formula (4.2) differs from the usual formulas for transformation under rotation of the co- ordinate axes in having hyperbolic functions in place of trigonometric functions. This is the difference between pseudo-euclidean and euclidean geometry. We try to find the formula of transformation from an inertial reference frame K to a system K' moving relative to iTwith velocity V along the x axis. In this case clearly only the coordinate x and the time t are subject to change. Therefore this transformation must have the form (4.2). Now it remains only to determine the angle \j/, which can depend only on the relative velocity Kf Let us consider the motion, in the K system, of the origin of the K' system. Then x' = and formulas (4.2) take the form: x = ct' sinh \{/, ct = ct' cosh \J/, or dividing one by the other, x , , — = tanh w. ct Y But xjt is clearly the velocity V of the K' system relative to K. So tanh y/ = — . c From this V sinh \J/ = — ■ ■ ■ , cosh \\i = V 1 c 2 V 1 c 2 Substituting in (4.2), we find: f+-*x' 7T '"'• ""• "7- x'+Vt' x = , — —^ y = y> z = z > t= f — — 2 - (4.3) This is the required transformation formula. It is called the Lorentz transformation, and is of fundamental importance for what follows. t Note that to avoid confusion we shall always use V to signify the constant relative velocity of two inertial systems, and v for the velocity of a moving particle, not necessarily constant. § 4 THE LORENTZ TRANSFORMATION 11 The inverse formulas, expressing x', y', z\ t' in terms of x, y, z, t, are most easily obtained by changing V to -V (since the K system moves with velocity - V relative to the K' system). The same formulas can be obtained directly by solving equations (4.3) for x', y', z', t'. It is easy to see from (4.3) that on making the transition to the limit c -» co and classical mechanics, the formula for the Lorentz transformation actually goes over into the Galileo transformation. For V > c in formula (4.3) the coordinates x, t are imaginary; this corresponds to the fact that motion with a velocity greater than the velocity of light is impossible. Moreover, one cannot use a reference system moving with the velocity of light— in that case the denominators in (4.3) would go to zero. For velocities V small compared with the velocity of light, we can use in place of (4.3) the approximate formulas : V x = x' + Vf, v = v\ z = z', t = t'+-^x'. (4.4) Suppose there is a rod at rest in the K system, parallel to the X axis. Let its length, measured in this system, be Ax = x 2 -x 1 (x 2 and Xj are the coordinates of the two ends of the rod in the K system). We now determine the length of tliis rod as measured in the K' system. To do this we must find the coordinates of the two ends of the rod (x' 2 and xi) in this system at one and the same time t'. From (4.3) we find: _ x[ + Vt' x' 2 + Vt' Xi — ^=« x 2 — V 1 -? J 1 - V The length of the rod in the K' system is Ax' = x^-x'j ; subtracting x x from x 2 , we find Ax' Ax = J ■-£ The proper length of a rod is its length in a reference system in which it is at rest. Let us denote it by l = Ax, and the length of the rod in any other reference frame K' by /. Then (=! 0N /l-J (4.5) Thus a rod has its greatest length in the reference system in which it is a t rest. Its l ength in a system in which it moves with velocity V is decreased by the factor VI - V 2 /c 2 . This result of the theory of relativity is called the Lorentz contraction. Since the transverse dimensions do not change because of its motion, the volume "T of a body decreases according to the similar formula / V 2 where y* is the proper volume of the body. From the Lorentz transformation we can obtain anew the results already known to us concerning the proper time (§ 3). Suppose a clock to be at rest in the K' system. We take two events occurring at one and the same point x', y', z' in space in the K' system. The time between these events in the K' system is Af' = t' 2 -t\. Now we find the time At which 12 THE PRINCIPLE OF RELATIVITY § 5 elapses between these two events in the K system. From (4.3), we have V V *i+-2*' '2+ -2*' C C t 2 = V 1 c 2 V 1 c 2 or, subtracting one from the other, t 7 -t< = At = 7- in complete agreement with (3.1). Finally we mention another general property of Lorentz transformations which distin- guishes them from Galilean transformations. The latter have the general property of com- mutativity, i.e. the combined result of two successive Galilean transformations (with different velocities V t and V 2 ) does not depend on the order in which the transformations are performed. On the other hand, the result of two successive Lorentz transformations does depend, in general, on their order. This is already apparent purely mathematically from our formal description of these transformations as rotations of the four-dimensional coordinate system: we know that the result of two rotations (about different axes) depends on the order in which they are carried out. The sole exception is the case of transformations with parallel vectors V ± and V 2 (which are equivalent to two rotations of the four-dimensional coordinate system about the same axis). § 5. Transformation of velocities In the preceding section we obtained formulas which enable us to find from the coordinates of an event in one reference frame, the coordinates of the same event in a second reference frame. Now we find formulas relating the velocity of a material particle in one reference system to its velocity in a second reference system. Let us suppose once again that the K' system moves relative to the K system with velocity V along the x axis. Let v x = dxjdt be the component of the particle velocity in the K system and v' x = dx'fdt' the velocity component of the same particle in the K' system. From (4.3), we have V J dx' + Vdt' , , dt'+- 2 dx' ax = — , ^ , dy = dy , dz = dz', dt = J 1 -? J l - „2 Dividing the first three equations by the fourth and introducing the velocities dr f dt' we find v' + V I V 2 I V 2 y x = y, v y = — , v z = — .. (5.1) l + v' x - 2 l + v' x j 2 l + tf § 5 TRANSFORMATION OF VELOCITIES 13 These formulas determine the transformation of velocities. They describe the law of com- position of velocities in the theory of relativity. In the limiting case of c -> oo, they go over into the formulas v x = v' x + V, v y = v' y , v z = v' z of classical mechanics. In the special case of motion of a particle parallel to the X axis, v x = v, v y = v x = 0. Then v' y = v' z = 0, v' x = i/, so that v +V v = V' 1 + v'-* (5.2) It is easy to convince oneself that the sum of two velocities each smaller than the velocity of light is again not greater than the light velocity. For a velocity V significantly smaller than the velocity of light (the velocity v can be arbitrary), we have approximately, to terms of order V/c: ( v' 2 \ V V v x = v' x + V yl--JL}> v y = V 'y- V 'A ^ v * = <-<»* ~v These three formulas can be written as a single vector formula v = v'+V- A (V *▼')▼'• ( 5 - 3 ) c We may point out that in the relativistic law of addition of velocities (5.1) the two velocities v' and V which are combined enter unsymmetrically (provided they are not both directed along the x axis). This fact is related to the noncommutativity of Lorentz trans- formations which we mentioned in the preceding Section. Let us choose our coordinate axes so that the velocity of the particle at the given moment lies in the XY plane. Then the velocity of the particle in the K system has components v x = v cos 0, v y = v sin 9, and in the K' system v' x = v' cos 6', v y = v' sin 6' (v, v', 9, 9' are the absolute values and the angles subtended with the X, X' axes respectively in the K, K' systems). With the help of formula (5.1), we then find "'V 1 - V 2 . — =- sin tan 9 = ; . (5.4) v cos +V This formula describes the change in the direction of the velocity on transforming from one reference system to another. Let us consider a very important special case of this formula, namely, the deviation of light in transforming to a new reference system — a phenomenon known as the aberration of light. In this case v = v' = c, so that the preceding formula goes over into J tan 9 = sin 9'. (5.5) - +cos0' 14 THE PRINCIPLE OF RELATIVITY § 6 From the same transformation formulas (5.1) it is easy to obtain for sin and cos 0: J< < V 2 V 1 2 cos 0' + - sin0 = — ^— sin0, cos 9 = — . (5.6) 1 + - cos 9' 1 + - cos 9' c c In case V < c, we find from this formula, correct to terms of order V/c: V sin 9 — sin 9' = sin 9' cos 9'. c Introducing the angle A0 = Q'-O (the aberration angle), we find to the same order of accuracy V A0 = - sin 9', (5.7) which is the well-known elementary formula for the aberration of light. § 6. Four-vectors The coordinates of an event (ct, x, y, z) can be considered as the components of a four- dimensional radius vector (or, for short, a four-radius vector) in a four-dimensional space. We shall denote its components by x\ where the index i takes on the values 0, 1,2, 3, and x° = ct, x 1 = x, x 2 = v, x 3 = z. The square of the "length" of the radius four- vector is given by (x ) 2 -^ 1 ) 2 -^ 2 ) 2 -^ 3 ) 2 . It does not change under any rotations of the four-dimensional coordinate system, in particular under Lorentz transformations. In general a set of four quantities A , A 1 , A 2 , A 3 which transform like the components of the radius four-vector x* under transformations of the four-dimensional coordinate system is called a. four-dimensional vector (four-vector) A 1 . Under Lorentz transformations, V V A' +-A" A fl + ~A'° A° = —jJL=-, A l = / ' , ^ 2 =^' 2 A 3 =A' 3 . (6.1) / xr7. I y 2 V 1 -? V 1 - The square magnitude of any four- vector is defined analogously to the square of the radius four-vector: (A ) 2 -(A 1 ) 2 -(A 2 ) 2 -(A 3 ) 2 . For convenience of notation, we introduce two "types" of components of four-vectors, denoting them by the symbols A 1 and A h with superscripts and subscripts. These are related by A = A , A x = -A 1 , A 2 = -A 2 , A 3 = -A 3 . (6.2) The quantities A 1 are called the contravariant, and the A t the covariant components of the four- vector. The square of the four- vector then appears in the form 3 I i = £ A% = A°A +A 1 A 1 +A 2 A 2 +A 3 A 3 . § 6 FOUR- VECTORS 15 Such sums are customarily written simply as A l A u omitting the summation sign. One agrees that one sums over any repeated index, and omits the summation sign. Of the pair of indices, one must be a superscript and the other a subscript. This convention for sum- mation over "dummy" indices is very convenient and considerably simplifies the writing of formulas. We shall use Latin letters i, k, I, . . . , for four-dimensional indices, taking on the values 0, 1, 2, 3. In analogy to the square of a four-vector, one forms the scalar product of two different four- vectors : A% = A°B + A 1 B 1 + A 2 B 2 + A 3 B 3 . It is clear that this can be written either as A l Bi or Afi— the result is the same. In general one can switch upper and lower indices in any pair of dummy indices. f The product A l B t is a. four-scalar — it is invariant under rotations of the four-dimensional coordinate system. This is easily verified directly,} but it is also apparent beforehand (from the analogy with the square A l A^) from the fact that all four-vectors transform according to the same rule. The component A° is called the time component, and A 1 , A 2 , A 3 the space components of the four- vector (in analogy to the radius four- vector). The square of a four- vector can be positive, negative, or zero; such vectors are called, timelike, spacelike, and null-vectors, respectively (again in analogy to the terminology for intervals). Under purely spatial rotations (i.e. transformations not affecting the time axis) the three space components of the four-vector A 1 form a three-dimensional vector A. The time component of the four-vector is a three-dimensional scalar (with respect to these trans- formations). In enumerating the components of a four- vector, we shall often write them as A 1 = (A , A). The co variant components of the same four- vector are A t = (A , —A), and the square of the four-vector is A l A t = (A ) 2 — A 2 . Thus, for the radius four-vector: x'" = (ct, r), x t = (ct, -r), x% = c 2 t 2 -r 2 . For three-dimensional vectors (with coordinates x, y, z) there is no need to distinguish between contra- and co variant components. Whenever this can be done without causing confusion, we shall write their components as A a (a = x, y, z) using Greek letters for sub- scripts. In particular we shall assume a summation over x, y, z for any repeated index (for example, A • B = A a i? a ). A four-dimensional tensor (four-tensor) of the second rank is a set of sixteen quantities A ik , which under coordinate transformations transform like the products of components of two four-vectors. We similarly define four-tensors of higher rank. t In the literature the indices are often omitted on four-vectors, and their squares and scalar products are written as A 2 , AB. We shall not use this notation in the present text. % One should remember that the law for transformation of a four- vector expressed in co variant components differs (in signs) from the same law expressed for contravariant components. Thus, instead of (6.1), one will have: V V A\--A\ A\--Ah A 2 = AL A* = A' 3 . 16 THE PRINCIPLE OF RELATIVITY § 6 The components of a second-rank tensor can be written in three forms: co variant, A ik , contravariant, A ik , and mixed, A\ (where, in the last case, one should distinguish between A\ and A k , i.e. one should be careful about which of the two is superscript and which a subscript). The connection between the different types of components is determined from the general rule: raising or lowering a space index (1, 2, 3) changes the sign of the com- ponent, while raising or lowering the time index (0) does not. Thus : a _ aoo a __^01 a —Ail A ° = A 00 , Al'=A°\ A\ = -A 0i , AS^-A 11 ,.... Under purely spatial transformations, the nine quantities A 11 , A 12 , . . . form a three- tensor. The three components A 01 , A 02 , A 03 and the three components A 10 , A 20 , A 30 constitute three-dimensional vectors, while the component A 00 is a three-dimensional scalar. A tensor A ik is said to be symmetric if A ik = A ki , and antisymmetric if A ik = —A ki . In an antisymmetric tensor, all the diagonal components (i.e. the components A 00 , A 11 , . . . ) are zero, since, for example, we must have A 00 = —A 00 . For a symmetric tensor A ik , the mixed components A\ and A k l obviously coincide; in such cases we shall simply write A k , putting the indices one above the other. In every tensor equation, the two sides must contain identical and identically placed (i.e. above or below) free indices (as distinguished from dummy indices). The free indices in tensor equations can be shifted up or down, but this must be done simultaneously in all terms in the equation. Equating covariant and contravariant components of different tensors is "illegal" ; such an equation, even if it happened by chance to be valid in a particular reference system, would be violated on going to another frame. From the tensor components A ik one can form a scalar by taking the sum A t i = A° +A\+A 2 2 + A 3 3 (where, of course, A? = A\). This sum is called the trace of the tensor, and the operation for obtaining it is called contraction. The formation of the scalar product of two vectors, considered earlier, is a contraction operation: it is the formation of the scalar A l Bi from the tensor A l B k . In general, contracting on any pair of indices reduces the rank of the tensor by 2. For example, A\ n is a tensor of second rank A\B k is a four- vector, A ik ik is a scalar, etc. The unit four-tensor 8 k satisfies the condition that for any four-vector A\ 51A 1 = A k . (6.3) It is clear that the components of this tensor are k fl, if i = k s ' = \o, if t + k - (6 - 4) Its trace is 5\ = 4. By raising the one index or lowering the other in 8 k , we can obtain the contra- or covariant tensor g lk or g ik , which is called the metric tensor. The tensors g lk and g ik have identical components, which can be written as a matrix : (<?") = G7*)= o ~o i «l (65) \0 § 6 FOUR- VECTORS 17 (the index i labels the rows, and k the columns, in the order 0, 1, 2, 3). It is clear that g ik A k = A h g ik A k = A l . (6.6) The scalar product of two four-vectors can therefore be written in the form: A i A i = g ik A i A k =g ik A i A k . (6.7) The tensors 5 l k , g ik , g* are special in the sense that their components are the same in all coordinate systems. The completely antisymmetric unit tensor of fourth rank, e Mm , has the same property. This is the tensor whose components change sign under interchange of any pair of indices, and whose nonzero components are ± 1. From the antisymmetry it follows that all components in which two indices are the same are zero, so that the only non- vanishing components are those for which all four indices are different. We set e 0123 = +l (6.8) (hence e 012 s = - !)• Then all the other nonvanishing components e iklm are equal to + 1 or - 1, according as the numbers i, k, I, m can be brought to the arrangement 0, 1, 2, 3 by an even or an odd number of transpositions. The number of such components is 4! = 24. Thus, e iklm e Mm =-24. (6.9) With respect to rotations of the coordinate system, the quantities e iklm behave like the components of a tensor; but if we change the sign of one or three of the coordinates the components e iklm , being defined as the same in all coordinate systems, do not change, whereas the components of a tensor should change sign. Thus e iklm is, strictly speaking, not a tensor, but rather a pseudotensor. Pseudotensors of any rank, in particular pseudoscalars, behave like tensors under all coordinate transformations except those that cannot be reduced to rotations, i.e. reflections, which are changes in sign of the coordinates that are not reducible to a rotation. The products e iklm e prst form a four-tensor of rank 8, which is a true tensor; by contracting on one or more pairs of indices, one obtains tensors of rank 6, 4, and 2. All these tensors have the same form in all coordinate systems. Thus their components must be expressed as combinations of products of components of the unit tensor 5 k — the only true tensor whose components are the same in all coordinate systems. These combinations can easily be found by starting from the symmetries that they must possess under permutation of indices.f If A ik is an antisymmetric tensor, the tensor A ik and the pseudotensor A* ik = \e iklm A lm are said to be dual to one another. Similarly, e iklm A m is an antisymmetric pseudotensor of rank 3, dual to the vector A 1 . The product A ik A* k of dual tensors is obviously a pseudoscalar. t For reference we give the following formulas: % S\ 31 S\ *w- S i S i f £, 3 i t 6™ 8? 8™ S? e mm e prlm = -2{8 l p S« - 8\8 k p ), e Mm e pklm = -68}. The overall coefficient in these formulas can be checked using the result of a complete contraction, which should give (6.9). As a consequence of these formulas we have: e prst A ip A kT A ls A mt = —Ae mm , e ikim e pru A{p Akr Ais Amt = 24 a , where A is the determinant formed from the quantities A (k . % V si K % SI K V *'. 18 THE PRINCIPLE OF RELATIVITY § 6 In this connection we note some analogous properties of three-dimensional vectors and tensors. The completely antisymmetric unit pseudotensor of rank 3 is the set of quantities e aPy which change sign under any transposition of a pair of indices. The only nonvanishing components of e aPy are those with three different indices. We set e xyz = 1 ; the others are 1 or - 1, depending on whether the sequence a, 0, y can be brought to the order x, y, z by an even or an odd number of transpositions, f The products e aPy e Atiy form a true three-dimensional tensor of rank 6, and are therefore expressible as combinations of products of components of the unit three-tensor 5 aP .% Under a reflection of the coordinate system, i.e. under a change in sign of all the co- ordinates, the components of an ordinary vector also change sign. Such vectors are said to be polar. The components of a vector that can be written as the cross product of two polar vectors do not change sign under inversion. Such vectors are said to be axial. The scalar product of a polar and an axial vector is not a true scalar, but rather a pseudoscalar : it changes sign under a coordinate inversion. An axial vector is a pseudovector, dual to some antisymmetric tensor. Thus, if C = A x B, then c « = i^pyCpy, where C Py = A p B y - A y B p . Now consider four-tensors. The space components (/, k, = 1, 2, 3) of the antisymmetric tensor A lk form a three-dimensional antisymmetric tensor with respect to purely spatial transformations; according to our statement its components can be expressed in terms of the components of a three-dimensional axial vector. With respect to these same trans- formations the components A 01 , A 02 , A 03 form a three-dimensional polar vector. Thus the components of an antisymmetric four-tensor can be written as a matrix: Px Py P 2 p x -a, a x 04**) = y Py a 2 -a x ■p 2 —a y a x (6.10) where, with respect to spatial transformations, p and a are polar and axial vectors, re- spectively. In enumerating the components of an antisymmetric four-tensor, we shall write them in the form ^' & = (p,a); then the covariant components of the same tensor are A-* = (-p,a). Finally we consider certain differential and integral operations of four-dimensional tensor analysis. t The fact that the components of the four-tensor e iklm are unchanged under rotations of the four- dimensional coordinate system, and that the components of the three-tensor e aSy are unchanged by rotations of the space axes are special cases of a general rule: any completely antisymmetric tensor of rank equal to the number of dimensions of the space in which it is defined is invariant under rotations of the coordinate system in the space. % For reference, we give the appropriate formulas: SaX. 8an S a taffy €auv = Sgx 8 'pu Sg OyS O yil Oy Contrasting this tensor on one, two and three pairs of indices, we get: taffy exuy = §a\ 8ff u —8 ali 8g\, Caffy Sxffy = 2S a x, ^affy taffy == ." - § 6 FOUR-VECTORS 19 The four-gradient of a scalar <j> is the four- vector d * f 1 d(j> vA We must remember that these derivatives are to be regarded as the covariant components of the four- vector. In fact, the differential of the scalar is also a scalar; from its form (scalar product of two four- vectors) our assertion is obvious. In general, the operators of differentiation with respect to the coordinates x\ d/dx\ should be regarded as the covariant components of the operator four-vector. Thus, for example, the divergence of a four-vector, the expression dAjdx 1 , in which we differentiate the contravariant components A\ is a scalar.f In three-dimensional space one can extend integrals over a volume, a surface or a curve. In four-dimensional space there are four types of integrations : (1) Integral over a curve in four-space. The element of integration is the line element, i.e. the four-vector dx\ (2) Integral over a (two-dimensional) surface in four-space. As we know, in three- space the projections of the area of the parallelogram formed from the vectors <h and dr' on the coordinate planes x a x p are dx a dx' p -dx p dx' x . Analogously, in four- space the infinitesimal element of surface is given by the antisymmetric tensor of second rank df ik = dx l dx' k - dx k dx' i ; its components are the projections of the element of area on the coordinate planes. In three-dimensional space, as we know, one uses as surface element in place of the tensor df aP the vector df x dual to the tensor df af} : df a = ^e aP7 df Pr Geometrically this is a vector normal to the surface element and equal in absolute mag- nitude to the area of the element. In four-space we cannot construct such a vector, but we can construct the tensor df* ik dual to the tensor df ik , df* ik = W klm df ln . (6.11) Geometrically it describes an element of surface equal to and "normal" to the element of surface df lk ; all segments lying in it are orthogonal to all segments in the element df ik . It is obvious that df ik dff k = 0. (3) Integral over a hypersurface, i.e. over a three-dimensional manifold. In three- dimensional space the volume of the parallelepiped spanned by three vectors is equal to the determinant of the third rank formed from the components of the vectors. One obtains analogously the projections of the volume of the parallelepiped (i.e. the "areas" of the t If we differentiate with respect to the "covariant coordinates" x u then the derivatives to, \c It v *7 form the contravariant components of a four-vector. We shall use this form only in exceptional cases [for example, for writing the square of the four-gradient (^/tU^/O^Xi)]. We note that in the literature partial derivatives with respect to the coordinates are often abbreviated using the symbols <>' = — , 7>= — In this form of writing of the differentiation operators, the co- or contravariant character of quantities formed with them is explicit. 20 THE PRINCIPLE OF RELATIVITY § 6 hypersurface) spanned by three four- vectors dx 1 , dx' 1 , dx"' 1 ; they are given by the deter- minants dx 1 dx n dx" 1 dS ikl = dx" dx' k dx" k , dx 1 dx' 1 dx" 1 which form a tensor of rank 3, antisymmetric in all three indices. As element of integration over the hypersurface, it is more convenient to use the four- vector dS\ dual to the tensor dS w : dS l = -ie iklm dS klm , dS klm = e nklm dS". (6.12) Here dS° = dS 12 \ dS 1 = dS 023 , .... Geometrically dS 1 is a four- vector equal in magnitude to the "areas" of the hypersurface element, and normal to this element (i.e. perpendicular to all lines drawn in the hyper- surface element). In particular, dS° = dxdydz, i.e. it is the element of three-dimensional volume dV, the projection of the hypersurface element on the hyperplane x° = const. (4) Integral over a four-dimensional volume; the element of integration is the scalar dQ = dx°dx 1 dx 2 dx 3 = cdtdV. (6.13) Analogous to the theorems of Gauss and Stokes in three-dimensional vector analysis, there are theorems that enable us to transform four-dimensional integrals. The integral over a closed hypersurface can be transformed into an integral over the four- volume contained within it by replacing the element of integration dS t by the operator d dSi-^dQ—.. (6.14) For example, for the integral of a vector A 1 we have : (fi^S f = f ^dQ. (6.15) This formula is the generalization of Gauss' theorem. An integral over a two-dimensional surface is transformed into an integral over the hyper- surface "spanning" it by replacing the element of integration df* k by the operator d f£-+ ds ii^- ds «i;r ( 6 - 16 > For example, for the integral of an antisymmetric tensor A ik we have : iS^a-iS{ iS -w-^wj-S is 'W (6 - 17) The integral over a four-dimensional closed curve is transformed into an integral over the surface spanning it by the substitution : ■a dx i -^dj ki — k . (6.18) Thus for the integral of a vector, we have: which is the generalization of Stokes' theorem. § 7 FOUR-DIMENSIONAL VELOCITY 21 PROBLEMS 1. Find the law of transformation of the components of a symmetric four-tensor A ik under Lorentz transformations (6.1). Solution: Considering the components of the tensor as products of components of two four- vectors, we get: 1 ^' 00 +2 --A' 01 + - 2 A'A, A" = —L-; (A' 11 *! -A'™+ ^ A' 00 <22 — J'22 ,<23 /<'23 i13 * / .„o . K y4 '22 > ^23 = ^23^ A 12 = / A , 12 . ^ ^'0 c 2 7' ^ 01 =- — 1-^ c 2 V 2 \ V V A ' 01 ( l + ~2)+^ A ' 00 +~ Anl A 02 = .J ( ^' 02 + - A' 12 ), ' F 2 V c /■ and analogous formulas for A 33 , A 13 and A 03 . 2. The same for the antisymmetric tensor A ik . Solution: Since the coordinates x 2 and x 3 do not change, the tensor component A 23 does not change, while the components A 12 , A 13 and A 02 , A 03 transform like x 1 and x°: A' 12 +~A'° 2 A '™ + Z A '12 A 23 = A' 23 , A 12 - ° *™ c J^ J c 2 and similarly for A 13 , A 03 . With respect to rotations of the two-dimensional coordinate system in the plane x°x x (which are the transformations we are considering) the components A 01 = —A 10 , A 00 = A 11 = 0, form an antisymmetric of tensor of rank two, equal to the number of dimensions of the space. Thus (see the remark on p. 18) these components are not changed by the transformations: A' 01 . § 7. Four-dimensional velocity From the ordinary three-dimensional velocity vector one can form a four-vector. This four-dimensional velocity {four-velocity) of a particle is the vector i _dx i ds (7.1) To find its components, we note that according to (3.1), / v~ 2 ds — cdt 1 1 5j 22 THE PRINCIPLE OF RELATIVITY § 7 where v is the ordinary three-dimensional velocity of the particle. Thus dx 1 dx v„ ds cd 'J 1 -? C J 1 -? etc. Thus «' = /-7=l. — 7=51- (7-2) Note that the four-velocity is a dimensionless quantity. The components of the four- velocity are not independent. Noting that dxidx 1 = ds 2 , we have u% = l. (7.3) Geometrically, w 1 is a unit four-vector tangent to the world line of the particle. Similarly to the definition of the four- velocity, the second derivative . d 2 x i du l ds 2 ds may be called the four-acceleration. Differentiating formula (7.3), we find: uiw 1 = 0, (7.4) i.e. the four-vectors of velocity and acceleration are "mutually perpendicular". PROBLEM Determine the relativistic uniformly accelerated motion, i.e. the rectilinear motion for which the acceleration w in the proper reference frame (at each instant of time) remains constant. Solution: In the reference frame in which the particle velocity is v = 0, the components of the four-acceleration w* = (0, w/c 2 , 0, 0) (where w is the ordinary three-dimensional acceleration, which is directed along the x axis). The relativistically invariant condition for uniform acceleration must be expressed by the constancy of the four-scalar which coincides with w 2 in the proper reference frame: w 2 w'H'i = const = r-. c* In the "fixed" frame, with respect to which the motion is observed, writing out the expression for w l w { gives the equation d v v ... — — , = w, or — , = wt +const. dt ' = ' J^ ' J>4 Setting v = for / = 0, we find that const = 0, so that wt v = J r.2 Integrating once more and setting x = for t = 0, we find: ~S(V 1+! ?-'} § 7 FOUR-DIMENSIONAL VELOCITY 23 For wt<^c, these formulas go over the classical expressions v = wt, x = wt 2 /2. For wt ->oo, the velocity tends toward the constant value c. The proper time of a uniformly accelerated particle is given by the integral As t-*oo, it increases much more slowly than t, according to the law c/w In (2wt/c). CHAPTER 2 RELATIVISTIC MECHANICS § 8. The principle of least action In studying the motion of material particles, we shall start from the Principle of Least Action. The principle of least action is defined, as we know, by the statement that for each mechanical system there exists a certain integral S, called the action, which has a minimum value for the actual motion, so that its variation 5S is zero.f To determine the action integral for a free material particle (a particle not under the influence of any external force), we note that this integral must not depend on our choice of reference system, that is, it must be invariant under Lorentz transformations. Then it follows that it must depend on a scalar. Furthermore, it is clear that the integrand must be a dif- ferential of the first order. But the only scalar of this kind that one can construct for a free particle is the interval ds, or a ds, where a is some constant. So for a free particle the action must have the form b S = —cc I ds, a where a \ b is an integral along the world line of the particle between the two particular events of the arrival of the particle at the initial position and at the final position at definite times t t and t 2 , i.e. between two given world points; and a is some constant characterizing the particle. It is easy to see that a must be a positive quantity for all particles. In fact, as we saw in § 3, a \ h ds has its maximum value along a straight world line; by integrating along a curved world line we can make the integral arbitrarily small. Thus the integral a \ b ds with the positive sign cannot have a minimum ; with the opposite sign it clearly has a minimum, along the straight world line. The action integral can be represented as an integral with respect to the time t2 S = ! Ldt. ti The coefficient L of dt represents the Lagrange function of the mechanical system. With the aid of (3.1), we find: »2 = -j« c J v 2 1 ~ dt, t Strictly speaking, the principle of least action asserts that the integral S must be a minimum only for infinitesimal lengths of the path of integration. For paths of arbitrary length we can say only that S must be an extremum, not necessarily a minimum. (See Mechanics, § 2.) 24 § 9 ENERGY AND MOMENTUM 25 where v is the velocity of the material particle. Consequently the Lagrangian for the particle is L = —ac\J\ — v 2 lc 2 . The quantity a, as already mentioned, characterizes the particle. In classical mechanics each particle is characterized by its mass m. Let us find the relation between a and m. It can be determined from the fact that in the limit as c -> oo, our expression for L must go over into the classical expression L = mv 2 J2. To carry out this transition we expand L in powers of v/c. Then, neglecting terms of higher order, we find ■/■ v 2 L = — ac l 1 1 — =x — ac+ -—. c 2c Constant terms in the Lagrangian do not affect the equation of motion and can be omitted. Omitting the constant ac in L and comparing with the classical expression L = mv 2 /2, we find that a = mc. Thus the action for a free material point is S=-mcjds (8.1) and the Lagrangian is L=-mc 2 Jl-^ 2 . (8.2) § 9. Energy and momentum By the momentum of a particle we can mean the vector p = 8L/d\ (dL/dx is the symbolic representation of the vector whose components are the derivatives of L with respect to the corresponding components of v). Using (8.2), we find: mv M For small velocities (v <^ c) or, in the limit as c-> oo, this expression goes over into the classical p = my. For v = c, the momentum becomes infinite. The time derivative of the momentum is the force acting on the particle. Suppose the velocity of the particle changes only in direction, that is, suppose the force is directed perpendicular to the velocity. Then dp m d\ d* I v dt 7 • ,. If the velocity changes only in magnitude, that is, if the force is parallel to the velocity, then dp m dv dt / i; 2 \* dt' We see that the ratio of force to acceleration is different in the two cases. (9.3) 26 RELATIVISTIC MECHANICS § 9 The energy $ of the particle is defined as the quantityf <f = pv-L. Substituting the expressions (8.2) and (9.1) for L and p, we find (9.4) V 1- V - c 2 This very important formula shows, in particular, that in relativistic mechanics the energy of a free particle does not go to zero for v = 0, but rather takes on a finite value & = mc 2 . (9.5) This quantity is called the rest energy of the particle. For small velocities (v/c < 1), we have, expanding (9.4) in series in powers of v/c, - mv 2 £ K mc +-—, which, except for the rest energy, is the classical expression for the kinetic energy of a particle. We emphasize that, although we speak of a "particle", we have nowhere made use of the fact that it is "elementary". Thus the formulas are equally applicable to any composite body consisting of many particles, where by m we mean the total mass of the body and by v the velocity of its motion as a whole. In particular, formula (9.5) is valid for any body which is at rest as a whole. We call attention to the fact that in relativistic mechanics the energy of a free body (i.e. the energy of any closed system) is a completely definite quantity which is always positive and is directly related to the mass of the body. In this connection we recall that in classical mechanics the energy of a body is defined only to within an arbitrary constant, and can be either positive or negative. The energy of a body at rest contains, in addition to the rest energies of its constituent particles, the kinetic energy of the particles and the energy of their interactions with one another. In other words, mc 2 is not equal to 2 m a c 2 (where m fl are the masses of the particles), and so m is not equal to Zm r Thus in relativistic mechanics the law of conservation of mass does not hold : the mass of a composite body is not equal to the sum of the masses of its parts. Instead only the law of conservation of energy, in which the rest energies of the particles are included, is valid. Squaring (9.1) and (9.4) and comparing the results, we get the following relation between the energy and momentum of a particle : ^ = p 2 + m 2 c 2 . (9.6) The energy expressed in terms of the momentum is called the Hamiltonian function 2tf\ tf = cjp 2 + m 2 c 2 . (9.7) For low velocities, p < mc, and we have approximately je x mc 2 + j— , 2m i.e., except for the rest energy we get the familiar classical expression for the Hamiltonian. t See Mechanics, § 6. § 9 ENERGY AND MOMENTUM 27 From (9.1) and (9.4) we get the following relation between the energy, momentum, and velocity of a free particle : P = <^- (9.8) For v = c, the momentum and energy of the particle become infinite. This means that a particle with mass m different from zero cannot move with the velocity of light. Nevertheless, in relativistic mechanics, particles of zero mass moving with the velocity of light can exist.f From (9.8) we have for such particles: P = ' (9.9) The same formula also holds approximately for particles with nonzero mass in the so-called ultrarelativistic case, when the particle energy $ is large compared to its rest energy mc 2 . We now write all our formulas in four-dimensional form. According to the principle of least action, b 5S = — mc8 ds = 0. a To set up the expression for 5S, we note that ds = y/dXidx 1 and therefore b m b co r dxfidx 1 r ie . ob = — mc — j=— = — mc Ufdox 1 . a a Integrating by parts, we obtain 5S — —mcUidx 1 C c idu, , + mc 5x l ~ds. (9.10) As we know, to get the equations of motion we compare different trajectories between the same two points, i.e. at the limits (<5x l ') fl = (5x% = 0. The actual trajectory is then deter- mined from the condition SS = 0. From (9.10) we thus obtain the equations dujds = 0; that is, a constant velocity for the free particle in four-dimensional form. To determine the variation of the action as a function of the coordinates, one must consider the point a as fixed, so that (<5x l ) a = 0. The second point is to be considered as variable, but only actual trajectories are admissible, i.e., those which satisfy the equations of motion. Therefore the integral in expression (9.10) for 8S is zero. In place of (5x\ we may write simply 8x\ and thus obtain SS= -mcUidx 1 . (9.11) The four-vector dS is called the momentum four-vector. As we know from mechanics, the derivatives dS/dx, dS/dy, dS/dz are the three components of the momentum vector p of the particle, while the derivative -dS/dt is the particle energy <f. Thus the covariant components of the four- t For example, light quanta and neutrinos. 28 RELATIVISTIC MECHANICS § 9 momentum are/? f = (^/c,-p), while the contravariant components aref p' = (*/c,p). (9.13) From (9.11) we see that the components of the four-momentum of a free particle are: p* = mcu\ (9.14) Substituting the components of the four-velocity from (7.2), we see that we actually get expressions (9.1) and (9.4) for p and 8. Thus, in relativistic mechanics, momentum and energy are the components of a single four-vector. From this we immediately get the formulas for transformation of momentum and energy from one inertial system to another. Substituting (9.13) in the general formulas (6.1) for transformation of four- vectors, we find: V P* = —j= v =? Py = Py> P* = P'» * = -7==f» < 9 ' 15 ) \ 1 ~V \l l ~~c 2 where p x , p y , p z are the components of the three-dimensional vector p. From the definition (9.14) of the four-momentum, and the identity u% = 1, we have, for the square of the four-momentum of a free particle: p i p i =m 2 c 2 . (9.1.6) Substituting the expressions (9.13), we get back (9.6). By analogy with the usual definition of the force, the force four-vector is defined as the derivative : dp ' du l g* = JL = mc (9.17) as as Its components satisfy the identity g t u l = 0. The components of this four- vector are expressed in terms of the usual three-dimensional force vector f = dp/dt: , , fv f 9 ' = / i=k> 7=*\- (9-18) SI 1 c 2 C \ 1 c 2 The time component is related to the work done by the force. The relativistic Hamilton-Jacobi equation is obtained by substituting the derivatives -dS/dx* for p t in (9.16): dS dS .. dS 8S , , £Ta?-« a?a?- BV ' (9 ' 19) or, writing the sum explicitly: The transition to the limiting case of classical mechanics in equation (9.19) is made as follows. First of all we must notice that just as in the corresponding transition with (9.7), t We call attention to a mnemonic for remembering the definition of the physical four-vectors: the contravariant components are related to the corresponding three-dimensional vectors (r for x\ p for /?') with the "right", positive sign. § 1° TRANSFORMATION OF DISTRIBUTION FUNCTIONS 29 the energy of a particle in relativistic mechanics contains the term mc 2 , which it does not in classical mechanics. Inasmuch as the action S is related to the energy by S - -(dS/dt), in making the transition to classical mechanics we must in place of S substitute a new action S' according to the relation: S = S'~mc 2 t. Substituting this in (9.20), we find 1 2m fdS'\ 2 (dS'\ 2 /dS'\ 2 l 1 foS'\ 2 dS' \T X ) + \Ty) + \T Z ) ^7 +~ 2mc 2 \dt J ' dt In the limit as c -» oo, this equation goes over into the classical Hamilton-Jacobi equatio § 10. Transformation of distribution functions In various physical problems we have to deal with distribution functions for the momenta of particles :f(p)dp x dp y dp z is the number of particles having momenta with components in given intervals dp x , dp y , dp z (or, as we say for brevity, the number of particles in a given volume element dp x dp y dp z in "momentum space"). We are then faced with the problem of finding the law of transformation of the distribution function /(p) when we transform from one reference system to another. To solve this problem, we first determine the properties of the "volume element" dp x dp y dp z with respect to Lorentz transformations. If we introduce a four-dimensional coordinate system, on whose axes are marked the components of the four-momentum of a particle, then dp x dp y dp z can be considered as the fourth component of an element of the hypersurface defined by the equation p l Pi = m 2 c 2 . The element of hypersurface is a four- vector directed along the normal to the hypersurface; in our case the direction of the normal obviously coincides with the direction of the four-vector Pi . From this it follows that the ratio dp x d p dp z —^~ (10.1) is an invariant quantity, since it is the ratio of corresponding components of two parallel four-vectors.f The number of particles, fdp x dp y dp z , is also obviously an invariant, since it does not depend on the choice of reference frame. Writing it in the form ( ^dp x d Py dp z S nft J h /f nte f 3ti ? n f l th f SpeCt t0 the element (101) can be ex P re ssed in four-dimensional form by means of the ^-function (cf. the footnote on p. 00) as an integration with respect to 2 - Sipip'+m^y^p, d 4 p = dp0dp*dp 2 dp*. (10. la) Jmf»?l C ^° ne T pl T Tu Sd aS inde P endent variable * (with/; taking on only positive values). Formula (10.1a) is obvious from the following representation of the delta function appearing in it : ?"*-"*■> - s {<"- ?) = £[' (»+ ~) +* {»- $\. oai» where * = cVp* +„?(*. This formula in turn follows from formula (V) of the footnote on p. 70. 30 RELATIVISTIC MECHANICS § 11 and using the invariance of the ratio (10.1), we conclude that the product/(p)^ is invariant. Thus the distribution function in the K' system is related to the distribution function in the K system by the formula /'(P') = ^> (10-2) where p and $ must be expressed in terms of p' and $' by using the transformation formulas (9.15). Let us now return to the invariant expression (10.1). If we introduce "spherical co- ordinates" in momentum space, the volume element dp x dp y dp z becomes p 2 dp do, where do is the element of solid angle around the direction of the vector p. Noting that pdp = SdSjc 2 [from (9.6)], we have: p 2 dpdo pdido J~ c 2 ' Thus we find that the expression pdSdo (10.3) is also invariant. If we are dealing with particles moving with the velocity of light, so that the relation <$ =pc (9.9) is valid, the invariant quantity (10.3) can be written as pdp do or SdSdo. §11. Decay of particles Let us consider the spontaneous decay of a body of mass M into two parts with masses m t and m 2 . The law of conservation of energy in the decay, applied in the system of reference in which the body is at rest, gives. f M = * 10 + *2o- (11-1) where <f 10 and S 2Q are the energies of the emerging particles. Since <f 10 > m 1 and S 2Q > m i-> the equality (11.1) can be satisfied only if M > m i +m 2 , i.e. a body can disintegrate spon- taneously into parts the sum of whose masses is less than the mass of the body. On the other hand, if M < m l + m 2 , the body is stable (with respect to the particular decay) and does not decay spontaneously. To cause the decay in this case, we would have to supply to the body from outside an amount of energy at least equal to its "binding energy" (m 1 +m 2 — M). Momentum as well as energy must be conserved in the decay process. Since the initial momentum of the body was zero, the sum of the momenta of the emerging particles must be zero: P10+P20 = 0- Consequently p\ Q = pj , or S\ Q -m\ = S 2 2Q -m 2 2 . (11.2) The two equations (11.1) and (11.2) uniquely determine the energies of the emerging t In §§ 11-13 we set c = 1. In other words the velocity of light is taken as the unit of velocity (so that the dimensions of length and time become the same). This choice is a natural one in relativistic mechanics and greatly simplifies the writing of formulas. However, in this book (which also contains a considerable amount of nonrelativistic theory) we shall not usually use this system of units, and will explicitly indicate when we do. If c has been set equal to unity in formulas, it is easy to convert back to ordinary units: the velocity is introduced to assure correct dimensions. § 11 DECAY OF PARTICLES 31 particles: M 2 + m 2 -m 2 2 M 2 -mj + m 2 2 Sxo = m ' /ao = m — ' (1L3) In a certain sense the inverse of this problem is the calculation of the total energy M of two colliding particles in the system of reference in which their total momentum is zero. (This is abbreviated as the "system of the center of inertia" or the "C-system".) The com- putation of this quantity gives a criterion for the possible occurrence of various inelastic collision processes, accompanied by a change in state of the colliding particles, or the "creation" of new particles. A process of this type can occur only if the sum of the masses of the "reaction products" does not exceed M. Suppose that in the initial reference system (the "laboratory" system) a particle with mass m 1 and energy S x collides with a particle of mass m 2 which is at rest. The total energy of the two particles is and their total momentum is p = p t +p 2 = Pi- Considering the two particles together as a single composite system, we find the velocity of its motion as a whole from (9.8): V = ^=-li— . (11.4) <T £ x + m 2 This quantity is the velocity of the C-system with respect to the laboratory system (the L- system). However, in determining the mass M, there is no need to transform from one reference frame to the other. Instead we can make direct use of formula (9.6), which is applicable to the composite system just as it is to each particle individually. We thus have M 2 = S 2 -p 2 = {S 1 + m 2 ) 2 -{S\-m 2 ) i from which M 2 = m\ + m 2 2 + 2m 2 £ x . (11.5) PROBLEMS 1. A particle moving with velocity V dissociates "in flight" into two particles. Determine the relation between the angles of emergence of these particles and their energies. Solution: Let «? be the energy of one of the decay particles in the C-system [i.e. ^ 10 or «? 2 o in (11.3)], & the energy of this same particle in the L-system, and its angle of emergence in the L- system (with respect to the direction of V). By using the transformation formulas we find: . #-VpcosO Vl-V 2 so that For the determination of «? from cos we then get the quadratic equation S\\ - v 2 cos 2 0)-2<W o Vl-K 2 +<^(l - V 2 )+V 2 m 2 cos 2 6 = 0, (2) which has one positive root (if the velocity v of the decay particle in the C-system satisfies v > V) or two positive roots (if v < V). The source of this ambiguity is clear from the following graphical construction. According to (9.15), the momentum components in the L-system are expressed in terms of quantities referring to 32 RELATIVISTIC MECHANICS § 11 the C-system by the formulas Eliminating O , we get (a) V<v p COS Oo + ^oV . . Px= 7==—, p y =Po sin 6» . V\-v 2 pl+(Px^l-V 2 -<?oV) 2 =pl (b) V>v Fig. 3. With respect to the variables p x , p y , this is the equation of an ellipse with semiaxes p /Vl — V 2 , p , whose center (the point O in Fig. 3) has been shifted a distance i Q V/Vl — V 2 from the point p = (point A in Fig. 3).f If V>po/£o = v , the point A lies outside the ellipse (Fig. 3b), so that for a fixed angle the vector p (and consequently the energy S) can have two different values. It is also clear from the construction that in this case the angle 6 cannot exceed a definite value max (corresponding to the position of the vector p in which it is tangent to the ellipse). The value of max is most easily determined analytically from the condition that the discriminant of the quadratic equation (2) go to zero : PoVl-V 2 sin 6L ax = mV 2. Find the energy distribution of the decay particles in the L-system. Solution: In the C-system the decay particles are distributed isotropically in direction, i.e. the number of particles within the element of solid angle do = In sin O d0 o is dN= — do = %d\cos O \. An The energy in the L-system is given in terms of quantities referring to the C-system by . So+poVcosOo e = Vi-v 2 0) to *o+Vp and runs through the range of values from *o-Vp Vi-v 2 Vi-v 2 Expressing d\cos 6 \ in terms of d<$, we obtain the normalized energy distribution (for each of the two types of decay particles) : 1 dN- 2V Po Vl-V 2 d£. 3. Determine the range of values in the L-system for the angle between the two decay particles (their separation angle) for the case of decay into two identical particles. Solution: In the C-system, the particles fly off in opposite directions, so that 9 10 = n-0 20 = O . According to (5.4), the connection between angles in the C- and L-systems is given by the formulas: cot 6 X = to cos 0o + V v sin t) Q Vl-V 2 ' COt 2 = —vq cos 0q + V Vq sin c o Vl-V 2 t In the classical limit, the ellipse reduces to a circle. (See Mechanics, § 16.) § 11 DECAY OF PARTICLES 33 (since v 10 = V2o = v in the present case). The required separation angle is = 0i + 2 , and a simple calculation gives : V 2 -v* + V 2 vl&m 2 d cot0 = 2Kt7 O Vl-F a sin0o An examination of the extreme for this expression gives the following ranges of possible values of 0: for V< v : 2 tan" 1 ( v -± VT^V 2 )<©<«; for v < V< , =: O<0 <sin _1 VI— i; 2 l l-V 2 n \l 1-«S < 2 ; for V>— ^=^:O<0<2tan- 1 (^Vr=T^l<?. Vl-v 2 \V VL / 2 4. Find the angular distribution in the L-system for decay particles of zero mass. Solution: According to (5.6) the connection between the angles of emergence in the C- and L-systems for particles with m = is cos0-F cos O = z — T/ 2- 1 — Fcos Substituting this expression in formula (1) of Problem 2, we find: (l-V 2 )do dN- 4n(l-Vcos9) 2 ' 5. Find the distribution of separation angles in the L-system for a decay into two particles of zero mass. Solution: The relation between the angles of emergence, 9 U 2 in the L-system and the angles 0io = 0o, 020 = 7T— O in the C-system is given by (5.6), so that we have for the separation angle = 0i+0 2 : 2F 2 -1-F 2 cos 2 o COS0 = l-^COS 2 0o and conversely, '»=V- L COS O = /l z 7r - cot 2 -. Substituting this expression in formula (1) of problem 2, we find : 1-F 2 do dN = J' 16nV sin 3 ^ IV 2 -cos 20 The angle takes on values from v to & mm = 2 cos -1 V. 6. Determine the maximum energy which can be carried off by one of the decay particles, when a particle of mass M at rest decays into three particles with masses m lt m 2 , and m 3 . Solution: The particle m 1 has its maximum energy if the system of the other two particles m 2 and m 3 has the least possible mass; the latter is equal to the sum m 2 +m 3 (and corresponds to the case where the two particles move together with the same velocity). Having thus reduced the problem to the decay of a body into two parts, we obtain from (11.3): _M 2 +ml— (m 2 +m 3 ) 2 * lw 2M ' 34 RELATIVISTIC MECHANICS § 12 § 12. Invariant cross-section Collision processes are characterized by their invariant cross-sections, which determine the number of collisions (of the particular type) occurring between beams of colliding particles. Suppose that we have two colliding beams; we denote by n t and n 2 the particle densities in them (i.e. the numbers of particles per unit volume) and by v x and v 2 the velocities of the particles. In the reference system in which particle 2 is at rest (or, as one says, in the rest frame of particle 2), we are dealing with the collision of the beam of particles 1 with a stationary target. Then according to the usual definition of the cross-section a, the number of collisions occurring in volume dV in time dt is dv = Gv xti n x n 2 dt, where v tel is the velocity of particle 1 in the rest system of particle 2 (which is just the definition of the relative velocity of two particles in relativistic mechanics). The number dv is by its very nature an invariant quantity. Let us try to express it in a form which is applicable in any reference system: dv = AniHzdVdt, (12.1) where A is a number to be determined, for which we know that its value in the rest frame of one of the particles is v rel <r. We shall always mean by a precisely the cross-section in the rest frame of one of the particles, i.e. by definition, an invariant quantity. From its definition, the relative velocity y rel is also invariant. In the expression (12.1) the product dVdt is an invariant. Therefore the product An ± n 2 must also be an invariant. The law of transformation of the particle density n is easily found by noting that the number of particles in a given volume element dV, ndV, is invariant. Writing ndV=n dV Q (the index refers to the rest frame of the particles) and using formula (4.6) for the trans- formation of the volume, we find: "=vfb (122) or n = n S\m, where £ is the energy and m the mass of the particles. Thus the statement that An± n 2 is invariant is equivalent to the invariance of the expression AS ! S 2 . This condition is more conveniently represented in the form A V-Ti = A „ , * n =mv > (12.3) where the denominator is an invariant — the product of the four-momenta of the two particles. In the rest frame of particle 2, we have $ 2 = m 2 , p 2 = 0, so that the invariant quantity (12.2) reduces to A. On the other hand, in this frame A = av tel . Thus in an arbitrary reference system, A = crv rel ^f. (12.4) 1©2 To give this expression its final form, we express v Tel in terms of the momenta or velocities of the particles in an arbitrary reference frame. To do this we note that in the rest frame of § 12 INVARIANT CROSS-SECTION 35 particle 2, PuPz = / r m 2 . Then 'rel L m\m\ V (PuPi) 2 ' (12.5) Expressing the quantity p u p 2 = <^i<^2~PrP2 m terms of the velocities y ± and v 2 by using formulas (9.1) and (9.4): l-vrv 2 p li pi = m 1 m 2 V(i-^Xi-^i)' and substituting in (12.5), after some simple transformations we get the following expression for the relative velocity : V(vi-v2) 2 -(vixv 2 ) 2 , 10 ~ r -" — r^vT^ — (12 ' 6) (we note that this expression is symmetric in \ t and v 2 , i.e. the magnitude of the relative velocity is independent of the choice of particle used in defining it). Substituting (12.5) or (12.6) in (12.4) and then in (12.1), we get the final formulas for solving our problem : y/(PuPl) 2 -mlml _ dv = a — — n 1 n 2 dVdt (12.7) & 1©2 or dv = ffV^i-Va^-^xva) 2 n 1 n 2 dVdt (12.8) (W. Pauli, 1933). If the velocities v x and v 2 are collinear, then v 1 xv 2 = 0, so that formula (12.8) takes the form: dv = a\y 1 -y 2 \n 1 n 2 dVdt. (12.9) PROBLEM Find the "element of length" in relativistic "velocity space". Solution: The required line element dl v is the relative velocity of two points with velocities v and v+</v. We therefore find from (12.6) (dy) 2 -(vxdv) 2 dv 2 v 2 . dl *= (1 -, 2)2 - (T ^ + iz^ 2 + sin2 ^ 2 >. where 0, 4> are the polar angle and azimuth of the direction of v. If in place of v we introduce the new variable x through the equation v = tanh x, the line element is expressed as : dl\ = cfr 2 +sinh 2 x (d9 2 +sin 2 6 • d(f> 2 ). From the geometrical point of view this is the line element in three-dimensional Lobachevskii space— the space of constant negative curvature (see (107.8)). 36 RELATIVISTIC MECHANICS § 13 § 13. Elastic collisions of particles Let us consider, from the point of view of relativistic mechanics, the elastic collision of particles. We denote the momenta and energies of the two colliding particles (with masses m i and m i) b y Pi> <?i and p 2 , S 2 \ we use primes for the corresponding quantities after collision. The laws of conservation of momentum and energy in the collision can be written together as the equation for conservation of the four-momentum: Pi+P2 = Pi+p'l (13.1) From this four-vector equation we construct invariant relations which will be helpful in further computations. To do this we rewrite (13.1) in the form: Pi+P2-Pi=p'l and square both sides (i.e. we write the scalar product of each side with itself). Noting that the squares of the four-momenta pi and p'[ are equal to m{, and the squares of pj and p'l are equal to mf, we get: ™ 2 i+PitP2-PitPi-P2iPi = 0. (13.2) Similarly, squaring the equation pt+pl-p'j = p'l, we get: ml + PiiPl-pip'i-PiiP'i = 0. (13.3) Let us consider the collision in a reference system (the L-system) in which one of the particles (m 2 ) was at rest before the collision. Then p 2 = 0, S 2 = m 2 , and the scalar products appearing in (13.2) are: PuPi = £im 2 , p 2i p'{ = m 2 g' u (13.4) PuPi = ^i^i-prpi = « x t'l-PtPl cos e lt where Q x is the angle of scettering of the incident particle m x . Substituting these expressions in (13.2) we get: cos 9, = -^ ' t i (13.5) Pi Pi ' Similarly, we find from (13.3): ^. C + mJM-mJ P1P2 where 2 is the angle between the transferred momentum p 2 and the momentum of the incident particle p x . The formulas (13.5-6) relate the angles of scattering of the two particles in the L-system to the changes in their energy in the collision. Inverting these formulas, we can express the energie s S' u S' 2 in terms of the angles ± or 2 . Thus, substituting in (13.6) p x = sfg\-m\, p 2 = yJS" 2 -ml and squaring both sides, we find after a simple computation: (^ + m 2 ) 2 + (^-m^)cos 2 2 ^ " mi (<f 1 + m 2 ) 2 -(<r 2 -m 2 )cos 2 2 - (13J) Inversion of formula (13.5) leads in the general case to a very complicated formula for $\ in terms of 9 t . We note that if m x > m 2 , i.e. if the incident particle is heavier than the target particle, the scattering angle X cannot exceed a certain maximum value. It is easy to find by elementary § 13 ELASTIC COLLISIONS OF PARTICLES 37 computations that this value is given by the equation sin0i ma x = ^, (13.8) which coincides with the familiar classical result. Formulas (13.5-6) simplify in the case when the incident particle has zero mass: m t — 0, and correspondingly p t = S u p\ = S\. For this case let us write the formula for the energy of the incident particle after the collision, expressed in terms of its angle of deflection : S\ = . (13.9) m 2 1-COS0 2 + — Let us now turn once again to the general case of collision of particles of arbitrary mass. The collision is most simply treated in the C-system. Designating quantities in this system by the additional subscript 0, we have p 10 = -P2o= Po- From the conservation of momen- tum, during the collision the momenta of the two particles merely rotate, remaining equal in magnitude and opposite in direction. From the conservation of energy, the value of each of the momenta remains unchanged. Let x be the angle of scattering in the C-system— the angle through which the momenta p 10 and p 20 are rotated by the collision. This quantity completely determines the scattering process in the C-system, and therefore also in any other reference system. It is also con- venient in describing the collision in the L-system and serves as the single parameter which remains undetermined after the conservation of momentum and energy are applied. We express the final energies of the two particles in the L-system in terms of this para- meter. To do this we return to (13.2), but this time write out the product p u p'{ in the C- system : PuPi = ^io^io-Pio "Pio = ^io-Po cos x = £0(1 -cos #) + m 2 (in the C-system the energies of the particles do not change in the collision: S' 10 = <^ 10 ). We write out the other two products in the L-system, i.e. we use (13.4). As a result we get: ^1-^1 = -(Po/n* 2 Xl-cos x). We must still express pi in terms of quantities referring to the L-system. This is easily done by equating the values of the invariant p u p± in the L- and C-systems : &10&20 — PlO'P20 = <^1 ™2> or ^{p 2 o + ml)(pl + m 2 2 ) = S x m 2 -pl. Solving the equation for pi, we get: P °-ml + m 2 2 + 2m 2 * 1 ' (13 ' 10 Thus, we finally have: The energy of the second particle is obtained from the conservation law: S x +m 2 = ^[+^' 2 . Therefore 38 RELATIVISTIC MECHANICS § 13 The second terms in these formulas represent the energy lost by the first particle and trans- ferred to the second particle. The maximum energy transfer occurs for % = n, and is equal to ^ 2m ax-m 2 - ^-^ lmin = ro 2 + m | +2m2 ^. (13-13) The ratio of the minimum kinetic energy of the incident particle after collision to its initial energy is : ^lmin-^i = (jni-Wi) 2 , 13 14) S x -m^ ml + ml + 2m 2 £i In the limiting case of low velocities (when $ « m+mv 2 /2), this relation tends to a constant limit, equal to / m 1 -m 2 \ 2 \m 1 + m 2 ) ' In the opposite limit of large energies $ l5 relation (13.14) tends to zero; the quantity $' x min tends to a constant limit. This limit is 6 1 min — « 2m 2 Let us assume that m 2 >m 1 , i.e. the mass of the incident particle is small compared to the mass of the particle at rest. According to classical mechanics the light particle could transfer only a negligible part of its energy (see Mechanics, § 17). This is not the case in relativistic mechanics. From formula (13.14) we see that for sufficiently large energies $ A the fraction of the energy transferred can reach the order of unity. For this it is not sufficient that the velocity of m 1 be of order 1, but one must have $ t ~ m 2 , i.e. the light particle must have an energy of the order of the rest energy of the heavy particle. A similar situation occurs for m 2 <^m 1 , i.e. when a heavy particle is incident on a light one. Here too, according to classical mechanics, the energy transfer would be insignificant. The fraction of the energy transferred begins to be significant only for energies i t ~ m\jm 2 . We note that we are not talking simply of velocities of the order of the light velocity, but of energies large compared to m lt i.e. we are dealing with the ultrarelativistic case. PROBLEMS 1. The triangle ABC in Fig. 4 is formed by the momentum vector p of the impinging particle and the momenta p' x , p' 2 of the two particles after the collision. Find the locus of the points C corresponding to all possible values of p'i, p' 2 . Solution: The required curve is an ellipse whose semiaxes can be found by using the formulas obtained in problem 1 of § 11. In fact, the construction given there determined the locus of the vectors p in the L-system which are obtained from arbitrarily directed vectors p with given length po in the C-system. (a) m 1 >m 2 (b) m, < m 2 Fig. 4. § 13 ELASTIC COLLISIONS OF PARTICLES 39 Since the absolute values of the momenta of the colliding particles are identical in the C-system, and do not change in the collision, we are dealing with a similar construction for the vector p'i, for which _ _ _ m 2 V Po = Pio — P20 — / - Vl-V 2 in the C-system where V is the velocity of particle m 2 in the X-system, coincides in magnitude with the velocity of the center of inertia, and is equal to V=p 1 /(<? 1 +m 2 ) (see (1 1.4)). As a result we find that the minor and major semiaxes of the ellipse are W2Pi Po m 2 pi(£i+m 2 ) Po Vm 2 1 +m 2 2 +2m 2 ^i Vl-V 2 m 2 +m\-\-2m 2 £ x (the first of these is, of course, the same as (13.10)). For 9 X = 0, the vector p' x coincides with p x , so that the distance AB is equal to p x . Comparing p x with the length of the major axis of the ellipse, it is easily shown that the point A lies outside the ellipse if m x > m 2 (Fig. 4a), and inside it if m x < m 2 (Fig. 4b). 2. Determine the minimum separation angle min of two particles after collision if the masses of the two particles are the same (mi = m 2 = m). Solution: If m x = m 2 , the point A of the diagram lies on the ellipse, while the minimum separation angle corresponds to the situation where point C is at the end of the minor axis (Fig. 5). From the construction it is clear that tan (0 min /2) is the ratio of the lengths of the semiaxes, and we find: or . ©mm / 2m tan— -=/—-— , 2 \] S-L+m cos & miJX = ———-. «?i+3m Fig. 5. 3. For the collision of two particles of equal mass m, express i\, S' 2 , x in terms of the angle 0i of scattering in the £-system. Solution: Inversion of formula (13.5) in this case gives: ^i+m)+(^ 1 ~m)cos 2 e 1 (*i*-m*)sm* 0i (*! +m) -(<?i-m) cos 2 e x ' 2 fT " 2m+{£ x -m) sin 2 ± ' Comparing with the expression for £\ in terms of x' ^i = ^i-^y^(l-cosz), we find the angle of scattering in the C-system: 2w-(<* , i+3/«)sin 2 1 cos# = 2m+(^ , 1 +m)sin 2 1 ' 40 RELATIVISTIC MECHANICS § 14 § 14. Angular momentum As is well known from classical mechanics, for a closed system, in addition to conserva- tion of energy and momentum, there is conservation of angular momentum, that is, of the vector M = £rxp where r and p are the radius vector and momentum of the particle; the summation runs over all the particles making up the system. The conservation of angular momentum is a con- sequence of the fact that because of the isotropy of space, the Lagrangian of a closed system does not change under a rotation of the system as a whole. By carrying through a similar derivation in four-dimensional form, we obtain the relativistic expression for the angular momentum. Let x* be the coordinates of one of the particles of the system. We make an infinitesimal rotation in the four-dimensional space. Under such a transformation, the coordinates x l take on new values x' 1 such that the differences x'^—x* are linear functions x"-x' = x k SQ. ik (14.1) with infinitesimal coefficients SQ ik . The components of the four-tensor SQ tk are connected to one another by the relations resulting from the requirement that, under a rotation, the length of the radius vector must remain unchanged, that is, x-x' £ = x f x'. Substituting for x n from (14.1) and dropping terms quadratic in SQ ik , as infinitesimals of higher order, we find x'x^Qtt = 0. This equation must be fulfilled for arbitrary x\ Since x*x k is a symmetric tensor, SQ. ik must be an antisymmetric tensor (the product of a symmetrical and an antisymmetrical tensor is clearly identically zero). Thus we find that SQ ki =-SQ ik . (14.2) The change 5S in the action S for an infinitesimal change in coordinates has the form (see 9.11): (the summation extends over all the particles of the system). In the case of rotation which we are now considering, 8x t = 5Q ik x k , and so 6S = *&*%?**. If we resolve the tensor l,p 1 x k into symmetric and antisymmetric parts, then the first of these when multiplied by an antisymmetric tensor gives identically zero. Therefore, taking the antisymmetric part of 'Lp'x 1 ', we can write the preceding equality in the form dS = 5Q ik -iZ(P i x k -P k x i )- (14-3) For a closed system, because of the isotropy of space and time, the Lagrangian does not change under a rotation in four-space, i.e. the parameters 5Q ik for the rotation are cyclic coordinates. Therefore the corresponding generalized momenta are conserved. These generalized momenta are the quantities dS/dQ ik . From (14.3), we have dQ ik 2^ KF y J Consequently we see that for a closed system the tensor M ik = Y,(x i p k -x k p i ) (14.4) § 14 ANGULAR MOMENTUM 41 is conserved. This antisymmetric tensor is called the four-tensor of angular momentum. The space components of this tensor are the components of the three-dimensional angular momentum vector M = Er x p: M 23 = M X , -M 13 = M y , M 12 = M Z . The components M 0i , M 02 , M 03 form a vector X(/p-<fr/c 2 ). Thus, we can write the components of the tensor M ik in the form: M lk = l(*-?}-M (14.5) (Compare (6.10).) Because of the conservation of M ik for a closed system, we have, in particular, z(*-J) = const. Since, on the other hand, the total energy S £ is also conserved, this equality can be written in the form ysx c 2 y P 4^r ^~ t = const. From this we see that the point with the radius vector moves uniformly with the velocity »-|5 d4.6) V-'-g? (14.7) which is none other than the velocity of motion of the system as a whole. [It relates the total energy and momentum, according to formula (9.8).] Formula (14.6) gives the relativistic definition of the coordinates of the center of inertia of the system. If the velocities of all the particles are small compared to c, we can approximately set £ « mc 2 so that (14.6) goes over into the usual classical expression We note that the components of the vector (14.6) do not constitute the space components of any four-vector, and therefore under a transformation of reference frame they do not transform like the coordinates of a point. Thus we get different points for the center of inertia of a given system with respect to different reference frames. PROBLEM Find the connection between the angular momentum M of a body (system of particles) in the reference frame K in which the body moves with velocity V, and its angular momentum M (0) in t We note that whereas the classical formula for the center of inertia applies equally well to interacting and non-interacting particles, formula (14.6) is valid only if we neglect interaction. In relativistic mechanics, the definition of the center of inertia of a system of interacting particles requires us to include explicitly the momentum and energy of the field produced by the particles. 42 RELATIVISTIC MECHANICS § 14 the frame K in which the body is at rest as a whole; in both cases the angular momentum is defined with respect to the same point— the center of inertia of the body in the system K .-\ Solution: The K system moves relative to the # system with velocity V; we choose its direction for the x axis. The components of M ik that we want transform according to the formulas (see problem 2 in § 6) : V V jV/(°> 12 -^ — M i0)02 jvf <0)13 + — A/ (0)03 M 12 = , M 13 = ° , M 23 =M m23 . \J 1 ~'c 2 V 1_ ^ Since the origin of coordinates was chosen at the center of inertia of the body (in the K Q system), in that system E#r = 0, and since in that system £p = 0, M (0)02 = M (0)03 = 0. Using the con- nection between the components of M ik and the vector M, we find for the latter: M (0) M (0) M S = M?\ My = -4=2=, M g =- ' 7^5 7 r f We remind the reader that although in the system K (in which Ep = 0) the angular momentum is independent of the choice of the point with respect to which it is defined, in the K system (in which Sp ^ 0) the angular momentum does depend on this choice (see Mechanics, § 9). CHAPTER 3 CHARGES IN ELECTROMAGNETIC FIELDS § 15. Elementary particles in the theory of relativity The interaction of particles can be described with the help of the concept of afield of force. Namely, instead of saying that one particle acts on another, we may say that the particle creates a field around itself; a certain force then acts on every other particle located in this field. In classical mechanics, the field is merely a mode of description of the physical phenomenon — the interaction of particles. In the theory of relativity, because of the finite velocity of propagation of interactions, the situation is changed fundamentally. The forces acting on a particle at a given moment are not determined by the positions at that same moment. A change in the position of one of the particles influences other particles only after the lapse of a certain time interval. This means that the field itself acquires physical reality. We cannot speak of a direct interaction of particles located at a distance from one another. Interactions can occur at any one moment only between neighboring points in space (contact interaction). Therefore we must speak of the interaction of the one particle with the field, and of the subsequent interaction of the field with the second particle. We shall consider two types of fields, gravitational and electromagnetic. The study of gravitational fields is left to Chapters 10 to 12 and in the other chapters we consider only electromagnetic fields. Before considering the interactions of particles with the electromagnetic field, we shall make some remarks concerning the concept of a "particle" in relativistic mechanics. In classical mechanics one can introduce the concept of a rigid body, i.e., a body which is not deformable under any conditions. In the theory of relativity it should follow similarly that we would consider as rigid those bodies whose dimensions all remain unchanged in the reference system in which they are at rest. However, it is easy to see that the theory of relativity makes the existence of rigid bodies impossible in general. Consider, for example, a circular disk rotating around its axis, and let us assumed that it is rigid. A reference frame fixed in the disk is clearly not inertial. It is possible, however, to introduce for each of the infinitesimal elements of the disk an inertial system in which this element would be at rest at the moment; for different elements of the disk, having different velocities, these systems will, of course, also be different. Let us consider a series of line elements, lying along a particular radius vector. Because of the rigidity of the disk, the length of each of these segments (in the corresponding inertial system of reference) will be the same as it was when the disk was at rest. This same length would be measured by an observer at rest, past whom this radius swings at the given moment, since each of its seg- ments is perpendicular to its velocity and consequently a Lorentz contraction does not 43 44 CHARGES IN ELECTROMAGNETIC FIELDS § 16 occur. Therefore the total length of the radius as measured by the observer at rest, being the sum of its segments, will be the same as when the disk was at rest. On the other hand, the length of each element of the circumference of the disk, passing by the observer at rest at a given moment, undergoes a Lorentz contraction, so that the length of the whole circum- ference (measured by the observer at rest as the sum of the lengths of its various segments) turns out to be smaller than the length of the circumference of the disk at rest. Thus we arrive at the result that due to the rotation of the disk, the ratio of circumference to radius (as measured by an observer at rest) must change, and not remain equal to 2n. The absurdity of this result shows that actually the disk cannot be rigid, and that in rotation it must necessarily undergo some complex deformation depending on the elastic properties of the material of the disk. The impossibility of the existence of rigid bodies can be demonstrated in another way. Suppose some solid body is set in motion by an external force acting at one of its points. If the body were rigid, all of its points would have to be set in motion at the same time as the point to which the force is applied; if this were not so the body would be deformed. How- ever, the theory of relativity makes this impossible, since the force at the particular point is transmitted to the others with a finite velocity, so that all the points cannot begin moving simultaneously. From this discussion we can draw certain conclusions concerning the treatment of "elementary" particles, i.e., particles whose state we assume to be described completely by giving its three coordinates and the three components of its velocity as a whole. It is obvious that if an elementary particle had finite dimensions, i.e. if it were extended in space, it could not be deformable, since the concept of deformability is related to the possibility of in- dependent motion of individual parts of the body. But, as we have seen, the theory of relativity shows that it is impossible for absolutely rigid bodies to exist. Thus we come to the conclusion that in classical (non-quantum) relativistic mechanics, we cannot ascribe finite dimensions to particles which we regard as elementary. In other words, within the framework of classical theory elementary particles must be treated as points.f §16. Four-potential of a field For a particle moving in a given electromagnetic field, the action is made up of two parts: the action (8.1) for the free particle, and a term describing the interaction of the particle with the field. The latter term must contain quantities characterizing the particle and quantities characterizing the field. It turns out J that the properties of a particle with respect to interaction with the electro- magnetic field are determined by a single parameter — the charge e of the particle, which can be either positive or negative (or equal to zero). The properties of the field are characterized t Quantum mechanics makes a fundamental change in this situation, but here again relativity theory makes it extremely difficult to introduce anything other than point interactions. t The assertions which follow should be regarded as being, to a certain extent, the consequence of experi- mental data. The form of the action for a particle in an electromagnetic field cannot be fixed on the basis of general considerations alone (such as, for example, the requirement of relativistic invariance). The latter would permit the occurrence in formula (16.1) of terms of the form J A ds, where A is a scalar function. To avoid any misunderstanding, we repeat that we are considering classical (and not quantum) theory, and therefore do not include effects which are related to the spins of particles. § 16 FOUR-POTENTIAL OF A FIELD 45 by a four- vector A h the four-potential, whose components are functions of the coordinates and time. These quantities appear in the action function in the term b e C A ( dx l , where the functions A { are taken at points on the world line of the particle. The factor Ijc has been introduced for convenience. It should be pointed out that, so long as we have no formulas relating the charge or the potentials with already known quantities, the units for measuring these new quantities can be chosen arbitrarily.! Thus the action function for a charge in an electromagnetic field has the form b S = l-mcds- - A t dx\ (16.1) a The three space components of the four-vector A 1 form a three-dimensional vector A called the vector potential of the field. The time component is called the scalar potential; we denote it by A° = 0. Thus A 1 = (</>, A). (16.2) Therefore the action integral can be written in the form b <S = l—mcds+ - A-dr—efidt). a Introducing dr/dt = v, and changing to an integration over t, t2 2 S= \l-tnc 2 Jl-- 2 +-k-y-eAdt. (16.3) ti The integrand is just the Lagrangian for a charge in an electromagnetic field: J i -i L= -mc 2 J l-^ + -A-\-e(f). (16.4) This function differs from the Lagrangian for a free particle (8.2) by the terms (e/c) A • v- e<f), which describe the interaction of the charge with the field. The derivative dL/dx is the generalized momentum of the particle; we denote it by P. Carrying out the differentiation, we find „ mv e e P = , -= + -A = p+-A. (16.5) J'4 Here we have denoted by p the ordinary momentum of the particle, which we shall refer to simply as its momentum. From the Lagrangian we can find the Hamiltonian function for a particle in a field from the general formula d\ t Concerning the establishment of these units, see § 27. 46 CHARGES IN ELECTROMAGNETIC FIELDS § 17 Substituting (16.4), we get mr 2 + e<f>. (16.6) J c z However, the Hamiltonian must be expressed not in terms of the velocity, but rather in terms of the generalized momentum of the particle. From (16.5) and (16.6) it is clear that the relation between W — efy and P— (ejc)A is the same as the relation between Jf and p in the absence of the field, i.e. or else = Jm 2 c 4 + c 2 (p--a) +e<f). (16.8) For low velocities, i.e. for classical mechanics, the Lagrangian (16.4) goes over into In this approximation mv 2 e L = — - + -A-v-e0. (16.9) 2 c e p = mv = P — A, c and we find the following expression for the Hamiltonian : jP = ^-(v--a) +e(f). (16.10) 2m \ c J Finally we write the Hamilton-Jacobi equation for a particle in an electromagnetic field. It is obtained by replacing, in the equation for the Hamiltonian, P by dS/dr, and 2tf by -(dS/dt). Thus we get from (16.7) ( vs -- A V-^(^+^) + m2 c 2 =°- (16-11) § 17. Equations of motion of a charge in a field A charge located in a field not only is subjected to a force exerted by the field, but also in turn acts on the field, changing it. However, if the charge e is not large, the action of the charge on the field can be neglected. In this case, when considering the motion of the charge in a given field, we may assume that the field itself does not depend on the coordinates or the velocity of the charge. The precise conditions which the charge must fulfil in order to be considered as small in the present sense, will be clarified later on (see § 75). In what follows we shall assume that this condition is fulfilled. So we must find the equations of motion of a charge in a given electromagnetic field. These equations are obtained by varying the action, i.e. they are given by the Lagrange § 17 EQUATIONS OF MOTION OF A CHARGE IN A FIELD 47 equations d /dL\ dL ._. *(*)-¥' (m) where L is given by formula (16.4). The derivative BL/dv is the generalized momentum of the particle (16.5). Further, we write — - = VL = - grad A • v— e grad <p. or c But from a formula of vector analysis. grad (a • b) = (a • V)b + (b • V)a + b x curl a + a x curl b, where a and b are two arbitrary vectors. Applying this formula to A • v, and remembering that differentiation with respect to r is carried out for constant v, we find 8L e e — = - (v • V)A+ - v x curl A— e grad <p. or c c So the Lagrange equation has the form : d / 6 \ 6 6 — ( p+ - A ) = - (v • V)A+ - v x curl A — e grad 0. at \ c J c c But the total differential (dA/dt)dt consists of two parts: the change (8Afdt)dt of the vector potential with time at a fixed point in space, and the change due to motion from one point in space to another at distance dr. This second part is equal to (dr • V)A. Thus dA dA , ^ k Substituting this in the previous equation, we find dp e dA e — =--- egmd(f)+- vxcurl A. (17.2) dt c ot c This is the equation of motion of a particle in an electromagnetic field. On the left side stands the derivative of the particle's momentum with respect to the time. Therefore the expression on the right of (17.2) is the force exerted on the charge in an electromagnetic field. We see that this force consists of two parts. The first part (first and second terms on the right side of 17.2) does not depend on the velocity of the particle. The second part (third term) depends on the velocity, being proportional to the velocity and perpendicular to it. The force of the first type, per unit charge, is called the electric field intensity; we denote it by E. So by definition, 1 dA E=----gradf (17.3) The factor of v/c in the force of the second type, per unit charge, is called the magnetic field intensity. We designate it by H. So by definition, H = curl A. (17.4) If in an electromagnetic field, E ^ but H = 0, then we speak of an electric field; if E = but H^O, then the field is said to be magnetic. In general, the electromagnetic field is a superposition of electric and magnetic fields. We note that E is a polar vector while H is an axial vector. 48 CHARGES IN ELECTROMAGNETIC FIELDS § 17 The equation of motion of a charge in an electromagnetic field can now be written as dp e ~ = eE+-vxH. (17.5) dt e v ' The expression on the right is called the Lor entz force. The first term (the force which the electric field exerts on the charge) does not depend on the velocity of the charge, and is along the direction of E. The second part (the force exerted by the magnetic field on the charge) is proportional to the velocity of the charge and is directed perpendicular to the velocity and to the magnetic field H. For velocities small compared with the velocity of light, the momentum p is approximately equal to its classical expression m\, and the equation of motion (17.5) becomes d\ e m- = eE+-vxH, (17.6) dt c Next we derive the equation for the rate of change of the kinetic energy of the particlef with time, i.e. the derivative a<5|,: n d dt dt It is easy to check that r kin = dp dt dt Substituting dpjdt from (17.5) and noting that v x H • v = 0, we have kin = eE-v. (17.7) dt The rate of change of the kinetic energy is the work done by the field on the particle per unit time. From (17.7) we see that this work is equal to the product of the velocity by the force which the electric field exerts on the charge. The work done by the field during a time dt, i.e. during a displacement of the charge by dt, is clearly equal to eE • dr. We emphasize the fact 1 lat work is done on the charge only by the electric field ; the mag- netic field does no work ( n a charge moving in it. This is connected with the fact that the force which the magnetic ield exerts on a charge is always perpendicular to the velocity of the charge. The equations of mecha lies are invariant with respect to a change in sign of the time, that is, with respect to intercha lge of future and past. In other words, in mechanics the two time directions are equivalent. This means that if a certain motion is possible according to the equations of mechanics, then the reverse motion is also possible, in which the system passes through the same states in reverse order. It is easy to see that this is also valid for the electromagnetic field in the theory of relativity. In this case, however, in addition to changing t into — t, we must reverse the sign of the mag- netic field. In fact it is easy to see that the equations of motion (17.5) are not altered if we make the changes t-+-t, E-+E, H-+-H. (17.8) f By "kinetic" we mean the energy (9.4), which includes the rest energy. § 18 GAUGE INVARIANCE 49 According to (17.3) and (17.4), this does not change the scalar potential, while the vector potential changes sign : $->(!>, A-+-A, (17.9) Thus, if a certain motion is possible in an electromagnetic field, then the reversed motion is possible in a field in which the direction of H is reversed. PROBLEM Express the acceleration of a particle in terms of its velocity and the electric and magnetic field intensities. Solution: Substitute in the equation of motion (17.5) p = v<^ kl n/c 2 , and take the expression for d<?kin/dt from (17.7). As a result, we get ■-sV i -?{ ,i - i v xh -?*-4 § 18. Gauge invariance Let us consider to what extent the potentials are uniquely determined. First of all we call attention to the fact that the field is characterized by the effect which it produces on the motion of a charge located in it. But in the equation of motion (17.5) there appear not the potentials, but the field intensities E and H. Therefore two fields are physically identical if they are characterized by the same vectors E and H. If we are given potentials A and </>, then these uniquely determine (according to (17.3) and (17.4)) the fields E and H. However, to one and the same field there can correspond different potentials. To show this, let us add to each component of the potential the quantity —df/dx k , where /is an arbitrary function of the coordinates and the time. Then the potential A k goes over into A ' k==Ak ~d?- (18 ' 1} As a result of this change there appears in the action integral (16.1) the additional term e iL** = d(zf\ (18.2) c dx \c which is a total differential and has no effect on the equations of motion. (See Mechanics, §2.) If in place of the four-potential we introduce the scalar and vector potentials, and in place of x\ the coordinates ct, x, y, z, then the four equations (18.1) can be written in the form 1 /-) A' = A+grad/, <j>' = $- ~-±. (18.3) cot It is easy to check that electric and magnetic fields determined from equations (17.3) and (17.4) actually do not change upon replacement of A and by A' and <£', defined by (18.3). Thus the transformation of potentials (18.1) does not change the fields. The potentials are therefore not uniquely defined; the vector potential is determined to within the gradient of an arbitrary function, and the scalar potential to within the time derivative of the same function. 50 CHARGES IN ELECTROMAGNETIC FIELDS § 19 In particular, we see that we can add an arbitrary constant vector to the vector potential, and an arbitrary constant to the scalar potential. This is also clear directly from the fact that the definitions of E and H contain only derivatives of A and 0, and therefore the addition of constants to the latter does not affect the field intensities. Only those quantities have physical meaning which are invariant with respect to the trans- formation (18.3) of the potentials; in particular all equations must be invariant under this transformation. This invariance is called gauge invariance (in German, eichinvarianz).\ This nonuniqueness of the potentials gives us the possibility of choosing them so that they fulfill one auxiliary condition chosen by us. We emphasize that we can set one condition, since we may choose the function/in (18.3) arbitrarily. In particular, it is always possible to choose the potentials so that the scalar potential 4> is zero. If the vector potential is not zero, then it is not generally possible to make it zero, since the condition A = represents three auxiliary conditions (for the three components of A). § 19. Constant electromagnetic field By a constant electromagnetic field we mean a field which does not depend on the time. Clearly the potentials of a constant field can be chosen so that they are functions only of the coordinates and not of the time. A constant magnetic field is equal, as before, to H = curl A. A constant electric field is equal to E=-grad0. (19.1) Thus a constant electric field is determined only by the scalar potential and a constant magnetic field only by the vector potential. We saw in the preceding section that the potentials are not uniquely determined. However, it is easy to convince oneself that if we describe the constant electromagnetic field in terms of potentials which do not depend on the time, then we can add to the scalar potential, without changing the fields, only an arbitrary constant (not depending on either the coordinates or the time). Usually is subjected to the additional requirement that it have a definite value at some particular point in space; most frequently (j> is chosen to be zero at infinity. Thus the arbitrary constant previously mentioned is determined, and the scalar potential of the con- stant field is thus determined uniquely. On the other hand, just as before, the vector potential is not uniquely determined even for the constant electromagnetic field; namely, we can add to it the gradient of an arbitrary function of the coordinates. We now determine the energy of a charge in a constant electromagnetic field. If the field is constant, then the Lagrangian for the charge also does not depend explicitly on the time. As we know, in this case the energy is conserved and coincides with the Hamiltonian. According to (16.6), we have mc 2 J + e<j>. (19.2) f We emphasize that this is related to the assumed constancy of e in (18.2). Thus the gauge invariance of the equations of electrodynamics (see below) and the conservation of charge are closely related to one another. § 19 CONSTANT ELECTROMAGNETIC FIELD 51 Thus the presence of the field adds to the energy of the particle the term e(j>, the potential energy of the charge in the field. We note the important fact that the energy depends only on the scalar and not on the vector potential. This means that the magnetic field does not affect the energy of the charge. Only the electric field can change the energy of the particle. This is related to the fact that the magnetic field, unlike the electric field, does no work on the charge. If the field intensities are the same at all points in space, then the field is said to be uniform. The scalar potential of a uniform electric field can be expressed in terms of the field intensity as #=-Et. (19.3) In fact, since E = const, V(E • r) = (E • V)r = E. The vector potential of a uniform magnetic field can be expressed in terms of its field intensity as A = ^Hxr. (19.4) In fact, recalling that H = const, we obtain with the aid of well-known formulas of vector analysis : curl (Hxr) = Hdiv r-(H V)r = 2H (noting that div r = 3). The vector potential of a uniform magnetic field can also be chosen in the form A x =-Hy, A y = A z = (19.5) (the z axis is along the direction of H). It is easily verified that with this choice for A we have H = curl A. In accordance with the transformation formulas (18.3), the potentials (19.4) and (19.5) differ from one another by the gradient of some function: formula (19.5) is obtained from (19.4) by adding V/, where/= —xyH/2. PROBLEM Give the variational principle for the trajectory of a particle (Maupertuis' principle) in a constant electromagnetic field in relativistic mechanics. Solution: Maupertuis' principle consists in the statement that if the energy of a particle is con- served (motion in a constant field), then its trajectory can be determined from the variational equation 5 j*P-dr = 0, where P is the generalized momentum of the particle, expressed in terms of the energy and the coordinate differentials, and the integral is taken along the trajectory of the particle, f Substituting P = p+(e/c)A and noting that the directions of p and dr coincide, we have <5 I (pdl+\k-dr)=0, c where dl = Vdr 2 is the element of arc. Determining p from p 2 +m 2 c 2 = we obtain finally '-e< ■SU(^ -m 2 c 2 dl+-A-dr)>=0. t See Mechanics, § 44. 52 CHARGES IN ELECTROMAGNETIC FIELDS § 20 § 20. Motion in a constant uniform electric field Let us consider the motion of a charge e in a uniform constant electric field E. We take the direction of the field as the Xaxis. The motion will obviously proceed in a plane, which we choose as the XY plane. Then the equations of motion (17.5) become Px = eE, p y = (where the dot denotes differentiation with respect to t), so that p x = eEt, p y = p . (20.1) The time reference point has been chosen at the moment when^ = 0; p is the momentum of the particle at that moment. The kin etic energ y of the particle (the energy omitting the potential energy in the field) is ^kin = c\lm 2 c 2 +p 2 . Substituting (20.1), we find in our case ^kin = \lm 2 c* + c 2 p 2 + (ceEtj 2 = Jg 2 + (ceEt) 2 , (20.2) where «f is the energy at t = 0. According to (9.8) the velocity of the particle is v = pc 2 /<f kln . For the velocity v x = x we have therefore dx p x c 2 c 2 eEt dt ^ kln Qg 2 + (ceEt) 2 ' Integrating, we find x = — yUl + (ceEt) 2 . (20.3) The constant of integration we set equal to zero.f For determining y, we have dy Vy c2 PoC 2 dt <f kin V^o + (ce£0 2 ' from which PnC . , , /ceEt s . — smiTM— - . (20.4) eE o We obtain the equation of the trajectory by expressing / in terms of y from (20.4) and sub- stituting in (20.3). This gives: fin eEy x= _?cosh— -. (20.5) eE p c Thus in a uniform electric field a charge moves along a catenary curve. If the velocity of the particle is v -4 c, then we can set p = mv , i Q = mc 2 , and expand (20.5) in series in powers of 1/c. Then we get, to within terms of higher order, eE x = - — 2 y + const, ,- 2 2mvQ that is, the charge moves along a parabola, a result well known from classical mechanics. f This result (for p Q = 0) coincides with the solution of the problem of relativistic motion with constant "proper acceleration" w = eEjm (see the problem in § 7). For the present case, the constancy of the accelera- tion is related to the fact that the electric field does not change for Lorentz transformations having velocities V along the direction of the field (see § 24). § 21 MOTION IN A CONSTANT UNIFORM MAGNETIC FIELD 53 § 21. Motion in a constant uniform magnetic field We now consider the motion of a charge e in a uniform magnetic field H. We choose the direction of the field as the Z axis. We rewrite the equation of motion e p = -vxH c in another form, by substituting for the momentum, from (9.8), Sy where $ is the energy of the particle, which is constant in the magnetic field. The equation of motion then goes over into the form $ d\ e ?ir~c yxH < 211 > or, expressed in terms of components, i> x = cov y , v y =-cov x , v z = 0, (21.2) where we have introduced the notation ecH co = (21.3) We multiply the second equation of (21.2) by i, and add it to the first: d_ dt — (v x + iv y ) = - ico(v x + ivX so that v x +iv y = ae~ i<ot , where a is a complex constant. This can be written in the form a = v 0t e~ ia where v Qt and a are real. Then v x +iv y = v 0t e-* cot+ * ) and, separating real and imaginary parts, we find v x = v 0t cos (cot + a), v y = -v ot sin (cot + a). (21.4) The constants v 0t and oc are determined by the initial conditions; a is the initial phase, and as for v 0t , from (21.4) it is clear that v 0t = \/v 2 x +v 2 y , that is, v 0t is the velocity of the particle in the XY plane, and stays constant throughout the motion. From (21.4) we find, integrating once more, x = x + r sin (cot +<x), y = y + r cos (cot+a), (21.5) where v 0t v 0t g cp t (p t is the projection of the momentum on the XY plane). From the third equation of (21.2), we find v z = v 0z and z = z + v 0z t. (21.7) C.T.F. 3 54 CHARGES IN ELECTROMAGNETIC FIELDS § 21 From (21.5) and (21.7), it is clear that the charge moves in a uniform magnetic field along a helix having its axis along the direction of the magnetic field and with a radius r given by (21.6). The velocity of the particle is constant. In the special case where v 0z = 0, that is, the charge has no velocity component along the field, it moves along a circle in the plane perpendicular to the field. The quantity w, as we see from the formulas, is the angular frequency of rotation of the particle in the plane perpendicular to the field. If the velocity of the particle is low, then we can approximately set $ = mc 2 . Then the frequency co is changed to ca = — . (21.8) mc We shall now assume that the magnetic field remains uniform but varies slowly in magnitude and direction. Let us see how the motion of a charged particle changes in this case. We know that when the conditions of the motion are changed slowly, certain quantities called adiabatic invariants remain constant. Since the motion in the plane perpendicular to the magnetic field is periodic, the adiabatic invariant is the integral t§ v - dt ' 2n taken over a complete period of the motion, i.e. over the circumference of a circle in the present case (P, is the projection of the generalized momentum on the plane perpendicular to Hf). Substituting P ( = p t + (e/c)A, we have: = 2^ P '- dr = ^ P '- dr+ 2^ A - dr In the first term we note that p t is constant in magnitude and directed along dr; we apply Stokes' theorem to the second term and write curl A = H : / = rp t + ± Hr\ where r is the radius of the orbit. Substituting the expression (21.6) for r, we find: From this we see that, for slow variation of H, the tangential momentum p t varies propor- tionally to v H. This result can also be applied to another case, when the particle moves along a helical path in a magnetic field that is not strictly homogeneous (so that the field varies little over distances comparable with the radius and step of the helix). Such a motion can be considered as a motion in a circular orbit that shifts in the course of time, while relative to the orbit the t See Mechanics, § 49. In general the integrals § p dq, taken over a period of the particular coordinate q, are adiabatic invariants. In the present case the periods for the two coordinates in the plane perpendicular to H coincide, and the integral / which we have written is the sum of the two corresponding adiabatic in- variants. However, each of these invariants individually has no special significance, since it depends on the (non-unique) choice of the vector potential of the field. The nonuniqueness of the adiabatic invariants which results from this is a reflection of the fact that, when we regard the magnetic field as uniform over all of space, we cannot in principle determine the electric field which results from changes in H, since it will actually depend on the specific conditions at infinity. § 22 MOTION OF A CHARGE IN CONSTANT UNIFORM ELECTRIC AND MAGNETIC FIELDS 55 field appears to change in time but remain uniform. One can then state that the component of the angular momentum transverse to the direction of the field varies according to the law: p t = \JCH, where C is a constant and #is a given function of the coordinates. On the other hand, just as for the motion in any constant magnetic field, the energy of the particle (and consequently the square of its momentum p 2 ) remains constant. Therefore the longitudinal component of the momentum varies according to the formula : pf = P 2 -Pf = P 2 ~CH(x, y, z). (21.10) Since we should always have pf ^ 0, we see that penetration of the particle into regions of sufficiently high field (CH > p 2 ) is impossible. During motion in the direction of increasing field, the radius of the helical trajectory decreases proportionally top t /H(i.Q. proportionally to 1/yJH), and the step proportionally to p x . On reaching the boundary where p t vanishes, the particle is reflected : while continuing to rotate in the same direction it begins to move opposite to the gradient of the field. Inhomogeneity of the field also leads to another phenomenon — a slow transverse shift (drift) of the guiding center of the helical trajectory of the particle (the name given to the center of the circular orbit); problem 3 of the next section deals with this question. PROBLEM Determine the frequency of vibration of a charged spatial oscillator, placed in a constant, uniform magnetic field; the proper frequency of vibration of the oscillator (in the absence of the field) is co . Solution: The equations of forced vibration of the oscillator in a magnetic field (directed along the z axis) are : •• , 2 eH . o eH x + co x = — y, y + ca 2 y= x, z+co 2 z = 0. mc mc ° Multiplying the second equation by i and combining with the first, we find eH ■ c+co 2 c = -i — i, mc where C=x+iy. From this we find that the frequency of vibration of the oscillator in a plane perpendicular to the field is J" t+ \(0)' eH 2mc' If the field H is weak, this formula goes over into co = co ±eH/2mc. The vibration along the direction of the field remains unchanged. § 22. Motion of a charge in constant uniform electric and magnetic fields Finally we consider the motion of a charge in the case where there are present both electric and magnetic fields, constant and uniform. We limit ourselves to the case where the velocity of the charge v < c, so that its momentum p = mv; as we shall see later, it is necessary for this that the electric field be small compared to the magnetic. 56 CHARGES IN ELECTROMAGNETIC FIELDS § 22 We choose the direction of H as the Z axis, and the plane passing through H and E as the YZ plane. Then the equation of motion e mv = eE+ - vxH c can be written in the form 6 6 mx = -yH, my = eE y — xH, m'z = eE z . (22.1) From the third equation we see that the charge moves with uniform acceleration in the Z direction, that is, eE Z = 2m t2 + V ° zt ' (22 ' 2) Multiplying the second equation of (22.1) by i and combining with the first, we find d e — (x + iy) + ico(x + iy) = i — E y at m (co = eH/mc). The integral of this equation, where x + iy is considered as the unknown, is equal to the sum of the integral of the same equation without the right-hand term and a particular integral of the equation with the right-hand term. The first of these is ae~ i(0t , the second is eE y /ma) = cE y (H. Thus cE x + iy = ae- imt +-f. H The constant a is in general complex. Writing it in the form a = be ict , with real b and a, we see that since a is multiplied by e~ imt , we can, by a suitable choice of the time origin, give the phase a any arbitrary value. We choose this so that a is real. Then breaking up x + iy into real and imaginary parts, we find cE x = a cos cot + — -, y=— a sin cot. (22.3) H At t = the velocity is along the X axis. We see that the components of the velocity of the particle are periodic functions of the time. Their average values are: - cE y ~ n *=-, 3-0. This average velocity of motion of a charge in crossed electric and magnetic fields is often called the electrical drift velocity. Its direction is perpendicular to both fields and independent of the sign of the charge. It can be written in vector form as: _ cExH _ .. v = -^f-- ( 22 - 4 ) All the formulas of this section assume that the velocity of the particle is small compared with the velocity of light; we see that for this to be so, it is necessary in particular that the electric and magnetic fields satisfy the condition ^<U, (22.5) XI while the absolute magnitudes of E y and H can be arbitrary. § 22 MOTION OF A CHARGE IN CONSTANT UNIFORM ELECTRIC AND MAGNETIC FIELDS 57 Fig. 6. Integrating equation (22.3) again, and choosing the constant of integration so that at t = 0, x = y = 0, we obtain cE r . (22.6) x = - sin cot-\ t: co H y = ~ (cos cor— 1). CO Considered as parametric equations of a curve, these equations define a trochoid. Depend- ing on whether a is larger or smaller in absolute value than the quantity cE y /H, the projection of the trajectory on the plane XY has the forms shown in Figs. 6a and 6b, respectively. If a = —cE y /H, then (22.6) becomes cE x = — -^ (cot — sin cot), coH cE y n y = ^ (1 - C0SCW ° (22.7) that is, the projection of the trajectory on the XY plane is a cycloid (Fig. 6c). PROBLEMS 1. Determine the relativistic motion of a charge in parallel uniform electric and magnetic fields. Solution: The magnetic field has no influence on the motion along the common direction of E and H (the z axis), which therefore occurs under the influence of the electric field alone; therefore according to § 20 we find: eE' Km ' For the motion in the xy plane we have the equation \+(ceEt) 2 . PX = - HVy, c Py = — ~ HV X c 58 CHARGES IN ELECTROMAGNETIC FIELDS § 22 or d r . . \ • e H , , . n ieHc , -jAPx+iPy) = -i-—(v x +Wy) = - — — (jp x +ipy). at c «r kln Consequently Px+ip y =Pte~ i * t whereat is the constant value of the projection of the momentum on the xy plane, and the auxiliary quantity (j> is defined by the relation d(f> = eHc -— , ®kin from which Furthermore we have: so that c ' = TE siDb p- ' (1 > P,+ip, = P ,e-<* = % (*+ W = <^±^' c/?( . , CD* x = -—sin<*, >>=— cos<£. (2) en en Formulas (1), (2) together with the formula Z = ^ COSh §<*' (3) determine the motion of the particle in parametric form. The trajectory is a helix with radius cptleH and monotonically increasing step, along which the particle moves with decreasing angular velocity <f> — eHc/£ kia and with a velocity along the z axis which tends toward the value c. 2. Determine the relativistic motion of a charge in electric and magnetic fields which are mutually perpendicular and equal in magnitude, f Solution: Choosing the z axis along H and the y axis along E and setting E = H, we write the equations of motion : d -^ = -Ev y , %L = eE (l- V A ^' = dt c dt \ c) dt and, as a consequence of them, formula (17.7), dt From these equations we have : p 3 = const, ^kin—cpx = const = a. Also using the equation Kin~C 2 p 2 x = (<? ki n + Cp x )(^ in -C Px ) = C 2 P 2 y +e 2 (where e 2 = trfc^+^p 2 = const), we find: <? k m + cp x = l -(c 2 p 2 y +e 2 ), a and so - _« ,c 2 p 2 +e 2 « , c 2 pl+s 2 Px= — ;r + 2c 2ac f The problem of motion in mutually perpendicular fields E and H which are not equal in magnitude can, by a suitable transformation of the reference system, be reduced to the problem of motion in a pure electric or a pure magnetic field (see § 25). -j t =eE\ * kln — J = eE{S kln - cp*) = e£a, § 22 MOTION OF A CHARGE IN CONSTANT UNIFORM ELECTRIC AND MAGNETIC FIELDS 59 Furthermore, we write <p fay ®kin from which *»- (»+;?)»+ £* a) To determine the trajectory, we make a transformation of variables in the equations dx _ c 2 p x to the variable /?„ by using the relation dt = £ kiD dp y /eEa, after which integration gives the formulas : c ( , £ 2 \ c 3 X= "leE\- l + ^)^ + 6^eE P - < 2 > <? 2 P*C 2 Formulas (1) and (2) completely determine the motion of the particle in parametric form (parameter p y ). We call attention to the fact that the velocity increases most rapidly in the direction per- pendicular to E and H (the x axis). 3. Determine the velocity of drift of the guiding center of the orbit of a nonrelativistic charged particle in a quasihomogeneous magnetic field (H. Alfven, 1940). Solution: We assume first that the particle is moving in a circular orbit, i.e. its velocity has no longitudinal component (along the field). We write the equation of the trajectory in the form r = R(/)+£(f ), where R(t) is the radius vector of the guiding center (a slowly varying function of the time), while £(0 is a rapidly oscillating quantity describing the rotational motion about the guiding center. We average the force (e/c)r x H(r) acting on the particle over a period of the oscil- latory (circular) motion (compare Mechanics, § 30). We expand the function H(r) in this expression in powers of e : H(r) = H(R)+(£-V)H(R). On averaging, the terms of first order in e(0 vanish, while the second-degree terms give rise to an additional force f=^x(C-V)H. For a circular orbit C = ^xn, C = ^, CO where n is a unit vector along H; the frequency m = eH/mc; v ± is the velocity of the particle in its circular motion. The average values of products of components of the vector C rotating in a plane (the plane perpendicular to n), are : where d ap is the unit tensor in this plane. As a result we find: f=-^(nxy)xH. Because of the equations div H = and curl H = which the constant field H(R) satisfies we have: ' (nx V)xH = -n div H+(n- V)H+n-(V xH) = (n- V)H = H(n- V)n+n(n- VH). We are interested in the force transverse to n, giving rise to a shift of the orbit; it is equal to f= -±(n-V)n = ^v, £ Zp where p is the radius of curvature of the force line of the field at the given point, and v is a unit vector directed from the center of curvature to this point. 60 CHARGES IN ELECTROMAGNETIC FIELDS § 23 The case where the particle also has a longitudinal velocity ^ ,( along n) reduces to the previous case if we go over to a reference frame which is rotating about the instantaneous center of curvature of the force line (which is the trajectory of the guiding center) with angular velocity v\\f P . In this reference system the particle has no longitudinal velocity, but there is an additional transverse force, the centrifugal force emvfj£. Thus the total transverse force is f i =v- M> P This force is equivalent to a constant electric field of strength ije. According to (22.40) it causes a drift of the guiding center of the orbit with a velocity The sign of this velocity depends on the sign of the charge. § 23. The electromagnetic field tensor In § 17, we derived the equation of motion of a charge in a field, starting from the Lagrangian (16.4) written in three-dimensional form. We now derive the same equation directly from the action (16.1) written in four-dimensional notation. The principle of least action states b dS = 8 \(-mcds--A t dx*\ = 0. (23.1) a Noting that ds = yjdxidx 1 , we find (the limits of integration a and b are omitted for brevity): Kdx { ddx l e , ,„ . e „ .\ mc — + - A t d5x l + - SAidx 1 ) = 0. 3S = We integrate the first two terms in the integrand by parts. Also, in the first term we set dxi/ds = u h where u t are the components of the four- velocity. Then /( e e A X ( e \ . mc du, 5x l + - 8x l dA, — 3 A, dx l ) — ( mcu,+ - A, J 5x l c c / L\ c J = 0. (23.2) The second term in this equation is zero, since the integral is varied with fixed coordinate values at the limits. Furthermore: and therefore C / e dA, e dA- \ (mcdiii Sx l +-—l dx l dx k — —\ d^dx*) = 0. In the first term we write du t = (dujds)ds, in the second and third, dx l = u l ds. In addition, in the third term we interchange the indices / and k (this changes nothing since the indices / and k are summed over). Then j[^-m-m^-°- § 23 THE ELECTROMAGNETIC FIELD TENSOR 61 In view of the arbitrariness of 5x\ it follows that the integrand is zero, that is, dui _ e /dA k dAA k ds c\ dx* dx k J We now introduce the notion dA k dA t Fik= M~d?- (233) The antisymmetric tensor F ik is called the electromagnetic field tensor. The equation of motion then takes the form : du* e .. mc ^- = " ir, V (23.4) These are the equations of motion of a charge in four-dimensional form. The meaning of the individual components of the tensor F ik is easily seen by substituting the values A t = (<£, -A) in the definition (23.3). The result can be written as a matrix in which the index i = 0, 1, 2, 3 labels the rows, and the index k the columns: / E x E y E 2 \ I -E x -E y ~E 2 \ ■ --h »: -; 4 -- [t : -s 4 « \-E z -H y H x 0/ \E Z -H y H x 0/ More briefly, we can write (see § 6): F ik = (E,H), F' fc = (-E,H). Thus the components of the electric and magnetic field strengths are components of the same electromagnetic field four-tensor. Changing to three-dimensional notation, it is easy to verify that the three space com- ponents (/ = 1, 2, 3) of (23.4) are identical with the vector equation of motion (17.5), while the time component (i = 0) gives the work equation (17.7). The latter is a consequence of the equations of motion; the fact that only three of the four equations are independent can also easily be found directly by multiplying both sides of (23.4) by u\ Then the left side of the equation vanishes because of the orthogonality of the four-vectors u l and dujds, while the right side vanishes because of the antisymmetry of F ik . If we admit only possible trajectories when we vary S, the first term in (23.2) vanishes identically. Then the second term, in which the upper limit is considered as variable, gives the differential of the action as a function of the coordinates. Thus 5S = - I mcu t + - Ai J 5xK (23.6) Then dS e e - — j = mcu^ - c A^ Vi ^- c A t . (23.7) The four-vector -dS/dx 1 is the four-vector P t of the generalized momentum of the particle. Substituting the values of the components p t and A t , we find that P ={—^-> P+-AJ. (23.8) As expected, the space components of the four- vector form the three-dimensional general- 62 CHARGES IN ELECTROMAGNETIC FIELDS § 24 ized momentum vector (16.5), while the time component is &/c, where £ is the total energy of the charge in the field. § 24. Lorentz transformation of the field In this section we find the transformation formulas for fields, that is, formulas by means of which we can determine the field in one inertial system of reference, knowing the same field in another system. The formulas for transformation of the potentials are obtained directly from the general formulas for transformation of four-vectors (6.1). Remembering that A 1 = ((f), A), we get easily V V <t>> +-<!>' A' x +-A' x 4> = , ° , , A X =-—JL=, A y = A' A Z = A' Z . (24.1) V 1 c 2 V 1 c 2 The transformation formulas for an antisymmetric second-rank tensor (like F lk ) were found in problem 2 of § 6: the components F 23 and F 01 do not change, while the com- ponents F 02 , F 03 , and F 12 , F 13 transform like x° and x 1 , respectively. Expressing the components of F ik in terms of the components of the fields E and H, according to (23.7), we then find the following formulas of transformation for the electric field : V E' y +-H' z E x = E' x , E v = ,_!__ E z = — =^=, (24.2) 1-^- ^ <r and for the magnetic field: V My— — E Z H x = H' H v = - , ° . , H z = ,— 1—. . (24.3) ■J 2 C V K--H' y J-? V h- + -e; \ c 2 Thus the electric and magnetic fields, like the majority of physical quantities, are relative ; that is, their properties are different in different reference systems. In particular, the electric or the magnetic field can be equal to zero in one reference system and at the same time be present in another system. The formulas (24.2), (24.3) simplify considerably for the case V < c. To terms of order V/c, we have : v v E x = E' x , E v = E'-\ — H' z , E z — E' z H y ; * " c c H x = H' x , H y = H y -- E' z , H z = H' z + - E y '. These formulas can be written in vector form E = E'+ -H' x V, H = H'- - E' x V. (24.4) c c § 25 INVARIANTS OF THE FIELD 63 The formulas for the inverse transformation from K' to K are obtained directly from (24.2)-(24.4) by changing the sign of V and shifting the prime. If the magnetic field H' = in the K' system, then, as we easily verify on the basis of (24.2) and (24.3), the following relation exists between the electric and magnetic fields in the K system: H = -VxE. (24.5) If in the K' system, E' = 0, then in the K system E=-ivxH. (24.6) Consequently, in both cases, in the K system the magnetic and electric fields are mutually perpendicular. These formulas also have a significance when used in the reverse direction: if the fields E and H are mutually perpendicular (but not equal in magnitude) in some reference system K, then there exists a reference system K' in which the field is pure electric or pure magnetic! The velocity V of this system (relative to K) is perpendicular to E and H and equal in magnitude to cH/E in the first case (where we must have H < E) and to cE/H in the second case (where E < H). § 25. Invariants of the field From the electric and magnetic field intensities we can form invariant quantities, which remain unchanged in the transition from one inertial reference system to another. The form of these invariants is easily found starting from the four-dimensional representa- tion of the field using the antisymmetric four-tensor F ik . It is obvious that we can form the following invariant quantities from the components of this tensor: F ik F ik = inv, (25.1) e iklm F ik F lm = mv, (25 .2) where e lklm is the completely antisymmetric unit tensor of the fourth rank (cf. § 6). The first quantity is a scalar, while the second is a pseudoscalar (the product of the tensor F ik with its dual tensor, f Expressing F ik in terms of the components of E and H using (23.5), it is easily shown that, in three-dimensional form, these invariants have the form: H 2 -E 2 = inv, (25.3) E-H = inv. (25.4) The pseudoscalar character of the second of these is here apparent from the fact that it is the product of the polar vector E with the axial vector H (whereas its square (E • H) 2 is a true scalar) . t We also note that the pseudoscalar (25.2) can also be expressed as a four-divergence: as can be easily verified by using the antisymmetry of e lklm . 64 CHARGES IN ELECTROMAGNETIC FIELDS § 25 From the invariance of the two expressions presented, we get the following theorems. If the electric and magnetic fields are mutually perpendicular in any reference system, that is, E • H = 0, then they are also perpendicular in every other inertial reference system. If the absolute values of E and H are equal to each other in any reference system, then they are the same in any other system. The following inequalities are also clearly valid. If in any reference system E > H (or H > E), then in every other system we will have E > H (or H > E). If in any system of reference the vectors E and H make an acute (or obtuse) angle, then they will make an acute (or obtuse) angle in every other reference system. By means of a Lorentz transformation we can always give E and H any arbitrary values, subject only to the condition that E 2 — H 2 and E-H have fixed values. In particular, we can always find an inertial system in which the electric and magnetic fields are parallel to each other at a given point. In this system E • H = EH, and from the two equations E — H = Eq — Hq, EH = Eo'Ho. we can find the values of E and H in this system of reference (E and H are the electric and magnetic fields in the original system of reference). The case where both invariants are zero is excluded. In this case, E and H are equal and mutually perpendicular in all reference systems. If E • H = 0, then we can always find a reference system in which E = or H = (accord- ing as E 2 — H 2 < or > 0), that is, the field is purely magnetic or purely electric. Con- versely, if in any reference system E = or H = 0, then they are mutually perpendicular in every other system, in accordance with the statement at the end of the preceding section. We shall give still another approach to the problem of finding the invariants of an anti- symmetric four-tensor. From this method we shall, in particular, see that (25.3-4) are actually the only two independent invariants and at the same time we will explain some instructive mathematical properties of the Lorentz transformations when applied to such a four-tensor. Let us consider the complex vector F = E+iH. (25.5) Using formulas (24.2-3), it is easy to see that a Lorentz transformation (along the x axis) for this vector has the form F x = F' x , F = F' cosh <j) - \F' Z sinh $ = F' y cos i(f>-F' z sin i(j>. V F z = F' z cos i(j)+F' y sin i(f), tanh (j> = -. (25.6) We see that a rotation in the x, t plane in four-space (which is what this Lorentz transforma- tion is) for the vector F is equivalent to a rotation in the y, z plane through an imaginary angle in three-dimensional space. The set of all possible rotations in four-space (including also the simple rotations around the x, y, and z axes) is equivalent to the set of all possible rotations, through complex angles in three-dimensional space (where the six angles of rotation in four-space correspond to the three complex angles of rotation of the three- dimensional system). The only invariant of a vector with respect to rotation is its square: F 2 = E 2 —H + +2i E-H; thus the real quantities E 2 -H 2 and E-H are the only two independent invariants of the tensor F ik . § 25 INVARIANTS OF THE FIELD 65 If F 2 # 0, the vector F can be written as F = a n, where n is a complex unit vector (n 2 = 1). By a suitable complex rotation we can point n along one of the coordinate axes; it is clear that then n becomes real and determines the directions of the two vectors E and H: F = (E+iH)n; in other words we get the result that E and H become parallel to one another. PROBLEM Determine the velocity of the system of reference in which the electric and magnetic fields are parallel. Solution: Systems of reference K', satisfying the required condition, exist in infinite numbers. If we have found one such, then the same property will be had by any other system moving relative to the first with its velocity directed along the common direction of E and H. Therefore it is sufficient to find one of these systems which has a velocity perpendicular to both fields. Choosing the direction of the velocity as the x axis, and making use of the fact that in K': E x = H' t = 0, E' y H' Z -E' Z H' V = 0, we obtain with the aid of formulas (24.2) and (24.3) for the velocity V of the K' system relative to the original system the following equation: y c _ ExH ~ \^~~E 2 +H 2 (we must choose that root of the quadratic equation for which V< c). CHAPTER 4 THE ELECTROMAGNETIC FIELD EQUATIONS § 26. The first pair of Maxwell's equations From the expressions 1 dA H = curl A, E = — grad <f> c dt it is easy to obtain equations containing only E and H. To do this we find curl E : 1 r\ curl E = — curl A — curl grad </>. c dt But the curl of any gradient is zero. Consequently, curlE= — . (26.1) c dt Taking the divergence of both sides of the equation curl A = H, and recalling that div curl = 0, we find div H = 0. (26.2) The equations (26.1) and (26.2) are called the first pair of Maxwell's equations. f We note that these two equations still do not completely determine the properties of the fields. This is clear from the fact that they determine the change of the magnetic field with time (the derivative dH/dt), but do not determine the derivative dE/dt. Equations (26.1) and (26.2) can be written in integral form. According to Gauss' theorem f divHdV = <$> H df, where the integral on the right goes over the entire closed surface surrounding the volume over which the integral on the left is extended. On the basis of (26.2), we have H-df = 0. (26.3) The integral of a vector over a surface is called the flux of the vector through the surface. Thus the flux of the magnetic field through every closed surface is zero. According to Stokes' theorem, f curl E • dt = i E • d\, where the integral on the right is taken over the closed contour bounding the surface over t Maxwell's equations (the fundamental equations of electrodynamics) were first formulated by him in the 1860's. 66 § 27 THE ACTION FUNCTION OF THE ELECTROMAGNETIC FIELD 67 which the left side is integrated. From (26.1) we find, integrating both sides for any surface, &E-dl=- 1 -j(H'dt. (26.4) The integral of a vector over a closed contour is called the circulation of the vector around the contour. The circulation of the electric field is also called the electromotive force in the given contour. Thus the electromotive force in any contour is equal to minus the time derivative of the magnetic flux through a surface bounded by this contour. The Maxwell equations (26.1) and (26.2) can be expressed in four-dimensional notation. Using the definition of the electromagnetic field tensor F ik = dA k /dx i -dA i ldx k , it is easy to verify that d_F* + dFja + d_F_ u dx l dx l dx' + ^« + ^»,0. (26.5) The expression on the left is a tensor of third rank, which is antisymmetric in all three indices. The only components which are not identically zero are those with i^k^ I Thus there are altogether four different equations which we can easily show [by substituting from (23.5)] coincide with equations (26.1) and (26.2). We can construct the four- vector which is dual to this antisymmetric four-tensor of rank three by multiplying the tensor by e iklm and contracting on three pairs of indices (see § 6). Thus (26.5) can be written in the form dFim = 0, (26.6) dx k which shows explicitly that there are only three independent equations. § 27. The action function of the electromagnetic field The action function S for the whole system, consisting of an electromagnetic field as well as the particles located in it, must consist of three parts : S = S f + S m + S mf , (27.1) where S m is that part of the action which depends only on the properties of the particles, that is, just the action for free particles. For a single free particle, it is given by (8.1). If there are several particles, then their total action is the sum of the actions for each of the individual particles. Thus, s m =-Y, mc \ ds - ( 27 - 2 > The quantity S mf is that part of the action which depends on the interaction between the particles and the field. According to § 16, we have for a system of particles: s M 7=-E ! [ A ^ k - ( 27 - 3 ) 68 THE ELECTROMAGNETIC FIELD EQUATIONS § 27 In each term of this sum, A k is the potential of the field at that point of spacetime at which the corresponding particle is located. The sum S m +S mf is already familiar to us as the action (16.1) for charges in a field. Finally S f is that part of the action which depends only on the properties of the field itself, that is, S f is the action for a field in the absence of charges. Up to now, because we were interested only in the motion of charges in a given electromagnetic field, the quantity S f , which does not depend on the particles, did not concern us, since this term cannot affect the motion of the particles. Nevertheless this term is necessary when we want to find equations determining the field itself. This corresponds to the fact that from the parts s m+ s mf of the action we found only two equations for the field, (26.1) and (26.2), which are not yet sufficient for complete determination of the field. To establish the form of the action S f for the field, we start from the following very important property of electromagnetic fields. As experiment shows, the electromagnetic field satisfies the so-called principle of superposition. This principle consists in the statement that the field produced by a system of charges is the result of a simple composition of the fields produced by each of the particles individually. This means that the resultant field intensity at each point is equal to the vector sum of the individual field intensities at that point. Every solution of the field equations gives a field that can exist in nature. According to the principle of superposition, the sum of any such fields must be a field that can exist in nature, that is, must satisfy the field equations. As is well known, linear differential equations have just this property, that the sum of any solutions is also a solution. Consequently the field equations must be linear differential equations. From the discussion, it follows that under the integral sign for the action S f there must stand an expression quadratic in the field. Only in this case will the field equations be linear; the field equations are obtained by varying the action, and in the variation the degree of the expression under the integral sign decreases by unity. The potentials cannot enter into the expression for the action S f , since they are not uniquely determined (in S mf this lack of uniqueness was not important). Therefore S f must be the integral of some function of the electromagnetic field tensor F ik . But the action must be a scalar and must therefore be the integral of some scalar. The only such quantity is the product F ik F ik .1[ Thus S f must have the form: S f = a J j F ik F ik dVdt, dV = dx dy dz, where the integral extends over all of space and the time between two given moments ; a is some constant. Under the integral stands F ik F ik =2(H 2 -E 2 ). The field E contains the derivative dA/dt; but it is easy to see that (dA/dt) 2 must appear in the action with the positive sign (and therefore E 2 must have a positive sign). For if (dA/t) 2 appeared in S f f The function in the integrand of S f must not include derivatives of F lk , since the Lagrangian can contain, aside from the coordinates, only their first time derivatives. The role of "coordinates" (i.e., parameters to be varied in the principle of least action) is in this case played by the field potential A k ; this is analogous to the situation in mechanics where the Lagrangian of a mechanical system contains only the coordinates of the particles and their first time derivatives. As for the quantity e mm F ik F lm (§ 25), as pointed out in the footnote on p. 63, it is a complete four- divergence, so that adding it to the integrand in S f would have no effect on the "equations of motion". It is interesting that this quantity is already excluded from the action for a reason independent of the fact that it is a pseudoscalar and not a true scalar. § 28 THE FOUR-DIMENSIONAL CURRENT VECTOR 69 with a minus sign, then sufficiently rapid change of the potential with time (in the time interval under consideration) could always make S f a negative quantity with arbitrarily large absolute value. Consequently S f could not have a minimum, as is required by the principle of least action. Thus, a must be negative. The numerical value of a depends on the choice of units for measurement of the field. We note that after the choice of a definite value for a and for the units of measurement of field, the units for measurement of all other electromagnetic quantities are determined. From now on we shall use the Gaussian system of units; in this system a is a dimension- less quantity, equal to — (l/167i).f Thus the action for the field has the form S f = f F ik F ik dCl, dQ = cdt dx dy dz. (27 A) J lone J In three-dimensional form : S f = i- f (E 2 -H 2 ) dVdt. (27.5) In other words, the Lagrangian for the field is L f = ^j(E 2 -H 2 )dV. (27.6) The action for field plus particles has the form S = - £ f meds- £ (- c A k dx k - ^ J* F ik F ik dQ. (27.7) We note that now the charges are not assumed to be small, as in the derivation of the equation of motion of a charge in a given field. Therefore A k and F ik refer to the actual field, that is, the external field plus the field produced by the particles themselves; A k and F ik now depend on the positions and velocities of the charges. § 28. The four-dimensional current vector Instead of treating charges as points, for mathematical convenience we frequently consider them to be distributed continuously in space. Then we can introduce the "charge density" q such that odVis the charge contained in the volume dV. The density q is in general a function of the coordinates and the time. The integral of q over a certain volume is the charge contained in that volume. Here we must remember that charges are actually pointlike, so that the density q is zero everywhere except at points where the point charges are located, and the integral J gdV must be equal to the sum of the charges contained in the given volume. Therefore q can be f In addition to the Gaussian system, one also uses the Heaviside system, in which a = — J. In this system of units the field equations have a more convenient form (4n does not appear) but on the other hand, n appears in the Coulomb law. Conversely, in the Gaussian system the field equations contain 4n, but the Coulomb law has a simple form. 70 THE ELECTROMAGNETIC FIELD EQUATIONS § 28 expressed with the help of the (5-function in the following form: = Ie«<5(r-r fl ) (28.1) a where the sum goes over all the charges and r a is the radius vector of the charge e a . The charge on a particle is, from its very definition, an invariant quantity, that is, it does not depend on the choice of reference system. On the other hand, the density q is not generally an invariant — only the product q dV is invariant. Multiplying the equality de = gdV on both sides with dx l : dx l de dx l = gdVdx 1 — gdVdt — . dt On the left stands a four-vector (since de is a scalar and dx l is a four-vector). This means that the right side must be a four-vector. But dVdt is a scalar, and so Q{dx l ldt) is a four- vector. This vector (we denote it by/) is called the current four-vector: . dx l J>= Q -. (28.2) The space components of this vector form a vector in ordinary space, J = Q\, (28.3) where v is the velocity of the charge at the given point. The vector j is called the current t The ^-function S(x) is defined as follows: 8(x) = 0, for all nonzero values of x; for x = 0, 3(0) = oo, in such a way that the integral + oo f S(x)dx = l. (I) — 00 From this definition there result the following properties: iff(x) is any continuous function, then + 00 j f(x) S(x-a) dx =f(a), (II) — 00 and in particular, + oo j f(x)d(x)dx = f(0). (Ill) (The limits of integration, it is understood, need not be ± co ; the range of integration can be arbitrary, provided it includes the point at which the <5-function does not vanish.) The meaning of the following equalities is that the left and right sides give the same result when introduced as factors under an integral sign: S(-x) = S(x), d(ax) = 1-. S(x). (IV) M The last equality is a special case of the more general relation Mx)] = V u ^.6(x-a t ), (V) where <l>(x) is a single-valued function (whose inverse need not be single-valued) and the a ( are the roots of the equation <j>(x) = 0. Just as S(x) was defined for one variable x, we can introduce a three-dimensional ^-function, d(r), equal to zero everywhere except at the origin of the three-dimensional coordinate system, and whose integral over all space is unity. As such a function we can clearly use the product S(x) d(y) 8(z). § 29 THE EQUATION OF CONTINUITY 71 density vector. The time component of the current four- vector is cq. Thus / = (q?,j). (28.4) The total charge present in all of space is equal to the integral J odV over all space. We can write this integral in four-dimensional form : f QdV = i jfdV = - c jj'dSt, (28.5) where the integral is taken over the entire four-dimensional hyperplane perpendicular to the x° axis (clearly this integration means integration over the whole three-dimensional space). Generally, the integral IJfdSi over an arbitrary hypersurface is the sum of the charges whose world lines pass through this surface. Let us introduce the current four- vector into the expression (27.7) for the action and transform the second term in that expression. Introducing in place of the point charges e a continuous distribution of charge with density q, we must write this term as - - J QA t dx*dV, replacing the sum over the charges by an integral over the whole volume. Rewriting in the form -M dx l q — A t dVdt, dt we see that this term is equal to Thus the action S takes the form S = - £ [mcds-\ f AifdQ- — — f F ik F ik dQ. (28.6) § 29. The equation of continuity The change with time of the charge contained in a certain volume is determined by the derivative dt] gdV. On the other hand, the change in unit time, say, is determined by the quantity of charge which in unit time leaves the volume and goes to the outside or, conversely, passes to its interior. The quantity of charge which passes in unit time though the element di of the surface bounding our volume is equal to oy • di, where v is the velocity of the charge at the point in space where the element di is located. The vector di is directed, as always, along the external normal to the surface, that is, along the normal toward the outside of the volume under consideration. Therefore q\ • di is positive if charge leaves the volume, and negative if charge enters the volume. The total amount of charge leaving the given volume per 72 THE ELECTROMAGNETIC FIELD EQUATIONS § 29 unit time is consequently § g\ ■ df, where the integral extends over the whole of the closed surface bounding the volume. From the equality of these two expressions, we get -J gdV = -<f>gvdf. (29.1) The minus sign appears on the right, since the left side is positive if the total charge in the given volume increases. The equation (29.1) is the so-called equation of continuity, expressing the conservation of charge in integral form. Noting that gv is the current density, we can rewrite (29.1) in the form -J QdV = -j>ydf. (29.2) We also write this equation in differential form. To do this we apply Gauss' theorem to (29.2): j>ydf= jdivjdV. and we find JH+g)"-* Since this must hold for integration© ver an arbitrary volume, the integrand must be zero : ,. . do. i+ dt = (29,3) This is the equation of continuity in differential form. It is easy to check that the expression (28.1) for g in <5-function form automatically satisfies the equation (29.3). For simplicity we assume that we have altogether only one charge, so that Q = ed(r-r ). The current j is then j = ev<5(r-r ), where v is the velocity of the charge. We determine the derivative dg/dt. During the motion of the charge its coordinates change, that is, the vector r changes. Therefore dg dg dr dt dr dt ' But dr /dt is just the velocity v of the charge. Furthermore, since q is a function of r— r , dg dg dr dr' Consequently — = - v • grad g = -div (gv) ot (the velocity v of the charge of course does not depend on r). Thus we arrive at the equation (29.3). It is easily verified that, in four-dimensional form, the continuity equation (29.3) is expressed by the statement that the four-divergence of the current four- vector is zero : S-°- (29 - 4) 30 THE SECOND PAIR OF MAXWELL EQUATIONS 73 In the preceding section we saw that the total charge present in all of space can be written as 1 r . l dS, -M where the integration is extended over the hyperplane x° = const. At each moment of time, the total charge is given by such an integral taken over a different hyperplane perpendicular to the x° axis. It is easy to verify that the equation (29.4) actually leads to conservation of charge, that is, to the result that the integral jfdSt is the same no matter what hyperplane x° = const we integrate over. The difference between the integrals jfdSi taken over two such hyperplanes can be written in the form § fdSi, where the integral is taken over the whole closed hypersurface surrounding the four-volume between the two hyperplanes under consideration (this integral differs from the required integral because of the presence of the integral over the infinitely distant "sides" of the hypersurface which, however, drop out, since there are no charges at infinity). Using Gauss' theorem (6.15) we can transform this to an integral over the four-volume between the two hyperplanes and verify that <£fdS i =(^ i dQ = 0. (29.5) The proof presented clearly remains valid also for any two integrals $j l dS it in which the integration is extended over any two infinite hypersurfaces (and not just the hyperplanes x° = const) which each contain all of three-dimensional space. From this it follows that the integral -JfdSt is actually identical in value (and equal to the total charge in space) no matter over what such hypersurface the integration is taken. We have already mentioned (see the footnote on p. 50) the close connection between the gauge invariance of the equations of electrodynamics and the law of conservation of charge. Let us show this once again using the expression for the action in the form (28.6). On re- placing A t by Ai-idfjdx 1 ), the integral df */' *>" is added to the second term in this expression. It is precisely the conservation of charge, as expressed in the continuity equation (29.4), that enables us to write the integrand as a four- divergence difj^/dx 1 , after which, using Gauss' theorem, the integral over the four-volume is transformed into an integral over the bounding hypersurface ; on varying the action, these integrals drop out and thus have no effect on the equations of motion. § 30. The second pair of Maxwell equations In finding the field equations with the aid of the principle of least action we must assume the motion of the charges to be given and vary only the potentials (which serve as the "coordinates" of the system) ; on the other hand, to find the equations of motion we assumed the field to be given and varied the trajectory of the particle. 74 THE ELECTROMAGNETIC FIELD EQUATIONS § 30 Therefore the variation of the first term in (28.6) is zero, and in the second we must not vary the current j \ Thus, 3s =-!l{l jisA < + l FiksF *} d *=°' (where we have used the fact that F ik 5F ik = F ik SF ik ). Substituting F ik = dA k jdx i -dA i l8x k we have ss —S\ {^< + 1 p,t h **- 1 F,k i? sa ) *>■ In the second term we interchange the indices i and k, over which the expressions are summed, and in addition replace F ik by -F ik . Then we obtain ts —S-c{^ SA --h F 'i? i 4 da - The second of these integrals we integrate by parts, that is, we apply Gauss' theorem: 1 r (1 1 dF ik "\ 1 r 5S = - -c J {c f+ fe &?} &A ' dCl - ^ J f ' t5A ' dS > ■ f 30 - 1 ) In the second term we must insert the values at the limits of integration. The limits for the coordinates are at infinity, where the field is zero. At the limits of the time integration, that is, at the given initial and final time values, the variation of the potentials is zero, since in accord with the principle of least action the potentials are given at these times. Thus the second term in (30.1) is zero, and we find Kl . 1 3F ik \ e -c i+ ^) 5A > dn = °- Since according to the principle of least action, the variations SA t are arbitrary, the co- efficients of the 8 A i must be set equal to zero: dF ik 4tt .. dx /• (30.2) Let us express these four (/ = 0, 1, 2, 3) equations in three-dimensional form. For i = 1 : dF 11 dF 12 8F 13 1 dF 10 An ^ dx dy dz c dt ~ c 3 Substituting the values for the components of F ik , we find dH z dH y 1 8E X _ An , dy dz c dt ~ c This together with the two succeeding equations (i = 2, 3) can be written as one vector equation : .__ IdE An. curlH = -- + -j. (30.3) Finally, the fourth equation (/ = 0) gives div E = AnQ. (30.4) Equations (30.3) and (30.4) are the second pair of Maxwell equations.f Together with the t The Maxwell equations in a form applicable to point charges in the electromagnetic field in vacuum were formulated by Lorentz. § 31 ENERGY DENSITY AND ENERGY FLUX 75 first pair of Maxwell equations they completely determine the electromagnetic field, and are the fundamental equations of the theory of such fields, i.e. of electrodynamics. Let us write these equations in integral form. Integrating (30.4) over a volume and applying Gauss' theorem J div EdV = j>E-df, we get JE-df = 4n j odV. (30.5) Thus the flux of the electric field through a closed surface is equal to 4n times the total charge contained in the volume bounded by the surface. Integrating (30.3) over an open surface and applying Stokes' theorem J curl H • di = j> H • d\, we find The quantity 1 dE (30.7) 4n dt is called the "displacement current". From (30.6) written in the form l-'-T/Kf)-* we see that the circulation of the magnetic field around any contour is equal to 4n/c times the sum of the true current and displacement current passing through a surface bounded by this contour. From the Maxwell equations we can obtain the already familiar continuity equation (29.3). Taking the divergence of both sides of (30.3), we find n r r 4;r ,. . div curl H = - — div E+ — div j. cdt c But div curl H = and div E = 4n Q , according to (30.4). Thus we arrive once more at equation (29.3). In four-dimensional form, from (30.2), we have: d 2 F ik ^ Andf^ dx l dx l = ~ 7 fa 1 ' But when the operator d 2 ldx i dx k , which is symmetric in the indices i and k, is applied to the antisymmetric tensor F ik , it gives zero identically and we arrive at the continuity equation (29.4) expressed in four-dimensional form. § 31. Energy density and energy flux Let us multiply both sides of (30.3) by E and both sides of (26.1) by H and combine the resultant equations. Then we get i E - — +-H- — = --j-E-(H-curlE-E-curlH). c dt c dt c 76 THE ELECTROMAGNETIC FIELD EQUATIONS § 31 Using the well-known formula of vector analysis, div (a x b) = b • curl a-a • curl b, we rewrite this relation in the form ^|(E* + H*)=-^j-E-div(ExH) or The vector d^/E 2 +H 2 /E z +H z \ r^/ = ~ j ' E ~ divS - (3L1) S = Z ExH (31-2) 4k is called the Poynting vector. We integrate (31.1) over a volume and apply Gauss' theorem to the second term on the right. Then we obtain d ce 2 +h 2 T r r etj^r dv =-)i- Edv -r' df ' < 31 - 3 > If the integral extends over all space, then the surface integral vanishes (the field is zero at infinity). Furthermore, we can express the integral Jj • EdV as a sum E e\ • E over all the charges, and substitute from (17.7): r /» 172 1 tt2 -J Then (31.3) becomes d_ f CE 2 +H 2 dt Thus for the closed system consisting of the electromagnetic field and particles present in it, the quantity in brackets in this equation is conserved. The second term in this expression is the kinetic energy (including the rest energy of all the particles; see the footnote on p. 48), the first term is consequently the energy of the field itself. We can therefore call the quantity w E 2 +H 2 W = ^n~ < 3L5 > the energy density of the electromagnetic field; it is the energy per unit volume of the field. If we integrate over any finite volume, then the surface integral in (31.3) generally does not vanish, so that we can write the equation in the form d c r f 2 4- ff 2 1 r *{j-»r dy+ Z'"'r -$*•*• (3L6) where now the second term in the brackets is summed only over the particles present in the volume under consideration. On the left stands the change in the total energy of field and particles per unit time. Therefore the integral j S • di must be interpreted as the flux of field energy across the surface bounding the given volume, so that the Poynting vector S is this flux density — the amount of field energy passing through unit area of the surface in unit time.f t We assume that at the given moment there are no charges on the surface itself. If this were not the case, then on the right we would have to include the energy flux transported by particles passing through the surface. § 32 THE ENERGY-MOMENTUM TENSOR 77 § 32. The energy-momentum tensor In the preceding section we derived an expression for the energy of the electromagnetic field. Now we derive this expression, together with one for the field momentum, in four- dimensional form. In doing this we shall for simplicity consider for the present an electro- magnetic field without charges. Having in mind later applications (to the gravitational field), and also to simplify the calculation, we present the derivation in a general form, not specializing the nature of the system. So we consider any system whose action integral has the form S = f A (q, p-\ dVdt = -[ AdQ, (32.1) where A is some function of the quantities q, describing the state of the system, and of their first derivatives with respect to coordinates and time (for the electromagnetic field the components of the four-potential are the quantities q) ; for brevity we write here only one of the #'s. We note that the space integral J A dVis the Lagrangian of the system, so that A can be considered as the Lagrangian "density'". The mathematical expression of the fact that the system is closed is the absence of any explicit dependence of A on the x\ similarly to the situation for a closed system in mechanics, where the Lagrangian does not depend explicitly on the time. The "equations of motion" (i.e. the field equations, if we are dealing with some field) are obtained in accordance with the principle of least action by varying S. We have (for brevity we write q t ,• = dqfdx 1 ), 1 C [dA c d (dA \ c d dA da = o. The second term in the integrand, after transformation by Gauss' theorem, vanishes upon integration over all space, and we then find the following "equations of motion" : d dA dA a?arr^"° (32 - 2) (it is, of course, understood that we sum over any repeated index). The remainder of the derivation is similar to the procedure in mechanics for deriving the conservation of energy. Namely, we write: dA _ dA dq dA dq^ k dx { dq dx l dq t k dx l ' Substituting (32.2) and noting that q, k ,i = Q, i, *> we find dA__d_ /dA\ dA_ dq^ _ d_ f dA\ dx l ~ dx k \dq t J q ' i+ dq tk dx k " dx k V' 1 dqj' On the other hand, we can write so that, introducing the notation dx l l dx" T k i=q,i^--d\A, (32.3) n = I^^)-^A. (32.5) 78 THE ELECTROMAGNETIC PlELt) EQUATIONS § 32 we can express the relation in the form dT k We note that if there is not one but several quantities q (l \ then in place of (32.3) we must write 8A But in § 29 we saw that an equation of the form dA k /dx k = 0, i.e. the vanishing of the four-divergence of a vector, is equivalent to the statement that the integral j A k dS k of the vector over a hypersurface which contains all of three-dimensional space is conserved. It is clear that an analogous result holds for the divergence of a tensor; the equation (32.4) asserts that the vector P l = const T lk dS k is conserved. This vector must be identified with the four-vector of momentum of the system. We choose the constant factor in front of the integral so that, in accord with our previous definition, the time component P° is equal to the energy of the system multiplied by l/c. To do this we note that P° = const j T 0k dS k = const j T 00 dV if the integration is extended over the hyperplane x° = const. On the other hand, according to (32.3), oo - dA * (,-H T oo = 4 __ A . ^ Comparing with the usual formulas relating the energy and the Lagrangian, we see that this quantity must be considered as the energy density of the system, and therefore \ T 00 dV is the total energy of the system. Thus we must set const = l/c, and we get finally for the four-momentum of the system the expression -\i T ik dS k . (32.6) The tensor T ik is called the energy-momentum tensor of the system. It is necessary to point out that the definition of the tensor T lk is not unique. In fact, if T ik is defined by (32.3), then any other tensor of the form T ik + —. xl/ m , \ji m = - y/ ilk (32.7) 8x will also satisfy equation (32.4), since we have identically d 2 ij/ ikl /dx k dx l = 0. The total four- momentum of the system does not change, since according to (6.17) we can write where the integration on the right side of the equation is extended over the (ordinary) surface which "bounds" the hypersurface over which the integration on the left is taken. This surface is clearly located at infinity in the three-dimensional space, and since neither field nor particles are present at infinity this integral is zero. Thus the four-momentum of the system is, as it must be, a uniquely determined quantity. To define the tensor T ik uniquely we can use the M § 32 THE ENERGY-MOMENTUM TENSOR 79 requirement that the four-tensor of angular momentum (see § 14) of the system be expressed in terms of the four-momentum by ik = J* (x i dP k -x k dP i ) = - f (x l T kl -x*T il )dS h (32.8) that is its "density" is expressed in terms of the "density" of momentum by the usual formula. It is easy to determine what conditions the energy-momentum tensor must satisfy in order that this be valid. We note that the law of conservation of angular momentum can be expressed, as we already know, by setting equal to zero the divergence of the expression under the integral sign in M ik . Thus — l (x i T kl -x k T il ) = 0. Noting that dx l jdx l = 5\ and that dT kl /dx l = 0, we find from this byr^-d^T" = T ki -T ik = o or T tk = T ki^ (32 9) that is, the energy-momentum tensor must be symmetric. We note that T lk , defined by formula (32.5), is generally speaking not symmetric, but can be made so by transformation (32.7) with suitable \j/ ikl . Later on (§ 94) we shall see that there is a direct method for obtaining a symmetric tensor T lk . As we mentioned above, if we carry out the integration in (32.6) over the hyperplane x° = const, then P ' takes on the form dV, (32.10) where the integration extends over the whole (three-dimensional) space. The space com- ponents of P ' form the three-dimensional momentum vector of the system and the time component is its energy multiplied by l/c. Thus the vector with components 111 - T 10 z. T" 20 - T 30 c c c may be called the "momentum density", and the quantity W = T 00 the "energy density". To clarify the meaning of the remaining components of T lk , we separate the conservation equation (32.4) into space and time parts : 1 dT 00 dT 0a 1 dT a0 dT aP ~ -^ + TT = °» ~ -T— + ^Tff- = 0. (32.11) c dt dx a c dt dx p v J We integrate these equations over a volume V in space. From the first equation \U T oo dv+ f^ dv = cdt J J dx« or, transforming the second integral by Gauss' theorem, - J T 00 dV =~c& T 0a df a , (32.12) 80 THE ELECTROMAGNETIC FIELD EQUATIONS § 33 where the integral on the right is taken over the surface surrounding the volume V {df x , df y , df z are the components of the three-vector of the surface element di). The expression on the left is the rate of change of the energy contained in the volume V; from this it is clear that the expression on the right is the amount of energy transferred across the boundary of the volume V, and the vector S with components cT 01 , cT°\ cT 03 is its flux density — the amount of energy passing through unit surface in unit time. Thus we arrive at the important conclusion that the requirements of relativistic invariance, as expressed by the tensor character of the quantities T ik , automatically lead to a definite connection between the energy flux and the momentum flux : the energy flux density is equal to the momentum flux density multiplied by c 2 . From the second equation in (32.11) we find similarly: d_ di f 1 T *o d y = - (f) T#df p . (32.13) On the left is the change of the momentum of the system in volume Fper unit time, therefore § T aP dfp is the momentum emerging from the volume Fper unit time. Thus the components T ap of the energy-momentum tensor constitute the three-dimensional tensor of momentum flux density — the so-called stress tensor; we denote it by a aP (a, /? = x, y, z). The energy flux density is a vector; the density of flux of momentum, which is itself a vector, must obviously be a tensor (the component <7 a/} of this tensor is the amount of the a-component of the momentum passing per unit time through unit surface perpendicular to the x p axis). We give a table indicating the meanings of the individual components of the energy- momentum tensor: (32.14) § 33. Energy-momentum tensor of the electromagnetic field We now apply the general relations obtained in the previous section to the electromagnetic field. For the electromagnetic field, the quantity standing under the integral sign in (32.1) is equal, according to (27.4), to F„F kl . The quantities q are the components of the four-potential of the field, A k , so that the definition (32.5) of the tensor T k becomes OX '»(§) § 33 ENERGY-MOMENTUM TENSOR OF THE ELECTROMAGNETIC FIELD 81 To calculate the derivatives of A which appear here, we find the variation <5A. We have 8tt *' 8tt \ dx k dx l J or, interchanging indices and making use of the fact that F kl = —F lk , 4n dx K From this we see that eA -If-. d (SA,\ 4n \dx k and therefore or, for the contravariant components : 1 8A l 1 But this tensor is not symmetric. To symmetrize it we add the quantity - — J*. 4tt dx t '* According to the field equation (30.2) in the absence of charges, dF\jdx l = 0, and therefore A%dx l l Andx lK h so that the change made in T ik is of the form (32.7) and is admissible. Since dA l /dx t — dA'/dXi = F a , we get finally the following expression for the energy-momentum tensor of the electromagnetic field : T ik = £ (-F il F * +i g ik F lm F lm ). (33.1) This tensor is obviously symmetric. In addition it has the property that T\ = 0, (33.2) i.e. the sum of its diagonal terms is zero. Let us express the components of the tensor T ik in terms of the electric and magnetic field intensities. By using the values (23.5) for the components F ik , we easily verify that the quantity T 00 coincides with the energy density (31.5), while the components cT 0a are the same as the components of the Poynting vector S (31.2). The space components T ap form a three-dimensional tensor with components a " = ^( E2 y + E *- E * + H2 y + H l- H2 x)> <r xy =--(E x E y +H x H y ), 82 THE ELECTROMAGNETIC FIELD EQUATIONS § 33 etc., or a « = ^{- E ^-H a H p + id al} (E 2 + H 2 )}. (33.3) This tensor is called the Maxwell stress tensor. To bring the tensor T ik to diagonal form, we must transform to a reference system in which the vectors E and H (at the given point in space and moment in time) are parallel to one another or where one of them is equal to zero; as we know (§ 25), such a transformation is always possible except when E and H are mutually perpendicular and equal in magnitude. It is easy to see that after the transformation the only non-zero components of T ik will be T 00 = — T 11 = T 22 = T 33 = W (the x axis has been taken along the direction of the field). But if the vectors E and H are mutually perpendicular and equal in magnitude, the tensor T lk cannot be brought to diagonal form.f The non-zero components in this case are T 00 = T 33 = T 30 = W (where the x axis is taken along the direction of E and the y axis along H). Up to now we have considered fields in the absence of charges. When charged particles are present, the energy-momentum tensor of the whole system is the sum of the energy- momentum tensors for the electromagnetic field and for the particles, where in the latter the particles are assumed not to interact with one another. To determine the form of the energy-momentum tensor of the particles we must describe their mass distribution in space by using a "mass density" in the same way as we describe a distribution of point charges in terms of their density. Analogously to formula (28.1) for the charge density, we can write the mass density in the form /x = Xm fl <5(r-r fl ), (33.4) a where r a are the radius-vectors of the particles, and the summation extends over all the particles of the system. The "four-momentum density" of the particles is given by pLcu t . We know that this density is the component T 0x /c of the energy-momentum tensor, i.e. T 0a = n c 2 u\a =1,2, 3). But the mass density is the time component of the four-vector pi/c (dx k /dt) (in analogy to the charge density; see § 28). Therefore the energy-momentum tensor of the system of non- interacting particles is . t dx l dx k , . ds T lk = nc — — = pi c u l u k -. (33.5) ds dt dt As expected, this tensor is symmetric. We verify by a direct computation that the energy and momentum of the system, defined as the sum of the energies and momenta of field and particles, are actually conserved. In other words we shall verify the equations. i_ (T (/)fe +T (P)/c )==0} (336) dx k which express these conservation laws. t The fact that the reduction of the symmetric tensor T ik to principal axes may be impossible is related to the fact that the four-space is pseudo-euclidean. (See also the problem in § 94.) § 33 ENERGY-MOMENTUM TENSOR OF THE ELECTROMAGNETIC FIELD 83 Differentiating (33.1), we write ST if)k t 1 n rlm 8F lm 5F, u dF kl 4n\2 dx l dx k dx k ll )' dx k Substituting from the Maxwell equations (26.5) and (30.2), aF w = 4^r, dF^^_dF^_dF« dx k c 3, dx l dx l dx m ' we have: dT<™ t 1 / 1 dF mi . 1 BF„ .. dF„ _ 4k dx k = L(_ ld J™ F lm_ 1?*U F lm__ ?Ftt F H_ 4* p ,\ 4n\ 2 dx l 2dx m 8x k c ilJ / By permuting the indices, we easily show that the first three terms on the right cancel one another, and we arrive at the result: J ~=-~F ik j\ (33.7) ox k c Differentiating the expression (33.5) for the energy-momentum tensor of the particles.gives dT {p) \ d ( dx k \ dx k du i )T {p)k t d ( dx k \ dx k du t -d^ = CUi d?{ li li) + ^ C ^d?- The first term in this expression is zero because of the conservation of mass for non- interacting particles. In fact, the quantities n(dx k /dt) constitute the "mass current" four- vector, analogous to the charge current four- vector (28.2); the conservation of mass is expressed by equating to zero the divergence of this four- vector: (33.8) d ( dx k \ rt just as the conservation of charge is expressed by equation (29.4). Thus we have : dT^ )k j _ (hfdUi_ dUi dx k ~ llC ~di~dx~ k ~ llC ~dt' Next we use the equation of motion of the charges in the field, expressed in the four- dimensional form (23.4). du t e k ds c Changing to continuous distributions of charge and mass, we have, from the definitions of the densities n and g : fi/m = gfe. We can therefore write the equation of motion in the form du { g n k or lxc-- = ~F ik u ds c n c ~n = - *ikQ u -j = ~ *ikJ ■ dt c ' dt c Thus, dT ( P )k x Combining this with (33.7), we find that we actually get zero, i.e. we arrive at equation (33.6). 84 THE ELECTROMAGNETIC FIELD EQUATIONS § 34 § 34. The virial theorem Since the sum of the diagonal terms of the energy-momentum tensor of the electro- magnetic field is equal to zero, the sum T\ for any system of interacting particles reduces to the trace of the energy-momentum tensor for the particles alone. Using (33.5), we therefore have: mi ~^ni; i ds ds , I V 2 Let us rewrite this result, shifting to a summation over the particles, i.e. writing /x as the sum (33.4). We then get finally: ri = Im fl c 2 7l-"f<5(r-r fl ). (34.1) a V C We note that, according to this formula, we have for every system: T\ > 0, (34.2) where the equality sign holds only for the electromagnetic field without charges. Let us consider a closed system of charged particles carrying out a finite motion, in which all the quantities (coordinates, momenta) characterizing the system vary over finite ranges, f We average the equation 1 dT aQ dT aP _ c dt dx p ~ [see (32.11)] with respect to the time. The average of the derivative dT a0 Jdt, like the average of the derivative of any bounded quantity, is zero. J Therefore we get — 7^ = We multiply this equation by x* and integrate over all space. We transform the integral by Gauss' theorem, keeping in mind that at infinity T p a = 0, and so the surface integral vanishes : ^3 dK =-JI?^ F =-Jv« dF = o ' or finally, j T a a dV = 0. (34.3) On the basis of this equality we can write for the integral of T\ = T%+T%: J T\dV = j* fldV = S, where $ is the total energy of the system. t Here we also assume that the electromagnetic field of the system also vanishes at infinity. This means that, if there is a radiation of electromagnetic waves by the system, it is assumed that special "reflecting walls" prevent these waves from going off to infinity. % Let/(/) be such a quantity. Then the average value of the derivative df/dt over a certain time interval Tis df_\ rdf f(T)-f(0) dt TJ dt T ' Since /(/) varies only within finite limits, then as T increases without limit, the average value of dffdt clearly goes to zero. § 35 THE ENERGY-MOMENTUM TENSOR FOR MACROSCOPIC BODIES 85 Finally, substituting (34.1) we get: <? = l™ a c 2 Jl-f 2 . (34.4) This relation is the relativistic generalization of the virial theorem of classical mechanics.! For low velocities, it becomes a a ■£ that is, the total energy (minus the rest energy) is equal to the negative of the average value of the kinetic energy — in agreement with the result given by the classical virial theorem for a system of charged particles (interacting according to the Coulomb law). § 35. The energy-momentum tensor for macroscopic bodies In addition to the energy-momentum tensor for a system of point particles (33.5), we shall also need the expression for this tensor for macroscopic bodies which are treated as being continuous. The flux of momentum through the element di of the surface of the body is just the force acting on this surface element. Therefore a aP df fi is the a-component of the force acting on the element. Now we introduce a reference system in which a given element of volume of the body is at rest. In such a reference system, Pascal's law is valid, that is, the pressure/? applied to a given portion of the body is transmitted equally in all directions and is everywhere perpendicular to the surface on which it acts.J Therefore we can write a a p df p = pdf a , so that the stress tensor is <r aP = pd a p. As for the components T a0 , which represent the momen- tum density, they are equal to zero for the given volume element in the reference system we are using. The component T 00 is as always the energy density of the body, which we denote by e; s/c 2 is then the mass density of the body, i.e. the mass per unit volume. We emphasize that we are talking here about the unit "proper" volume, that is, the volume in the reference system in which the given portion of the body is at rest. Thus, in the reference system under consideration, the energy-momentum tensor (for the given portion of the body) has the form: r - L ; - J (35.D Now it is easy to find the expression for the energy-momentum tensor in an arbitrary reference system. To do this we introduce the four-velocity u l for the macroscopic motion of an element of volume of the body. In the rest frame of the particular element, u l = (1, 0). The expression for T ik must be chosen so that in this reference system it takes on the form t See Mechanics, § 10. t Strictly speaking, Pascal's law is valid only for liquids and gases. However, for solid bodies the maximum possible difference in the stress in different directions is negligible in comparison with the stresses which can play a role in the theory of relativity, so that its consideration is of no interest. 86 THE ELECTROMAGNETIC FIELD EQUATIONS § 35 (35.1). It is easy to verify that this is T ik = (p+e)u i u k -pg ik , (35.2) or, for the mixed components, T k i =(p + e)u l u k -p%. This expression gives the energy-momentum tensor for a macroscopic body. The expressions for the energy density W, energy flow vector S and stress tensor <r aP are : v 2 W v 2 ' _v 2 ' 1_ ? 1_ ? (35.3) K) If the velocity v of the macroscopic motion is small compared with the velocity of light, then we have approximately: S = (p + e)v. Since S/c 2 is the momentum density, we see that in this case the sum (p + e)/c 2 plays the role of the mass density of the body. The expression for T ik simplifies in the case where the velocities of all the particles making up the body are small compared with the velocity of light (the velocity of the macroscopic motion itself can be arbitrary). In this case we can neglect, in the energy density e, all terms small compared with the rest energy, that is, we can write fi c 2 in place of e, where fi is the sum of the masses of the particles present in unit (proper) volume of the body (we emphasize that in the general case, n must differ from the actual mass density e/c 2 of the body, which includes also the mass corresponding to the energy of microscopic motion of the particles in the body and the energy of their interactions). As for the pressure determined by the energy of microscopic motion of the molecules, in the case under consideration it is also clearly small compared with the rest energy n c 2 . Thus we find T* = j* cVu\ (35.4) From the expression (35.2), we get T\ = e-3p. (35.5) The general property (34.2) of the energy-momentum tensor of an arbitrary system now shows that the following inequality is always valid for the pressure and density of a macro- scopic body: P<\. (35.6) Let us compare the relation (35.5) with the general formula (34.1) which we saw was valid for an arbitrary system. Since we are at present considering a macroscopic body, the expres- sion (34.1) must be averaged over all the values of r in unit volume. We obtain the result s-lp = l™ a c 2 Jl-4 (35.7) a 11 c (the summation extends over all particles in unit volume). § 35 THE ENERGY-MOMENTUM TENSOR FOR MACROSCOPIC BODIES 87 We apply our formula to an ideal gas, which we assume to consist of identical particles, since the particles of an ideal gas do not interact with one another, we can use formula (33.5) after averaging it. Thus for an ideal gas, T lk = nmc — - — , at as where n is the number of particles in unit volume and the dash means an average over all the particles. If there is no macroscopic motion in the gas then we can use for T lk the expression (35.1). Comparing the two formulas, we arrive at the equations: YlYYl e = nm / — \, p = —— I — \. (35.8) M These equations determine the density and pressure of a relativistic ideal gas in terms of the velocity of its particles ; the second of these replaces the well-known formula p = nmv 2 {2> of the nonrelativistic kinetic theory of gases. CHAPTER 5 CONSTANT ELECTROMAGNETIC FIELDS § 36. Coulomb's law For a constant electric, or as it is usually called, electrostatic field, the Maxwell equations have the form: divE = 47rp, (36.1) curl E = 0. (36.2) The electric field E is expressed in terms of the scalar potential alone by the relation E=-grad</>. (36.3) Substituting (36.3) in (36.1), we get the equation which is satisfied by the potential of a constant electric field : Acf)=-47ip. (36.4) This equation is called the Poisson equation. In particular, in vacuum, i.e., for q = 0, the potential satisfies the Laplace equation A<£ = 0. (36.5) From the last equation it follows, in particular, that the potential of the electric field can nowhere have a maximum or a minimum. For in order that <p have an extreme value, it would be necessary that the first derivatives of with respect to the coordinates be zero, and that the second derivatives d 2 <j)Jdx 2 , d 2 <j>/dy 2 , d 2 $\dz 2 all have the same sign. The last is impossible, since in that case (36.5) could not be satisfied. We now determine the field produced by a point charge. From symmetry considerations, it is clear that it is directed along the radius-vector from the point at which the charge e is located. From the same consideration it is clear that the value E of the field depends only on the distance jR from the charge. To find this absolute value, we apply equation (36.1) in the integral form (30.5). The flux of the electric field through a spherical surface of radius R circumscribed around the charge e is equal to 4nR 2 E; this flux must equal 4ne. From this we get In vector notation: R 2 E = — (36.6) R* Thus the field produced by a point charge is inversely proportional to the square of the 88 § 37 ELECTROSTATIC ENERGY OF CHARGES 89 distance from the charge. This is the Coulomb law. The potential of this field is, clearly, <£ = |. (36.7) If we have a system of charges, then the field produced by this system is equal, according to the principle of superposition, to the sum of the fields produced by each of the particles individually. In particular, the potential of such a field is a K a where R a is the distance from the charge e a to the point at which we are determining the potential. If we introduce the charge density q, this formula takes on the form -/ | dV, (36.8) where R is the distance from the volume element dV to the given point of the field. We note a mathematical relation which is obtained from (36.4) by substituting the values of q and 4> for a point charge, i.e. q = e <5(R) and = e/R. We then find aQ)=-4tt<5(R). (36.9) § 37. Electrostatic energy of charges We determine the energy of a system of charges. We start from the energy of the field, that is, from the expression (31.5) for the energy density. Namely, the energy of the system of charges must be equal to U = J- I E 2 dV, where E is the field produced by these charges, and the integral goes over all space. Sub- stituting E = — grad 0, U can be changed to the following form: U = - — J E-grad $ dV = - — J div (E</>) dV+ — f <f> div E dV. According to Gauss' theorem, the first integral is equal to the integral of E<£ over the surface bounding the volume of integration, but since the integral is taken over all space and since the field is zero at infinity, this integral vanishes. Substituting in the second integral, div E = 4tiq, we find the following expression for the energy of a system of charges: U = ijo(f>dV. (37.1) For a system of point charges, e a , we can write in place of the integral a sum over the charges tf = i5>.0.. (37.2) a where 4> a is the potential of the field produced by all the charges, at the point where the charge e a is located. 90 CONSTANT ELECTROMAGNETIC FIELDS § 37 If we apply our formula to a single elementary charged particle (say, an electron), and the field which the charge itself produces, we arrive at the result that the charge must have a certain "self-potential energy equal to ecj)/2, where (f) is the potential of the field produced by the charge at the point where it is located. But we know that in the theory of relativity every elementary particle must be considered as pointlike. The potential cj) = ejR of its field becomes infinite at the point R = 0. Thus according to electrodynamics, the electron would have to have an infinite "self-energy", and consequently also an infinite mass. The physical absurdity of this result shows that the basic principles of electrodynamics itself lead to the result that its application must be restricted to definite limits. We note that in view of the infinity obtained from electrodynamics for the self-energy and mass, it is impossible within the framework of classical electrodynamics itself to pose the question whether the total mass of the electron is electrodynamic (that is, associated with the electromagnetic self-energy of the particle). f Since the occurrence of the physically meaningless infinite self-energy of the elementary particle is related to the fact that such a particle must be considered as pointlike, we can conclude that electrodynamics as a logically closed physical theory presents internal con- tradictions when we go to sufficiently small distances. We can pose the question as to the order of magnitude of such distances. We can answer this question by noting that for the electromagnetic self-energy of the electron we should obtain a value of the order of the rest energy mc 2 . If, on the other hand, we consider an electron as possessing a certain radius R , then its self-potential energy would be of order e 2 /R . From the requirement that these two quantities be of the same order, e 2 /R ~ mc 2 , we find Ro ~ — 2 - (37.3) mc This dimension (the "radius" of the electron) determines the limit of applicability of electrodynamics to the electron, and follows already from its fundamental principles. We must, however, keep in mind that actually the limits of applicability of the classical electro- dynamics which is presented here lie much higher, because of the occurrence of quantum phenomena. J We now turn again to formula (37.2). The potentials cf) a which appear there are equal, from Coulomb's law, to *. = Eir» (37 - 4) K ab where R ab is the distance between the charges e a , e b . The expression for the energy (37.2) consists of two parts. First, it contains an infinite constant, the self-energy of the charges, not depending on their mutual separations. The second part is the energy of interaction of the charges, depending on their separations. Only this part has physical interest. It is equal to t/' = iZ e «&> (37.5) where &=Iir ( 37 - 6 ) b±a K*ab t From the purely formal point of view, the finiteness of the electron mass can be handled by introducing an infinite negative mass of nonelectromagnetic origin which compensates the infinity of the electromagnetic mass (mass "renormalization"). However, we shall see later (§ 75) that this does not eliminate all the internal contradictions of classical electrodynamics. % Quantum effects become important for distances of the order of h/mc, where h is Planck's constant. § 38 THE FIELD OF A UNIFORMLY MOVING CHARGE 91 is the potential at the point of location of e a , produced by all the charges other than e a . In other words, we can write ^' = ~Z~. (37.7) In particular, the energy of interaction of two charges is U' = - 1 . (37.8) *M2 § 38. The field of a uniformly moving charge We determine the field produced by a charge e, moving uniformly with velocity V. We call the laboratory frame the system K; the system of reference moving with the charge is the K' system. Let the charge be located at the origin of coordinates of the K' system. The system K' moves relative to K along the X axis ; the axes Y and Z are parallel to Y' and Z '. At the time / = the origins of the two systems coincide. The coordinates of the charge in the K system are consequently x = Vt, y = z = 0. In the K' system, we have a constant electric field with vector potential A' = 0, and scalar potential equal to 4>' = e/R', where R' 2 = x '2+y' 2 + z '2, in the K system, according to (24.1) for A' = 0, 6' e <f>= , = . (38.1) V 1 -^ "'J 1 ,2 We must now express R' in terms of the coordinates x, y, z, in the K system. According to the formulas for the Lorentz transformation from which x = ' — T72 ' y = y > z =z > J' (x-F0 2 +(l-^rW + z 2 ) R' 2 = ^ °J- (38.2) Substituting this in (38.1) we find R* where we have introduced the notation 1-^ c 2 4> = ^i (38.3) (-?) R* 2 = (x- Vt) 2 + 1- -y (y 2 + z 2 ). (38.4) The vector potential in the K system is equal to e c 7r*' V e\ A = <£- = -^. (38.5) 92 CONSTANT ELECTROMAGNETIC FIELDS § 38 In the K' system the magnetic field H' is absent and the electric field is From formula (24.2), we find ex' ^ E' y ey' Hd -w. — Hd -*. — , o i -Hi », — 2 V'-5 R,i J x - E = c ez' R» ./i-^ Substituting for R', x' , y', z\ their expressions in terms of x, y, z, we obtain / V 2 \ eR where R is the radius vector from the charge e to the field point with coordinates x, y, z (its components are x— Vt, y, z). This expression for E can be written in another form by introducing the angle 9 between the direction of motion and the radius vector R. It is clear that y 2 + z 2 = R 2 sin 2 0, and there- fore R* 2 can be written in the form: R* 2 Then we have for E, = R 2 ( 1 - ^j sin 2 e\ (38.7) v eR c 2 » 3 / v 2 \ 3 / 2 ' (38.8) R ^-^rsin 2 ^ For a fixed distance R from the charge, the value of the field E increases as 9 increases from to 7i/2 (or as 9 decreases from n to n/2). The field along the direction of motion ((9 = 0, 7t) has the smallest value ; it is equal to HI V Ey R 2 \ c 2 )' The largest field is that perpendicular to the velocity (9 = n/2), equal to v- R z I v 2 2 We note that as the velocity increases, the field E\\ decreases, while E x increases. We can describe this pictorially by saying that the electric field of a moving charge is "contracted" in the direction of motion. For velocities V close to the velocity of light, the denominator in formula (38.8) is close to zero in a narrow interval of values 9 around the value 9 = n/2. The "width" of this interval is, in order of magnitude, -v-s § 39 MOTION IN THE COULOMB FIELD 93 Thus the electric field of a rapidly moving charge at a given distance from it is large only in a narrow range of angles in the neighborhood of the equatorial plane, and the width of this interval decreases with increasing V like V 1 — (V 2 /c 2 ). The magnetic field in the K system is H = -VxE (38.9) c [see (24.5)]. In particular, for V <^c the electric field is given approximately by the usual formula for the Coulomb law, E = eR/R 3 , and the magnetic field is „ eVxR H = --^-, (38.10) PROBLEM Determine the force (in the K system) between two charges moving with the same velocity V. Solution: We shall determine the force F by computing the force acting on one of the charges (e^) in the field produced by the other (e 2 ). Using (38.9), we have F = e 1 E a +^VxH a = e 1 n-^E a +^VCV-E a ). Substituting for E 2 from (38.8), we get for the components of the force in the direction of motion (F x ) and perpendicular to it (F y ) : V 2 \ ( V 2 \ 2 1 jr- J cos I 1 =- ] sin R 2 / v 2 \ 3/2 ' R 2 / V 2 \ 3/2 ' 1 - -a sin 2 J ( ! - ~2 sin2 e where R is the radius vector from e 2 to ex, and is the angle between R and V. § 39. Motion in the Coulomb field We consider the motion of a particle with mass m and charge e in the field produced by a second charge e' ; we assume that the mass of this second charge is so large that it can be considered as fixed. Then our problem becomes the study of the motion of a charge e in a centrally symmetric electric field with potential </> = e'/r. The total energy $ of the particle is equal to = c\lp 2 + m 2 c 2 + -, a 5 r where a = ee'. If we use polar coordinates in the plane of motion of the particle, then as we know from mechanics, p* = (M 2 lr 2 ) + p 2 r , where p r is the radial component of the momentum, and M is the constant angular momen- tum of the particle. Then $ = c s Jp 2 +- ?r +m 2 c 2 +~. (39.1) 94 CONSTANT ELECTROMAGNETIC FIELDS § 39 We discuss the question whether the particle during its motion can approach arbitrarily close to the center. First of all, it is clear that this is never possible if the charges e and e' repel each other, that is, if e and e' have the same sign. Furthermore, in the case of attraction (e and e' of opposite sign), arbitrarily close approach to the center is not possible if Mc > |a|, for in this case the first term in (39.1) is always larger than the second, and for r-»0 the right side of the equation would approach infinity. On the other hand, if Mc < |a|, then as r -»• 0, this expression can remain finite (here it is understood that p r approaches infinity). Thus, if cM < |a|, (39.2) the particle during its motion "falls in" toward the charge attracting it, in contrast to non- relativistic mechanics, where for the Coulomb field such a collapse is generally impossible (with the exception of the one case M = 0, where the particle e moves on a line toward the particle e'). A complete determination of the motion of a charge in a Coulomb field starts most conveniently from the Hamilton-Jacobi equation. We choose polar coordinates r, (f>, in the plane of the motion. The Hamilton-Jacobi equation (16.11) has the form 1 fdS a\ 2 /3S\ 2 1 /dS\ 2 , , n -?{Yt + r) + U) + ?{ei) +mc=0 - We seek an S of the form S= -£t+M<t>+f{r\ where $ and M are the constant energy and angular momentum of the moving particle. The result is C / 1 / ^A 2 Jp S= -£t + M(j)+ \ J-2[£--) --2-m 2 c 2 dr. (39.3) The trajectory is determined by the equation dS/dM = const. Integration of (39.3) leads to the following results for the trajectory: (a) If Mc > \ot\, (c 2 M 2 - a 2 ) - = cJ(M£) 2 - m 2 c\M 2 c 2 - a 2 ) cos (<f> J 1 - -^—^\ - Sol. (39.4) (b) If Mc < |a|, (a 2 -MV) - = ±cV(M<f) 2 + m 2 c 2 (a 2 -M 2 c 2 ) cosh U J~^2~A +#<*- (39.5) (c) If Mc = |a|, s /2_ ro 2 c 4_^/'^\ m (39<6) r \cMJ The integration constant is contained in the arbitrary choice of the reference line for measurement of the angle $. In (39.4) the ambiguity of sign in front of the square root is unimportant, since it already contains the arbitrary reference origin of the angle $ under the cos. In the case of attraction (a < 0) the trajectory corresponding to this equation lies entirely at finite values of r (finite motion), if $ < mc 2 . If $ > mc 2 , then r can go to infinity (infinite motion). The finite motion corresponds to motion in a closed orbit (ellipse) in nonrelativistic mechanics. From (39.4) it is clear that in relativistic mechanics the trajectory can never be closed; when the angle $ § 39 MOTION IN THE COULOMB FIELD 95 changes by 2n, the distance r from the center does not return to its initial value. In place of ellipses we here get orbits in the form of open "rosettes". Thus, whereas in nonrelativistic mechanics the finite motion in a Coulomb field leads to a closed orbit, in relativistic mechanics the Coulomb field loses this property. In (39.5) we must choose the positive sign for the root in case a < 0, and the negative sign if a > [the opposite choice of sign would correspond to a reversal of the sign of the root in (39.1)]. For a < the trajectories (39.5) and (39.6) are spirals in which the distance r approaches as -^ oo. The time required for the "falling in" of the charge to the coordinate origin is finite. This can be verified by noting that the dependence of the coordinate r on the time is determined by the equation dS\dS = const; substituting (39.3), we see that the time is determined by an integral which converges for r -* 0. PROBLEMS 1. Determine the angle of deflection of a charge passing through a repulsive Coulomb field (a > 0). Solution: The angle of deflection x equals x = n—2fa, where ^ is the angle between the two asymptotes of the trajectory (39.4). We find 2cM , fv Vc 2 M 2 -a 2 \ X = n tan" 1 ( ), Vc 2 M 2 -a 2 \ ca J' where v is the velocity of the charge at infinity. 2. Determine the effective scattering cross section at small angles for the scattering of particles in a Coulomb field. Solution: The effective cross section da is the ratio of the number of particles scattered per second into a given element do of solid angle to the flux density of impinging particles (i.e., to the number of particles crossing one square centimeter, per second, of a surface perpendicular to the beam of particles). Since the angle of deflection x of the particle during its passage through the field is determined by the impact parameter q (i.e. the distance from the center to the line along which the particle would move in the absence of the field), da = InodQ = 2tiq ~-dx = Q dx dx sin x where do = In sin xdxA The angle of deflection (for small angles) can be taken equal to the ratio of the change in momentum to its initial value. The change in momentum is equal to the time integral of the force acting on the charge, in the direction perpendicular to the direction of motion- it is approximately (a/r 2 )/fe/r). Thus we have — CO oq dt 2a da 2 +v 2 t 2 ) 3 ! 2 P QV G (v is the velocity of the particles). From this we find the effective cross section for small x' \PvJ X* In the nonrelativistic case, p a* mv, and the expression coincides with the one obtained from the Rutherford formula:): for small x- t See Mechanics, § 18. t See Mechanics, § 19. 96 CONSTANT ELECTROMAGNETIC FIELDS §40 § 40. The dipole moment We consider the field produced by a system of charges at large distances, that is, at distances large compared with the dimensions of the system. We introduce a coordinate system with origin anywhere within the system of charges. Let the radius vectors of the various charges be r a . The potential of the field produced by all the charges at the point having the radius vector R is (the summation goes over all charges); here R -r a are the radius vectors from the charges e a to the point where we are finding the potential. We must investigate this expansion for large R (R > r a ). To do this, we expand it in powers of rJR , using the formula /(R -r)=/(R )-r-grad/(R ) (in the grad, the differentiation applies to the coordinates of the vector R ). To terms of first order, Ye 1 <£ = V ~ Z e « r a • grad — . (40.2) The sum d = £ e a r a (40.3) is called the dipole moment of the system of charges. It is important to note that if the sum of all the charges, X e a , is zero, then the dipole moment does not depend on the choice of the origin of coordinates, for the radius vectors r a and r' a of one and the same charge in two different coordinate systems are related by r' a = r a + a, where a is some constant vector. Therefore if E e a = 0, the dipole moment is the same in both systems : " d' = I e a r' a = Z ^4-a £ e a = d. If we denote by e+, r a + and e~, r~ the positive and negative charges of the system and their radius vectors, then we can write the dipole moment in the form d = Z « - £ e~ a x- a = R a + £ e fl + -R; £ e~ a (40.4) where R ~t^p R ~T^r (40 - 5) are the radius vectors of the "charge centers" for the positive and negative charges. If £ e„ = E e~ — e, then d = eR + _, (40.6) where R+_ = R + -R" is the radius vector from the center of negative to the center of positive charge. In particular, if we have altogether two charges, then R + _ is the radius vector between them. If the total charge of the system is zero, then the potential of the field of this system at large distances is , „ 1 d-R 4>=-d.V- = — J- . (40.7) § 41 MULTIPOLE MOMENTS 97 The field intensity is : or finally, E= _grad^?= -^grad(d-R )-(d-R )grad ^ K o *<o Ko _ 3(nd)n-d e=sJl -r| ' (40.8) where n is a unit vector along R . Another useful expression for the field is E = (d-V)V— . (40.9) Thus the potential of the field at large distances produced by a system of charges with total charge equal to zero is inversely proportional to the square of the distance, and the field intensity is inversely proportional to the cube of the distance. This field has axial symmetry around the direction of d. In a plane passing through this direction (which we choose as the z axis), the components of the vector E are : 3 cos 2 0-1 3sin0cos0 * - ' ~, , E x = d -3 . (40.10) R The radial and tangential components in this plane are , 2 cos sin E R = d-~ r -, E e ^-d— r . (40.11) § 41. Multipole moments In the expansion of the potential in powers of l/R , = 0<°> + 0< 1 > + (2) +..., (41.1) the term (j) (n) is proportional to 1/#S +1 . We saw that the first term, $ (0) , is determined by the sum of all the charges; the second term, <£ (1) , sometimes called the dipole potential of the system, is determined by the dipole moment of the system. The third term in the expansion is ^-^"•"'axfeGy- (4L2) where the sum goes over all charges; we here drop the index numbering the charges; x x are the components of the vector r, and X a those of the vector R . This part of the potential is usually called the quadrupole potential. If the sum of the charges and the dipole moment of the system are both equal to zero, the expansion begins with (f> (2) . In the expression (41.2) there enter the six quantities £ ex a x p . However, it is easy to see that the field depends not on six independent quantities, but only on five. This follows from the fact that the function \jR satisfies the Laplace equation, that is, A Uo/ 3 « dX.dX f \R ) = 0. aid of D ap , we can write 4> m = ^J^(U, (41.5) 98 CONSTANT ELECTROMAGNETIC FIELDS § 41 We can therefore write (2) in the form The tensor D aP = ^e(3x a x p -r 2 S ap ) (41.3) is called the quadrupole moment of the system. From the definition of D aP it is clear that the sum of its diagonal elements is zero : D aa = 0. (41.4) Therefore the symmetric tensor D xP has altogether five independent components. With the %e & ( 1 6 8X a dX p \R^ or, performing the differentiation, a 2 1 = 3X a X, S aP dX a aX p R R R and using the fact that S aP D ap = D aa = 0, * m - ^P- (41.6) ZK Like every symmetric three-dimensional tensor, the tensor D aP can be brought to principal axes. Because of (41.4), in general only two of the three principal values will be independent. If it happens that the system of charges is symmetric around some axis (the z axis)f then this axis must be one of the principal axes of the tensor D aP , the location of the other two axes in the x, y plane is arbitrary, and the three principal values are related to one another: D xx = D yy =-iD z2 . (41.7) Denoting the component D zz by D (in this case it is simply called the quadrupole moment), we get for the potential ^ (2> = i§3 ( 3 co§2 e ~ 1) = ^3 ^(cos 6), (41.8) 4K ZK where 6 is the angle between R and the z axis, and P 2 is a Legendre polynomial. Just as we did for the dipole moment in the preceding section, we can easily show that the quadrupole moment of a system does not depend on the choice of the coordinate origin, if both the total charge and the dipole moment of the system are equal to zero. In similar fashion we could also write the succeeding terms of the expansion (41.1). The /'th term of the expansion defines a tensor (which is called the tensor of the 2 z -pole moment) of rank /, symmetric in all its indices and vanishing when contracted on any pair of indices ; it can be shown that such a tensor has 21+ 1 independent components. J We shall express the general term in the expansion of the potential in another form, by using the well-known formula of the theory of spherical harmonics fir~7l = / D2 2 \ = =E^I A(cos xl (41-9) | R o-*| %/R% + r 2 -2rR cosx i=o R t We are assuming a symmetry axis of any order higher than the second. j Such a tensor is said to be irreducible. The vanishing on contraction means that no tensor of lower rank can be formed from the components. §41 MULTIPOLE MOMENTS 99 where i is the angle between R and r. We introduce the spherical angles ®, <D and 6, (j), formed by the vectors R and r, respectively, with the fixed coordinate axes, and use the addition theorem for the spherical harmonics : P l (cos X )= £ ^ I1 !^ p J m| ( cos0 ) P i W, ( cos ^ e " im(O "* ) ' ( 4L1 °) m =_i(/ + |m|)! where the P, M are the associated Legendre polynomials. We also introduce the spherical functionst yjp, & = (- m l s] 2 ^- l^j ^r(cos ey m \ m > o, Then the expansion (41.9) takes the form: 1 =E £ ^^^*(0,o)r, m (M). (41.11) |R -r| ^ 0m e_^' +1 2/+l Carrying out this expansion in each term of (40.1), we finally get the following expression for the /'th term of the expansion of the potential: ^^iV&Oe,®), (4i.i2) where QX^eat J —Yae., •!>.)■ (41-13) The set of 21+ 1 quantities Qj? form the 2'-pole moment of the system of charges The quantities Q^ defined in this way are related to the components of the dipole moment vector d by the formulas QP = id„ e ( ± 1} = + ^ (4± id y ). (41.14) The quantities Q™ are related to the tensor components D ajj by the relations Q ( o 2) = - \ D zz , Qg\ = ± 4 (A«± ">„). 1 V6 (41.15) Q { ±\ = " A (D xx -D yy ±2iD xy ). PROBLEM Determine the quadrupole moment of a uniformly charged ellipsoid with respect to its center. Solution: Replacing the summation in (41.3) by an integration over the volume of the ellipsoid, we have : Dxx = p j / i (2 * 2 ~ y2 ~ z2)dx dy dz ' et °' Let us choose the coordinate axes along the axes of the ellipsoid with the origin at its center; from symmetry considerations it is obvious that these axes are the principal axes of the tensor D aB . By means of the transformation x = x'a, y = y'b, z = z'c t In accordance with the definition used in quantum mechanics. 100 CONSTANT ELECTROMAGNETIC FIELDS § 42 the integration over the volume of the ellipsoid a 2 ^ b 2 ^ c 2 is reduced to integration over the volume of the unit sphere jc ,a +/ a +z ,2 = l. As a result we obtain : D zx =| (2a 2 -b 2 -c 2 ), D yy = ~ (2b 2 -a 2 -c 2 ), D 3Z = e -(2c 2 -a 2 -b 2 ), where e = (4n/3)abce is the total charge of the ellipsoid. § 42. System of charges in an external field We now consider a system of charges located in an external electric field. We designate the potential of this external field by <£(r). The potential energy of each of the charges is e a <KO, and the total potential energy of the system is tf = !>«#■«)■ (42.1) a We introduce another coordinate system with its origin anywhere within the system of charges; r fl is the radius vector of the charge e a in these coordinates. Let us assume that the external field changes slowly over the region of the system of charges, i.e. is quasiuniform with respect to the system. Then we can expand the energy U in powers of r a . In this expansion, U=U™+U™+U™+ ..., (42.2) the first term is ^ (0) = ^oE^ (42.3) where </> is the value of the potential at the origin. In this approximation, the energy of the system is the same as it would be if all the charges were located at one point (the origin). The second term in the expansion is l^> = (gradtf>) -2> fl r a Introducing the field intensity E at the origin and the dipole moment d of the system, we have l/ (1) =-d-E . (42.4) The total force acting on the system in the external quasiuniform field is, to the order we are considering, F = Eo5> fl + (Vd-E) . If the total charge is zero, the first term vanishes, and F = (d-V)E, (42.5) i.e. the force is determined by the derivatives of the field intensity (taken at the origin). The total moment of the forces acting on the system is K = £ (r a x e a E ) = d x E , (42.6) i.e. it is determined by the field intensity itself. § 43 CONSTANT MAGNETIC FIELD 101 Let us assume that there are two systems, each having total charge zero, and with dipole moments d t and d 2 , respectively. Their mutual distance is assumed to be large in comparison with their internal dimensions. Let us determine their potential energy of interaction, U. To do this we regard one of the systems as being in the field of the other. Then t/=-d 2 -E 1 . where E x is the field of the first system. Substituting (40.8) for E l5 we find: TT (d 1 -d 2 )/? 2 -3(d 1 -R)(d 2 -R) U = ^— ^ ^ — ^ — J -, (42 J) where R is the vector separation between the two systems. For the case where one of the systems has a total charge different from zero (and equal to e), we obtain similarly d R U = e-^-, (42.8) where R is the vector directed from the dipole to the charge. The next term in the expansion (42.1) is tt(2) 1 v 5 ^° Here, as in § 41, we omit the index numbering the charge; the value of the second derivative of the potential is taken at the origin; but the potential (f> satisfies Laplace's equation, ^ = <5 ** -0 dxl aP dx a dxp Therefore we can write or, finally, 6 ox a oxp ' The general term in the series (42.2) can be expressed in terms of the 2'-pole moments D^ defined in the preceding section. To do this, we first expand the potential 0(r) in spherical harmonics; the general form of this expansion is 00 ' / 4k #r)= J r l £ J—-a lm Y lm (0,<l>), (42.10 Z = m=-l V U+ 1 where r, 9, § are the spherical coordinates of a point and the a lm are constants. Forming the sum (42.1) and using the definition (41.13), we obtain: tf (0 = I fltaQi - (42.11) § 43. Constant magnetic field Let us consider the magnetic field produced by charges which perform a finite motion, in which the particles are always within a finite region of space and the momenta also always remain finite. Such a motion has a "stationary" character, and it is of interest to consider 102 CONSTANT ELECTROMAGNETIC FIELDS §43 the time average magnetic field H, produced by the charges ; this field will now be a function only of the coordinates and not of the time, that is, it will be constant. In order to find equations for the average magnetic field H, we take the time average of the Maxwell equations 1 dE 4tt . div H = 0, curl H = -—-+— j. c at c The first of these gives simply divH = 0. (43.1) In the second equation the average value of the derivative dE/dt, like the derivative of any quantity which varies over a finite range, is zero (cf. the footnote on p. 84). Therefore the second Maxwell equation becomes curl H = — j. (43.2) These two equations determine the constant field H. We introduce the average vector potential A in accordance with curl A = H. We substitute this in equation (43.2). We find _ _ 4n grad div A — A A = — j. c But we know that the vector potential of a field is not uniquely defined, and we can impose an arbitrary auxiliary condition on it. On this basis, we choose the potential A so that div A = 0. (43.3) Then the equation defining the vector potential of the constant magnetic field becomes AX=-yj. (43.4) It is easy to find the solution of this equation by noting that (43.4) is completely analogous to the Poisson equation (36.4) for the scalar potential of a constant electric field, where in place of the charge density q We here have the current density ]/c. By analogy with the solution (36.8) of the Poisson equation, we can write -1(1 c] R R dV, (43.5) where R is the distance from the field point to the volume element dV. In formula (43.5) we can go over from the integral to a sum over the charges, by sub- stituting in place of j the product qx, and recalling that all the charges are pointlike. In this we must keep in mind that in the integral (43.5), R is simply an integration variable, and is therefore not subject to the averaging process. If we write in place of the integral / ±dV, the sum £^-« then R a here are the radius vectors of the various particles, which change during the motion of the charges. Therefore we must write A-=-I^, (43.6) c R a where we average the whole expression under the summation sign. § 44 MAGNETIC MOMENTS 103 Knowing A, we can also find the magnetic field, H = curl A = curl - | ~dV. c] R The curl operator refers to the coordinates of the field point. Therefore the curl can be brought under the integral sign and j can be treated as constant in the differentiation. Applying the well known formula curl fa=f curl a + grad /xa, where /and a are an arbitrary scalar and vector, to the product j. 1/jR, we get and consequently, E = lJ l W dV (43J) (the radius vector R is directed from dV to the field point). This is the law of Biot and Savart. § 44. Magnetic moments Let us consider the average magnetic field produced by a system of charges in stationary motion, at large distances from the system. We introduce a coordinate system with its origin anywhere within the system of charges, just as we did in § 40. Again we denote the radius vectors of the various charges by r fl , and the radius vector of the point at which we calculate the field by R . Then R — r fl is the radius vector from the charge e a to the field point. According to (43.6), we have for the vector potential : C | K — T a\ As in § 40, we expand this expression in powers of r fl . To terms of first order (we omit the index a), we have *=k^-^('-v-k) In the first term we can write z*-|e«- But the average value of the derivative of a quantity changing within a finite interval (like S er) is zero. Thus there remains for A the expression We transform this expression as follows. Noting that v = f , we can write (remembering 104 CONSTANT ELECTROMAGNETIC FIELDS § 44 that R is a constant vector) 1/7 1 J e(R T)v = -- X er(r -R )+ - £ e [v(r -R )-r(vR )]. Upon substitution of this expression in A, the average of the first term (containing the time derivative) again goes to zero, and we get S = 2^1 £ e[v(r -R )-r(vR )]. We introduce the vector 1 v m = — £ er x v, (44.2) which is called the magnetic moment of the system. Then we get for A: _ mxR 1 A = — 3- = V-xm (44.3 Knowing the vector potential, it is easy to find the magnetic field. With the aid of the formula we find Furthermore, and curl (a x b) = (b • V)a-(a • V)b + a div b-b div a, ,/mxR \ ,. R R n H = curlA = curl^-^J = mdiv^-(m-V)^|. ,. Ro 11 dlv ^3 = Ro'grad -3 + -3 divR = K K K Ro 1 ,— „.„ „ ,_ _- 1 m 3R (TTt-R ) ( m * V > S§ = ^3 (m- V)R +R (m- V) -3 = Thus, Ro Rq Ro Ro Rq „ 3n(m-n) — m H = ~S^ , (44.4) K o where n is again the unit vector along R . We see that the magnetic field is expressed in terms of the magnetic moment by the same formula by which the electric field was expressed in terms of the dipole moment [see (40.8)]. If all the charges of the system have the same ratio of charge to mass, then we can write 1 v e r m = — •>erxv = - — > mrxv. 2c 2mc If the velocities of all the charges v ^ c, then m\ is the momentum p of the charge and we get m = ^' XV = 4nc M ' (44 - 5) where M = E r x p is the mechanical angular momentum of the system. Thus in this case, the ratio of magnetic moment to the angular momentum is constant and equal to e/2mc. § 45 larmor's theorem 105 PROBLEM Find the ratio of the magnetic moment to the angular momentum for a system of two charges (velocities v <^ c). Solution: Choosing the origin of coordinates as the center of mass of the two particles we have /Miri+m 2 r 2 = and pi = — p 2 = p, where p is the momentum of the relative motion. With the aid of these relations, we find m = — (— + —\ Wima M 2c \ml m\) mx+m?. § 45. Larmor's theorem Let us consider a system of charges in an external constant uniform magnetic field. The time average of the force acting on the system, F = Y-7xH = ^£-rxH, ^ c dt^c is zero, as is the time average of the time derivative of any quantity which varies over a finite range. The average value of the moment of the forces is K = I-((rx(vxH)) ^ c and is different from zero. It can be expressed in terms of the magnetic moment of the system, by expanding the vector triple product: K = X^Wi-H)-H(vr)}=S^{v(r-H)-^H|r The second term gives zero after averaging, so that 1 K = £ - v(r H) = - £ e{v(r -H)-r(vH)} [the last transformation is analogous to the one used in deriving (44.3)], or finally K = mxH. (45.1) We call attention to the analogy with formula (42.6) for the electrical case. The Lagrangian for a system of charges in an external constant uniform magnetic field contains (compared with the Lagrangian for a closed system) the additional term L H = ^A-v = £^Hxr-v = £^rxv-H (45 2) [where we have used the expression (19.4) for the vector potential of a uniform field]. Introducing the magnetic moment of the system, we have : L H = m-H. (45.3) We call attention to the analogy with the electric field; in a uniform electric field, the Lagrangian of a system of charges with total charge zero contains the term L £ = dE, which in that case is the negative of the potential energy of the charge system (see § 42). 106 CONSTANT ELECTROMAGNETIC FIELDS § 45 We now consider a system of charges performing a finite motion (with velocities v < c) in the centrally symmetric electric field produced by a certain fixed charge. We transform from the laboratory coordinate system to a system rotating uniformly around an axis passing through the fixed particle. From the well-known formula, the velocity v of the particle in the new coordinate system is related to its velocity v' in the old system bv the relation v' = v+fixr, where r is the radius vector of the particle and il is the angular velocity of the rotating co- ordinate system. In the fixed system the Lagrangian of the system of charges is _, mv' 2 where U is the potential energy of the charges in the external field plus the energy of their mutual interactions. The quantity U is a function of the distances of the charges from the fixed particle and of their mutual separations; when transformed to the rotating system it obviously remains unchanged. Therefore in the new system the Lagrangian is m L = Z-(y + Slxr) 2 -U. Let us assume that all the charges have the same charge-to-mass ratio ejm, and set n = ^H. (45.4) Then for sufficiently small H (when we can neglect terms in H 2 ) the Lagrangian becomes: „ mv 2 1 r-, „ L-E — + -£eHxr-v-l7. We see that it coincides with the Lagrangian which would have described the motion of the charges in the laboratory system of coordinates in the presence of a constant magnetic field (see (45.2)). Thus we arrive at the result that, in the nonrelativistic case, the behavior of a system of charges all having the same elm, performing a finite motion in a centrally symmetric electric field and in a weak uniform magnetic field H, is equivalent to the behavior of the same system of charges in the same electric field in a coordinate system rotating uniformly with the angular velocity (45.4). This assertion is the content of the Larmor theorem, and the angular velocity Q == eHjlmc is called the Larmor frequency. We can approach this same problem from a different point of view. If the magnetic field H is sufficiently weak, the Larmor frequency will be small compared to the frequencies of the finite motion of the system of charges. Then we may consider the averages, over times small compared to the period 2n/Q, of quantities describing the system. These new quantities will vary slowly in time (with frequency Q). Let us consider the change in the average angular momentum M of the system. According to a well-known equation of mechanics, the derivative of M is equal to the moment K of the forces acting on the system. We therefore have, using (45.1): ^ = K = mxH. dt If the e/m ratio is the same for all particles of the system, the angular momentum and §45 larmor's theorem 107 magnetic moment are proportional to one another, and we find by using formulas (44.5) and (45.3): —-=-QxM. (45.5) dt This equation states that the vector M (and with it the magnetic moment m) rotates with angular velocity -O around the direction of the field, while its absolute magnitude and the angle which it makes with this direction remain fixed. (This motion is called the Larmor precession.) CHAPTER 6 ELECTROMAGNETIC WAVES § 46. The wave equation The electromagnetic field in vacuum is determined by the Maxwell equations in which we must set p = 0, j = 0. We write them once more : curlE = — , divH = 0, (46.1) c ot v J curl H = - — , div E = 0. (46.2) c ot K ' These equations possess nonzero solutions. This means that an electromagnetic field can exist even in the absence of any charges. Electromagnetic fields occurring in vacuum in the absence of charges are called electro- magnetic waves. We now take up the study of the properties of such waves. First of all we note that such fields must necessarily be time-varying. In fact, in the contrary case, dH/dt = dE/dt = and the equations (46.1) and (46.2) go over into the equations (36.1), (36.2) and (43.1), (43.2) of a constant field in which, however, we now have p = 0, j = 0. But the solution of these equations which is given by formulas (36.8) and (43.5) becomes zero for p = 0, j = 0. We derive the equations determining the potentials of electromagnetic waves. As we already know, because of the ambiguity in the potentials we can always subject them to an auxiliary condition. For this reason, we choose the potentials of the electro- magnetic wave so that the scalar potential is zero : 4> = 0. (46.3) Then IdA E = - - — , H = curl A. (46.4) c ot Substituting these two expressions in the first of equations (46.2), we get 1 d 2 A curl curl A = - AA + grad div A = =■ — =-. (46.5) c 2 dt 2 y ' Despite the fact that we have already imposed one auxiliary condition on the potentials, the potential A is still not completely unique. Namely, we can add to it the gradient of an arbitrary function which does not depend on the time (meantime leaving <f) unchanged). In particular, we can choose the potentials of the electromagnetic wave so that div A = 0. (46.6) 108 AA-^-^=0. (46.7) § 46 THE WAVE EQUATION 109 In fact, substituting for E from (46.4) in div E = 0, we have ,- dA d ,- 4 ^ dlv a7 = a* dlvA = ' that is, div A is a function only of the coordinates. This function can always be made zero by adding to A the gradient of a suitable time-independent function. The equation (46.5) now becomes 1_<^A c 2 dt 2 This is the equation which determines the potentials of electromagnetic waves. It is called the d'Alembert equation, or the wave equation.^ Applying to (46.7) the operators curl and d/dt, we can verify that the electric and magnetic fields E and H satisfy the same wave equation. We repeat the derivation of the wave equation in four-dimensional form. We write the second pair of Maxwell equations for the field in the absence of charges in the form (This is equation (30.2) with 7' = 0.) Substituting F ik , expressed in terms of the potentials, dx t dx k we get d 2 A k d 2 A l n ^ ON ^~n - 5— r» = °- (46 - 8) OXiOX ox k ox We impose on the potentials the auxiliary condition : f)A k ^ = 0. (46.9) (This condition is called the Lorentz condition, and potentials that satisfy it are said to be in the Lorentz gauge) Then the first term in (46.8) drops out and there remains d 2 A 1 r) 2 A 1 = g»°A l = . (46.10) dx k dx k dx k dx This is the wave equation expressed in four-dimensional form.f In three-dimensional form, the condition (46.9) is: ld<b — ^-+divA = 0. (46.11) c dt It is more general than the conditions <fi = and div A = that were used earlier; potentials that satisfy those conditions also satisfy (46.1 1). But unlike them the Lorentz condition has a relativistically invariant character : potentials satisfying it in one frame satisfy it in any other frame (whereas condition (46.6) is generally violated if the frame is changed). t The wave equation is sometimes written in the form DA = 0, where n = -— — = a -I — d Xl dx l c 2 dt 2 is called the d'Alembertian operator. t It should be mentioned that the condition (46.9) still does not determine the choice of the potentials uniquely. We can add to A a term grad/, and subtract a term 1/c (df/dt) from #, where the function /is not arbitrary but must satisfy the wave equation Of=0. 110 ELECTROMAGNETIC WAVES § 47 § 47. Plane waves We consider the special case of electromagnetic waves in which the field depends only on one coordinate, say x (and on the time). Such waves are said to be plane. In this case the equation for the field becomes dt 2 c dx 2 -c 2 ^, = 0, (47.1) where by /is understood any component of the vectors E or H. To solve this equation, we rewrite it in the form and introduce new variables (d d\/d d\ r n {dt- C Tx){dt +C dx) f=0 > x x £ = t--, ri = t+ c c so that t = i(n + 0, x = c/2(r}-£). Then d__\(o__ d_\ d 1/d d d£~2\dt~ C dx =- { -- c -) i = \(l +c E)> so that the equation for /becomes d 2 f = o. d£dri The solution obviously has the formf=f 1 (^)+f 2 (ri), where/! and/ 2 are arbitrary functions. Thus /=/ 'H) +/ *(' + ;;)' (47 - 2) Suppose, for example, f 2 = 0, so that Cj Let us clarify the meaning of this solution. In each plane x = const, the field changes with the time; at each given moment the field is different for different x. It is clear that the field has the same values for coordinates x and times t which satisfy the relation t—(xjc) = const, that is, x = const +ct. This means that if, at some time t = 0, the field at a certain point x in space had some definite value, then after an interval of time t the field has that same value at a distance ct along the Zaxis from the original place. We can say that all the values of the electromagnetic field are propagated in space along the X axis with a velocity equal to the velocity of light, c. Thus, ■H) represents a plane wave moving in the positive direction along the Jif axis. It is easy to show that represents a wave moving in the opposite, negative, direction along the X axis. § 47 PLANE WAVES 111 In § 46 we showed that the potentials of the electromagnetic wave can be chosen so that <J> = 0, and div A = 0. We choose the potentials of the plane wave which we are now con- sidering in this same way. The condition div A = gives in this case dx since all quantities are independent of y and z. According to (47.1) we then have also d 2 AJdt 2 = 0, that is, dAJdt = const. But the derivative dA/dt determines the electric field, and we see that the nonzero component A x represents in this case the presence of a constant longitudinal electric field. Since such a field has no relation to the electromagnetic wave, we can set A x = 0. Thus the vector potential of the plane wave can always be chosen perpendicular to the X axis, i.e. to the direction of propagation of that wave. We consider a plane wave moving in the positive direction of the Zaxis; in this wave, all quantities, in particular also A, are functions only of t—(x/c). From the formulas 1 dA E = , H = curl A, c dt we therefore obtain E = - - A', H = Vx A = V (t- -) x A' = - - nx A', (47.3) c \ cj c where the prime denotes differentiation with respect to t—(x/c) and n is a unit vector along the direction of propagation of the wave. Substituting the first equation in the second, we obtain H = n x E. (47.4) We see that the electric and magnetic fields E and H of a plane wave are directed perpen- dicular to the direction of propagation of the wave. For this reason, electromagnetic waves are said to be transverse. From (47.4) it is clear also that the electric and magnetic fields of the plane wave are perpendicular to each other and equal to each other in absolute value. The energy flux in the plane wave, i.e. its Poynting vector is and since E • n = 0, S = -^-ExH = ^-Ex(nxE), An An S = ^E 2 n = ~ H 2 n. An An Thus the energy flux is directed along the direction of propagation of the wave. Since W = ^(E 2 +H 2 ) = ^ Sn An is the energy density of the wave, we can write S = cWn, (47.5) in accordance with the fact that the field propagates with the velocity of light. The momentum per unit volume of the electromagnetic field is S/c 2 . For a plane wave this gives (W/c)n. We call attention to the fact that the relation between energy Wand momen- tum W/c for the electromagnetic wave is the same as for a particle moving with the velocity of light [see (9.9)]. 112 ELECTROMAGNETIC WAVES § 47 The flux of momentum of the field is determined by the components a ap of the Maxwell stress tensor. Choosing the direction of propagation of the wave as the X axis, we find that the only nonzero component of a aP is <y xx = W. (47.6) As it must be, the flux of momentum is along the direction of propagation of the wave, and is equal in magnitude to the energy density. Let us find the law of transformation of the energy density of a plane electromagnetic wave when we change from one inertial reference system to another. To do this we start from the formula 1 / V V 2 W = 1 / V V 2 \ c 2 (see problem 1 in § 6) and substitute S' x = cW cos a', o' xx = W cos 2 a', where a' is the angle (in the K' system) between the X' axis (along which the velocity V is directed) and the direction of propagation of the wave. We find: V \ 2 1 -\ — cos a' I W=W - C —yi-*-- (47.7) Since W = E^/4n = H 2 /4n, the absolute values of the field intensities in the wave trans- form like V W. PROBLEMS 1 . Determine the force exerted on a wall from which an incident plane electromagnetic wave is reflected (with reflection coefficient R). Solution: The force f acting on unit area of the wall is given by the flux of momentum through this area, i.e., it is the vector with components f a ^ ar ae N + a' a0 N B , where N is the vector normal to the surface of the wall, and a a0 and a' a& are the components of the energy-momentum tensors for the incident and reflected waves. Using (47.6), we obtain: f - JFh(N • n)+ 0"n'(N ■ n'). From the definition of the reflection coefficient, we have: W = RW. Also introducing the angle of incidence 6 (which is equal to the reflection angle) and writing out components, we find the normal force ("light pressure") ^^^(l+^cos 2 ^ and the tangential force ft = W(\ -R) sin 9 cos 6. 2. Use the Hamilton- Jacobi method to find the motion of a charge in the field of a plane electro- magnetic wave with vector potential X[t~ (x/c)]. Solution: We write the Hamilton- Jacobi equation in four-dimensional form: 8S e \ / 8S e \ dx l c } V 8x k c § 47 PLANE WAVES 113 The fact that the field is a plane wave means that the A 1 are functions of one independent variable, which can be written in the form £ = k t x\ where k l is a constant four- vector with its square equal to zero, k t k* = (see the following section). We subject the potentials to the Lorentz condition 8A l dA l , for the variables field this is equivalent to the condition A l ki — 0. We seek a solution of equation (1) in the form S=-f t x l +F(0, where / i: =(/ , f) is a constant vector satisfying the condition f t f t = m 2 c 2 (S = —fix 1 is the solution of the Hamilton- Jacobi equation for a free particle with four-momentum p l =/'). Sub- stitution in (1) gives the equation e 2 „, „ dF 2e „ A , -A,A'-2y T( --f,A<=0, where the constant y = k t f l . Having determined F from this equation, we get (2) S—fi*~\fiM+^A<Atdt. Changing to three-dimensional notation with a fixed reference frame, we choose the direction of propagation of the wave as the x axis. Then £ = ct—x, while the constant y =/°— f 1 . Denoting the two-dimensional vector/^, f z by k, we find from the condition / ( f { — (/°) 2 — (Z 1 ) 2 — k 2 = m 2 c 2 , We choose the potentials in the gauge in which <j> = 0, while A(£) lies in the yz plane. Then equation (2) takes the form: According to the general rules (Mechanics, § 47), to determine the motion we must equate the derivatives dS/dic, dS/dy to certain new constants, which can be made to vanish by a suitable choice of the coordinate and time origins. We thus obtain the parametric equations in f: y = -K y £, AydZ, z=-k 3 £ A z d£, y cy J y cy J The generalized momentum P =p+(e/c)A and the energy £ are found by differentiating the action with respect to the coordinates and the time; this gives: e e Py = K y Ay, Pz = K z ■ A z , c c _ y m 2 c 2 +K 2 e e 2 2 2y cy 2yc* £ = (y+p x )c If we average these over the time, the terms of first degree in the periodic function A(£) vanish. We assume that the reference system has been chosen so that the particle is at rest in it on the average, i.e. so that its averaged momentum is zero. Then k = 0, y 2 — m 2 c 2 + e 2 A 2 . 114 ELECTROMAGNETIC WAVES § 48 The final formulas for determining the motion have the form: 2yV JV-A*)*, y=- e -^A y dt, *=- e -\ A z di, e" Ct==i+ W) ( A2 - A2 >^5 ( 3 ) Px = 2^ (A 2 -A 2 ), p y =--A y , P*=-- A * <? = cy+£ c (A*-A>). (4) § 48. Monochromatic plane waves A very important special case of electromagnetic waves is a wave in which the field is a simply periodic function of the time. Such a wave is said to be monochromatic. All quantities (potentials, field components) in a monochromatic wave depend on the time through a factor of the form cos (cot+a). The quantity co is called the cyclic frequency of the wave (we shall simply call it the frequency). In the wave equation, the second derivative of the field with respect to the time is now d 2 f/dt 2 = —co 2 f, so that the distribution of the field in space is determined for a mono- chromatic wave by the equation co 2 Af+-tf=0. (48.1) In a plane wave (propagating along the x axis), the field is a function only of t-(x/c). Therefore, if the plane wave is monochromatic, its field is a simply periodic function of t—(xfc). The vector potential of such a wave is most conveniently written as the real part of a complex expression : A = Re{A e~ to H)} (48<2 ) Here A is a certain constant complex vector. Obviously, the fields E and H of such a wave have analogous forms with the same frequency co. The quantity 2nc X = — (48.3) co is called the wavelength; it is the period of variation of the field with the coordinate x at a fixed time t. The vector k = - n (48.4) c (where n is a unit vector along the direction of propagation of the wave) is called the wave vector. In terms of it we can write (48.2) in the form A = Re{A e i(k - r - wt) }, (48.5) which is independent of the choice of coordinate axes. The quantity which appears multiplied by / in the exponent is called the phase of the wave. § 48 MONOCHROMATIC PLANE WAVES 115 So long as we perform only linear operations, we can omit the sign Re for taking the real part, and operate with complex quantities as such.f Thus, substituting A = A e'' (kr - wf) in (47.3), we find the relation between the intensities and the vector potential of a plane monochromatic wave in the form E = ifcA, H = ikxA. (48.6) We now treat in more detail the direction of the field of a monochromatic wave. To be specific, we shall talk of the electric field E = Re{E e , ' (k - r - C0 °} (everything stated below applies equally well, of course, to the magnetic field). The quantity E is a certain complex vector. Its square E 2 , is (in general) a complex number. If the argument of this number is -2a (i.e. E 2 , = |E^|e" 2ia ), the vector b defined by E = be~ ia (48.7) will have its square real, b 2 = |E | 2 . With this definition, we write: E = Re{be i(k - r - wt - a) }. (48.8) We write b in the form b = bi + ib 2 , where b A and b 2 are real vectors. Since b 2 = b 2 -b| + 2ib 1 b 2 must be a real quantity, \) 1 • b 2 = 0, i.e. the vectors b t and b 2 are mutually perpendicular. We choose the direction of b t as the y axis (and the x axis along the direction of propagation of the wave). We then have from (48.8): E y = b t cos ((ot-k-r + a), E z — ±b 2 sin (cof — k-r + a), where we use the plus (minus) sign if b 2 is along the positive (negative) z axis. From (48.9) it follows that f 2 F 2 rli + ti = i (48.10) Thus we see that, at each point in space, the electric field vector rotates in a plane perpen- dicular to the direction of propagation of the wave, while its endpoint describes the ellipse (48.10). Such a wave is said to be elliptically polarized. The rotation occurs in the direction of (opposite to) a right-hand screw rotating along the x axis, if we have the plus (minus) sign in (48.9). jfb 2 = b 2 , the ellipse (48.10) reduces to a circle, i.e. the vector E rotates while remaining constant in magnitude. In this case we say that the wave is circularly polarized. The choice of the directions of the y and z axes is now obviously arbitrary. We note that in such a wave t If two quantities kit) and B(f ) are written in complex form A(/) = A e- tu ", B(O = B e- ,a ", then in forming their product we must first, of course, separate out the real part. But if, as it frequently happens, we are interested only in the time average of this product, it can be computed as £Re {A-B*}. In fact, we have: Re A • Re B = i(A <r"°'+ A*e t(0t ) • (B e- m +BttT r ). When we average, the terms containing factors e ±2fa>t vanish, so that we are left with ReAReB = i(A • B* + A* • B ) = \ Re (A • B*). 0z =±i (48.11) Oy H6 ELECTROMAGNETIC WAVES § 48 the ratio of the y and z components of the complex amplitude E is E for rotation in the same (opposite) direction as that of a right-hand screw {right and left polarizations), f Finally, if b x or b 2 equals zero, the field of the wave is everywhere and always parallel (or antiparallel) to one and the same direction. In this case the wave is said to be linearly polarized, or plane polarized. An elliptically polarized wave can clearly be treated as the superposition of two plane polarized waves. Now let us turn to the definition of the wave vector and introduce the four-dimensional wave vector with components ** = (".*)■ (48.12) That these quantities actually form a four-vector is obvious from the fact that we get a scalar (the phase of the wave) when we multiply by x l : k i x i = (ot-k-T. (48.13) From the definitions (48.4) and (48.12) we see that the square of the wave four-vector is zero: k% = 0. (48.14) This relation also follows directly from the fact that the expression A = A e~ ikiXi must be a solution of the wave equation (46.10). As is the case for every plane wave, in a monochromatic wave propagating along the x axis only the following components of the energy-momentum tensor are different from zero (see §47): T 00 = T 01 = T 11 = W By means of the wave four- vector, these equations can be written in tensor form as Wc 2 T ik = — k l k k . (48.15) or v Finally, by using the law of transformation of the wave four- vector we can easily treat the so-called Doppler effect — the change in frequency co of the wave emitted by a source moving with respect to the observer, as compared to the "true" frequency co of the same source in the reference system (K ) in which it is at rest. Let Vbe the velocity of the source, i.e. the velocity of the K system relative to K. Accord- ing to the general formula for transformation of four- vectors, we have : v k°- - k x /c (0)0 = C J i-X (the velocity of the K system relative to K is —V). Substituting &° = co/c, k 1 = k cos a = co/c cos a, where a is the angle (in the K system) between the direction of f We assume that the coordinate axes form a right-handed system. § 48 MONOCHROMATIC PLANE WAVES 117 emission of the wave and the direction of motion of the source, and expressing co in terms of co , we obtain: J ■-C co = co . (48.16) 1 cos a c This is the required formula. For V <^ c, and if the angle a is not too close to n/2, it gives : For a = n/2, we have : co ^ co ( 1 + - cos a ). (48.17) <° = ffl o J 1- ^f = w o (l- £2); (48.18) in this case the relative change in frequency is proportional to the square of the ratio V/c. PROBLEMS 1. Determine the direction and magnitude of the axes of the polarization ellipse in terms of the complex amplitude E . Solution: The problem consists in determining the vector b =bi+/b 2 , whose square is real. We have from (48.7): Eo-Ei =b*+b*, E xE* = -2ib 1 xb a , (1) or bl+bl=A 2 +B 2 , bibz^ABsmd, where we have introduced the notation \Eoy\=A, \Eo.\=B, % = ^V* B A for the absolute values of E 0y and £"03 and for the phase difference 5 between them. Then bi, 2 = VA 2 +B 2 +2ABsin 3 ±Va 2 +B 2 -2AB sin 8, (2) from which we get the magnitudes of the semiaxes of the polarization ellipse. To determine their directions (relative to the arbitrary initial axes y and z) we start from the equality RelfEo-bxXEJ-ba)}^, which is easily verified by substituting E = (bi + /b 2 )e""' a . Writing out this equality in the y, z co- ordinates, we get for the angle 9 between the direction of bi and the y axis: „ n 2AB cos <5 tan 26 = — — . f3^ A 2 -B 2 K } The direction of rotation of the field is determined by the sign of the x component of the vector biXb 2 . Taking its expression from (1) VQ>iXb*X = Eo a E* y -* - _..*. .af/M (E QZ we see that the direction of b x x b 2 (whether it is along or opposite to the positive direction of the x axis), and the sign of the rotation (whether in the same direction, or opposite to the direction of a right-hand screw along the x axis) are given by the sign of the imaginary part of the ratio E 0z /E 0y (plus for the first case and minus for the second). This is a generalization of the rule (48.1 1) for the case of circular polarization. 118 ELECTROMAGNETIC WAVES § 49 2. Determine the motion of a charge in the field of a plane monochromatic linearly polarized wave. Solution: Choosing the direction of the field E of the wave as the y axis, we write: cE E y =E = E cos co$ , A y = A — sin ©<* CO (f = t—xjc). From formulas (3) and (4) of problem 2, § 47, we find (in the reference system in which the particle is at rest on the average) the following representation of the motion in terms of the parameter n = cog) : e 2 Ecl . „ eE c — — sin 2fj, y = 2>y z co 6 yco 2 x=- 5-3-3 sin 217, j = - — ,- cos ?/, z = 0, ,2P2 , * g2 ^o • . a 2 2 , e 2 E_ t== 5^"^ sm 2? 7' ^ = /w 2 c 2 + — - T to 8rco 3 2ct> 2 e 2 ^^ . eE . Px= — - A — a cos 2»7, />„ = — sin n, p z = 0. Ayco 2, co The charge moves in the x, y plane in a symmetric figure-8 curve with its longitudinal axis along the y axis. 3. Determine the motion of a charge in the field of a circularly polarized wave. Solution: For the field of the wave we have: E y = E Q cos co£, E z — E sin co£, The motion is given by the formulas : a cE Q . cE Ay = — ■ — sm coq, A z — — cos coq. CO CO ecE ecE . x = 0, y— =- cos cot, z= ^ sin cot, yco A yco* a eE Q . eE Px = 0, p y — — sin cot, Pz= cos cot, co co r 2 F 2 9 9 9 i y 2 = wrrH =-. CO A Thus the charge moves in the y, z plane along a circle of radius ecE lyco 2 with a momentum having the constant magnitude p = eE /co; at each instant the direction of the momentum p coincides with the direction of the magnetic field H of the wave. § 49. Spectral resolution Every wave can be subjected to the process of spectral resolution, i.e. can be represented as a superposition of monochromatic waves with various frequencies. The character of this expansion varies according to the character of the time dependence of the field. One category consists of those cases where the expansion contains frequencies forming a discrete sequence of values. The simplest case of this type arises in the resolution of a purely periodic (though not monochromatic) field. This is the usual expansion in Fourier series; it contains the frequencies which are integral multiples of the "fundamental" frequency co = InjT, where T is the period of the field. We write it in the form /= Z f n e- i<aont (49.1) n= — oo (where /is any of the quantities describing the field). The quantities /„ are defined in terms § 50 PARTIALLY POLARIZED LIGHT 119 of the function /by the integrals r/2 - = f J f(t)e in <°*dt. (49.2) -r/2 /* Because f(t) must be real, -n =fn (49.3) In more complicated cases, the expansion may contain integral multiples (and sums of integral multiples) of several different incommensurable fundamental frequencies. When the sum (49.1) is squared and averaged over the time, the products of terms with different frequencies give zero because they contain oscillating factors. Only terms of the form /„,/_„ = |/„| 2 remain. Thus the average of the square of the field, i.e. the average intensity of the wave, is the sum of the intensities of its monochromatic components: oo oo f 2 = S \L\ 2 = 2I \f n \\ (49 .4) «= -00 71= 1 (where it is assumed that the average of the function /over a period is zero, i.e./ =/= 0). Another category consists of fields which are expandable in a Fourier integral containing a continuous sequence of different frequencies. For this to be possible, the function f(t ) must satisfy certain definite conditions; usually we consider functions which vanish for t -> ± oo. Such an expansion has the form 00 (0= \ f m e-^ d ^ (49.5) — oo where the Fourier components are given in terms of the function f(t) by the integrals 00 L= j f(t)e ilot dt. (49.6) — oo Analogously to (49.3), /-«,=/*• (49.7) Let us express the total intensity of the wave, i.e. the integral of/ 2 over all time, in terms of the intensity of the Fourier components. Using (49.5) and (49.6), we have: 00 °° 00 00 00 00 //•*-/{/ J/..-t}*-/{/.J/.-*}S-j/^. -co -oo -oo -oo -oo -oo or, using (49.7), oo oo oo J>*-JW'!-2jW'£. (-8) § 50. Partially polarized light Every monochromatic wave is, by definition, necessarily polarized. However we usually have to deal with waves which are only approximately monochromatic, and which contain frequencies in a small interval Acq. We consider such a wave, and let co be some average 120 ELECTROMAGNETIC WAVES § 50 frequency for it. Then its field (to be specific we shall consider the electric field E) at a fixed point in space can be written in the form where the complex amplitude E (t) is some slowly varying function of the time (for a strictly monochromatic wave E would be constant). Since E determines the polarization of the wave, this means that at each point of the wave, its polarization changes with time; such a wave is said to be partially polarized. The polarization properties of electromagnetic waves, and of light in particular, are observed experimentally by passing the light to be investigated through various bodies f and then observing the intensity of the transmitted light. From the mathematical point of view this means that we draw conclusions concerning the polarization properties of the light from the values of certain quadratic functions of its field. Here of course we are considering the time averages of such functions. Quadratic functions of the field are made up of terms proportional to the products E a E p , E*E$ or E a Ef. Products of the form E a Ep = E 0a E pe l< ° , E a Ep = E 0a E op e lui , which contain the rapidly oscillating factors e ±2icot give zero when the time average is taken. The products E a E p = E 0oL E* p do not contain such factors, and so their averages are net zero. Thus we see that the polarization properties of the light are completely characterized by the tensor J aP = EZE%. (50.1) Since the vector E always lies in a plane perpendicular to the direction of the wave, the tensor J aP has altogether four components (in this section the indices a, /? are understood to take on only two values : a, /? = 1 , 2, corresponding to the y and z axes ; the x axis is along the direction of propagation of the wave). The sum of the diagonal elements of the tensor J af} (we denote it by /) is a real quantity — the average value of the square modulus of the vector E (or E) : J^J aa = E^E*. (50.2) This quantity determines the intensity of the wave, as measured by the energy flux density. To eliminate this quantity which is not directly related to the polarization properties, we introduce in place of J aP the tensor P aP = J f, (50.3) for which p aa = 1 ; we call it the polarization tensor. From the definition (50.1) we see that the components of the tensor J a/} , and consequently also p aP , are related by p aP = P% (50.4) (i.e. the tensor is hermitian). Consequently the diagonal components p xl and p 22 are real (with P11 + P22 = 1) while p 21 = p* 2 - Thus the polarization is characterized by three real parameters. Let us study the conditions that the tensor p aP must satisfy for completely polarized light. In this case E = const, and so we have simply J« = Jp*e = E o«E%e (50-5) t For example, through a Nicol prism. § 50 PARTIALLY POLARIZED LIGHT 121 (without averaging), i.e. the components of the tensor can be written as products of com- ponents of some constant vector. The necessary and sufficient condition for this is that the determinant vanish : \Pap\ = P11P22-P12P21 =0- (50.6) The opposite case is that of unpolarized or natural light. Complete absence of polarization means that all directions (in the y 2 plane) are equivalent. In other words the polarization tensor must have the form: P«/»=iV (50.7) The determinant is \p ap \ = £. In the general case of arbitrary polarization the determinant has values between and \:\ By the degree of polarization we mean the positive quantity P, defined from \pat\ = X\-P 2 ). (50.8) It runs from the value for unpolarized to 1 for polarized light. An arbitrary tensor p af} can be split into two parts— a symmetric and an antisymmetric part. Of these, the first is real because of the hermiticity of p af> . The antisymmetric part is pure imaginary. Like any antisymmetric tensor of rank equal to the number of dimensions, it reduces to a pseudo- scalar (see the footnote on p. 17): 2.{p a p-ppo)= ~2 e «P A > where A is a real pseudoscalar, e afi is the unit antisymmetric tensor (with components *i2 = -^21 = !)• Thus the polarization tensor has the form: Pat = S aP -^ e aP A, S aP = S Pa , (50.9) i.e. it reduces to one real symmetric tensor and one pseudoscalar. For a circularly polarized wave, the vector E = const, where E 02 = ±iE 01 . It is easy to see that then S af = S ap , while A = ± 1 . On the other hand, for a linearly polarized wave the constant vector E can be chosen to be real, so that A = 0. In the general case the quantity A may be called the degree of circular polarization; it runs through values from + 1 to -1, where the limiting values correspond to right- and left-circularly polarized waves, respectively. The real symmetric tensor S aP , like any symmetric tensor, can be brought to principal axes, with different principal values which we denote by X t and X 2 . The directions of the principal axes are mutually perpendicular. Denoting the unit vectors along these directions by n (1) and n (2) , we can write S aP in the form: S ap = X in ^n^ + X 2 n^nf\ X,+X 2 = 1. (50.10) The quantities X i and X 2 are positive and take on values from to 1. f The fact that the determinant is positive for any tensor of the form (50.1) is easily seen by considering the averaging, for simplicity, as a summation over discrete values, and using the well-known algebraic inequality I Z x a y b \ 2 < Z |x a | 2 Z \y b \*. 122 ELECTROMAGNETIC WAVES § 50 Suppose that A = 0, so that p aP = S aP . Each of the two terms in £50. 10) has the form of a product of two components of a constant vector (VAin (1) or VA 2 n (2) ). In other words, each of the terms corresponds to linearly polarized light. Furthermore, we see that there is no term in (50.10) containing products of components of the two waves. This means that the two parts can be regarded as physically independent of one another, or, as one says, they are incoherent. In fact, if two waves are independent, the average value of the product E^ 1) £j 2) is equal to the product of the averages of each of the factors, and since each of them is zero, £p£f> = 0. Thus we arrive at the result that in this case (A = 0) the partially polarized light can be represented as a superposition of two incoherent waves (with intensities proportional to X x and A 2 ), linearly polarized along mutually perpendicular directions.f (In the general case of a complex tensor p a/} one can show that the light can be represented as a superposition of two incoherent elliptically polarized waves, whose polarization ellipses are similar and mutually perpendicular (see problem 2).) Let be the angle between the axis 1 (the y axis) and the unit vector n (1) ; then n (1) = (cos 0, sin 0), n (2) = (-sin 0, cos 0). Introducing the quantity l=l^-X 2 (assume A 2 > A 2 ), we write the components of the tensor (50.10) in the following form: 1 /l + /cos20 /sin 20 \ aP 2\ /sin20 l-/cos20/ Thus, for an arbitrary choice of the axes y and z, the polarization properties of the wave can be characterized by the following three real parameters: A— the degree of circular polariza- tion, /—the degree of maximum linear polarization, and 0— the angle between the direction n (1) of maximum polarization and the y axis. In place of these parameters one can choose another set of three parameters: $! = / sin 20, Zi=A, £ 3 = /cos 20 (50.12) (the Stokes parameters). The polarization tensor is expressed in terms of them as p 1( 1 + ^ *i-**Y (50.13) All three parameters run through values from -1 to +1. The parameter £3 characterizes the linear polarization along the y and z axes: the value £ 3 = 1 corresponds to complete linear polarization along the y axis, and £ 3 = - 1 to complete polarization along the z axis. The parameter ^ characterizes the linear polarization along directions making an angle of 45° with the y axis: the value £ 2 = 1 means complete polarization at an angle = 7t/4, while £ 2 = — 1 means complete polarization at = — n/4. J f The determinant \S af \ = X 1 X a ; suppose that X 1 > X 2 ; then the degree of polarization, as denned in (50.8), is P = 1 -2/1,2. In the present case {A = 0) one frequently characterizes the degree of polarization by using the depolarization coefficient, defined as the ratio /L 2 Mi- % For a completely elliptically polarized wave with axes of the ellipse bi and b 2 (see § 48), the Stokes parameters are : £ 1= 0, & = ±2Z>i6 2 , ^ = bl-b%. Here the y axis is along b x , while the two signs in £ 2 correspond to directions of b 2 along and opposite to the direction of the z axis. § 50 PARTIALLY POLARIZED LIGHT 123 The determinant of (50.13) is equal to k,| = Ki-tf-«-£i). (50-14) Comparing with (50.8), we see that P = V« + £l + fr (50.15) Thus, for a given overall degree of polarization P, different types of polarization are possible, characterized by the values of the three quantities £ 2 , £,i, £3, the sum of whose squares is fixed ; they form a sort of vector of fixed length. We note that the quantities £ 2 = A and vCi+£i = I are invariant under Lorentz trans- formations. This remark is already almost obvious from the very meaning of these quantities as degrees of circular and linear polarization. f PROBLEMS 1. Resolve an arbitrary partially polarized light wave into its "natural" and "polarized" parts. Solution: This resolution means the representation of the tensor J aB in the form J a0 — 2 J a al3 \ ^ Oa -^ Off • The first term corresponds to the natural, and the second to the polarized parts of the light. To determine the intensities of the parts we note that the determinant \Ja,-iJ (n) S a0 \ = \E™E<*f\ =0. Writing J a/} =Jp a p in the form (50.13) and solving the equation, we get y(»)=/(l-P). The intensity of the polarized part is /<*> = |E<, P) | 2 = J-jw=JP. The polarized part of the light is in general an elliptically polarized wave, where the directions of the axes of the ellipse coincide with the principal axes of the tensor S aB . The lengths Z>i and b 2 of the axes of the ellipse and the angle ^ formed by the axis bi and the y axis are given by the equations : bl+bl=JP, 2b ± b 2 = J£ 2 , tan 2«5 = p. 2. Represent an arbitrary partially polarized wave as a superposition of two incoherent ellip- tically polarized waves. Solution: For the hermitian tensor p a0 the "principal axes" are determined by two unit complex vectors n (n n* = 1), satisfying the equations Pann = An a . (1) The principal values X x and A 2 are the roots of the equation \pap — ^dae\ =0. Multiplying (1) on both sides by n*, we have: ^■ = Pann*n & = j\E 0a n*\ 2 , from which we see that X u X 2 are real and positive. Multiplying the equations n M (i) _ ; W (D -,* w (2)* _ ;„ w (2)* f For a direct proof, we note that since the field of the wave is transverse in any reference frame, it is clear from the start that the tensor p aB remains two-dimensional in any new frame. The transformation of p aB into Pae leaves unchanged the sum of absolute squares p a0 pt (in fact, the form of the transformation does not depend on the specific polarization properties of the light, while for a completely polarized wave this sum is 1 in any reference system). Because this transformation is real, the real and imaginary parts of the tensor Pap (50.9) transform independently, so that the sums of the squares of the components of each separately remain constant, and are expressed in terms of / and A. 124 ELECTROMAGNETIC WAVES § 51 for the first by n (2) * and for the second by n™, taking the difference of the results and using the hermiticity of p a0 , we get : It then follows that n (1) • n (2) * = 0, i.e. the unit vectors n (1) and n <2) are mutually orthogonal. The expansion of the wave is provided by the formula One can always choose the complex amplitude so that, of the two mutually perpendicular com- ponents, one is real and the other imaginary (compare § 48). Setting « ( i> - bi, /# = ib 2 (where now b t and b 2 are understood to be normalized by the condition b\+bl=\\ we get from the equation n (1) • n (2) * =0: n™ = ib 2 , rf* = b x . We then see that the ellipses of the two elliptically polarized vibrations are similar (have equal axis ratio), and one of them is turned through 90° relative to the other. 3. Find the law of transformation of the Stokes parameters for a rotation of the y, z axes through an angle $. Solution: The law is determined by the connection of the Stokes parameters to the components of the two-dimensional tensor in the yz plane, and is given by the formulas £i = £icos20— £ 3 sin2& ^ = ii sin 2^+£ 3 cos 20, & = &. § 51. The Fourier resolution of the electrostatic field The field produced by charges can also be formally expanded in plane waves (in a Fourier integral). This expansion, however, is essentially different from the expansion of electro- magnetic waves in vacuum, for the field produced by charges does not satisfy the homo- geneous wave equation, and therefore each term of this expansion does not satisfy the equation. From this it follows that for the plane waves into which the field of charges can be expanded, the relation k 2 = a> 2 /c 2 , which holds for plane monochromatic electromagnetic waves, is not fulfilled. In particular, if we formally represent the electrostatic field as a superposition of plane waves, then the "frequency" of these waves is clearly zero, since the field under consideration does not depend on the time. The wave vectors themselves are, of course, different from zero. We consider the field produced by a point charge e, located at the origin of coordinates. The potential 4> of this field is determined by the equation (see § 36) A0 = -4ne5(T). (51.1) We expand $ in a Fourier integral, i.e., we represent it in the form + 00 * -/**'*<& (5L2) — oo where d 3 k denotes dk x dk y dk z . In this formula <£ k = J #>"' k ' r ^. Applying the Laplace operator to both sides of (51.2), we obtain + 00 § 52 CHARACTERISTIC VIBRATIONS OF THE FIELD 125 so that the Fourier component of the expression A<j> is (A<£) k =-/c 2 <£ k . On the other hand, we can find (A<£) k by taking Fourier components of both sides of equation (51.1), (A(f>) k =- j 4ne5(i)e-* r dV= - Ane. — 00 Equating the two expressions obtained for (A</>) k , we find Ane <t>* = ^- (51-3) This formula solves our problem. Just as for the potential (f>, we can expand the field + 00 ■■_/ *•*'!£• (5i - 4) With the aid of (51.2), we have + 00 *--*«! ***■'§>— !**>+■'§?• — oo Comparing with (51.4), we obtain „ ., . 4nek E k =-ik(f> k =-i^ r . (51.5) From this we see that the field of the waves, into which we have resolved the Coulomb field, is directed along the wave vector. Therefore these waves can be said to be longitudinal. § 52. Characteristic vibrations of the field We consider an electromagnetic field (in the absence of charges) in some finite volume of space. To simplify further calculations we assume that this volume has the form of a rect- angular parallelepiped with sides A, B, C, respectively. Then we can expand all quantities characterizing the field in this parallelepiped in a triple Fourier series (for the three co- ordinates). This expansion can be written (e.g. for the vector potential) in the form: A = X(a ke '' k -'+a* e - ik -'), (52.1) k explicitly indicating that A is real. The summation extends here over all possible values of the vector k whose components run through the values 2nn x 2nn v , 2nn, fe - = -^-> ^ = V' K = ~C^' (52 - 2) where n x , n y , n z are positive and negative integers/From the equation div A = it follows that for each k, k'a k = 0, (52.3) i.e., the complex vectors a k are "perpendicular" to the corresponding wave vectors k. The 126 ELECTROMAGNETIC WAVES § 52 vectors a k are, of course, functions of the time; they satisfy the equation a k + c 2 ic 2 a k = 0. (52.4) If the dimensions A, B, C of the volume are sufficiently large, then neighboring values of k x , k y , k z (for which n x , n y , n z differ by unity) are very close to one another. In this case we may speak of the number of possible values of k x , k y , k z in the small intervals Ak x , Ak y , Ak z . Since to neighboring values of, say, k x , there correspond values of n x differing by unity, the number An x of possible values of k x in the interval Ak x is equal simply to the number of values of n x in the corresponding interval. Thus, we obtain A„, = ^Afc„ A„, = £Afc„ A„ 2 = |Afc r . The total number An of possible values of the vector k with components in the intervals Ak x , Ak y , Ak z is equal to the product An x An y An z , that is, An = -^ Ak x Ak y Ak z , (52.5) (27T) where V = ABC is the volume of the field. It is easy to determine from this the number of possible values of the wave vector having absolute values in the interval Ak, and directed into the element of solid angle Ao. To get this we need only transform to polar coordinates in the "k space" and write in place of Ak x Ak y Ak z the element of volume in these coordinates. Thus An = ^k 2 AkAo. (52.6) (2tt) 3 Finally, the number of possible values of the wave vector with absolute value k in the interval Ak and pointing in all directions is (we write An in place of Ao) An = -^ k 2 Ak. (52.7) 2n 2 The vectors a k as functions of the time behave like simply periodic functions with frequencies co k = ck (see 52.4). We present the expansion of the field in such a form that it appears as an expansion in propagating plane waves. To do this we assume that each of the a k depends on the time through the factor e~ iWkt : a k ~<T tofcf , <o k =c/c. (52.8) Then each individual term in the sum (52.1) is a function only of the difference k-r -co k t which corresponds to a wave propagating in the direction of the vector k. We calculate the total energy Sn] (E 2 + H 2 )dV of the field in the volume V, expressing it in terms of the quantities a k . For the electric field we have E = _ 1 A = - ix(%e ikr + a^- £kr ), c c k or, keeping in mind (52.8), E = i £ /c(a k e ,kr -a^- ikr ). (52.9) § 52 CHARACTERISTIC VIBRATIONS OF THE FIELD 127 For the magnetic field H = curl A, we obtain H = i £ (k x a k e +ikr -k x aje"*"). (52.10 k When calculating the squares of these sums, we must keep in mind that all products of terms with wave vectors k # k' give zero on integration over the whole volume. In fact, such terms contain factors of the form e +/q ' r , q = k+k', and the integral, e.g. of A .In r l ~A nxX J e dx, o with integer n x different from zero, gives zero. For the same reason, products containing the factors e ±2ik ' r vanish. In those terms from which the exponentials drop out, integration over dV gives just the volume V. As a result, we obtain ' = f I { fc2 "k * a * +( k x a *) ' ( k x a **)l- 47T k Since a k • k = 0, we have (k x a k ) • (k x aif) = fc 2 a k • a£, and we obtain finally k 2 V $ — Z <^k> <^k = ir~ a k ' a k k 2.TI (52.11) Thus the total energy of the field is expressed as a sum of the energies <f k , associated with each of the plane waves individually. In a completely analogous fashion, we can calculate the total momentum of the field, 4 f SdV = -J- f ExHdV, c 2 J Arte J for which we obtain ?5-t- (52 - l2) This result could have been anticipated in view of the relation between the energy and momentum of a plane wave (see § 47). The expansion succeeds in expressing the field in terms of a series of discrete parameters (the vectors a k ), in place of the description in terms of a continuous series of parameters, which is essentially what is done when we give the potential A(x, y,z,t) at all points of space. We now make a transformation of the variables a k , which has the result that the equations of the field take on a form similar to the canonical equations (Hamilton equations) of mechanics. We introduce the real "canonical variables" Q k and P k according to the relations Qk = J ^T2 (% + %*), (52.13) 4nc = "' COfe V4^ (ak ~ ak * ) = 0k - 128 ELECTROMAGNETIC WAVES § 52 The Hamiltonian of the field is obtained by substituting these expressions in the energy (52.11): * = Z ^k = Z i(P k 2 + o>jQ2). (52.14) k k Then the Hamilton equations dj^/8F k = k coincide with P k = k , which is thus a con- sequence of the equations of motion. (This was achieved by an appropriate choice of the coefficient in (52.13).) The equations of motion, dJ^jdQ k = -P k , become the equations Qk + o>*Qk = 0, (52.15) that is, they are identical with the equations of the field. Each of the vectors Q k and P k is perpendicular to the wave vector k, i.e. has two in- dependent components. The direction of these vectors determines the direction of polariza- tion of the corresponding travelling wave. Denoting the two components of the vector Q k (in the plane perpendicular to k) by Q kj ,j = 1, 2, we have Qk 2 = iek> and similarly for P k . Then JT = £ Jf ki , M> kj = KPlj + colQtj). (52.16) kj We see that the Hamiltonian splits into a sum of independent terms 3f kj , each of which contains only one pair of the quantities Q kj , P kJ . Each such term corresponds to a traveling wave with a definite wave vector and polarization. The quantity 34? kJ has the form of the Hamiltonian of a one-dimensional "oscillator", performing a simple harmonic vibration. For this reason, one sometimes refers to this result as the expansion of the field in terms of oscillators. We give the formulas which express the field explicitly in terms of the variables P k , Q k . From (52.13), we have * k = iJf ( p k~ ^4), ^ = ~ljf ( P " + fft) *2k). (52.17) Substituting these expressions in (52.1), we obtain for the vector potential of the field: A = 2 Jf Z I ( cfe Qk cos k-r-P k sin k-r). (52.18) For the electric and magnetic fields, we find E= -2^£(cfcQ k sink-r + P k cosk-r), In k 1 < 52 - 19 > H = -2 J - £ - {ck(k x Q k ) sin k • r+(k x P k ) cos k • r}. CHAPTER 7 THE PROPAGATION OF LIGHT § 53. Geometrical optics A plane wave is characterized by the property that its direction of propagation and amplitude are the same everywhere. Arbitrary electromagnetic waves, of course, do not have this property. Nevertheless, a great many electromagnetic waves, which are not plane, have the property that within each small region of space they can be considered to be plane. For this, it is clearly necessary that the amplitude and direction of the wave remain practically constant over distances of the order of the wavelength. If this condition is satisfied, we can introduce the so-called wave surface, i.e. a surface at all of whose points the phase of the wave is the same (at a given time). (The wave surfaces of a plane wave are obviously planes perpendicular to the direction of propagation of the wave.) In each small region of space we can speak of a direction of propagation of the wave, normal to the wave surface. In this way we can introduce the concept of rays — curves whose tangents at each point coincide with the direction of propagation of the wave. The study of the laws of propagation of waves in this case constitutes the domain of geometrical optics. Consequently, geometrical optics considers the propagation of waves, in particular of light, as the propagation of rays, completely divorced from their wave properties. In other words, geometrical optics corresponds to the limiting case of small wavelength, A -> 0. We now take up the derivation of the fundamental equation of geometrical optics — the equation determining the direction of the rays. Let / be any quantity describing the field of the wave (any component of E or H). For a plane monochromatic wave, /has the form /= fle «k-'-«*+«> = fle *(-*i*'+«) (53.1) (we omit the Re; it is understood that we take the real part of all expressions). We write the expression for the field in the form f=atf*. (53.2) In case the wave is not plane, but geometrical optics is applicable, the amplitude a is, generally speaking, a function of the coordinates and time, and the phase ^, which is called the eikonal, does not have a simple form, as in (53.1). It is essential, however, that ^ be a large quantity. This is clear immediately from the fact that it changes by In when we move through one wavelength, and geometrical optics corresponds to the limit X -* 0. Over small space regions and time intervals the eikonal \j/ can be expanded in series; to 129 130 THE PROPAGATION OF LIGHT § 53 terms of first order, we have (the origin for coordinates and time has been chosen within the space region and time interval under consideration; the derivatives are evaluated at the origin). Comparing this expression with (53.1), we can write k = -^sgradtfr, <*>=-f t , (53.3) which corresponds to the fact that in each small region of space (and each small interval of time) the wave can be considered as plane. In four-dimensional form, the relation (53.3) is expressed as k >=-£» < 53 - 4 > where k t is the wave four-vector. We saw in § 48 that the components of the four- vector k l are related by k t k l = 0. Sub- stituting (53.4), we obtain the equation dXi dx l ' ' ' This equation, the eikonal equation, is the fundamental equation of geometrical optics. The eikonal equation can also be derived by direct transition to the limit X -* in the wave equation. The field / satisfies the wave equation d 2 f dx t dx l Substituting /= ae^, we obtain -. = 0. d 2 a .. ^ da dxj/ ,. p d 2 ^ d\b M dx t dx l dxidx 1 dxfix 1 dx t dx l But the eikonal if/, as we pointed out above, is a large quantity; therefore we can neglect the first three terms compared with the fourth, and we arrive once more at equation (53.5). We shall give certain relations which, in their application to the propagation of light in vacuum, lead only to completely obvious results. Nevertheless, they are important because, in their general form, these derivations apply also to the propagation of light in material media. From the form of the eikonal equation there results a remarkable analogy between geometrical optics and the mechanics of material particles. The motion of a material particle is determined by the Hamilton- Jacobi equation (16.11). This equation, like the eikonal equation, is an equation in the first partial derivatives and is of second degree. As we know, the action S is related to the momentum p and the Hamiltonian «?f of the particle by the relations 8S „ dS '-*• * = -*■ Comparing these formulas with the formulas (53.3), we see that the wave vector plays the same role in geometrical optics as the momentum of the particle in mechanics, while the frequency plays the role of the Hamiltonian, i.e., the energy of the particle. The absolute § 53 GEOMETRICAL OPTICS 131 magnitude k of the wave vector is related to the frequency by the formula k = co/c. This relation is analogous to the relation/? = $/c between the momentum and energy of a particle with zero mass and velocity equal to the velocity of light. For a particle, we have the Hamilton equations In view of the analogy we have pointed out, we can immediately write the corresponding equations for rays : dco dco In vacuum, co = ck, so that k = 0, v = en (n is a unit vector along the direction of propaga- tion) ; in other words, as it must be, in vacuum the rays are straight lines, along which the light travels with velocity c. The analogy between the wave vector of a wave and the momentum of a particle is made especially clear by the following consideration. Let us consider a wave which is a super- position of monochromatic waves with frequencies in a certain small interval and occupying some finite region in space (this is called a wave packet). We calculate the four-momentum of the field of this wave, using formula (32.6) with the energy-momentum tensor (48.15) (for each monochromatic component). Replacing k l in this formula by some average value, we obtain an expression of the form P l = Ak\ (53.8) where the coefficient of proportionality A between the two four- vectors P ' and k l is some scalar. In three-dimensional form this relation gives : P = Ak, £ = Acq. (53.9) Thus we see that the momentum and energy of a wave packet transform, when we go from one reference system to another, like the wave vector and the frequency Pursuing the analogy, we can establish for geometrical optics a principle analogous to the principle of least action in mechanics. However, it cannot be written in Hamiltonian form as 5 \Ldt = 0, since it turns out to be impossible to introduce, for rays, a function analogous to the Lagrangian of a particle. Since the Lagrangian of a particle is related to the Hamil- tonian #? by the equation L = p • dJf/dp — 34?, replacing the Hamiltonian 3f by the frequency co and the momentum by the wave vector k, we should have to write for the Lagrangian in optics k • dco/dk — co. But this expression is equal to zero, since co = ck. The impossibility of introducing a Lagrangian for rays h also clear directly from the considera- tion mentioned earlier that the propagation of rays is analogous to the motion of particles with zero mass. If the wave has a definite constant frequency co, then the time dependence of its field is given by a factor of the form e~ imt . Therefore for the eikonal of such a wave we can write xj/ = -cot + ij/ (x, y, z), (53.10) where i/f is a function only of the coordinates. The eikonal equation (53.5) now takes the form (grad ^ ) 2 = ^- (53.11) c The wave surfaces are the surfaces of constant eikonal, i.e. the family of surfaces of the form 132 THE PROPAGATION OF LIGHT § 54 ij/ (x, y, z) — const. The rays themselves are at each point normal to the corresponding wave surface; their direction is determined by the gradient V^ - As is well known, in the case where the energy is constant, the principle of least action for particles can also be written in the form of the so-called principle of Maupertuis: 5S = 8 Jp-rfl = 0, where the integration extends over the trajectory of the particle between two of its points. In this expression the momentum is assumed to be a function of the energy and the co- ordinate differentials. The analogous principle for rays is called Fermafs principle. In this case, we can write by analogy: dij/ = 8 !k-dl = 0. (53.12) In vacuum, k = (co/c)n, and we obtain (dl-n = dl): dfdl = 0, (53.13) which corresponds to rectilinear propagation of the rays. § 54. Intensity In geometrical optics, the light wave can be considered as a bundle of rays. The rays themselves, however, determine only the direction of propagation of the light at each point; there remains the question of the distribution of the light intensity in space. On some wave surface of the bundle of rays under consideration, we isolate an in- finitesimal surface element. From differential geometry it is known that every surface has, at each of its points, two (generally different) principal radii of curvature. Let ac and bd (Fig. 7) be elements of the principal circles of curvature, constructed at a given element of •-B Fig. 7. the wave surface. Then the rays passing through a and c meet at the corresponding center of curvature lt while the rays passing through b and d meet at the other center of curvature o 2 . For fixed angular openings of the beams starting from O t and 2 , the lengths of the arcs ac and bd are, clearly, proportional to the corresponding radii of curvature R t and R 2 (i.e. to the lengths O t O and 2 O). The area of the surface element is proportional to the product § 54 INTENSITY 133 of the lengths ac and bd, i.e., proportional to R ± R 2 . In other words, if we consider the element of the wave surface bounded by a definite set of rays, then as we move along them the area of the element will change proportionally to R t R 2 . On the other hand, the intensity, i.e. the energy flux density, is inversely proportional to the surface area through which a given amount of light energy passes. Thus we arrive at the result that the intensity is '-£ This formula must be understood as follows. On each ray (AB in Fig. 7) there are definite points O t and 2 , which are the centers of curvature of all the wave surfaces intersecting the given ray. The distances OO t and 00 2 from the point O where the wave surface inter- sects the ray, to the points O x and 2 , are the radii of curvature R x and R 2 of the wave surface at the point O. Thus formula (54.1) determines the change in intensity of the light along a given ray as a function of the distances from definite points on this ray. We emphasize that this formula cannot be used to compare intensities at different points on a single wave surface. Since the intensity is determined by the square modulus of the field, we can write for the change of the field itself along the ray /= ^L e - (542) where in the phase factor e ikR we can write either e ikRl or e ikRl . The quantities e ikRl and e ikR 2 (f or a given ray) differ from each other only by a constant factor, since the difference R 1 —R 2 , the distance between the two centers of curvature, is a constant. If the two radii of curvature of the wave surface coincide, then (54.1) and (54.2) have the form: comt const ^ R 2 R ' This happens always when the light is emitted from a point source (the wave surfaces are then concentric spheres and R is the distance from the light source). From (54.1) we see that the intensity becomes infinite at the points R x = 0, R 2 = 0, i.e. at the centers of curvature of the wave surface. Applying this to all the rays in a bundle, we find that the intensity of the light in the given bundle becomes infinite, generally, on two surfaces — the geometrical loci of all the centers of curvature of the wave surfaces. These surfaces are called caustics. In the special case of a beam of rays with spherical wave surfaces, the two caustics fuse into a single point (focus). We note from well-known results of differential geometry concerning the properties of the loci of centers of curvature of a family of surfaces, that the rays are tangent to the caustic. It is necessary to keep in mind that (for convex wave surfaces) the centers of curvature of the wave surfaces can turn out to lie not on the rays themselves, but on their extensions beyond the optical system from which they emerge. In such cases we speak of imaginary caustics (or foci). In this case the intensity of the light does not become infinite anywhere. As for the increase of intensity to infinity, in actuality we must understand that the intensity does become large at points on the caustic, but it remains finite (see the problem in § 59). The formal increase to infinity means that the approximation of geometrical optics is never applicable in the neighborhood of the caustic. To this is related the fact that the 134 THE PROPAGATION OF LIGHT § 55 change in phase along the ray can be determined from formula (54.2) only over sections of the ray which do not include its point of tangency to the caustic. Later (in § 59), we shall show that actually in passing through the caustic the phase of the field decreases by njl. This means that if, on the section of the ray before its first intersection with the caustic, the field is proportional to the factor e lkx (x is the coordinate along the ray), then after passage through the caustic the field will be proportional to e ^ kx -( n / 2 )\ The same thing occurs in the neighborhood of the point of tangency to the second caustic, and beyond that point the field is proportional to e l '(fc*-«) t -j- § 55. The angular eikonal A light ray traveling in vacuum and impinging on a transparent body will, on its emergence from this body, generally have a direction different from its initial direction. This change in direction will, of course, depend on the specific properties of the body and on its form. However, it turns out that one can derive general laws relating to the change in direction of a light ray on passage through an arbitrary material body. In this it is assumed only that geometrical optics is applicable to rays propagating in the interior of the body under con- sideration. As is customary, we shall call such transparent bodies, through which rays of light propagate, optical systems. Because of the analogy mentioned in § 53, between the propagation of rays and the motion of particles, the same general laws are valid for the change in direction of motion of a particle, initially moving in a straight line in vacuum, then passing through some electromagnetic field, and once more emerging into vacuum. For definiteness, we shall, however, always speak later of the propagation of light rays. We saw in a previous section that the eikonal equation, describing the propagation of the rays, can be written in the form (53.11) (for light of a definite frequency). From now on we shall, for convenience, designate by xj/ the eikonal i]/ divided by the constant co/c. Then the basic equation of geometrical optics has the form : (VtfO 2 = 1. (55.1) Each solution of this equation describes a definite beam of rays, in which the direction of the rays passing through a given point in space is determined by the gradient of \j/ at that point. However, for our purposes this description is insufficient, since we are seeking general relations determining the passage through an optical system not of a single definite bundle of rays, but of arbitrary rays. Therefore we must use an eikonal expressed in such a form that it describes all the generally possible rays of light, i.e. rays passing through any pair of points in space. In its usual form the eikonal i/<r) is the phase of the rays in a certain bundle passing through the point r. Now we must introduce the eikonal as a function i/^r, r') of the coordinates of two points (r, r' are the radius vectors of the initial and end points of the ray). A ray can pass through each pair of points r, r', and i/<r, r') is the phase difference (or, as it is called, the optical path length) of this ray between the points r and r'. From now on we shall always understand by r and r' the radius vectors to points on the ray before and after its passage through the optical system. t Although formula (54.2) itself is not valid near the caustic, the change in phase of the field corresponds formally to a change in sign (i.e. multiplication by e in ) of Rt or R 2 in this formula. § 55 THE ANGULAR EIKONAL 135 If in \l/(r, r') one of the radius vectors, say r', is fixed, then i// as a function of r describes a definite bundle of rays, namely, the bundle of rays passing through the point r'. Then \{/ must satisfy equation (55.1), where the differentiations are applied to the components of r. Similarly, if r is assumed fixed, we again obtain an equation for if/(r, r'), so that (V r i/0 2 = 1, (V r ,</0 2 =1. (55.2) The direction of the ray is determined by the gradient of its phase. Since \j/(r, r') is the difference in phase at the points r and r', the direction of the ray at the point r' is given by the vector n' = d\j//8r', and at the point r by the vector n = -8\j//dr. From (55.2) it is clear that n and n' are unit vectors : n 2 = n' 2 = l. (55.3) The four vectors r, r', n, n' are interrelated, since two of them (n, n') are derivatives of a certain function \]/ with respect to the other two (r, r'). The function \\i itself satisfies the auxiliary conditions (55.2). To obtain the relation between n, n', r, r', it is convenient to introduce, in place of \//, another quantity, on which no auxiliary condition is imposed (i.e., is not required to satisfy any differential equations). This can be done as follows. In the function \J/ the independent variables are r and r', so that for the differential dij/ we have dil/ d\l/ , , , dij/ = -r--dr+ ^-/dr' = -n- dr + ri • dx' . Y dr dr' We now make a Legendre transformation from r, r' to the new independent variables n, n', that is, we write di]/ = — d(n-r) + r- dn + d(ri -r') — r' -dri, from which, introducing the function X = n'T'-n-r-^, (55.4) we have dx= -r-dn + r'-dn'. (55.5) The function x is called the angular eikonal; as we see from (55.5), the independent variables in it are n and n\ No auxiliary conditions are imposed on x- I n f act > equation (55.3) now states only a condition referring to the independent variables : of the three components n x , n y , n z , of the vector n (and similarly for n'), only two are independent. As independent variables we shall use n y , n z , n' y , n' z ; then n 3C = Vl-«J-«z, n' x = \ll-n' y 2 -n z 2 . Substituting these expressions in dx— —xdn x — ydn y — zdn z + x'dn' x + y'dn' y +z'dn z , dn' we obtain for the differential dx : dx= ~[y x)dn y - (z x) dn z +(y'-^ f Adn' y +(z' - ~ x From this we obtain, finally, the following equations : n y dx y- — x=-~, n x on y n z dx z x = -t— , n x dn z , n' y , dx n x dn y , K , dx z - — x = — , n x dn z (55.6) 136 THE PROPAGATION OF LIGHT § 56 which is the relation sought between n, n', r, r'. The function x characterizes the special properties of the body through which the rays pass (or the properties of the field, in the case of the motion of a charged particle). For fixed values of n, n', each of the two pairs of equations (55.6) represent a straight line. These lines are precisely the rays before and after passage through the optical system. Thus the equation (55.6) directly determines the path of the ray on the two sides of the optical system. § 56. Narrow bundles of rays In studying the passage of beams of rays through optical systems, special interest attaches to bundles whose rays all pass through one point (such bundles are said to be homocentric). After passage through an optical system, homocentric bundles in general cease to be homocentric, i.e. after passing through a body the rays no longer come together in any one point. Only in exceptional cases will the rays starting from a luminous point come together after passage through an optical system and all meet at one point (the image of the luminous point).f One can show (see § 57) that the only case for which all homocentric bundles remain strictly homocentric after passage through the optical system is the case of identical imaging, i.e., the case where the image differs from the object only in its position or orientation, or is mirror inverted. Thus no optical system can give a completely sharp image of an object (having finite dimensions) except in the trivial case of identical imaging. $ Only approximate, but not completely sharp images can be produced of an extended body, in any case other than for identical imaging. The most important case where there is approximate transition of homocentric bundles into homocentric bundles is that of sufficiently narrow beams (i.e. beams with a small opening angle) passing close to a particular line (for a given optical system). This line is called the optic axis of the system. Nevertheless, we must note that even infinitely narrow bundles of rays (in the three- dimensional case) are in general not homocentric; we have seen (Fig. 7) that even in such a bundle different rays intersect at different points (this phenomenon is called astigmatism). Exceptions are those points of the wave surface at which the two principal radii of curvature are equal — a small region of the surface in the neighborhood of such points can be considered as spherical, and the corresponding narrow bundle of rays is homocentric. We consider an optical system having axial symmetry. § The axis of symmetry of the system is also its optical axis. The wave surface of a bundle of rays traveling along this axis also has axial symmetry; as we know, surfaces of rotation have equal radii of curvature at their points of intersection with the symmetry axis. Therefore a narrow bundle moving in this direction remains homocentric. t The point of intersection can lie either on the rays themselves or on their continuations; depending on this, the image is said to be real or virtual. % Such imaging can be produced with a plane mirror. § It can be shown that the problem of image formation with the aid of narrow bundles, moving in the neighborhood of the optical axis in a nonaxially-symmetric system, can be reduced to image formation in an axially-symmetric system plus a subsequent rotation of the image thus obtained, relative to the object. § 56 NARROW BUNDLES OF RAYS 137 To obtain general quantitative relations, determining image formation with the aid of narrow bundles, passing through an axially-symmetric optical system, we use the general equations (55.6) after determining first of all the form of the function x in the case under consideration. Since the bundles of rays are narrow and move in the neighborhood of the optical axis, the vectors n, n' for each bundle are directed almost along this axis. If we choose the optical axis as the X axis, then the components n y , n z , ri y , ri z will be small compared with unity. As for the components n x , ri x ;n x &l and n' x can be approximately equal to either +1 or — 1. In the first case the rays continue to travel almost in their original direction, emerging into the space on the other side of the optical system, which in this case is called a lens. In the second the rays change their direction to almost the reverse; such an optical system is called a mirror. Making use of the smallness of n y , n z , n' y , n' z , we expand the angular eikonal x (« y , n z , ri y , n' z ) in series and stop at the first terms. Because of the axial symmetry of the whole system, X must be invariant with respect to rotations of the coordinate system around the optical axis. From this it is clear that in the expansion of x there can be no terms of first order, proportional to the first powers of the y- and z-components of the vectors n and n' ; such terms would not have the required invariance. The terms of second order which have the required property are the squares n 2 and n' 2 and the scalar product n n'. Thus, to terms of second order, the angular eikonal of an axially-symmetric optical system has the form X = const+ | (n 2 y + n 2 z )+f(n y ri y +n z ri z ) + ± {ri 2 + ri 2 ), (56.1) where/, g, h are constants. For definiteness, we now consider a lens, so that we set n' x « 1 ; for a mirror, as we shall show later, all the formulas have a similar appearance. Now substituting the expression (56.1) in the general equations (55.6), we obtain: np - g) -fn' y = y, fn y + rip' + h) = y', n z (x-g)-fri 2 = Z , fn z + ri z (x' + h) = z'. ( ' } We consider a homocentric bundle emanating from the point x,y,z; let the point x', y' z' be the point in which all the rays of the bundle intersect after passing through the lens. If the first and second pairs of equations (56.2) were independent, then these four equations, for given x, y, z, x', y', z', would determine one definite set of values n y , n z , ri y , ri z , that is, there would be just one ray starting from the point x, y, z, which would pass through the point x', y', z'. In order that all rays starting from x, y, z shall pass through x', y', z', it is consequently necessary that the equations (56.2) not be independent, that is, one pair of these equations must be a consequence of the other. The necessary condition for this dependence is that the coefficients in the one pair of equations be proportional to the coefficients of the other pair. Thus we must have *_^ = __/_ = Z = i. (56.3) / x' + h y' z' In particular, (x-g)(x' + h)=-f 2 . (56.4) The equations we have obtained give the required connection between the coordinates of the image and object for image formation using narrow bundles. 138 THE PROPAGATION OF LIGHT § 56 The points x = g and x = —A on the optical axis are called the principal foci of the optical system. Let us consider bundles of rays parallel to the optical axis. The source point of such rays is, clearly, located at infinity on the optical axis, that is, x = oo. From (56.3) we see that in this case, x' = —A. Thus a parallel bundle of rays, after passage through the optical system, intersects at the principal focus. Conversely, a bundle of rays emerging from the principal focus becomes parallel after passage through the system. In the equation (56.3) the coordinates x and x' are measured from the same origin of co- ordinates, lying on the optical axis. It is, however, more convenient to measure the co- ordinates of object and image from different origins, choosing them at the corresponding principal foci. As positive direction of the coordinates we choose the direction from the corresponding focus toward the side to which the light travels. Designating the new co- ordinates of object and image by capital letters, we have X = x-g, X' = x' + h, Y = y, Y' = y', Z = z, Z' = z'. The equations of image formation (56.3) and (56.4) in the new coordinates take the form XX' = -f 2 , (56.5) r z' f x' The quantity /is called the principal focal length of the system. The ratio Y'/Y is called the lateral magnification. As for the longitudinal magnification, since the coordinates are not simply proportional to each other, it must be written in dif- ferential form, comparing the length of an element of the object (along the direction of the axis) with the length of the corresponding element in the image. From (56.5) we get for the "longitudinal magnification" dX' f^ = /T X 2 \Y 2 ~ \ v i • (56.7) dX We see from this that even for an infinitely small object, it is impossible to obtain a geometrically similar image. The longitudinal magnification is never equal to the transverse (except in the trivial case of identical imaging). A bundle passing through the point X =/on the optical axis intersects once more at the point X' = —/on the axis; these two points are called principal points. From equation (56.2) {n y X-fny = Y, n z X-fn' z = Z) it is clear that in this case (X=f Y=Z = 0), we have the equations n y = n' y , n z = n' z . Thus every ray starting from a principal point crosses the optical axis again at the other principal point in a direction parallel to its original direction. If the coordinates of object and image are measured from the principal points (and not from the principal foci), then for these coordinates £ and £', we have ? = X'+f, Z = X-f. Substituting in (56.5) it is easy to obtain the equations of image formation in the form i-i=-l. (56.8) One can show that for an optical system with small thickness (for example, a mirror or a thin lens), the two principal points almost coincide. In this case the equation (56.8) is particularly convenient, since in it £ and £' are then measured practically from one and the same point. § 56 NARROW BUNDLES OF RAYS 139 If the focal distance is positive, then objects located in front of the focus (X > 0) are imaged erect (Y'/Y> 0); such optical systems are said to be converging. Iff< 0, then for I>0we have Y'/Y<0, that is, the object is imaged in inverted form; such systems are said to be diverging. There is one limiting case of image formation which is not contained in the formulas (56.8) ; this is the case where all three coefficients/, g, h are infinite (i.e. the optical system has an infinite focal distance and its principal foci are located at infinity). Going to the limit of infinite f, g, h in (56.4) we obtain . h , f 2 -9 h x = - x-\ . 9 9 Since we are interested only in the case where the object and its image are located at finite distances from the optical system, /, g, h must approach infinity in such fashion that the ratios h/g, (f 2 -gh)/g are finite. Denoting them, respectively, by a 2 and /?, we have x' = a 2 x + p. For the other two coordinates we now have from the general equation (56.7) : v' z' y - = - = + a . V z Finally, again measuring the coordinates x and x' from different origins, namely from some arbitrary point on the axis and from the image of this point, respectively, we finally obtain the equations of image formation in the simple form X' = a 2 X, Y' = +<xY, Z' = ±aZ. (56.9) Thus the longitudinal and transverse magnifications are constants (but not equal to each other). This case of image formation is called telescopic. All the equations (56.5) through (56.9), derived by us for lenses, apply equally to mirrors, and even to an optical system without axial symmetry, if only the image formation occurs by means of narrow bundles of rays traveling near the optical axis. In this, the reference points for the x coordinates of object and image must always be chosen along the optical axis from corresponding points (principal foci or principal points) in the direction of propaga- tion of the ray. In doing this, we must keep in mind that for an optical system not possessing axial symmetry, the directions of the optical axis in front of and beyond the system do not lie in the same plane. PROBLEMS 1. Find the focal distance for image formation with the aid of two axially-symmetric optical systems whose optical axes coincide. Solution: Let/i and/ 2 be the focal lengths of the two systems. For each system separately, we have X\ X \ = — / 1 , X2X2 = —f<2, • Since the image produced by the first system acts as the object for the second, then denoting by / the distance between the rear principal focus of the first system and the front focus of the second, we have X 2 = X x ' —I; expressing X 2 ' in terms of X u we obtain yl _ Xif 2 2 f 1 2 +ix 1 140 THE PROPAGATION OF LIGHT § 56 or * + T")(* i -fl"(¥)'. from which it is clear that the principal foci of the composite system are located at the points *i = ~fi 2 /l, X* = A 2 // and the focal length is f— —(ill J i (to choose the sign of this expression, we must write the corresponding equation for the transverse magnification). x = x= Fig. 8. In case / — 0, the focal length /= oo, that is, the composite system gives telescopic image for- mation. In this case we have X 2 ' = Xitfalftf, that is, the parameter a in the general formula (56.9)isa = / 2 //i. 2. Find the focal length for charged particles of a "magnetic lens" in the form of a longitudinal homogeneous field in the section of length / (Fig. 8).f Solution: The kinetic energy of the particle is conserved during its motion in a magnetic field; therefore the Hamilton-Jacobi equation for the reduced action S (r) (where the total action is S= -£t+S )is (v*-?a) Af=p 2 , where p = -£ —m 2 c 2 = const. c Using formula (19.4) for the vector potential of the homogeneous magnetic field, choosing the x axis along the field direction and considering this axis as the optical axis of an axially-symmetric optical system, we get the Hamilton-Jacobi equation in the form: /dS \ 2 ZdSoY e 2 where r is the distance from the x axis, and S is a function of x and r. For narrow beams of particles propagating close to the optical axis, the coordinate r is small, so that accordingly we try to find 5*0 as a power series in r. The first two terms of this series are S =px+ia(x)r 2 , (2) where a(x) satisfies the equation pa'(x) + a 2 +^- 2 H 2 = 0. (3) In region 1 in front of the lens, we have: „cd _ P X — Xx where x x < is a constant. This solution corresponds to a free beam of particles, emerging along straight line rays from the point x = x x on the optical axis in region 1. In fact, the action function t This might be the field inside a long solenoid, when we neglect the disturbance of the homogeneity of the field near the ends of the solenoid. § 57 IMAGE FORMATION WITH BROAD BUNDLES OF RAYS 141 for the free motion of a particle with a momentum p in a direction out from the point x = Xi is S =p^r 2 +{x-x x ) 2 g*p(x-x 1 ) + pr* 2(jc— xi)" Similarly, in region 2 behind the lens we write: X — X-2. where the constant x 2 is the coordinate of the image of the point x x . In region 3 inside the lens, the solution of equation (3) is obtained by separation of variables, and gives : CT c3> = ^cot(^*+c\ 2c \2cp J where C is an arbitrary constant. The constant C and x 2 (for given x x ) are determined by the requirements of continuity of o(x) for x = and x = l: p eH p eH A I ' eH , , _ - — = — cot C, r^— = -=- cot ( — l+C x x 2c l—x 2 2c \2cp Eliminating the constant C from these equations, we find : (xi—g)(x a +h)= -f 2 , wheref 2cp eHl , , g== -eH COt W H=:g - l > f ^_2cp „ . eHl' eH sin - — 2cp § 57. Image formation with broad bundles of rays The formation of images with the aid of narrow bundles of rays, which was considered in the previous section, is approximate; it is the more exact (i.e. the sharper) the narrower the bundles. We now go over to the question of image formation with bundles of rays of arbitrary breadth. In contrast to the formation of an image of an object by narrow beams, which can be achieved for any optical system having axial symmetry, image formation with broad beams is possible only for specially constituted optical systems. Even with this limitation, as already pointed out in § 56, image formation is not possible for all points in space. The later derivations are based on the following essential remark. Suppose that all rays, starting from a certain point O and traveling through the optical system, intersect again at some other point O'. It is easy to see that the optical path length \j/ is the same for all these rays. In the neighborhood of each of the points O, O', the wave surfaces for the rays inter- secting in them are spheres with centers at O and O', respectively, and, in the limit as we approach O and O', degenerate to these points. But the wave surfaces are the surfaces of constant phase, and therefore the change in phase along different rays, between their points of intersection with two given wave surfaces, is the same. From what has been said, it follows that the total change in phase between the points O and O' is the same (for the different rays). t The value of /is given with the correct sign. However, to show this requires additional investigation. 142 THE PROPAGATION OF LIGHT § 57 Let us consider the conditions which must be fulfilled in order to have formation of an image of a small line segment using broad beams ; the image is then also a small line segment. We choose the directions of these segments as the directions of the £ and £' axes, with origins at any two corresponding points O and O' of the object and image. Let \j/ be the optical path length for the rays starting from O and reaching O'. For the rays starting from a point infinitely near to O with coordinate d£, and arriving at a point of the image with coordinate d£', the optical path length is ^ + #, where We introduce the "magnification" d? as the ratio of the length d% of the element of the image to the length d£ of the imaged element. Because of the smallness of the line segment which is being imaged, the quantity a can be considered constant along the line segment. Writing, as usual, di]//d£ = -n^, #/d£' = n'z (n^, n\ are the cosines of the angles between the directions of the ray and the corresponding axes £ and £'), we obtain dty = (a^n'^ — n^dt;. As for every pair of corresponding points of object and image, the optical path length ip + diff must be the same for all rays starting from the point d£ and arriving at the point d£'. From this we obtain the condition: a 5 n\ — n% = const. (57. 1) This is the condition we have been seeking, which the paths of the rays in the optical system must satisfy in order to have image formation for a small line segment using broad beams. The relation (57.1) must be fulfilled for all rays starting from the point O. Let us apply this condition to image formation by means of an axially-symmetric optical system. We start with the image of a line segment coinciding with the optical axis (x axis) ; clearly the image also coincides with the axis. A ray moving along the optical axis (n x = 1), because of the axial symmetry of the system, does not change its direction after passing through it, that is, n' x is also 1. From this it follows that const in (57.1) is equal in this case toa^-1, and we can rewrite (57.1) in the form l-n x l-n' x "*• Denoting by 9 and 6' the angles subtended by the rays with the optical axis at points of the object and image, we have R' i — n x = 1 — cos 6 = 2 sin 2 -, 1 — n' x = 1 — cos 6' = 2 sin 2 -. Thus we obtain the condition for image formation in the form . d sin - 2 /- = const = Va,. (57.2) sin — 2 § 58 THE LIMITS OF GEOMETRICAL OPTICS 143 Next, let us consider the imaging of a small portion of a plane perpendicular to the optical axis of an axially symmetric system; the image will obviously also be perpendicular to this axis. Applying (57. 1) to an arbitrary segment lying in the plane which is to be imaged, we get : a r sin 9' — sin 9 = const, where 9 and 9' are again the angles made by the beam with the optical axis. For rays emerging from the point of intersection of the object plane with the optical axis, and directed along this axis (9 = 0), we must have 9' = 0, because of symmetry. Therefore const is zero, and we obtain the condition for imaging in the form = const = a r . (57.3) sin 9' As for the formation of an image of a three-dimensional object using broad beams, it is easy to see that this is impossible even for a small volume, since the conditions (57.2) and (57.3) are incompatible. § 58. The limits of geometrical optics From the definition of a monochromatic plane wave, its amplitude is the same everywhere and at all times. Such a wave is infinite in extent in all directions in space, and exists over the whole range of time from - oo to + oo. Any wave whose amplitude is not constant every- where at all times can only be more or less monochromatic. We now take up the question of the "degree of non-monochromaticity" of a wave. Let us consider an electromagnetic wave whose amplitude at each point is a function of the time. Let w be some average frequency of the wave. Then the field of the wave, for example the electric field, at a given point has the form E (t)e~ i(0ot . This field, although it is of course not monochromatic, can be expanded in monochromatic waves, that is, in a Fourier integral. The amplitude of the component in this expansion, with frequency co, is propor- tional to the integral + oo [ E o (0e i(to " Wo)f ^- — oo The factor e Ka ~ ao)t is a periodic function whose average value is zero. If E were exactly constant, then the integral would be exactly zero, for co # co . If, however, E (0 is variable, but hardly changes over a time interval of order l/(co-co ), then the integral is almost equal to zero, the more exactly the slower the variation of E . In order for the integral to be sig- nificantly different from zero, it is necessary that E (f ) vary significantly over a time interval of the order of l/(co — co ). We denote by At the order of magnitude of the time interval during which the amplitude of the wave at a given point in space changes significantly. From these considerations, it now follows that the frequencies deviating most from w , which appear with reasonable intensity in the spectral resolution of this wave, are determined by the condition l/(co-co ) ~ At. If we denote by Aco the frequency interval (around the average frequency co ) which enters in the spectral resolution of the wave, then we have the relation AeoA*~l. (58.1) 144 THE PROPAGATION OF LIGHT § 58 We see that a wave is the more monochromatic (i.e. the smaller Aco) the larger At, i.e. the slower the variation of the amplitude at a given point in space. Relations similar to (58.1) are easily derived for the wave vector. Let Ax, Ay, Az be the orders of magnitude of distances along the X, Y, Z axes, in which the wave amplitude changes significantly. At a given time, the field of the wave as a function of the coordinates has the form E (ry' k °- r , where k is some average value of the wave vector. By a completely analogous derivation to that for (58.1) we can obtain the interval Ak of values contained in the expansion of the wave into a Fourier integral : Afc^Ax - 1, Ak y Ay ~ 1, Ak z Az ~ 1. (58.2) Let us consider, in particular, a wave which is radiated during a finite time interval. We denote by At the order of magnitude of this interval. The amplitude at a given point in space changes significantly during the time At in the course of which the wave travels completely past the point. Because of the relations (58.1) we can now say that the "lack of mono- chromaticity" of such a wave, Aco, cannot be smaller than 1/Af (it can of course be larger): 1 Aco ^ — . (58.3) Similarly, if Ax, Ay, Az are the orders of magnitude of the extension of the wave in space, then for the spread in the values of components of the wave vector, entering in the resolution of the wave, we obtain M,>i. Afc y »i-, A*,>1. (58.4 From these formulas it follows that if we have a beam of light of finite width, then the direction of propagation of the light in such a beam cannot be strictly constant. Taking the X axis along the (average) direction of light in the beam, we obtain 1 k '^kAy-A-y' (58 " 5) where 9 y is the order of magnitude of the deviation of the beam from its average direction in the XY plane and k is the wavelength. On the other hand, the formula (58.5) answers the question of the limit of sharpness of optical image formation. A beam of light whose rays, according to geometrical optics, would all intersect in a point, actually gives an image not in the form of a point but in the form of a spot. For the width A of this spot, we obtain, according to (58.5), where 9 is the opening angle of the beam. This formula can be applied not only to the image but also to the object. Namely, we can state that in observing a beam of light emerging from a luminous point, this point cannot be distinguished from a body of dimensions k/9. In this way formula (58.6) determines the limiting resolving power of a microscope. The minimum value of A, which is reached for 9 ~ 1, is k, in complete agreement with the fact that the limit of geometrical optics is determined by the wavelength of the light. § 59 DIFFRACTION 145 PROBLEM Determine the order of magnitude of the smallest width of a light beam produced from a parallel beam at a distance / from a diaphragm. Solution: Denoting the size of the aperture in the diaphragm by d, we have from (58.5) for the angle of deflection of the beam (the "diffraction angle"), Md, so that the width of the beam is of order d+(A/d)l. The smallest value of this quantity ~ VaL § 59. Diffraction The laws of geometrical optics are strictly correct only in the ideal case when the wave- length can be considered to be infinitely small. The more poorly this condition is fulfilled, the greater are the deviations from geometrical optics. Phenomenon which are the con- sequence of such deviations are called diffraction phenomena. Diffraction phenomena can be observed, for example, if along the path of propagation of the light f there is an obstacle — an opaque body (we call it a screen) of arbitrary form or, for example, if the light passes through holes in opaque screens. If the laws of geometrical optics were strictly satisfied, there would be beyond the screen regions of "shadow" sharply delineated from regions where light falls. The diffraction has the consequence that, instead of a sharp boundary between light and shadow, there is a quite complex distribution of the intensity of the light. These diffraction phenomena appear the more strongly the smaller the dimensions of the screens and the apertures in them, or the greater the wavelength. The problem of the theory of diffraction consists in determining, for given positions and shapes of the objects (and locations of the light sources), the distribution of the light, that is, the electromagnetic field over all space. The exact solution of this problem is possible only through solution of the wave equation with suitable boundary conditions at the surface of the body, these conditions being determined also by the optical properties of the material. Such a solution usually presents great mathematical difficulties. However, there is an approximate method which for many cases is a satisfactory solution of the problem of the distribution of light near the boundary between light and shadow. This method is applicable to cases of small deviation from geometrical optics, i.e. when firstly, the dimensions of all bodies are large compared with the wavelength (this requirement applies both to the dimensions of screens and apertures and also to the distances from the bodies to the points of emission and observation of the light) ; and secondly when there are only small deviations of the light from the directions of the rays given by geometrical optics. Let us consider a screen with an aperture through which the light passes from given sources. Figure 9 shows the screen in profile (the heavy line); the light travels from left to right. We denote by u some one of the components of E or H. Here we shall understand u to mean a function only of the coordinates, i.e. without the factor e~ i<ot determining the time dependence. Our problem is to determine the light intensity, that is, the field u, at any point of observation P beyond the screen. For an approximate solution of this problem in cases where the deviations from geometrical optics are small, we may assume that at the points of the aperture the field is the same as it would have been in the absence of the screen. In other words, the values of the field here are those which follow directly from geometrical t In what follows, in discussing diffraction we shall talk of the diffraction of light; all these same con- siderations also apply, of course, to any electromagnetic wave. 146 THE PROPAGATION OF LIGHT § 59 Fig. 9. optics. At all points immediately behind the screen, the field can be set equal to zero. In this the properties of the screen (i.e. of the screen material) obviously play no part. It is also obvious that in the cases we are considering, what is important for the diffraction is only the shape of the edge of the aperture, while the shape of the opaque screen is unimportant. We introduce some surface which covers the aperture in the screen and is bounded by its edges (a profile of such a surface is shown in Fig. 9 as a dashed line). We break up this surface into sections with area df whose dimensions are small compared with the size of the aperture, but large compared with the wavelength of the light. We can then consider each of these sections through which the light passes as if it were itself a source of light waves spreading out on all sides from this section. We shall consider the field at the point P to be the result of superposition of the fields produced by all the sections dfof the surface covering the aperture. (This is called Huygens' principle.) The field produced at the point P by the section <^is obviously proportional to the value u of the field at the section ^itself (we recall that the field at ^is assumed to be the same as it would have been in the absence of the screen). In addition, it is proportional to the projection df„ of the area df on the plane perpendicular to the direction n of the ray coming from the light source at df. This follows from the fact that no matter what shape the element ^has, the same rays will pass through it provided its projection df n remains fixed, and therefore its effect on the field at P will be the same. Thus the field produced at the point P by the section dfis proportional to udf n . Further- more, we must still take into account the change in the amplitude and phase of the wave during its propagation from df to P. The law of this change is determined by formula (54.3). Therefore udf n must be multiplied by (\/R)e ikR (where R is the distance from df to P, and k is the absolute value of the wave vector of the light), and we find that the required field is au R dfn, where a is an as yet unknown constant. The field at the point P, being the result of the addition of the fields produced by all the elements df, is consequently equal to u p = a u R df„ (59.1) where the integral extends over the surface bounded by the edge of the aperture. In the approximation we are considering, this integral cannot, of course, depend on the form of this surface. Formula (59.1) is, obviously, applicable not only to diffraction by an aperture § 59 DIFFRACTION 147 in a screen, but also to diffraction by a screen around which the light passes freely. In that case the surface of integration in (59.1) extends on all sides from the edge of the screen. To determine the constant a, we consider a plane wave propagating along the X axis ; the wave surfaces are parallel to the plane YZ. Let u be the value of the field in the YZ plane. Then at the point P, which we choose on the X axis, the field is equal to u p = ue lkx . On the other hand, the field at the point P can be determined starting from formula (59.1), choosing as surface of integration, for example, the YZ plane. In doing this, because of the smallness of the angle of diffraction, only those points of the YZ plane are important in the integral which lie close to the origin, i.e. the points for which y, z<x(x'\s the coordinate of the point P). Then /-» — 5 — 5 y 2 + z 2 R = jx 2 + y 2 + z 2 xx + 2x and (59.1) gives + 00 + 00 7 ikx p -Icy* /• .fcz^ — J e 2x dy J e 2x dz, where u is a constant (the field in the YZ plane) ; in the factor 1/R, we can put R ^ x = const. By the substitution y = ^2x/k these two integrals can be transformed to the integral + oo +oo +oo ikx 2lK u D = aue ——. k and we get On the other hand, u p = ue ikx , and consequently k 2ni' Substituting in (59.1), we obtain the solution to our problem in the form "'-J"^*""- (59 - 2) In deriving formula (59.2), the light source was assumed to be essentially a point, and the light was assumed to be strictly monochromatic. The case of a real, extended source, which emits non-monochromatic light, does not, however, require special treatment. Because of the complete independence (incoherence) of the light emitted by different points of the source, and the incoherence of the different spectral components of the emitted light, the total diffraction pattern is simply the sum of the intensity distributions obtained from the diffrac- tion of the independent components of the light. Let us apply formula (59.2) to the solution of the problem of the change in phase of a ray on passing through its point of tangency to the caustic (see the end of § 54). We choose as our surface of integration in (59.2) any wave surface, and determine the field u p at a point P, lying on some given ray at a distance x from its point of intersection with the wave surface we have chosen (we choose this point as coordinate origin O, and as YZ plane the plane tangent to the wave surface at the point O). In the integration of (59.2) only a small area of the wave surface in the neighborhood of O is important. If the XY and A'Zplanesare chosen 148 THE PROPAGATION OF LIGHT § 59 to coincide with the principal planes of curvature of the wave surface at the point O, then near this point the equation of the surface is V 2 Z 2 X _ * j 2.R i 2R 2 where R t and R 2 are the radii of curvature. The distance R from the point on the wave surface with coordinates X, y, z, to the point P with coordinates x, 0, 0, is On the wave surface, the field u can be considered constant; the same applies to the factor l/R. Since we are interested only in changes in the phase of the wave, we drop coefficients and write simply + 00 y 2 /l 1 \ +00 z 2 /l 1 \ u p~~i) e dfn--r \ dye \ dze . (59.3) - 00 — 00 The centers of curvature of the wave surface lie on the ray we are considering, at the points x = R t and x = R 2 ; these are the points where the ray is tangent to the caustic. Suppose R 2 < R v For x < R 2 , the coefficients of i in the exponentials appearing in the two integrands are positive, and each of these integrals is proportional to (1+0- Therefore on the part of the ray before its first tangency to the caustic, we have u p ~ e ikx . For R 2 < x < R u that is, on the segment of the ray between its two points of tangency, the integral over y is proportional to 1 + 1, but the integral over z is proportional to 1 - i, so that their product does not contain i. Thus we have here u p - -ie ikx = g'(**-(*/2)) s that is, as the ray passes in the neighborhood of the first caustic, its phase undergoes an additional change of —n/2. Finally, for x>R 1 ,vre have u p ~ -e ikx = e i(kx ~ n \ that is, on passing in the neighborhood of the second caustic, the phase once more changes by — njl. PROBLEM Determine the distribution of the light intensity in the neighborhood of the point where the ray is tangent to the caustic. Solution: To solve the problem, we use formula (59.2), taking the integral in it over any wave surface which is sufficiently far from the point of tangency of the ray to the caustic. In Fig. 10, ab is a section of this wave surface, and a'b' is a section of the caustic; a'b' is the evolute of the curve Fig. 10. § 59 DIFFRACTION 149 ab. We are interested in the intensity distribution in the neighborhood of the point O where the ray QO is tangent to the caustic; we assume the length D of the segment QO of the ray to be large. We denote by x the distance from the point O along the normal to the caustic, and assume positive values x for points on the normal in the direction of the center of curvature. The integrand in (59.2) is a function of the distance R from the arbitrary point Q' on the wave surface to the point P. From a well-known property of the evolute, the sum of the length of the seg- ment Q'O' of the tangent at the point O' and the length of the arc OO' is equal to the length QO of the tangent at the point O. For points O and O' which are near to each other we have OO' = Oq (q is the radius of curvature of the caustic at the point O). Therefore the length Q'O' = D — Oq. The distance Q'O (along a straight line) is approximately (the angle 9 is assumed to be small) 3 Q'O^ Q'O'+gsm 9 = D-Gq+q sin 9 ^ D-Q-. 6 Finally, the distance R = Q'P is equal to R ^ Q'O— x sin 9 ^ Q'O— x9, that is, Rc±D-x9-\q9 3 . Substituting this expression in (59.2), we obtain + 00 00 «p~ \ e e d9 = 2\ cos I kx9+-^-9 3 \d9 - oo o (the slowly varying factor 1/D in the integrand is unimportant compared with the exponential factor, so we assume it constant). Introducing the new integration variable £= (kg/l) 113 9, we get where 0(0 is the Airy function.f For the intensity / ~ |w p | 2 , we write: (concerning the choice of the constant factor, cf. below). For large positive values of x, we have from this the asymptotic formula "'~*( x { 2j fyy 2Vx f The Airy function <D(/) is defined as 4x 3 exp M <wn = -j= fcos(|+<^W (i) o (see Quantum Mechanics, Mathematical Appendices, § b). For large positive values of the argument, the asymptotic expression for <E>(f ) is 1 / 2 3/2 \ (2) that is, 0(0 goes exponentially to zero. For large negative values of t, the function 0(t) oscillates with decreasing amplitude according to the law: «KO* ( - = ^ 8 in(f(-/)3/'+^. (3) The Airy function is related to the MacDonald function (modified Hankel function) of order 1/3 : <t>(t) = V}J3nK ll3 (it 3 ' 2 ). (4) Formula (2) corresponds to the asymptotic expansion of K v (t): **)«Jl 150 THE PROPAGATION OF LIGHT § 60 that is, the intensity drops exponentially (shadow region). For large negative values of x, we have 2A V- sin 2 2(-x) 312 /2k* n\ 3 V q + 4/' that is, the intensity oscillates rapidly; its average value over these oscillations is A V-x From this meaning of the constant A is clear — it is the intensity far from the caustic which would be obtained from geometrical optics neglecting diffraction effects. The function <£(/) attains its largest value, 0.949, for t = -1.02; correspondingly, the maximum intensity is reached at x^lk^lo) 1 ' 3 = —1.02, where I = 2.02>Ak 1,3 Q- 1 > e . At the point where the ray is tangent to the caustic (x = 0), we have I = 0.S9 Ak ll3 q- 1 ' 6 [since 0(0) = 0.629]. Thus near the caustic the intensity is proportional to k 113 , that is, to X~ 1I3 (X is the wavelength). For A->0, the intensity goes to infinity, as it should (see § 54). § 60. Fresnel diffraction If the light source and the point P at which we determine the intensity of the light are located at finite distances from the screen, then in determining the intensity at the point P, only those points are important which lie in a small region of the wave surface over which we integrate in (59.2) — the region which lies near the line joining the source and the point P. In fact, since the deviations from geometrical optics are small, the intensity of the light arriving at P from various points of the wave surface decreases very rapidly as we move away from this line. Diffraction phenomena in which only a small portion of the wave surface plays a role are called Fresnel diffraction phenomena. Let us consider the Fresnel diffraction by a screen. From what we have just said, for a given point P only a small region at the edge of the screen is important for this diffraction. But over sufficiently small regions, the edge of the screen can always be considered to be a straight line. We shall therefore, from now on, understand the edge of the screen to mean just such a small straight line segment. We choose as the XY plane a plane passing through the light source Q (Fig. 1 1) and through the line of the edge of the screen. Perpendicular to this, we choose the plane XZ so that it passes through the point Q and the point of observation P, at which we try to deter- mine the light intensity. Finally, we choose the origin of coordinates O on the line of the edge of the screen, after which the positions of all three axes are completely determined. Fig. 11. §60 FRESNEL DIFFRACTION 151 p and D q . A negative d Let the distance from the light source Q to the origin be D q . We denote the x-coordinate of the point of observation P by D p , and its z-coordinate, i.e. its distance from the XY plane, by d. According to geometrical optics, the light should pass only through points lying above the X Y plane; the region below the XY plane is the region which according to geometrical optics should be in shadow (region of geometrical shadow). We now determine the distribution of light intensity on the screen near the edge of the geometrical shadow, i.e. for values of d small compared with D means that the point P is located within the geometrical shadow As the surface of integration in (59.2) we choose the half-plane passing through the line of the edge of the screen and perpendicular to the XY plane. The coordinates jc and y of points on this surface are related by the equation x = y tan a (a is the angle between the line of the edge of the screen and the Y axis), and the z-coordinate is positive. The field of the wave produced by the source Q, at the distance R q from it, is proportional to the factor e ikR q Therefore the field u on the surface of integration is proportional to u ~ exp {iky/y 2 + z 2 + (D q + y tan a) 2 }. In the integral (59.2) we must now substitute for R, R = y/ y 2 + ( z - df + (D p -y tan a) 2 . The slowly varying factors in the integrand are unimportant compared with the exponential. Therefore we may consider l/R constant, and write dy dz in place of df n . We then find that the field at the point P is + 00 00 m p ~ J* Jexp{/fc(V(Aj+J ; tana) 2 -l-v 2 + z 2 -oo + V(£p - y tan a) 2 + (z - d) 2 + v 2 )} dy dz. (60.1) As we have already said, the light passing through the point P comes mainly from points of the plane of integration which are in the neighborhood of O. Therefore in the integral (60.1) only values of y and z which are small (compared with D q and D p ) are important. For this reason we can write /■ v sec oc-t-z y/(D q +y tan a) 2 + y 2 + z 2 ~ D+ — +y tan a, y/(D p - y tan a) 2 + (z - d) 2 + y 2 D P + (z-d) 2 + y 2 2D, ■ v tan a. We substitute this in (60.1). Since we are interested only in the field as a function of the distance d, the constant factor exp {ik(D p +D q )} can be omitted; the integral over y also gives an expression not containing d, so we omit it also. We then find Jexp{//c( ik[ — 2D„ z 2 + 1 2D, (z-d) 2 )}dz. This expression can also be written in the form exp < ik 2(D P + D q )\ J exp < ik 1 K 1 + n 1*- d' 2 2 LW *v »p\ 1 1 dz. (60.2) 152 THE PROPAGATION OF LIGHT § 60 The light intensity is determined by the square of the field, that is, by the square modulus \u p \ 2 . Therefore, when calculating the intensity, the factor standing in front of the integral is irrelevant, since when multiplied by the complex conjugate expression it gives unity. An obvious substitution reduces the integral to | e^drj, e l " drj, (60.3) where — w kD { 2D/D^+D P ) Thus, the intensity / at the point P is : ™ = dJ 7 „Z q ,„ v (60.4) ' = 'i _ oo Jll *>r = ^° (f<V)+ \) 2 + (s(w 2 )+ iVl (60.5) where 2 [\ K ' 2) V 2 C(z) = I - cos n 2 dt], S(z) = I - sin r\ 2 dr\ o are called the Fresnel integrals. Formula (60.5) solves our problem of determining the light intensity as a function of d. The quantity I is the intensity in the illuminated region at points not too near the edge of the shadow ; more precisely, at those points with w P 1 (C(oo) = 5(oo) = \ in the limit w -> oo). The region of geometrical shadow corresponds to negative w. It is easy to find the asymptotic form of the function / (w) for large negative values of w. To do this we proceed as follows. Integrating by parts, we have J 2i\w\ 2i J n 2 ' \w\ \w\ Integrating by parts once more on the right side of the equation and repeating this process, we obtain an expansion in powers of l/|vv|: f e i,,2 drj 1 1 + 2i|w| 4|w| 3 (60.6) Although an infinite series of this type does not converge, nevertheless, because the suc- cessive terms decrease very rapidly for large values of \w\, the first term already gives a good representation of the function on the left for sufficiently large \w\ (such a series is said to be asymptotic). Thus, for the intensity I(w), (60.5), we obtain the following asymptotic formula, valid for large negative values of w : 1 = 1—2- ( 60 - 7 ) We see that in the region of geometric shadow, far from its edge, the intensity goes to zero as the inverse square of the distance from the edge of the shadow. § 61 FRAUNHOFER DIFFRACTION 153 We now consider positive values of w, that is, the region above the XY plane. We write 00 +00 — W CO f e in2 dn = f e^dn- f J" 2 dn = (l + i) J?- ( e ir > 2 drj. For sufficiently large w, we can use an asymptotic representation for the integral standing on the right side of the equation, and we have 00 j>> s(1+0 ^ + _i — w Substituting this expression in (60.5), we obtain / = Mi + \l n 2iw - sin | w — - (60.8) w (60.9 Thus in the illuminated region, far from the edge of the shadow, the intensity has an infinite sequence of maxima and minima, so that the ratio I/I oscillates on both sides of unity. With increasing w, the amplitude of these oscillations decreases inversely with the distance from the edge of the geometric shadow, and the positions of the maxima and minima steadily approach one another. For small w, the function I(w) has qualitatively this same character (Fig. 12). In the region of the geometric shadow, the intensity decreases monotonically as we move away from the boundary of the shadow. (On the boundary itself, I/I = £.) For positive w, the intensity has alternating maxima and minima. At the first (largest) maximum, I/I = 1.37. § 61. Fraunhofer diffraction Of special interest for physical applications are those diffraction phenomena which occur when a plane parallel bundle of rays is incident on a screen. As a result of the diffraction, the beam ceases to be parallel, and there is light propagation along directions other than the initial one. Let us consider the problem of determining the distribution over direction of the intensity of the diffracted light at large distances beyond the screen (this formulation of the problem corresponds to Fraunhofer diffraction). Here we shall again restrict ourselves to the case of small deviations from geometrical optics, i.e. we shall assume that the angles of deviation of the rays from the initial direction (the diffraction angles) are small. Fig. 12. 154 THE PROPAGATION OF LIGHT § 61 This problem can be solved by starting from the general formula (59.2) and passing to the limit where the light source and the point of observation are at infinite distances from the screen. A characteristic feature of the case we are considering is that, in the integral which determines the intensity of the diffracted light, the whole wave surface over which the integral is taken is important (in contrast to the case of Fresnel diffraction, where only the portions of the wave surface near the edge of the screens are important).f However, it is simpler to treat this problem anew, without recourse to the general formula (59.2). Let us denote by u Q the field which would exist beyond the screens if geometrical optics were rigorously valid. This field is a plane wave, but its cross section has certain regions (corresponding to the "shadows" of opaque screens) in which the field is zero. We denote by S the part of the plane cross-section on which the field u is different from zero ; since each such plane is a wave surface of the plane wave, u = const over the whole surface S. Actually, however, a wave with a limited cross-sectional area cannot be strictly plane (see § 58). In its spatial Fourier expansion there appear components with wave vectors having different directions, and this is precisely the origin of the diffraction. Let us expand the field u into a two-dimensional Fourier integral with respect to the co- ordinates y, z in the plane of the transverse cross-section of the wave. For the Fourier components, we have : w q = I I u e~ iq -'dydz, (61.1) where the vectors q are constant vectors in the y, z plane ; the integration actually extends only over that portion S of the y, z plane on which u is different from zero. If k is the wave vector of the incident wave, the field component u q e l qT gives the wave vector k' = k+q. Thus the vector q = k' — k determines the change in the wave vector of the light in the diffrac- tion. Since the absolute values k = k' = co/c, the small diffraction angles 6 y , 6 Z in the xy- and xz-planes are related to the components of the vector q by the equations q y = ~ 9 y , q z = ~ 6 Z . (61.2) c c For small deviations from geometrical optics, the components in the expansion of the field u can be assumed to be identical with the components of the actual field of the dif- fracted light, so that formula (61.1) solves our problem. The intensity distribution of the diffracted light is given by the square |wj 2 as a function of the vector q. The quantitative connection with the intensity of the incident light is established by the formula \ \< d y dz = \ \\»«\ 2d j0 ( 61 - 3 > t The criteria for Fresnel and Fraunhofer diffraction are easily found by returning to formula (60.2) and applying it, for example, to a slit of width a (instead of to the edge of an isolated screen). The integration over z in (60.2) should then be taken between the limits from to a. Fresnel diffraction corresponds to the case when the term containing z 2 in the exponent of the integrand is important, and the upper limit of the integral can be replaced by oo. For this to be the case, we must have \D P D q J^ On the other hand, if this inequality is reversed, the term in z 2 can be dropped; this corresponds to the case of Fraunhofer diffraction. § 61 FRAUNHOFER DIFFRACTION 155 [compare (49.8)]. From this we see that the relative intensity diffracted into the solid angle do = dO y d6 z is given by u I 2 dq y dq z ( a> \ 2 -G - 2 do. (61.4) u 2 (2n) 2 \2ncJ Let us consider the Fraunhofer diffraction from two screens which are "complementary" : the first screen has holes where the second is opaque and conversely. We denote by m (1) and u {2) the field of the light diffracted by these screens (when the same light is incident in both cases). Since w q (1) and u q (2) are expressed by integrals (61.1) taken over the surfaces of the apertures in the screens, and since the apertures in the two screens complement one another to give the whole plane, the sum u q (1) + w q (2) is the Fourier component of the field obtained in the absence of the screens, i.e. it is simply the incident light. But the incident light is a rigorously plane wave with definite direction of propagation, so that w q (1) + w q (2) = for all nonzero values of q. Thus we have w q (1) = -w q (2) , or for the corresponding intensities, |w q (1 T = K (2) | 2 forq^0. (61.5) This means that complementary screens give the same distribution of intensity of the diffracted light (this is called Babinefs principle). We call attention here to one interesting consequence of the Babinet principle. Let us consider a blackbody, i.e. one which absorbs completely all the light falling on it. According to geometrical optics, when such a body is illuminated, there is produced behind it a region of geometrical shadow, whose cross-sectional area is equal to the area of the body in the direction perpendicular to the direction of incidence of the light. However, the presence of diffraction causes the light passing by the body to be partially deflected from its initial direction. As a result, at large distances behind the body there will not be complete shadow but, in addition to the light propagating in the original direction, there will also be a certain amount of light propagating at small angles to the original direction. It is easy to determine the intensity of this scattered light. To do this, we point out that according to Babinet's principle, the amount of light deviated because of diffraction by the body under considera- tion is equal to the amount of light which would be deviated by diffraction from an aperture cut in an opaque screen, the shape and size of the aperture being the same as that of the transverse section of the body. But in Fraunhofer diffraction from an aperture all the light passing through the aperture is deflected. From this it follows that the total amount of light scattered by a blackbody is equal to the amount of light falling on its surface and absorbed by it. PROBLEMS 1. Calculate the Fraunhofer diffraction of a plane wave normally incident on an infinite slit (of width 2d) with parallel sides cut in an opaque screen. Solution: We choose the plane of the slit as the yz plane, with the z axis along the slit (Fig. 13 shows a section of the screen). For normally incident light, the plane of the slit is one of the wave surfaces, and we choose it as the surface of integration in (61.1). Since the slit is infinitely long, the light is deflected only in the xy plane [since the integral (61.1) becomes zero for g 3 =£ 0]. Therefore the field should be expanded only in the y coordinate: a U q = Uq J <- {,iy dy= — sinqa. 156 THE PROPAGATION OF LIGHT § 61 "t +a~ \ ^ V x Fig. 13. The intensity of the diffracted light in the angular range d9 is dl = h \Uc \ 2 dq 2k I sin 2 kaO de, 2a \uq\ 2k nak 9 2 where k = cole, and I is the total intensity of the light incident on the slit. dl/dO as a function of diffraction angle has the form shown in Fig. 14. As 9 increases toward either side from = 0, the intensity goes through a series of maxima with rapidly decreasing height. The successive maxima are separated by minima at the points 9 = nnjka (where n is an integer); at the minima, the intensity falls to zero. Fig. 14. 2. Calculate the Fraunhofer diffraction by a diffraction grating — a plane screen in which are cut a series of identical parallel slits (the width of the slits is 2a, the width of opaque screen between neighboring slits is 2b, and the number of slits is N). Solution: We choose the plane of the grating as the yz plane, with the z axis parallel to the slits- Diffraction occurs only in the xy plane, and integration of (61.1) gives: "s'Z 1 - ZiNqd n = J. c where d = a+b, and u' q is the result of the integration over a single slit. Using the results of problem 1, we get: . _ I Q a /sin Ngd\ 2 /sin qa\ 2 , _ h /sin Nk9d\ 2 sin 2 ka9 Nn \ sin qd J \ qa J Nnak \ sin k9d J 9 2 dl- d9 (Io is the total intensity of the light passing through all the slits). § 61 FRAUNHOFER DIFFRACTION 157 For the case of a large number of slits (N-> co), this formula can be written in another form. For values q — nn/d, where n is an integer, dl/dq has a maximum; near such a maximum (i.e. for qd = nn+E, with e small) (sin^aVsir qa ) n But for N-* oo, we have the formula t N-»oo 7tNx 2 We therefore have, in the neighborhood of each maximum: ! sin^ Ns a /sin qa\ 2 dI=Io-[ — S(e)de, d\ qa J i.e., in the limit the widths of the maxima are infinitely narrow and the total light intensity in the «'th maximum is roo _ r d sin2 i nna l d ) 3. Find the distribution of intensity over direction for the diffraction of light which is incident normal to the plane of a circular aperture of radius a. Solution: We introduce cylindrical coordinates z, r, <j> with the z axis passing through the center of the aperture and perpendicular to its plane. It is obvious that the diffraction is symmetric about the z axis, so that the vector q has only a radial component q r =q = k9. Measuring the angle <f> from the direction q, and integrating in (61.1) over the plane of the aperture, we find: a 2n a u q = u Q f (e- i9rcos '"rd^dr = 2nuo f J (qr)rdr, where J is the zero'th order Bessel function. Using the well-known formula a I Jo(.qr)rdr = -Jx(aq), we then have u a u q — 2% Ji(aq), and according to (61.4) we obtain for the intensity of the light diffracted into the element of solid angle do: dI = I -^-do, where I is the total intensity of the light incident on the aperture. t For x # o the function on the left side of the equation is zero, while according to a well-known formula of the theory of Fourier series, ^(s/^^*)-^ From this we see that the properties of this function actually coincide with those of the <5-function (see the footnote on p. 7). CHAPTER 8 THE FIELD OF MOVING CHARGES § 62. The retarded potentials In Chapter 5 we studied the constant field, produced by charges at rest, and in Chapter 6, the variable field in the absence of charges. Now we take up the study of varying fields in the presence of arbitrarily moving charges. We derive equations determining the potentials for arbitrarily moving charges. This derivation is most conveniently done in four-dimensional form, repeating the derivation at the end of § 46, with the one change that we use the second pair of Maxwell equations in the form (30.2) dF ik _ 4ti .. a?~ ~~c j ' The same right-hand side also appears in (46.8), and after imposing the Lorentz condition a*'_ft i.e. !^+divA = 0, (62.1) OX c ot on the potentials, we get * A ' A *}. (62.2) A#-4£*--4«* (62.4) dx k dx This is the equation which determines the potentials of an arbitrary electromagnetic field In three-dimensional form it is written as two equations, for A and for $ : 1 d 2 A An ?~dt c 2 dt For constant fields, these reduce to the already familiar equations (36.4) and (43.4), and for variable fields without charges, to the homogeneous wave equation. As we know, the solution of the inhomogeneous linear equations (62.3) and (62.4) can be represented as the sum of the solution of these equations without the right-hand side, and a particular integral of these equations with the right-hand side. To find the particular solution, we divide the whole space into infinitely small regions and determine the field produced by the charges located in one of these volume elements. Because of the linearity of the field equations, the actual field will be the sum of the fields produced by all such elements. The charge de in a given volume element is, generally speaking, a function of the time. If we choose the origin of coordinates in the volume element under consideration, then the 158 § 62 THE RETARDED POTENTIALS 159 charge density is q = de(t ) <5(R), where R is the distance from the origin. Thus we must solve the equation 1 rfi" A\ ^<t>~-2^= -4nde(t) SQL). (62.5) Everywhere, except at the origin, <5(R) = 0, and we have the equation 1 d 2 <b A0-- 2 -^ = O. (62.6) It is clear that in the case we are considering $ has central symmetry, i.e., (j> is a function only of R. Therefore if we write the Laplace operator in spherical coordinates, (62.6) reduces to R 2 dR \ 8RJ c 2 8t 2 ~ To solve this equation, we make the substitution <£ = %(i?, t )jR. Then, we find for x dR 2 c 2 dt 2 ~ But this is the equation of plane waves, whose solution has the form (see § 47) : *-''('-7) +/ »(' + 7 Since we only want a particular solution of the equation, it is sufficient to choose only one of the functions /i and/ 2 - Usually it turns out to be convenient to take/2 = (concerning this, see below). Then, everywhere except at the origin, (f) has the form H) <t> = ^A (62.7) So far the function x is arbitrary; we now choose it so that we also obtain the correct value for the potential at the origin. In other words, we must select x so that at the origin equation (62.5) is satisfied. This is easily done noting that as R -> 0, the potential increases to infinity, and therefore its derivatives with respect to the coordinates increase more rapidly than its time derivative. Consequently as R -» 0, we can, in equation (62.5), neglect the term (\/c 2 )/(d 2 (l)/dt 2 ) compared with A<£. Then (62.5) goes over into the familiar equation (36.9) leading to the Coulomb law. Thus, near the origin, (62.7) must go over into the Coulomb law, from which it follows that x(t) = de(t), that is, dett ) Y R From this it is easy to get to the solution of equation (62.4) for an arbitrary distribution of charges q(x, y, z, t). To do this, it is sufficient to write de = q dV(dV is the volume element) and integrate over the whole space. To this solution of the inhomogeneous equation (6.24) we can still add the solution (f> of the same equation without the right-hand side. Thus, 160 THE FIELD OF MOVING CHARGES § 63 the general solution has the form: <Kr, *) = j^Q U, t- fj dV' + <t> , (62.8) R = r-r', dV = dx' dy' dz' where r = (x,y,z), r' = (x',y', z'); R is the distance from the volume element dVto the "field point" at which we determine the potential. We shall write this expression briefly as ^Jfo±p> dK + 0o> (62 .9) where the subscript means that the quantity q is to be taken at the time t — (R/c), and the prime on dV has been omitted. Similarly we have for the vector potential: = 1 ffc-jj, c] R -Wei dV+A , (62.10) where A is the solution of equation (62.3) without the right-hand term. The potentials (62.9) and (62.10) (without <j) and A ) are called the retarded potentials. In case the charges are at rest (i.e. density q independent of the time), formula (62.9) goes over into the well-known formula (36.8) for the electrostatic field; for the case of stationary motion of the charges, formula (62.10), after averaging, goes over into formula (43.5) for the vector potential of a constant magnetic field. The quantities A and <f> in (62.9) and 62.10) are to be determined so that the conditions of the problem are fulfilled. To do this it is clearly sufficient to impose initial conditions, that is, to fix the values of the field at the initial time. However we do not usually have to deal with such initial conditions. Instead we are usually given conditions at large distances from the system of charges throughout all of time. Thus, we may be told that radiation is incident on the system from outside. Corresponding to this, the field which is developed as a result of the interaction of this radiation with the system can differ from the external field only by the radiation originating from the system. This radiation emitted by the system must, at large distances, have the form of waves spreading out from the system, that is, in the direction of increasing R. But precisely this condition is satisfied by the retarded potentials. Thus these solutions represent the field produced by the system, while </> and A must be set equal to the external field acting on the system. § 63. The Lienard-Wiechert potentials Let us determine the potentials for the field produced by a charge carrying out an assigned motion along a trajectory r = r (t ). According to the formulas for the retarded potentials, the field at the point of observation P(x, y, z) at time t is determined by the state of motion of the charge at the earlier time t ', for which the time of propagation of the light signal from the point r (t ' ), where the charge was located,to the field point P just coincides with the difference t—t'. Let R(?) = r— r (t) be the radius vector from the charge e to the point P; like r (0 it is a given function of the § 63 THE LIENARD-WIECHERT POTENTIALS 161 time. Then the time t ' is determined by the equation t'+ R ^ = t. (63.1) c For each value of t this equation has just one root t'.\ In the system of reference in which the particle is at rest at time t ', the potential at the point of observation at time t is just the Coulomb potential, 6 = -£-, A = 0. (63.2) v R(t'y v ' The expressions for the potentials in an arbitrary reference system can be found directly by finding a four- vector which for v = coincides with the expressions just given for <$> and A. Noting that, according to (63.1), <f> in (63.2) can also be written in the form #- e c(t-t'y we find that the required four- vector is : A l = e-f- k , (63.3) K k U where u k is the four- velocity of the charge, R k = [c(t— t '), r— r'], where x', y', z', t' are related by the equation (63.1), which in four-dimensional form is R k R k =0. (63.4) Now once more transforming to three-dimensional notation, we obtain, for the potentials of the field produced by an arbitrarily moving point charge, the following expressions: <j> = — , A = — — , (63.5) ('-?i ■(-'-?) where R is the radius vector, taken from the point where the charge is located to the point of observation P, and all the quantities on the right sides of the equations must be evaluated at the time t', determined from (63.1). The potentials of the field, in the form (63.5), are called the Lienor d-Wiechert potentials. To calculate the intensities of the electric and magnetic fields from the formulas 1 dA E = — grad 6, H = cud A, c at we must differentiate <£ and A with respect to the coordinates x, y, z of the point, and the time t of observation. But the formulas (63.5) express the potentials as functions of t ', and only through the relation (63.1) as implicit functions of x, y, z, t. Therefore to calculate the t This point is obvious but it can be verified directly. To do this we choose the field point P and the time of observation t as the origin O of the four-dimensional coordinate system and construct the light cone (§2) with its vertex at O. The lower half of the cone, containing the absolute past (with respect to the event O), is the geometrical locus of world points such that signals sent from them reach O. The points in which this hypersurface intersects the world line of the charge are precisely the roots of (63. 1). But since the velocity of a particle is always less than the velocity of light, the inclination of its world line relative to the time axis is everywhere less than the slope of the light cone. It then follows that the world line of the particle can inter- sect the lower half of the light cone in only one point. 162 THE FIELD OF MOVING CHARGES § 63 required derivatives we must first calculate the derivatives of t '. Differentiating the relation R(t ') = c{t-t') with respect to t, we get dR _ dR <tf_ _ _ R-jdf _ f df\ dt~ dt'~di~ ~~R~dt~ C V~ ft)' (The value of dR/dt ' is obtained by differentiating the identity R 2 = R 2 and substituting dR(t ')/dt ' = - \(t '). The minus sign is present because R is the radius vector from the charge e to the point P, and not the reverse.) Thus, df 1 8t „ v • R' (63.6) Similarly, differentiating the same relation with respect to the coordinates, we find grad ?=-- grad R(t') = - - f — gra d ?+-), so that R grad t'= — — . (63.7) (- c With the aid of these formulas, there is no difficulty in carrying out the calculation of the fields E and H. Omitting the intermediate calculations, we give the final results: 1- E=e (^^( R "" R )\T7^y Rx {( R -^) x *}' (63 - 8) H = -RxE. (63.9) Here, v = dv/dt'; all quantities on the right sides of the equations refer to the time t '. It is interesting to note that the magnetic field turns out to be everywhere perpendicular to the electric. The electric field (63.8) consists of two parts of different type. The first term depends only on the velocity of the particle (and not on its acceleration) and varies at large distances like 1/R 2 . The second term depends on the acceleration, and for large R it varies like l/R. Later (§ 66) we shall see that this latter term is related to the electromagnetic waves radiated by the particle. As for the first term, since it is independent of the acceleration it must correspond to the field produced by a uniformly moving charge. In fact, for constant velocity the difference R f ---i^ = R t ,-v(*-0 c is the distance R t from the charge to the point of observation at precisely the moment of observation. It is also easy to show directly that R t '-- c R t '-y=jRf-^(^^t) 2 = R t Jl- V ^m 2 e t , § 64 SPECTRAL RESOLUTION OF THE RETARDED POTENTIALS 163 where 9 t is the angle between R t and v. Consequently the first term in (63.8) is identical with the expression (38.8). PROBLEM Derive the Lienard-Wiechert potentials by integrating (62.9-10). Solution: We write formula (62.8) in the form: ^ (r ' 0= f \y^ S ( T ~ t+ ~c lr ~ r ' l ) dTdV ' (and similarly for A(r, t)), introducing the additional delta function and thus eliminating the implicit arguments in the function q. For a point charge, moving in a trajectory r = r (t), we have: o(r', x) = e8[r'— r (r)]. Substituting this expression and integrating over dV\ we get : *-'>-Jf£f' r—t+- |r-r (r) c The t integration is done using the formula s[f(t)] = -fWT [where f ' is the root of Fit') = 0], and gives formula (63.5). § 64. Spectral resolution of the retarded potentials The field produced by moving charges can be expanded into monochromatic waves. The potentials of the different monochromatic components of the field have the form ^e" to , Aa,e~ l(0t . The charge and current densities of the system of charges producing the field can also be expanded in a Fourier series or integral. It is clear that each Fourier component of q and j is responsible for the creation of the corresponding monochromatic component of the field. In order to express the Fourier components of the field in terms of the Fourier components of the charge density and current, we substitute in (62.9) for $ and q respectively, $ m e~ i<ot and Q m e~ i<ot . We then obtain 4>»e- u " = ]Q a -— ir -dV. Factoring e~ imt and introducing the absolute value of the wave vector k = co/c, we have: ,ikR 0co = J Q.^rdV. (64.1) Similarly, for A w we get "J" 1, y R , cR dV. (64.2) 164 THE FIELD OF MOVING CHARGES § 64 We note that formula (64.1) represents a generalization of the solution of the Poisson equation to a more general equation of the form &<t><o + k 2 (f>< = -4n Qa (64.3) (obtained from equations (62.4) for q, $ depending on the time through the factor e~ iwt ). If we were dealing with expansion into a Fourier integral, then the Fourier components of the charge density would be Q m = / Qe icot dt. — 00 Substituting this expression in (64.1), we get + oo 4> a = jjy«" +m dVdt. (64.4) — oo We must still go over from the continuous distribution of charge density to the point charges whose motion we are actually considering. Thus, if there is just one point charge, we set q = <?<5[r-r (0], where r (? ) is the radius vector of the charge, and is a given function of the time. Substituting this expression in (64.4) and carrying out the space integration [which reduces to replacing rb yro(OL weget: oo <t>m==e I j^ efa,[t+ * (0/Cl ^' ( 64 - 5 ) — 00 where now R(t) is the distance from the moving particle to the point of observation. Similarly we find for the vector potential: oo °>~ c) R{f) 6 dt > (64 - 6) — 00 where v = i (t) is the velocity of the particle. Formulas analogous to (64.5), (64.6) can also be written for the case where the spectral resolution of the charge and current densities contains a discrete series of frequencies. Thus, for a periodic motion of a point charge (with period T = 2n/co ) the spectral resolution of the field contains only frequencies of the form nco , and the corresponding components of the vector potential are T X n = -^[ y -~ e in <°° lt+R ^ dt (64.7) cT J R(l) o (and similarly for (f> n ). In both (64.6) and (64.7) the Fourier components are defined in accordance with § 49. PROBLEM Find the expansion in plane waves of the field of a charge in uniform rectilinear motion. Solution: We proceed in similar fashion to that used in § 51. We write the charge density in the form q = eS(r— \t), where v is the velocity of the particle. Taking Fourier components of the equation Q^ = — Ane S(r—\t), we find (D^) k = — 47ree -i<vk)f . § 65 THE LAGRANGIAN TO TERMS OF SECOND ORDER 165 On the other hand, from we have (D0 k = -k 2 fa--^ ..a • c 2 dt' Thus, c 2 9f 2 from which, finally e K = 47re - i(k • v)t -M From this it follows that the wave with wave vector k has the frequency co = k • v. Similarly, we obtain for the vector potential, yg-i(k-V)J A k — 47ie- * 2 -' k ' V c Finally, we have for the fields, k • v r E k = — ik^ k +i A k = bite i c - i(k • \» "-<?)' 4ne kx v ... ,, H k =/kxAk = — / r, — r-o e" ,(kv)f . »k = «* aA|j ("?)' § 65. The Lagrangian to terms of second order In ordinary classical mechanics, we can describe a system of particles interacting with each other with the aid of a Lagrangian which depends only on the coordinates and velocities of these particles (at one and the same time). The possibility of doing this is, in the last analysis, dependent on the fact that in mechanics the velocity of propagation of interactions is assumed to be infinite. We already know that because of the finite velocity of propagation, the field must be considered as an independent system with its own "degrees of freedom". From this it follows that if we have a system of interacting particles (charges), then to describe it we must consider the system consisting of these particles and the field. Therefore, when we take into account the finite velocity of propagation of interactions, it is impossible to describe the system of interacting particles rigorously with the aid of a Lagrangian, depending only on the co- ordinates and velocities of the particles and containing no quantities related to the internal "degrees of freedom" of the field. However, if the velocity v of all the particles is small compared with the velocity of light, then the system can be described by a certain approximate Lagrangian. It turns out to be possible to introduce a Lagrangian describing the system, not only when all powers of v/c are neglected (classical Lagrangian), but also to terms of second order, v 2 /c 2 . This last 166 THE FIELD OF MOVING CHARGES § 65 remark is related to the fact that the radiation of electromagnetic waves by moving charges (and consequently, the appearance of a "self "-field) occurs only in the third approximation in v/c (see later, in § 67).f As a preliminary, we note that in zero'th approximation, that is, when we completely neglect the retardation of the potentials, the Lagrangian for a system of charges has the form L^-^mA-Y— (65.1) a a>b i\ a b (the summation extends over the charges which make up the system). The second term is the potential energy of interaction as it would be for charges at rest. To get the next approximation, we proceed in the following fashion. The Lagrangian for a charge e a in an external field is / v 2 e L a = - mc 2 / ^l--!_^ + J?A-v (65.2) Choosing any one of the charges of the system, we determine the potentials of the field produced by all the other charges at the position of the first, and express them in terms of the coordinates and velocities of the charges which produce this field (this can be done only approximately— for <j), to terms of order v 2 /c 2 , and for A, to terms in v/c). Substituting the expressions for the potentials obtained in this way in (65.2), we get the Lagrangian for one of the charges of the system (for a given motion of the other charges). From this, one can then easily find the Lagrangian for the whole system. We start from the expressions for the retarded potentials + -! e -T*r- H/¥^ If the velocities of all the charges are small compared with the velocity of light, then the charge distribution does not change significantly during the time R/c. Therefore we can expand Q t _ Rjc and j t _ R/c in series of powers of R/c. For the scalar potential we thus find, to terms of second order: , C QdV 13 f , 1 d 2 r (q without indices is the value of q at time t ; the time differentiations can clearly be taken out from under the integral sign). But I QdV is the constant total charge of the system. Therefore the second term in our expression is zero, so that ± r gdv i e 2 r We can proceed similarly with A. But the expression for the vector potential in terms of the current density already contains l/c, and when substituted in the Lagrangian is multiplied once more by 1/c. Since we are looking for a Lagrangian which is correct only to terms of second order, we can limit ourselves to the first term in the expansion of A, that is, ^dV (65.4) (we have substituted j = qv). -u t In special cases the appearance of the radiation terms can even be put off until the fifth approximation in v/c; in this case a Lagrangian even exists up to terms of order (v/c)*. (See problem 2 of § 75.) § 65 THE LAGRANGIAN TO TERMS OF SECOND ORDER 167 Let us first assume that there is only a single point charge e. Then we obtain from (65.3) and (65.4), ± e e d 2 R e\ * = R + 2?W A = c"R' (65 - 5) where R is the distance from the charge. We choose in place of <£ and A other potentials <£' and A', making the transformation (see §18): 4>' = <£- 15/ cdf A' = A+grady, in which we choose for /the function e dR f = 2c dt' Then we gett V e = R' A' 2c dt' To calculate A' we note first of all that V(dR/dt) = (d/dt)VR. The grad operator here means differentiation with respect to the coordinates of the field point at which we seek the value of A'. Therefore WR is the unit vector n, directed from the charge e to the field point, so that ev e . A = -^ + ;r n - cR 2c We also write: RR 7i n " dt \r) ~ R ~ R : But the derivative —ft for a given field point is the velocity v of the charge, and the derivative R is easily determined by differentiating R 2 = R 2 , that is, by writing RR = Rti= -Rv. Thus, — v+n(n-v) „= — - — . Substituting this in the expression for A;, we get finally : r R 2cR v ' If there are several charges then we must, clearly, sum these expressions over all the charges. Substituting these expressions in (65.2), we obtain the Lagrangian L a for the charge e a (for a fixed motion of the other charges). In doing this we must also expand the first term in (65.2) in powers of vjc, retaining terms up to the second order. Thus we find : m a v 2 1 m a v1: „, e b e a _, e 6 r , w N _ L a = -^r + 7>^r--e a Y}r + T^ XV" l Y a-y b +(y a -n ab )(y b -n ab )] Z o C b K ab ZC b K ah (the summation goes over all the charges except e a ; n ab is the unit vector from e b to e a ). t These potentials no longer satisfy the Lorentz condition (62.1), nor the equations (62.3-4). 168 THE FIELD OF MOVING CHARGES § 65 From this, it is no longer difficult to get the Lagrangian for the whole system. It is easy to convince oneself that this function is not the sum of the L a for all the charges, but has the form T ^m a v 2 m a vt e a e b e a e b L = L ~y- + L -^r - L -W~ + L y^y- ba ' v fc + (v a • n ab )(y b • n a6 )]. (65. /) a £ a OL a>b J\ ab a>b ZC l\ ab Actually, for each of the charges under a given motion of all the others, this function L goes over into L a as given above. The expression (65.7) determines the Lagrangian of a system of charges correctly to terms of second order. (It was first obtained by Darwin, 1920.) Finally we find the Hamiltonian of a system of charges in this same approximation. This could be done by the general rule for calculating Jf from L; however it is simpler to proceed as follows. The second and fourth terms in (65.7) are small corrections to L (0) (65.1). On the other hand, we know from mechanics that for small changes of L and Jt? , the additions to them are equal in magnitude and opposite in sign (here the variations of L are considered for constant coordinates and velocities, while the changes in j4? refer to constant coordinates and momenta). f Therefore we can at once write Jf , subtracting from ^ (0) = <£ — +% — a 2m a a >b R ab the second and fourth terms of (65.7), replacing the velocities in them by the first approxima- tion v fl = yjm a . Thus, 2 4 ~W = V ^ a — V ^ a ■+- v gqgft — ~2m fl iSc 2 m 3 a kh R„ b ~ £ -,„2 W " p [P«'Pfc + (P«-n fl6 Xp 6 -n fl6 )] (65.8) a >b2c m a m b R ab PROBLEMS 1. Determine (correctly to terms of second order) the center of inertia of a system of interacting particles. Solution: The problem is solved most simply by using the formula U _ _? a [see (14.6)], where S a is the kinetic energy of the particle (including its rest energy), and W is the energy density of the field produced by the particles. Since the & a contain the large quantities m a c 2 , it is sufficient, in obtaining the next approximation, to consider only those terms in & a and W which do not contain c, i.e. we need consider only the nonrelativistic kinetic energy of the particles and the energy of the electrostatic field. We then have: JuW-iJlPrrfK (SvfrdV t See Mechanics, § 40. § 65 THE LAGRANGIAN TO TERMS OF SECOND ORDER 169 the integral over the infinitely distant surface vanishes; the second integral also is transformed into a surface integral and vanishes, while we substitute A<p = — 4tiq in the third integral and obtain: WrdV=- P <prdV=-Y j ea<Par a where <p a is the potential produced at the point r a by all the charges other than e a .f Finally, we get : (with a summation over all b except b = a), where p% , ^ e a e b ?( W ° 2 "*a &» R al a>b ^ab, is the total energy of the system. Thus in this approximation the coordinates of the center of inertia can actually be expressed in terms of quantities referring only to the particles. 2. Write the Hamiltonian in second approximation for a system of two particles, omitting the motion of the system as a whole. Solution: We choose a system of reference in which the total momentum of the two particles is zero. Expressing the momenta as derivatives of the action, we have Pi +Pa = W/ dr x + dS/ dr 2 = 0. From this it is clear that in the reference system chosen the action is a function of r = r 2 — r x , the difference of the radius vectors of the two particles. Therefore we have p 2 = — Pi = p, where p = cS/ or is the momentum of the relative motion of the particles. The Hamiltonian is * _ \ (1 + L) „*. i (± + 1\ pt+ >j« + _^v ^ +to .„ n 2 \mi m 2 J 8c 2 y Wx 3 m 2 3 / r Irrii m 2 c 2 r t The elimination of the self-field of the particles corresponds to the mass "renormalization" mentioned in the footnote on p. 90). CHAPTER 9 RADIATION OF ELECTROMAGNETIC WAVES § 66. The field of a system of charges at large distances We consider the field produced by a system of moving charges at distances large compared with the dimensions of the system. We choose the origin of coordinates O anywhere in the interior of the system of charges. The radius vector from O to the point P, where we determine the field, we denote by R , and the unit vector in this direction by n. Let the radius vector of the charge element de = odV be r, and the radius vector from de to the point P be R. Obviously R = R -r. At large distances from the system of charges, R > r, and we have approximately, K = |Ro-r|^i? -r-n. We substitute this in formulas (62.9), (62.10) for the retarded potentials. In the denominator of the integrands we can neglect rn compared with R . In t-(R/c), however, this is generally not possible; whether it is possible to neglect these terms is determined not by the relative values of R /c and r • (n/c), but by how much the quantities q and j change during the time r • (n/c). Since R is constant in the integration and can be taken out from under the integral sign, we get for the potentials of the field at large distances from the system of charges the expressions : *"£/*-?—:'"'• (6<u) A-5f J'i-S + ,. ! <»'- (6") At sufficiently large distances from the system of charges, the field over small regions of space can be considered to be a plane wave. For this it is necessary that the distance be large compared not only with the dimensions of the system, but also with the wavelength of the electromagnetic waves radiated by the system. We refer to this region of space as the wave zone of the radiation. In a plane wave, the fields E and H are related to each other by (47.4), E = Hxn. Since H = curl A, it is sufficient for a complete determination of the field in the wave zone to calculate only the vector potential. In a plane wave we have H = (l/c)Axn [see (47.3)], where the dot indicates differentiation with respect to time.f Thus, knowing A, we find H t In the present case, this formula is easily verified also by direct computation of the curl of the expression (66.2), and dropping terms in IjRl in comparison with terms ~ l/i? . 170 § 66 THE FIELD OF A SYSTEM OF CHARGES AT LARGE DISTANCES 171 and E from the formulas:! H = -Axn, E = - (Axn)xn. (66.3) c c We note that the field at large distances is inversely proportional to the first power of the distance R from the radiating system. We also note that the time t enters into the expressions (66.1) to (66.3) always in the combination t-(R lc). For the radiation produced by a single arbitrarily moving point charge, it turns out to be convenient to use the Lienard-Wiechert potentials. At large distances, we can replace the radius vector R in formula (63.5) by the constant vector R , and in the condition (63.1) determining t ', we must set R = R -r -n(r (r) is the radius vector of the charge). Thus, J A= e ^l , (66.4) where t ' is determined from the equality f- T ^-n = t-^. (66.5) c c The radiated electromagnetic waves carry off energv. The energy flux is given by the Poynting vector which, for a plane wave, is S = c-n. An The intensity dl of radiation into the element of solid angle do is defined as the amount of energy passing in unit time through the element df= R 2 do of the spherical surface with center at the origin and radius R . This quantity is clearly equal to the energy flux density S multiplied by df, i.e. dI = c~R 2 do. (66.6) 47T Since the field H is inversely proportional to R , we see that the amount of energy radiated by the system in unit time into the element of solid angle do is the same for all distances (if the values of t-(R /c) are the same for them). This is, of course, as it should be, since the energy radiated from the system spreads out with velocity c into the surrounding space, not accumulating or disappearing anywhere. We derive the formulas for the spectral resolution of the field of the waves radiated by the system. These formulas can be obtained directly from those in § 64. Substituting in (64.2) R = R -t • n (in which we can set R = R in the denominator of the integrand), we get for the Fourier components of the vector potential : JkRo f A ^kS Le " t " dV (66J) (where k = An). The components H w and E w are determined using formula (66.3). Sub- stituting in it for H, E, A, respectively, H^e"^', E^e"^, Ke~ imt , and then dividing by t The formula E = -(l/c)A [see (47.3)] is here not applicable to the potentials <j>, A, since they do not satisfy the same auxiliary condition as was imposed on them in § 47. % In formula (63.8) for the electric field, the present approximation corresponds to dropping the first term in comparisonwith the second. 172 RADIATION OF ELECTROMAGNETIC WAVES § 66 e~ i(0t , we find ic H £0 = ikxA a) , E t0 = -(kxAJxk. (66.8) When speaking of the spectral distribution of the intensity of radiation, we must dis- tinguish between expansions in Fourier series and Fourier integrals. We deal with the expan- sion into a Fourier integral in the case of the radiation accompanying the collision of charged particles. In this case the quantity of interest is the total amount of energy radiated during the time of the collision (and correspondingly lost by the colliding particles). Suppose d£ na) is the energy radiated into the element of solid angle do in the form of waves with frequencies in the interval dco. According to the general formula (49.8), the part of the total radiation lying in the frequency interval dco/2n is obtained from the usual formula for the intensity by replacing the square of the field by the square modulus of its Fourier component and multi- plying by 2. Therefore we have in place of (66.6): Mum = y % W^o do ~. (66.9) If the charges carry out a periodic motion, then the radiation field must be expanded in a Fourier series. According to the general formula (49.4) the intensities of the various com- ponents of the Fourier resolution are obtained from the usual formula for the intensity by replacing the field by the Fourier components and then multiplying by two. Thus the intensity of the radiation into the element of solid angle do, with frequency co = nco Q equals c dI„ = —\H„\ 2 Rldo. (66.10) Finally, we give the formulas for determining the Fourier components of the radiation field directly from the given motion of the radiating charges. For the Fourier integral expansion, we have: 00 L= j ie^dt. — 00 Substituting this in (66.7) and changing from the continuous distribution of currents to a point charge moving along a trajectory r = r (t) (see § 64), we obtain: + 00 A w = — J ey{ty^-^m dt> (66U) — oo Since v = drjdt, vdt = dr and this formula can also be written in the form of a line integral taken along the trajectory of the charge : gikRo /• A °> = e ^j e i(cot - k ' ro) dr . (66.12) According to (66.8), the Fourier components of the magnetic field have the form: icoe ikR ° C H. = e -j£- J e ^-*-'o) n x dr ^ (66 13) If the charge carries out a periodic motion in a closed trajectory, then the field must be expanded in a Fourier series. The components of the Fourier series expansion are obtained by replacing the integration over all times in formulas (66.1 1) to (66.13) by an average over § 67 DIPOLE RADIATION 173 the period T of the motion (see § 49). For the Fourier component of the magnetic field with frequency co = nco = n(2n/T), we have T 2nine ikRo H = e ,IKK /• e i[ n o,ot-kT O (0] nxv (A^ R J c 2 T 2 „ 2nine ikRo f In the second integral, the integration goes over the closed orbit of the particle = e i-viry <P e ,( " COof - kro) n x dr . (66.14) c T R J PROBLEM Find the four-dimensional expression for the spectral resolution of the four-momentum radiated by a charge moving along a given trajectory. Solution: Substituting (66.8) in (66.9), and using the fact that, because of the condition (62.1), Ar^o = k • A a , we find: d* w = £■ (A^IA^-Ik • A.,1 2 ) Rl do f° Zn An Lit In Ln In Representing the four-potential A lm in a form analogous to (66.12), we get: k 2 e 2 d&nm = - -^ Xt X 1 * do dk, where x l denotes the four-vector X 1 = exp {—ikix l )dx { and the integration is performed along the world line of the trajectory of the particle. Finally, changing to four-dimensional notation [including the four-dimensional "volume element" in fc-space, as in (10.1a)], we find for the radiated four-momentum: 2jtc § 67. Dipole radiation The time r -(n/c) in the integrands of the expressions (66.1) and (66.2) for the retarded potentials can be neglected in cases where the distribution of charge changes little during this time. It is easy to find the conditions for satisfying this requirement. Let T denote the order of magnitude of the time during which the distribution of the charges in the system changes significantly. The radiation of the system will obviously contain periods of order T (i.e. frequencies of order l/T). We further denote by a the order of magnitude of the dimensions of the system. Then the time r • (n/c) ~ a/c. In order that the distribution of the charges in the system shall not undergo a significant change during this time, it is necessary that a/c <^ T. But cT is just the wavelength X of the radiation. Thus the condition a<cT can be written in the form a < A, (67.1) that is, the dimensions of the system must be small compared with the radiated wavelength. 174 RADIATION OF ELECTROMAGNETIC WAVES § 67 We note that this same condition (67.1) can also be obtained from (66.7). In the integrand, r goes through values in an interval of the order of the dimensions of the system, since outside the system j is zero. Therefore the exponent zk • r is small, and can be neglected for those waves in which ka <^ 1, which is equivalent to (67.1). This condition can be written in still another form by noting that T ~ a/v, so that X ~ ca/v, if v is of the order of magnitude of the velocities of the charges. From a <^ A, we then find v<c, (67.2) that is, the velocities of the charges must be small compared with the velocity of light. We shall assume that this condition is fulfilled, and take up the study of the radiation at distances from the radiating system large compared with the wavelength (and consequently, in any case, large compared with the dimensions of the system). As was pointed out in § 66, at such distances the field can be considered as a plane wave, and therefore in determining the field it is sufficient to calculate only the vector potential. The vector potential (66.2) of the field now has the form =ii j " dK ' (67.3) where the time t ' = t — (R lc) now no longer depends on the variable of integration. Sub- stituting j = q\, we rewrite (67.3) in the form cR 1 Q>) (the summation goes over all the charges of the system ; for brevity, we omit the index t ' — all quantities on the right side of the equation refer to time t '). But where d is the dipole moment of the system. Thus, a = 4- a - ( 67 ' 4 > cR With the aid of formula (66.3) we find that the magnetic field is equal to 1 c r R and the electric field to H=-y— axn, (67.5) E = 4— (^ x n) x n. (67.6) c R We note that in the approximation considered here, the radiation is determined by the second derivative of the dipole moment of the system. Radiation of this kind is called dipole radiation. Since d = 2 er, A = Z ev. Thus the charges can radiate only if they move with acceleration. Charges in uniform motion do not radiate. This also follows directly from the principle of relativity, since a charge in uniform motion can be considered in the inertial system in which it is at rest, and a charge at rest does not radiate. § 67 DIPOLE RADIATION 175 Substituting (67.5) in (66.6), we get the intensity of the dipole radiation: dl = — -i<ixa) 2 do = — , sin 2 do, (67.7) where is the angle between d and n. This is the amount of energy radiated by the system in unit time into the element of solid angle do. We note that the angular distribution of the radiation is given by the factor sin 2 9. Substituting do — 2n sin 6 dd and integrating over 6 from to n, we find for the total radiation "&*■ (67 - 8) If we have just one charge moving in the external field, then d = ex and 3 = ew, where w is the acceleration of the charge. Thus the total radiation of the moving charge is 2e 2 w 2 1 —&-• (67 - 9 > We note that a closed system of particles, for all of which the ratio of charge to mass is the same, cannot radiate (by dipole radiation). In fact, for such a system, the dipole moment g d = yer = Y— mr = const Y mr, m where const is the charge-to-mass ratio common to all the charges. But S mr = R 2 m, where R is the radius vector of the center of inertia of the system (remember that all of the velocities are small, v <^ c, so that non-relativistic mechanics is applicable). Therefore d is proportional to the acceleration of the center of inertia, which is zero, since the center of inertia moves uniformly. Finally, we give the formula for the spectral resolution of the intensity of dipole radiation. For radiation accompanying a collision, we introduce the quantity dS a of energy radiated throughout the time of the collision in the form of waves with frequencies in the interval d(oj2% (see § 66). It is obtained by replacing the vector d in (67.8) by its Fourier component A a and multiplying by 2 : 4 d£ (0 = — 3 (&J 2 dco. For determining the Fourier components, we have d 2 dt from which 5 W == — co 2 d a) . Thus, we get For periodic motion of the particles, we obtain in similar fashion the intensity of radiation with frequency co = nco in the form h= 3c 3 l d »| • (67.11) l(Ot =^2(d 0i e- l(Ot )=-<o 2 a (O e- 176 RADIATION OF ELECTROMAGNETIC WAVES § 67 PROBLEMS 1. Find the radiation from a dipole d, rotating in a plane with constant angular velocity Q.f Solution: Choosing the plane of the rotation as the x, y plane, we have: d x — do cos Clt, d y — d sin €lt. Since these functions are monochromatic, the radiation is also monochromatic, with frequency co = Q. From formula (67.7) we find for the angular distribution of the radiation (averaged over the period of the rotation): dl= d ^(l+cos 2 G)do, Snc 3 where 9 is the angle between the direction n of the radiation and the z axis. The total radiation is 2d%& 1 = 3c 3 The polarization of the radiation is along the vector A x n = co 2 n x d. Resolving it into com- ponents in the n, z plane and perpendicular to it, we find that the radiation is elliptically polarized, and that the ratio of the axes of the ellipse is equal to n 3 = cos 9; in particular, the radiation along the z axis is circularly polarized. 2. Determine the angular distribution of the radiation from a system of charges, moving as a whole (with velocity v), if the distribution of the radiation is known in the reference system in which the system is at rest as a whole. Solution: Let dl ' =/(cos 9', fi) do', do' = d(cos 9') d</>' be the intensity of the radiation in the K' frame which is attached to the moving charge system (9', </>' are the polar coordinates; the polar axis is along the direction of motion of the system). The energy d£ radiated during a time interval dt in the fixed (laboratory) reference frame K, is related to the energy d£' radiated in the K' system by the transformation formula V j* \r jd * cos e dS—y-dY ,„ c J>~$ J l- V - c 2 (the momentum of radiation propagating in a given direction is related to its energy by the equation \dP\ = dtf/c). The polar angles, 9, 9' of the direction of the radiation in the K and K' frames are related by formulas (5.6), and the azimuths $ and </>' are equal. Finally, the time interval dt' in the K' system corresponds to the time dt' dt = i y - in the K system. As a result, we find for the intensity dl = d#/dt in the K system: J' Thus, for a dipole moving'along the direction of its own axis, /= const • sin 2 9', and by using the cos 9 dl= , x T , " 7 x 3 / I v ° , i I do. 1 cos 9 f The radiation from a rotator or a symmetric top which has a dipole moment is of this type. In the first case, d is the total dipole moment of the rotator; in the second case d is the projection of the dipole moment of the top on a plane perpendicular to its axis of precession (i.e. the direction of the total angular momentum). § 68 DIPOLE RADIATION DURING COLLISIONS 177 formula just obtained, we find: ( 1 -?) sin2 * / V V dl = const • K tt- rv do. § 68. Dipole radiation during collisions In problems of radiation during collisions, one is seldom interested in the radiation accompanying the collision of two particles moving along definite trajectories. Usually we have to consider the scattering of a whole beam of particles moving parallel to each other, and the problem consists in determining the total radiation per unit current density of particles. If the current density is unity, i.e. if one particle passes per unit time across unit area of the cross-section of the beam, then the number of particles in the flux which have "impact parameters" between g and g + dg is 2ng dg (the area of the ring bounded by the circles of radius g and g + dg). Therefore the required total radiation is gotten by multiplying the total radiation A£ from a single particle (with given impact parameter) by 2tiq dg and integrat- ing over g from to oo. The quantity determined in this way has the dimensions of energy times area. We call it the effective radiation (in analogy to the effective cross-section for scattering) and denote it by x : x- f M-2ngdg. (68.1 We can determine in completely analogous manner the effective radiation in a given solid angle element do, in a given frequency interval dco, etc.f We derive the general formula for the angular distribution of radiation emitted in the scattering of a beam of particles by a centrally symmetric field, assuming dipole radiation. The intensity of the radiation (at a given time) from each of the particles of the beam under consideration is determined by formula (67.7), in which d is the dipole moment of the particle relative to the scattering center.} First of all we average this expression over all directions of the vectors d in the plane perpendicular to the beam direction. Since (3 x n) 2 = 3 2 — (n • fl) 2 , the averaging affects only (n • d) 2 . Because the scattering field is centrally symmetric and the incident beam is parallel, the scattering, and also the radiation, has axial symmetry around an axis passing through the center. We choose this axis as x axis. From symmetry, it is obvious that the first powers J y , d z give zero on averaging, and since d x is not subjected to the averaging process, d x d y = d x d z = 0. The average values of d 2 y and d z are equal to each other, so that ? = 5 = i[(d) 2 -^]. t If the expression to be integrated depends on the angle of orientation of the projection of the dipole moment of the particle on the plane transverse to the beam, then we must first average over all directions in this plane and only then multiply by 2ng dg and integrate. % Actually one usually deals with the dipole moment of two particles — the scatterer and the scattered particle — relative to their common center of inertia. 178 RADIATION OF ELECTROMAGNETIC WAVES § 68 Keeping all this in mind, we find without difficulty: (3 x n) 2 = i(3 2 + ell) + i@ 2 ~ 3^) cos 2 9, where 9 is the angle between the direction n of the radiation and the x axis. Integrating the intensity over the time and over all impact parameters, we obtain the following final expression giving the effective radiation as a function of the direction of radiation: , do dx„ = A + B 3 cos 2 0-1 (68.2) Anc where 00+00 00+00 A = 3 I \ ^ dtln Q d Q> B = ? I I @ 2 -3dl)dt2nQdQ. (68.3) -oo -oo The second term in (68.2) is written in such a form that it gives zero when averaged over all directions, so that the total effective radiation is x = A/c 3 . We call attention to the fact that the angular distribution of the radiation is symmetric with respect to the plane passing through the scattering center and perpendicular to the beam, since the expression (68.2) is unchanged if we replace 9 by n - 9. This property is specific to dipole radiation, and is no longer true for higher approximations in v/c. The intensity of the radiation accompanying the scattering can be separated into two parts — radiation polarized in the plane passing through the x axis and the direction n (we choose this plane as the xy plane), and radiation polarized in the perpendicular plane xz. The vector of the electric field has the direction of the vector n x (A x n) = n(n • d) — fl [see (67.6)]. The component of this vector in the direction perpendicular to the xy plane is — d z , and its projection on the xy plane is jsin 9d x — cos 9d y \. This latter quantity is most con- veniently determined from the z-component of the magnetic field which has the direction dxn. Squaring E and averaging over all directions of the vector A in the yz plane, we see first of all that the product of the projections of the field on the xy plane and perpendicular to it, vanishes. This means that the intensity can actually be represented as the sum of two independent parts — the intensities of the radiation polarized in the two mutually per- pendicular planes. The intensity of the radiation with its electric vector perpendicular to the xy plane is determined by the mean square d\ = ^(d 2 — d 2 ). For the corresponding part of the effective radiation, we obtain the expression 00 +00 dxi = -^- I I (A 2 -d 2 x )dt2nQdQ. (68.4) - oo We note that this part of the radiation is isotropic. It is unnecessary to give the expression for the effective radiation with electric vector in the xy plane since it is clear that dx\+dx^ = dx a . In a similar way we can get the expression for the angular distribution of the effective radiation in a given frequency interval dco : § 69 where RADIATION OF LOW FREQUENCY IN COLLISIONS "^n, co ~~ A(co) + B(co) 3 cos 2 0-1 do dco 2nc 3 lit 179 (68.5) 00 00 ^(co) = — J dl2n Q d e , 5(a)) = y J (di-3dL)2jrede. (68.6) § 69. Radiation of low frequency in collisions In the spectral distribution of the radiation accompanying a collision, the main part of the intensity is contained in frequencies co ~ 1/t, where t is the order of magnitude of the dura- tion of the collision. However, we shall here not consider this region of the spectrum (for which one cannot obtain any general formulas) but rather the "tail" of the distribution at low frequencies, satisfying the condition (ox<\. (69.1) We shall not assume that the velocities of the colliding particles are small compared to the velocity of light, as we did in the preceding section; the formulas which follow will be valid for arbitrary velocities. In the integral H t0 = j He icot dt, the field H of the radiation is significantly different from zero only during a time interval of the order of x. Therefore, in accord with condition (69.1), we can assume that cox <^ 1 in the integral, so that we can replace e iwt by unity; then oo H,= JH dt. Substituting H = Axn/c and carrying out the time integration, we get : c (69.2) where A 2 — A t is the change in the vector potential produced by the colliding particles during the time of the collision. The total radiation (with frequency co) during the time of the collision is found by sub- stituting (69.2) in (66.9): R 2 ° [(Aa-AJxnpdoJo). (69.3) am 4cn 2 We can use the Lienard-Wiechert expression (66.4) for the vector potential, and obtain I v 2 xn v x xn I — n-v, 1 — nv. c c l do dco, (69.4) **'" nco a 2 3 4n z c 6 < where v x and v 2 are the velocities of the particle before and after the collision, and the sum is 180 RADIATION OF ELECTROMAGNETIC WAVES § 69 taken over the two colliding particles. We note that the coefficient of dco is independent of frequency. In other words, at low frequencies [condition (69.1)], the spectral distribution is independent of frequency, i.e. dS n Jda> tends toward a constant limit as co -> O.f If the velocities of the colliding particles are small compared with the velocity of light, then (69.4) becomes An c 2 3 \^e{y 1 — y 1 )xvL\ 2 dod(o. (69.5) This expression corresponds to the case of dipole radiation, with the vector potential given by formula (67.4). An interesting application of these formulas is to the radiation produced in the emission of a new charged particle (e.g. the emergence of a /^-particle from a nucleus). This process is to be treated as an instantaneous change in the velocity of the particle from zero to its actual value. [Because of the symmetry of formula (69.5) with respect to interchange of v t and v 2 , the radiation originating in this process is identical with the radiation which would be produced in the inverse process — the instantaneous stopping of the particle.] The essential point is that, since the "time" for the process is x -» 0, condition (69.1) is actually satisfied for all frequencies.! PROBLEM Find the spectral distribution of the total radiation produced when a charged particle is emitted which moves with velocity v. Solution: According to formula (69.4) (in which we set v 2 = v, Vi = 0), we have: e 2 v 2 C sin 2 9 •o [ l - c cos6 ) d g (o = e l( c \ n C +l-?\ d(0 . (1) nc \v c—v ) Evaluation of the integral gives :§ For v<^.c, this formula goes over into 2eV 5TZC which can also be obtained directly from (69.5). 3 dco, f By integrating over the impact parameters, we can obtain an analogous result for the effective radiation in the scattering of a beam of particles. However it must be remembered that this result is not valid for the effective radiation when there is a Coulomb interaction of the colliding particles, because then the integral over q is divergent (logarithmically) for large q. We shall see in the next section that in this case the effective radiation at low frequencies depends logarithmically on frequency and does not remain constant. J However, the applicability of these formulas is limited by the quantum condition that too be small compared with the total kinetic energy of the particle. § Even though, as we have already pointed out, condition (69.1) is satisfied for all frequencies, because the process is "instantaneous" we cannot find the total radiated energy by integrating (1) over co — the integral diverges at high frequencies. We mention that, aside from the violation of the conditions for classical behavior at high frequencies, in the present case the cause of the divergence lies in the incorrect formulation of the classical problem, in which the particle has an infinite acceleration at the initial time. § 70 RADIATION IN THE CASE OF COLOUMB INTERACTION 181 § 70. Radiation in the case of Coulomb interaction In this section we present, for reference purposes, a series of formulas relating to the dipole radiation of a system of two charged particles; it is assumed that the velocities of the particles are small compared with the velocity of light. Uniform motion of the system as a whole, i.e. motion of its center of mass, is not of interest, since it does not lead to radiation, therefore we need only consider the relative motion of the particles. We choose the origin of coordinates at the center of mass. Then the dipole moment of the system d = e 1 r 1 + e 2 r 2 has the form e.m,- e 2 m 1 ( e x e 2 \ d = _1_J l_k r = ^ (_!__!) r (70.1) m 1 + m 2 \m 1 m 2 f where the indices 1 and 2 refer to the two particles, and r = r l — r 2 is the radius vector between them, and m 1 m 2 P = i m 1 + m 2 is the reduced mass. We start with the radiation accompanying the elliptical motion of two particles attracting each other according to the Coulomb law. As we know from mechanicsf , this motion can be expressed as the motion of a particle with mass \i in the ellipse whose equation in polar coordinates is , a(l-e 2 ) 1 + s cos = — \ (70.2) where the semimajor axis a and the eccentricity e are Here $ is the total energy of the particles (omitting their rest energy !) and is negative for a finite motion; M = \ir 2 § is the angular momentum, and a is the constant in the Coulomb law: a = \e x e 2 \. The time dependence of the coordinates can be expressed in terms of the parametric equations r = fl (l - e cos 0, t= . /— (£ - e sin £). (70.4) yj a. One full revolution in the ellipse corresponds to a change of the parameter £ from to 2n; the period of the motion is /^ T = 2k V a We calculate the Fourier components of the dipole moment. Since the motion is periodic we are dealing with an expansion in Fourier series. Since the dipole moment is proportional to the radius vector r, the problem reduces to the calculation of the Fourier components of the coordinates x = r cos 0, y = r sin <f>. The time dependence of x and y is given by the | See Mechanics, § 15. 182 RADIATION OF ELECTROMAGNETIC WAVES § 70 parametric equations (70.5) x = a(cos £ — e), y = a\J 1 — e 2 sin £, o t = £ — s sin £. Here we have introduced the frequency IT /-T~~* (2 \^ Instead of the Fourier components of the coordinates, it is more convenient to calculate the Fourier components of the velocities, using the fact that x n = — ia> nx n ; y n = — ico ny„. We have T — ico n co ni J f e iB «- esin{) sinfd£. o But xdt — dx = — as'm£d£; transforming from an integral over t to one over £, we have la 2nn o Similarly, we find 2n 2k iayl — e 2 f . ,. . .. ia^/l—8 2 f . ,. . .. v „ = — g'»«- esin «cos^^ = — e'»«-" in «^ 27m J 27ine J o o (in going from the first to the second integral, we write the integrand as cos £ = (cos £ — 1/g) + 1/e; then the integral with cos £— 1/e can be done, and gives identically zero). Finally, we use a formula of the theory of Bessel functions, 2n n f e w«-xsin«)^ = 1 _f CQS ( n £- X sin Q d £ = JJ^^ ( 70 6 ) where J n (x) is the Bessel function of integral order n. As a final result, we obtain the follow- ing expression for the required Fourier components: a iayl — e 2 x H = - J ' n {ne\ y n = J n (ne) (70.7) n ne (the prime on the Bessel function means differentiation with respect to its argument). The expression for the intensity of the monochromatic components of the radiation is obtained by substituting x a and y a into the formula 4. ,„4 / „ „ \ 2 Action 3c 2 M_M (W2+W2) [see (67.11)]. Expressing a and co in terms of the characteristics of the particles, we obtain finally : 64nWd e,\ 2 X _, 2( l-e 2 2 5 a 2 Vm! m 2 / j; 2 (ne)+— 2- J^(ne) (70.8) " 3c' In particular, we shall give the asymptotic formula for the intensity of very high harmonics (large n) for motion in an orbit which is close to a parabola (s close to 1). For this purpose, § 70 RADIATION IN THE CASE OF COULOMB INTERACTION we use the formula 1 /2\ 1/3 UnB)*-r- (-) O Vn W 2/3 (l-£ 2 ) 183 (70.9) n ^> 1, 1 — e <^ 1, where O is the Airy function defined on p. 179.f Substituting in (70.8) gives: + =) a-* 2N 2/3 + d>" 2/3 (1-8 2 ) (70.10) This result can also be expressed in terms of the MacDonald function K /„ = 64 n 2 <f v / e x e 2 9n c a \Trii m 2 K\ /3 (l-e 2 ) 3/2 + K 2/3 (l-s 2 ) 3/2 (the necessary formulas are given in the footnote on p. 201). Next, we consider the collision of two attracting charged particles. Their relative motion is described as the motion of a particle with mass /n in the hyperbola l+£ cos (f) = fl(£ 2 -l) where a = -J 1 + 2$M A 2$' "V fia' (now $ > 0). The time dependence of r is given by the parametric equations r = a(ecosh £— 1), t /iur V a (ssinh£-£), (70.11) (70.12) (70.13) where the parameter £ runs through values from — oo to + oo. For the coordinates x, j,we have x = a(s — cosh <jj), y = a\J& 2 — 1 sinh £. (70.14) t For w :> 1 , the main contributions to the integral % /„(«£) = - cos [«(£— e sin £)] *#; 7T J come from small values of £, (for larger values of £, the integrand oscillates rapidly). In accordance with this, we expand the argument of the cosine in powers of £, : y*.) = lja»[.(l^{+f)]« ; because of the rapid convergence of the integral, the upper limit has been replaced by oo ; the term in £ 3 must be kept because the first order term contains the small coefficient 1 — e ^ (1 — e 2 )/2. The integral above is reduced to the form (70.9) by an obvious substitution. 184 RADIATION OF ELECTROMAGNETIC WAVES § 70 The calculation of the Fourier components (we are now dealing with expansion in a Fourier integral) proceeds in complete analogy to the preceding case. We find the result: na ,,. TtaJs 2 — \ ,,. x„ = — H^'(ivs), y w = - HiPQve). (70.15) CO COS where H\^ is the Hankel function of the first kind, of order iv, and we have introduced the notation v = — 1 = = -^ (70.16) N ua fia 3 (v is the relative velocity of the particles at infinity; the energy $ = fivl/2).-\ In the calcula- tion we have used the formula from the theory of Bessel functions : + 00 J" ePt-to^id^inHPiix). (70.17) — oo Substituting (70.15) in the formula A # 4o>V /ci e 2 \ 2 , |2 . |2 dco 3c \m l m 2 J 2n [see (67.10)], we get: "• - w 2 fe - ^) 2 { W " ,,(, ' V6)]2 - E ^ w * >0v£)]2 } *»• (70 - 18) A quantity of greater interest is the "effective radiation" during the scattering of a parallel beam of particles (see § 68). To calculate it, we multiply dS^ by 2nQdq and integrate over all q from zero to infinity. We transform from an integral over q to one over e (between the limits 1 and go) using the fact that 2tzq dg = 2na 2 s ds ; this relation follows from the definition (70.12), in which the angular momentum M and the energy i are related to the impact parameter q and the velocity v by M = hqv , S-\l~. The resultant integral can be directly integrated with the aid of the formula d ? + (H = S (zZ pZ p, where Z p (z) is an arbitrary solution of the Bessel equation of order p. J Keeping in mind that for s-)- oo, the Hankel function H^Qvs) goes to zero, we get as our result the following formula : 4n 2 a 3 co f - - x 2 dx^ = 3c 3 [!Vq (l±_^l\ \H\?{iv)\H\?\iv)da>. (70.19). \m 1 m 2 J t Note that the function H^iive) is purely imaginary, while its derivative H\V' (ive) is real. % This formula is a direct consequence of the Bessel equation z"+|z'+(i-^)z=o. § 70 RADIATION IN THE CASE OF COLOUMB INTERACTION 185 Let us consider the limiting cases of low and high frequencies. In the integral + 00 J e iH4-sinh4) d £ = inH M( iv ) (70.20) — oo defining the Hankel function, the only important range of the integration parameter £ is that ift which the exponent is of order unity. For low frequencies (v <^ 1), only the region of large f is important. But for large £ we have sinh f P £. Thus, approximately, + oo Htfiiv)*-- f e _l ' vsinh? ^ = ^ 1) (/v). — oo Similarly, we find that H}»'(iv)*HP'(iv)- Using the approximate expression (for small x) from the theory of Bessel functions : 2 2 iH^ (ix)^- In — n yx (y = e c , where C is the Euler constant; y = 1.781 .. .), we get the following expression for the effective radiation at low frequencies : dXtt . »&!l /£. _ «. y ta (*e&) dco for m < fa3. (70 . 21) 3vqC Vwii m 2 / \ycoct/ a. It depends logarithmically on the frequency. For high frequencies (v > 1), on the other hand, the region of small £ is important in the integral (70.20). In accordance with this, we expand the exponent of the integrand in powers of £ and get, approximately, + 00 oo H\l\iv)^~- e «^=— Re e 6 dn. -oo By the substitution iv£ 3 /6 = rj, the integral goes over into the T-function, and we obtain the result : i /6V' 3 _/r Similarly, we find Next, using the formula of the theory of the T-function, r(*)r(i-x) = -A-, sin nx we obtain for the effective radiation at high frequencies : d *» " W%? fe " £) d <°' for <° * T- (m22) that is, an expression which is independent of the frequency. We now proceed to the radiation accompanying the collision of two particles repelling 186 RADIATION OF ELECTROMAGNETIC WAVES § 70 each other according to the Coulomb law U = ot/r(<x > 0). The motion occurs in a hyper- bola, - 1 + e cos <£ = - '-, (70.23) r x = a(s + cosh £), y = «v e 2 — 1 sinh £, ' = vT" (e sinh ^ + ^ (70 - 24) [a and e as in (70.12)]. All the calculations for this case reduce immediately to those given above, so it is not necessary to present them. Namely, the integral + oo x a = - f e iv(£sinh?+?) sinh^^ co J — 00 for the Fourier component of the coordinate x reduces, by making the substitution £ -> in — %, to the integral for the case of attraction, multiplied by e~ nv ; the same holds for yco- Thus the expressions for the Fourier components x^, y a in the case of repulsion differ from the corresponding expressions for the case of attraction by the factor e~ Kv . So the only change in the formulas for the radiation is an additional factor e~ 2nv . In particular, for low frequencies we get the previous formula (70.21) (since for v <^ 1, e~ 2nv ^ 1). For high frequencies, the effective radiation has the form 16na 2 ( e\ e 2 \ 2 ( 2ncoa\ 7 „ uvl dx ° - wk? U " t) exp ( ~ I*) da ' for m > -?■ <70 - 25) It drops exponentially with increasing frequency. PROBLEMS 1. Calculate the average total intensity of the radiation for elliptical motion of two attracting charges. Solution: From the expression (70.1) for the dipole moment, we have for the total intensity of the radiation : 3c 3 l/Wi w 2 / 3c 3 V Wi m 2j where we have used the equation of motion fir = — ar/r 3 . We express the coordinate r in terms of <j> from the orbit equation (70.2) and, by using the equation dt = fir 2 d(f>/M, we replace the time in- tegration by an integration over the angle 4> (from to In). As a result, we find for the average intensity : 7== L [ idt = 23/2 ( ei g 2 V ^ 5/2 « 3 K1 3/2 / 2Kjm 2 \ TJ 3c 3 \wi m 2 ) M 5 \ fix 2 /' o 2. Calculate the total radiation A<f for the collision of two charged particles. Solution: In the case of attraction the trajectory is the hyperbola (70.11) and in the case of repulsion, (70.23). The angle between the asymptotes of the hyperbola and its axis is ^ , determined from ±cos (j> — 1/e, and the angle of deflection of the particles (in the system of coordinates in which the center of mass is at rest) is x = I n — 2<f>o\ • The calculation proceeds the same as in Problem § 70 RADIATION IN THE CASE OF COLOUMB INTERACTION 187 1 (the integral over <j> is taken between the limits — <t> and +<t> ). The result for the case of attraction and for the case of repulsion: A ^« tan 3|{ ( „_ x) ( 1+3tani )_ 6tan ^(|-^ In both, x is understood to be a positive angle, determined from the relation , X HvIq cot - = . 2 a Thus for a head-on collision 0?— >0, x— >rc) of charges repelling each other: 45c 3 a \Wi 3. Calculate the total effective radiation in the scattering of a beam of particles in a repulsive Coulomb field. Solution: The required quantity is 00+00 00+00 0-oo 0-00 We replace the time integration by integration over r along the trajectory of the charge, writing dt = drjvr, where the radial velocity v r = r is expressed in terms of r by the formula -VH'-^-H-^ 2a The integration over r goes between the limits from oo to the distance of closest approach r — r (e) (the point at which v r = 0), and then from r Q once again to infinity; this reduces to twice the integral from r to oo. The calculation of the double integral is conveniently done by changing the order of integration — integrating first over q and then over r. The result of the calculation is : 8tt 4. Calculate the angular distribution of the total radiation emitted when one charge passes by another, if the velocity is so large (though still small compared with the velocity of light) that the deviation from straight-line motion can be considered small. Solution: The angle of deflection is small if the kinetic energy fiv 2 /2 is large compared to the potential energy, which is of order a/g (}iv 2 > a/g). We choose the plane of the motion as the x, y plane, with the origin at the center of inertia and the x axis along the direction of the velocity. In first approximation, the trajectory is the straight line x = vt, y = q. In the next approximation, the equations of motion give .. _ a x ocvt .. ay <xq with r = Vx 2 +y 2 ^ V~g 2 +v 2 t 2 . Using formula (67.7), we have: 00 188 RADIATION OF ELECTROMAGNETIC WAVES § 71 ection of do. Expi Ivc 3 q 3 \mi m 2 J where n is the unit vector in the direction of do. Expressing the integrand in terms of / and per- forming the integration, we get : 32yc 3„3l.„ _ .'(M-^A. §71. Quadrupole and magnetic dipole radiation We now consider the radiation associated with the succeeding terms in the expansion of the vector potential in powers of the ratio a\X of the dimensions of the system to the wave length. Since a]X is assumed to be small, these terms are generally small compared with the first (dipole) term, but they are important in those cases where the dipole moment of the system is zero, so that dipole radiation does not occur. Expanding the integrand in (66.2), A = „ cR ( in powers of r-n/c, we find, correct to terms of first order: Substituting j = q\ and changing to point charges, we obtain : A =k + ^"o^ Zcv(r,,,) - (7L1) (From now on, as in § 67, we drop the index t ' in all quantities). In the second term we write v(r-n) = --r(n-r)+-v(n-r)--r(n-v) = 2af r(n ' r)+ 2 (rxv)xn * We then find for A the expression A = ^ + 2kS le ' {a ' l)+ ^ Xn) ' (7t2) where d is the dipole moment of the system, and 1 v m = — > erx v 2c ^ is its magnetic moment. For further transformation, we note that we can, without changing the field, add to A any vector proportional to n, since according to formula (66.3), H and E are unchanged by this. For this reason we can replace (71.2) by d 1 d 2 „ _„ . . 2 _ 1 . A = h „ - —^ V e[3r(n • r)-nr 2 ] + -— m x n. cR 6c 2 R dt 2 ^ L v cR But the expression under the summation sign is just the product n p D ap of the vector n and the quadrupole moment tensor D aP = L e(3x a x p -d ap r 2 ) (see § 41). We introduce the vector § 71 QUADRUPOLE AND MAGNETIC DIPOLE RADIATION 189 D with components D a = D ap n p , and get the final expression for the vector potential : ^^^^i*™- (7U) Knowing A, we can now determine the fields H and E of the radiation, using the general formula (66.3): H = -y— <itxii+ — Cxn + (mxn)xn c z R I 6c E = i^- -1(3 x n) x n+ — (D x n) x n + nx mi. (71.4) The intensity dl of the radiation in the solid angle do is given by the general formula (66.6). We calculate here the total radiation, i.e., the energy radiated by the system in unit time in all directions. To do this, we average dl over all directions of n; the total radiation is equal to this average multiplied by An. In averaging the square of the magnetic field, all the cross-products of the three terms in H vanish, so that there remain only the mean squares of the three. A simple calculation! gives the following result for /: Thus the total radiation consists of three independent parts ; they are called, respectively, dipole, quadrupole, and magnetic dipole radiation. We note that the magnetic dipole radiation is actually not present for many systems. Thus it is not present for a system in which the charge-to-mass ratio is the same for all the moving charges (in this case the dipole radiation also vanishes, as already shown in § 67). Namely, for such a system the magnetic moment is proportional to the angular momentum (see § 44) and therefore, since the latter is conserved, m = 0. For the same reason, magnetic dipole radiation does not occur for a system consisting of just two particles (cf. the problem in § 44. In this case we cannot draw any conclusion concerning the dipole radiation). PROBLEM Calculate the total effective radiation in the scattering of a beam of charged particles by particles identical with them. Solution: In the collision of identical particles, dipole radiation (and also magnetic dipole radiation) does not occur, so that we must calculate the quadrupole radiation. The quadrupole moment tensor of a system of two identical particles (relative to their center of mass) is g where x a are the components of the radius vector r between the particles. After threefold differentia- t We present a convenient method for averaging the products of components of a unit vector. Since the tensor n a tt is symmetric, it can be expressed in terms of the unit tensor S a0 . Also noting that its trace is 1 we have: ' The average value of the product of four components is: n a n n y n 6 = Ts(d ae S yi + d ay 8 ed + d ad S 0y ). The right side is constructed from unit tensors to give a fourth-rank tensor that is symmetric in all its indices; the overall coefficient is determined by contracting on two pairs of indices, which must give unity. 190 RADIATION OF ELECTROMAGNETIC WAVES § 72 tion of D a p, we express the first, second, and third derivatives with respect to time of x a in terms of the relative velocity of the particles v a as : . _ .. _m .. _e 2 x a m ... _ 2 v a r—3x a v T where v r = \-rjr is the radial component of the velocity (the second equality is the equation of motion of the charge, and the third is obtained by differentiating the second). The calculation leads to the following expression for the intensity : Dip _ & 1 180c 5 15m 2 c 5 H 2 = v* + v£) ; v and v^ are expressible in terms of r by using the equalities 2 2 4e2 Q v o v 2 = vZ , v 4> = — . mr r We replace the time integration by an integration over r in the same way as was done in Problem 3 of § 70, namely, we write dr dr dt = — = '=Tis& = rw«;^ 2 +iK) " fi Q 2 vl 4e 2 r 2 mr In the double integral (over q and r), we first carry out the integration over q and then over r. The result of the calculation is : An eH% Y. = — ;. 9 mc 3 § 72. The field of the radiation at near distances The formulas for the dipole radiation were derived by us for the field at distances large compared with the wavelength (and, all the more, large compared with the dimensions of the radiating system). In this section we shall assume, as before, that the wavelength is large compared with the dimensions of the system, but shall consider the field at distances which are not large compared with, but of the same order as, the wavelength. The formula (67.4) for the vector potential A=-j-d (72.1) cR is still valid, since in deriving it we used only the fact that R was large compared with the dimensions of the system. However, now the field cannot be considered to be a plane wave even over small regions. Therefore the formulas (67.5) and (67.6) for the electric and mag- netic fields are no longer applicable, so that to calculate them, we must first determine both A and (f>. The formula for the scalar potential can be derived directly from that for the vector potential, using the general condition (62.1), divA+i^-0, c dt imposed on the potentials. Substituting (72.1) in this, and integrating over the time, we get 0=_div^-. (72.2) § 72 THE FIELD OF THE RADIATION AT NEAR DISTANCES 191 The integration constant (an arbitrary function of the coordinates) is omitted, since we are interested only in the variable part of the potential. We recall that in the formula (72.2) as well as in (72.1) the value of d must be taken at the time t' — t — (i? /c).f Now it is no longer difficult to calculate the electric and magnetic field. From the usual formulas, relating E and H to the potentials, 1 , d H = curl - c R, , ,. d Id E = graddiv----2 — . K C K The expression for E can be rewritten in another form, noting that d t /R (72.3) (72.4) just as any 1 / R function of coordinates and time of the form — fit R \ c satisfies the wave equation: i f? (A. c^dt^Ro A R t Also using the formula we find that curl curl a = grad div a — Aa, d E = curl curl Ro (72.5) The results obtained determine the field at distances of the order of the wavelength. It is understood that in all these formulas it is not permissible to take l/R out from under the differentiation sign, since the ratio of terms containing \jR% to terms with l/R is just of the same order as X/R . Finally, we give the formulas for the Fourier components of the field. To determine H CT we substitute in (72.3) for H and d their monochromatic components H (O e~ i0Jt and d w e~ itot , respectively. However, we must remember that the quantities on the right sides of equations (72.1) to (72.5) refer to the time t' = t—(R /c). Therefore we must substitute in place of d the expression — i<o/Ro\ ftp ^ c > = ft p-i<»t + ikRo Making the substitution and dividing by e~ lmt , we get e ikR o\ JkRo -) = *d„xV--. ik Rn k e ikR H co = -;/ccurl(d a or, performing the differentiation, H w = ikft a x n where n is a unit vector along R . t Sometimes one introduces the so-called Hertz vector, defined by Ro \ c J A = - l Z, ^ = div Z. (72.6) Then 192 RADIATION OF ELECTROMAGNETIC WAVES § 72 In similar fashion, we find from (72.4): JikRo JkR V a = k 2 d a — +(d (U -V)V-— , and differentiation gives At distances large compared to the wave length (kR > 1), we can neglect the terms in IjRl and 1/Rl in formulas (72.5) and (72.6), and we arrive at the field in the "wave zone", E (0 =*-nx(d (0 x n)e ikR °, H 0} =-^d (O x ne ikRo . At distances which are small compared to the wave length (JcR <| 1), we neglect the terms in l/R and 1/jRg and set e ikR s 1 ; then E co = ^3( 3n ( d ,-n)-d w }, which corresponds to the static field of an electric dipole (§40); in this approximation, the magnetic field vanishes. PROBLEM Calculate the quadrupole and magnetic dipole radiation fields at near distances. Solution: Assuming, for brevity, that dipole radiation is not present, we have (see the calculation carried out in § 71) If dV If J'-- C J *~c R C J R where we have expanded in powers of r = R — R. In contrast to what was done in § 71, the factor 1/Ro cannot here be taken out from under the differentiation sign. We take the differential operator out of the integral and rewrite the integral in tensor notation: A a = c dX p J R (Xp are the components of the radius vector R ). Transforming from the integral to a sum over the charges, we find _ 1 8 ( I>^*A ' cdXp R ' In the same way as in § 71, this expression breaks up into a quadrupole part and a magnetic dipole part. The corresponding scalar potentials are calculated from the vector potentials in the same way as in the text. As a result, we obtain for the quadrupole radiation : 6c dX p Rq' v 6 dX a dXp R ' and for the magnetic dipole radiation : A = curl—-, # = Ro [all quantities on the right sides of the equations refer as usual to the time /' = t—(R /c)]. The field intensities for magnetic dipole radiation are: E = — curl — , H = curl curl ~. c Ro' Rq § 73 RADIATION FROM A RAPIDLY MOVING CHARGE 193 Comparing with (72.3), (72.4), we see that in the magnetic dipole case, E and H are expressed in terms of m in the same way as — H and E are expressed in terms of d for the electric dipole case. The spectral components of the potentials of the quadrupole radiation are : 6 aB dX p R ' * 6'" dX a dX p R • Because of their complexity, we shall not give the expressions for the field. § 73. Radiation from a rapidly moving charge Now we consider a charged particle moving with a velocity which is not small compared with the velocity of light. The formulas of § 67, derived under the assumption that v <^c, are not immediately applicable to this case. We can, however, consider the particle in that system of reference in which the particle is at rest at a given moment; in this system of reference the formulas referred to are of course valid (we call attention to the fact that this can be done only for the case of a single moving particle ; for a system of several particles there is generally no system of reference in which all the particles are at rest simultaneously). Thus in this particular system of reference the particle radiates, in time dt, the energy 2e z dS = — w 2 dt (73.1) [in accordance with formula (67.9], where w is the acceleration of the particle in this system of reference. In this system of reference, the "radiated" momentum is zero: dP = 0. (73.2) In fact, the radiated momentum is given by the integral of the momentum flux density in the radiation field over a closed surface surrounding the particle. But because of the symmetry of the dipole radiation, the momenta carried off in opposite directions are equal in magnitude and opposite in direction; therefore the integral is identically zero. For the transformation to an arbitrary reference system, we rewrite formulas (73.1) and (73.2) in four-dimensional form. It is easy to see that the "radiated four-momentum" dP t must be written as 2e 2 du k du k , . 2e 2 du k du k . dP ' ~ 17 *"* dx = - Tc Ts-Ts " *• (73 - 3) In fact, in the reference frame in which the particle is at rest, the space components of the four-velocity «' are equal to zero, and du k du k w 2 ds ds c 4 ' therefore the space components of dP 1 become zero and the time component gives equation (73.1). The total four-momentum radiated during the time of passage of the particle through a given electromagnetic field is equal to the integral of (73.3), that is, 2e 2 C du k diiir AP l =~^- — — fc dx\ (73.4) 3c J ds ds We rewrite this formula in another form, expressing the four-acceleration du'/ds in terms of 194 RADIATION OF ELECTROMAGNETIC WAVES the electromagnetic field tensor, using the equation of motion (23.4): § 73 mc du k ds We then obtain AP l = - 2e A 3m V J F hl u l (F kl u l ){F km u m )dx\ (73.5) The time component of (73.4) or (73.5) gives the total radiated energy A<f . Substituting for all the four-dimensional quantities their expressions in terms of three-dimensional quantities, we find , (v x w) 2 oo w 2_V J 2e 2 C c 2 J — Zi^dt (73.6) LS 3r 1- (w = v is the acceleration of the particle), or, in terms of the external electric and magnetic fields: M = 2e A 3m c s E+-vxh1 --4(E-v) 2 dt. (73.7) 1- The expressions for the total radiated momentum differ by having an extra factor v in the integrand. It is clear from formula (73.7) that for velocities close to the velocity of light, the total energy radiated per unit time varies with the velocity essentially like [1 — (y 2 /c 2 )] -1 , that is, proportionally to the square of the energy of the moving particle. The only exception is motion in an electric field, along the direction of the field. In this case the factor [1 — (v 2 /c 2 )] standing in the denominator is cancelled by an identical factor in the numerator, and the radiation does not depend on the energy of the particle. Finally there is the question of the angular distribution of the radiation from a rapidly moving charge. To solve this problem, it is convenient to use the Lienard-Wiechert expres- sions for the fields, (63.8) and (63.9). At large distances we must retain only the term of lowest order in l/R [the second term in (63.8)]. Introducing the unit vector n in the direction of the radiation (R = nR), we get the formulas nx E = n — x w c 2 R 1- n-v H = n x E, (73.8) where all the quantities on the right sides of the equations refer to the retarded time t' = t-(jR/c). The intensity radiated into the solid angle do is dl = (c/4n)E 2 R 2 do. Expanding E 2 , we get l-4Ww) 2 dl = 4nc~ 2(n • w)(v • w) vn c + w vn c 1- vn do. (73.9) §73 RADIATION FROM A RAPIDLY MOVING CHARGE 195 If we want to determine the angular distribution of the total radiation throughout the whole motion of the particle, we must integrate the intensity over the time. In doing this, it is important to remember that the integrand is a function of t ' ; therefore we must write dt dt = — dt ' = ( 1 (73.10) [see (63.6)], after which the integration over t' is immediately done. Thus we have the following expression for the total radiation into the solid angle do : Anc- do I ' 2(n- w)(v w) [•(■ V c ■)' + w (n-w) 2 vn c 1- n-v > dt'. (73.11) As we see from (73.9), in the general case the angular distribution of the radiation is quite complicated. In the ultrarelativistic case, (l—(v/c)-4l) it has a characteristic appearance, which is related to the presence of high powers of the difference 1 — (v • n/c) in the denominators of the various terms in this expression. Thus, the intensity is large within the narrow range of angles in which the difference 1 — (v -n/c) is small. Denoting by 6 the small angle between n and v, we have : 1 - - cos s 1 c ■+ ^r = B- 2 > this difference is small for J 1- (73.12) Thus an ultrarelativistic particle radiates mainly along the direction of its own motion, within the small range (73.12) of angles around the direction of its velocity. We also point out that, for arbitrary velocity and acceleration of the particle, there are always two directions for which the radiated intensity is zero. These are the directions for which the vector n— (v/c) is parallel to the vector w, so that the field (73.8) becomes zero. (See also problem 2 of this section.) Finally, we give the simpler formulas to which (73.9) reduces in two special cases. If the velocity and acceleration of the particle are parallel, H wxn n- v c and the intensity is dl = 4nr 1 — cos do. (73.13) It is naturally, symmetric around the common direction of v and w, and vanishes along (6 = 0) and opposite to (9 = n) the direction of the velocity. In the ultrarelativistic case, the intensity as a function of 6 has a sharp double maximum in the region (73.12), with a steep drop to zero for = 0. 196 RADIATION OF ELECTROMAGNETIC WAVES § 73 If the velocity and acceleration are perpendicular to one another, we have from (73.9): dl = 4nc 3 ('-?)•■ ~2 ) sin 2 9 cos (p (l-^cos9) M--cos^ do, (73.14) where 9 is again the angle between n and v, and <f> is the azimuthal angle of the vector n relative to the plane passing through v and w. This intensity is symmetric only with respect to the plane of v and w, and vanishes along the two directions in this plane which form the angle 9 - cos" x (v/c) with the velocity. PROBLEMS 1. Find the total radiation from a relativistic particle with charge e u which passes with impact parameter q through the Coulomb field of a fixed center (with potential <l> = e 2 /r). Solution: In passing through the field, the relativistic particle is hardly deflected at all.f We may therefore regard the velocity v in (73.7) as constant, so that the field at the position of the particle is E = e 2 r e 2 r (Q 2 +V 2 t 2 ) ,2,2\3/2' Fig. 15. with x = vt, y = q. Performing the time integration in (73.7), we obtain: Acf _ Tte\e% 4c 2 — v 2 12m 2 c 3 Q 3 v c 2 —v 2 ' 2. Find the directions along which the intensity of the radiation from a moving particle vanishes. Solution: From the geometrical construction (Fig. 15) we find that the required directions n lie in the plane passing through v and w, and form an angle x with the direction of w where and a is the angle between v and w. sin x — - sin a, c t For y~c, deviations through sizable angles can occur only for impact parameters Q~e 2 /mc 2 , which cannot in general be treated classically. §74 SYNCHROTRON RADIATION (MAGNETIC BREMSSTRAHLUNG) 197 § 74. Synchrotron radiation (magnetic bremsstrahlung) We consider the radiation from a charge moving with arbitrary velocity in a circle in a uniform constant magnetic field; such radiation is called magnetic bremsstrahlung. The radius of the orbit r and the cyclic frequency of the motion co H are expressible in terms of the field intensity H and the velocity of the particle v, by the formulas (see § 21): mcv v eH r = eH J^i co„ = - = mc j-i (74.1) The total intensity of the radiation over all directions is given directly by (73.7), omitting the time integration, in which we must set E = and H J_v: / = 2e 4 H 2 v 2 (-3 (74.2) We see that the total intensity is proportional to the square of the momentum of the particle. If we are interested in the angular distribution of the radiation, then we must use formula (73.11). One quantity of interest is the average intensity during a period of the motion. For this we integrate (73.11) over the time of revolution of the particle in the circle and divide the result by the period T = 27i/co H . We choose the plane of the orbit as the XY plane (the origin is at the center of the circle), and we draw the X Y plane to pass through the direction n of the radiation (Fig. 16). The Fig. 16. magnetic field is along the negative Z axis (the direction of motion of the particle in Fig. 16 corresponds to a positive charge e). Further, let 9 be the angle between the direction k of the radiation and the Faxis, and (j> = a> H t be the angle between the radius vector of the particle and the X axis. Then the cosine of the angle between k and the velocity v is cos 6 cos (f) (the vector v lies in the XY plane, and at each moment is perpendicular to the radius vector of the particle). We express the acceleration w of the particle in terms of the field H and the velocity v by means of the equation of motion [see (21.1)]: w = — Jl mc J< v -~ vxH. c z 198 RADIATION OF ELECTROMAGNETIC WAVES § 74 After a simple calculation, we get: o (1 — cos 9 cos <f> 1 (the time integration has been converted into integration over $ = co H t). The integration is elementary, though rather lengthy. As a result one finds the following formula: „ 2 \ r ,,2 / 2\ / ..2 dl = do Snm c 2„5 (-?K 2 + -^ cos 2 (l-^)(4+^cos 2 0)cos 2 ,2 WS " I x „2 A^" 1 " Zi / u 2 \ 5/2 / I) 2 \ 7 ' 2 (l- ? cos 2 0J 4 (l-- 2 cos 2 0J (74.4) The ratio of the intensity of radiation for 9 = n/2 (perpendicular to the plane of the orbit) to the intensity for = (in the plane of the orbit) is (-) \doJ v 2 4 + 3-2 s-"'(-r As v -*■ 0, this ratio approaches \, but for velocities close to the velocity of light, it becomes very large. Next we consider the spectral distribution of the radiation. Since the motion of the charge is periodic, we are dealing with expansion in a Fourier series. The calculation starts con- veniently with the vector potential. For the Fourier components of the vector potential we have the formula [see (66.12)]: ifc.Ro /"» A„ = e — - (J) e l <"" , "- k "></!•, cR Tj where the integration is taken along the trajectory of the particle (the circle). For the co- ordinates of the particle we have x = r cos a> H t, y = r sin co H t. As integration variable we choose the angle = co H t. Noting that k • r = kr cos sin <p = (nv[c) cos 9 sin (k = nco H /c = nv/cr), we find for the Fourier components of the x-component of the vector potential : In Axn = ~ 2~Sr eikRo f e '"^~ c " cosflsin *) sin <t> #• We have already had to deal with such an integral in § 70. It can be expressed in terms of the derivative of a Bessel function : iev ., o /nv \ A ---w a ^' r -{7 eo "'} (746) Similarly, one calculates A yn : A y» = IT^r-fi eikR ° J » (~ cos e \ ( 74 - 7 > R cos 9 \ c ) The component along the Z axis obviously vanishes. § 74 SYNCHROTRON RADIATION (MAGNETIC BREMSSTRAHLUNG) 199 From the formulas of § 66 we have for the intensity of radiation with frequency co = nco H , in the element of solid angle do : dl n = -~ |H„| 2 R 2 do = ~ |k x A n \ 2 R 2 do. 2% In Noting that lAxkl 2 = Alk 2 + A 2 k 2 sin and substituting (74.6) and (74.7), we get for the intensity of radiation the following formula (Schott, 1912): 2„4rr2 / „2\ r / m» \ n 2 dI " ~ 2nc 3 m 7 { nv \ v ,,. nv Ian 2 0-J 2 A "— osG) + \J' n 2 [ — ™sQ c / c 2 \ c do. (74.8) To determine the total intensity over all directions of the radiation with frequency co = na> H , this expression must be integrated over all angles. However, the integration cannot be carried out in finite form. By a series of transformations, making use of certain relations from the theory of Bessel functions, the required integral can be written in the following form:f o (74.9) We consider in more detail the ultrarelativistic case where the velocity of motion of the particle is close to the velocity of light. Setting v = c in the numerator of (74.2), we find that in the ultrarelativistic case the total intensity of the synchrotron radiation is proportional to the square of the particle energy S: "?£(A)'. C74.10) 3m c \mc / The angular distribution of the radiation is highly anisotropic. The radiation is con- centrated mainly in the plane of the orbit. The "width" A0 of the angular range within which most of the radiation is included is easily evaluated from the condition 1 — (v 2 /c 2 ) cos 2 6 - 1 ~(v 2 /c 2 ), writing = n/2±A9, sin 6 ^ 1 -(A0) 2 /2. It is clear that J / v 2 mc 2 .„. . _ A0 ~V 1 -?=^- ( ' 4 - n) We shall see below that in the ultrarelativistic case the main role in the radiation is played by frequencies with large n (Arzimovich and Pomeranchuk, 1945). We can therefore use the asymptotic formula (70.9), according to which : J 2n (2nO £ y i- fc[n*(l - e)l (74.12) \Jnn 3 Substituting in (74.9), we get the following formula for the spectral distribution of the f The computations can be found in the book by G. A. Schott, Electromagnetic Radiation, § 84, Cam- bridge, 1912. % This result is, of course, in agreement with the angular distribution of the instantaneous intensity which we found in the preceding section [see (73.12)]; however, the reader should not confuse the angle 6 of this section with the angle between n and v in § 73 ! 200 RADIATION OF ELECTROMAGNETIC WAVES § 74 radiation for large values of «:f 2e 4 H 2 — u* h = ~ jJc* r (u)+ U 2 S * (u)du }> (74 - 13) ^/mc 2 \ 2 u = n 3 1 — j . V For w-»0 the function in the curly brackets approaches the constant limit O'(0/ = -0.4587 . . .{ Therefore for u < 1, we have / - BBa52 ^v(- r ;f. K»^(^a). (74-14) For u > 1, we can use the asymptotic expression for the Airy function (see the footnote on p. 179, and obtain): /.= H 2 (aa^Y n i \ * J f 2 fmc 2 \ 3 } ( $ \ 3 v««v exp r 3 n (t) j' w ► (w) < 74 - 15 > that is, the intensity drops exponentially for large n. Consequently the spectrum has a maximum for / (O 1 mc ' and the main part of the radiation is concentrated in the region of frequencies for which / & \ 3 eH ( & \ 2 <o~co H ( — -A = — ( _ . ( 74#16 ) \mc J mc \mc / v / These values of co are very large, compared to the distance co H between neighboring fre- quencies. We may say that the spectrum has a "quasicontinuous" character, consisting of a large number of closely spaced lines. In place of the distribution function I n we can therefore introduce a distribution over the continuous series of frequencies co = nco H , writing dl = I n dn = I n — . For numerical computations it is convenient to express this distribution in terms of the t In making the substitution, the limit « 2 ' 3 of the integral can be changed to infinity, to within the required accuracy; we have also set v = c wherever possible. Even though values of £ close to 1 are important in the integral (74.9), the use of formula (74.12) is still permissible, since the integral converges rapidly at the lower limit. t From the definition of the Airy function, we have: °' (0) "-^J^f*-- v^ J x ~ w sin xdx 3 1/6 r(f) 2Vn ' § 74 SYNCHROTRON RADIATION (MAGNETIC BREMSSTRAHLUNG) 201 MacDonald function K v .-\ After some simple transformations of formula (74.13), it can be written as _ on ^3e 3 H dl = dco where we use the notation 2n mc' f{^\ F{Q = tJKtf)dZ, (74.17) a>„ = 3eH / 8 2mc \mc 2 Figure 17 shows a graph of the function F(£). )' (74.18) F(£) 1 0-92 0-5 n.?P 12 3 4 Fig. 17. Finally, a few comments on the case when the particle moves, not in a plane orbit, but in a helical trajectory, i.e. has a longitudinal velocity (along the field) v ]} = v cos / (where / is the angle between H and v). The frequency of the rotational motion is given by the same formula (74.1), but now the vector v moves, not in a circle, but on the surface of a cone with its axis along H and with vertex angle 2/. The total intensity of the radiation (defined as the total energy loss per sec from the particle) will differ from (74.2) in having H replaced by H L = H sin x- In the ultrarelativistic case the radiation is concentrated in directions near the generators of the "velocity cone". The spectral distribution and the total intensity (in the same sense as above) are obtained from (74.17) and (74.10) by the substitution H-* H x . If we are talking about the intensity as seen in these directions by an observer at rest, then we must introduce an additional factor (1 — (v^/c) cos x) _1 — sin -2 %, which takes into account the approach of the radiator to the observer, which occurs at a velocity v§ cos x- f The connection between the Airy function and the function K x / 3 is given by formula (4) of the footnote on p. 179. In the further transformations one uses the recursion relations K v - i(x) —K v + i(a:) 2v K v , 2K' v (x) — — Ky-^x)— K v +i(x), where K. v (pc) = K v (x). In particular, it is easy to show that •™ — vWf'"} 202 RADIATION OF ELECTROMAGNETIC WAVES PROBLEMS § 74 1 . Find the law of variation of energy with time for a charge moving in a circular orbit in a constant uniform magnetic field, and losing energy by radiation. Solution: According to (74.2), we have for the energy loss per unit time: dS 2e*H 2 -*-5Sv<'"-'" v > (<? is the energy of the particle). From this we find : « , (2e i H 2 — = = coth . „ . t+ const m<r \ 3m d c 5 As / increases, the energy decreases monotonically, approaching the value e = mc 2 (for complete stopping of the particle) asymptotically as /->oo. 2. Find the asymptotic formula for the spectral distribution of the radiation at large values of n for a particle moving in a circle with a velocity which is not close to the velocity of light. Solution: We use the well known asymptotic formula of the theory of Bessel functions J n (H£) 1 \/27m(l-e 2 ) 1/4 1+Vl- ,vr- which is valid for n{\ — e 2 ) 312 e 4 H 2 n 112 In = > 1. Using this formula, we find from (74.9): n "1 2n 2V nm c J 5/4 J^ 1 + This formula is applicable for /i[l-(» a /c 2 )] 3/2 » 1; if in addition l-(v 2 /c 2 ) is small, the formula goes over into (74.15). 3. Find the polarization of the synchrotron radiation. Solution: The electric field E n is calculated from the vector potential A n (74.6-7) according to the formula E n 1 (k x A„) x k = - \ k(k • A„) + ikk n . Let d, e 2 be unit vectors in the plane perpendicular to k, where ei is parallel to the x axis and e 2 lies in the yz plane (their components are d = (1, 0, 0), e 2 = (0, sin 9, -cos 0); the vectors e u e 2 and k form a right-hand system. Then the electric field will be : E„ = ikA^e-L+ik sin 0A yn e 2 , or, dropping the unimportant common factors : E n ~ - J'n ( — cos 9 | d +tan 9 J n I — cos 9 J /e 2 . The wave is elliptically polarized (see § 48). In the ultrarelativistic case, for large n and small angles 9, the functions /„ and J^ are expressed in terms of K 1/3 and K 2j3 , where we set 1 cos 2 9x1 + 9 2 in the arguments. As a result we get : (wc 2 V + 9 2 E„ = e iy ,K 2/3 ( f ~ ¥ A +/e a 9K 1/3 ( ^ r For 9 = the elliptical polarization degenerates into linear polarization along d. For large 0(| 0| > mc 2 /«?, «0 3 >1), we have K 1/3 (x) a K 2 , 3 {x) a V^/2xe~ x , and the polarization tends to become circular: E n i>d+ze 2 ; the intensity of the radiation, however, also becomes exponentially small. In the intermediate range of angles the minor axis of the ellipse lies along e 2 and the major axis along d. The direction of rotation depends on the sign of the angle 9 (9 > if the directions of H and k lie on opposite sides of the orbit plane, as shown in Fig. 16). § 75 RADIATION DAMPING 203 § 75. Radiation damping In § 65 we showed that the expansion of the potentials of the field of a system of charges in a series of powers of v/c leads in the second approximation to a Lagrangian completely describing (in this approximation) the motion of the charges. We now continue the expansion of the field to terms of higher order and discuss the effects to which these terms lead. In the expansion of the scalar potential the term of third order in 1/c is For the same reason as in the derivation following (65.3), in the expansion of the vector potential we need only take the term of second order in 1/c, that is, A<2, = -r4J>- (75.2) We make a transformation of the potentials: 4>' = 4>--T> A'=A + grad/, c ot choosing the function /so that the scalar potential (£ (3) becomes zero: J 6c 2 dt 2 J * Then the new vector potential is equal to A '(2) = _J| f ldv _ 1 |! V [jPqdV c 2 dt) J 6c 2 dt 2 J c 2 dt J J 3c 2 dt 2 J Making the transition from the integral to a sum over individual charges, we get for the first term on the right the expression In the second term, we write R = R -r, where R and r have their usual meaning (see § 66) ; then R = — f = — v and the second term takes the form 3c 2 ^ Thus, A' (2) =-^Iev. (75.3) The magnetic field corresponding to this potential is zero (H = curl A' (2) = 0), since 204 RADIATION OF ELECTROMAGNETIC WAVES § 75 A' (2) does not contain the coordinates eplicitly. The electric field E = -(l/c)A' (2) is E = ^5 3 . (75.4) where d is the dipole moment of the system. Thus the third order terms in the expansion of the field lead to certain additional forces acting on the charges, not contained in the Lagrangian (65.7); these forces depend on the time derivatives of the accelerations of the charges. Let us consider a system of charges carrying out a stationary motionf and calculate the average work done by the field (75.4) per unit time. The force acting on each charge e is f = eE, that is, f =3^- (75.5) The work done by this force in unit time is f • v, so that the total work performed on all the charges is equal to the sum, taken over all the charges: When we average over the time, the first term vanishes, so that the average work is equal to I^-Js* 2 - (75.6) The expression standing on the right is (except for a sign reversal) just the average energy radiated by the system in unit time [see (67.8)]. Thus, the forces (75.5) appearing in third approximation, describe the reaction of the radiation on the charges. These forces are called radiation damping or Lorentz frictional forces. Simultaneously with the energy loss from a radiating system of charges, there also occurs a certain loss of angular momentum. The decrease in angular momentum per unit time, dM/dt, is easily calculated with the aid of the expression for the damping forces. Taking the time derivative of the angular momentum M = I r x p, we have M = E r x p, since Sf x p = 2 m(v x v) = 0. We replace the time derivative of the momentum of the particle by the friction force (75.5) acting on it, and find We are interested in the time average of the loss of angular momentum for a stationary motion, just as before, we considered the time average of the energy loss. Writing dx3 = — (dxfl)-dxci at and noting that the time derivative (first term) vanishes on averaging, we finally obtain the following expression for the average loss of angular momentum of a radiating system: dM 2 -r—v t More precisely, a motion which, although it would have been stationary if radiation were neglected, proceeds with continual slowing down. § 75 RADIATION DAMPING 205 Radiation damping occurs also for a single charge moving in an external field. It is equal to 2e 2 „ 3c f = ^v. (75.8) For a single charge, we can always choose such a system of reference that the charge at the given moment is at rest in it. If, in this reference frame, we calculate the higher terms in the expansion of the field produced by the charge, it turns out that they have the following property. As the radius vector R from the charge to the field point approaches zero, all these terms become zero. Thus in the case of a single charge, formula (75.8) is an exact formula for the reaction of the radiation, in the system of reference in which the charge is at rest. Nevertheless, we must keep in mind that the description of the action of the charge "on itself" with the aid of the damping force is unsatisfactory in general, and contains contradic- tions. The equation of motion of a charge, in the absence of an external field, on which only the force (75.8) acts, has the form 2e 2 mv = — -j v. 3c 3 This equation has, in addition to the trivial solution v = const, another solution in which the acceleration v is proportional to exp(3mc 3 t/2e 2 ), that is, increases indefinitely with the time. This means, for example, that a charge passing through any field, upon emergence from the field, would have to be infinitely "self-accelerated". The absurdity of this result is evidence for the limited applicability of formula (75.8). One can raise the question of how electrodynamics, which satisfies the law of conservation of energy, can lead to the absurd result that a free charge increases its energy without limit. Actually the root of this difficulty lies in the earlier remarks (§ 37) concerning the infinite electromagnetic "intrinsic mass" of elementary particles. When in the equation of motion we write a finite mass for the charge, then in doing this we essentially assign to it formally an infinite negative "intrinsic mass" of nonelectromagnetic origin, which together with the electromagnetic mass should result in a finite mass for the particle. Since, however, the sub- traction of one infinity from another is not an entirely correct mathematical operation,this leads to a series of further difficulties, among which is the one mentioned here. In a system of coordinates in which the velocity of the particle is small, the equation of motion when we include the radiation damping has the form e „ 2c 2 3 c 3 From our discussion, this equation is applicable only to the extent that the damping force is small compared with the force exerted on the charge by the external field. To clarify the physical meaning of this condition, we proceed as follows. In the system of reference in which the charge is at rest at a given moment, the second time derivative of the velocity is equal, neglecting the damping force, to v = —EH vxH. m mc mv = eE+-vxH+ --3V. (75.9) In the second term we substitute (to the same order of accuracy) v = (e/w)E, and obtain e . e 2 v = -E+-rExH. 206 RADIATION OF ELECTROMAGNETIC WAVES § 75 Corresponding to this, the damping force consists of two terms : 2e 3 . 2e 4 f=~ — 3E+— -y^ExH. (75.10) If w is the frequency of the motion, then E is proportional to coE and, consequently, the first term is of order (e 3 co/mc 3 )E; the second is of order (e 4 /m 2 c 4 )EH. Therefore the condition for the damping force to be small compared with the force eE exerted by the external field on the charge gives, first of all, e 2 — 3 co < 1, mc or, introducing the wavelength X ~ c/co, k > — 2 . (75.11) mc z Thus formula (75.8) for the radiation damping is applicable only if the wavelength of the radiation incident on the charge is large compared with the "radius" of the charge e z /mc 2 . We see that once more a distance of order e 2 /mc 2 appears as the limit at which electro- dynamics leads to internal contradictions (see § 37). Secondly, comparing the second term in the damping force to the force eE, we find the condition m 2 c 4 H < —3-. (75.12) e 5 Thus it is also necessary that the field itself be not too large. A field of order m 2 c 4 /e 3 also represents a limit at which classical electrodynamics leads to internal contradictions. Also we must remember here that actually, because of quantum effects, electrodynamics is already not applicable for considerably smaller fields.! To avoid misunderstanding, we remind the reader that the wavelength in (75.11) and the field value in (75.12) refer to the system of reference in which the particle is at rest at the given moment. PROBLEMS 1. Calculate the time in which two attracting charges, performing an elliptic motion (with velocity small compared with the velocity of light) and losing energy due to radiation, "fall in" toward each other. Solution: Assuming that the relative energy loss in one revolution is small, we can equate the time derivative of the energy to the average intensity of the radiation (which was determined in problem 1 of § 70): d\*\ (2K|) 3/ V /2 a 3 /*>i e 2 \ 2 /„ 2|«#|M 2> dt 3c 3 M 5 where a = |e 1 e 2 |. Together with the energy, the particles lose angular momentum. The loss of angular momentum per unit time is given by formula (75.7); substituting the expression (70.1) for d, and noting that nx = — ar/r 3 and M = /irx v, we find: dM _ 2a ~d7~~3c 3 (e 1 _e 2 \ 2 M \mi m 2 ) r 3 ' f For fields of order m 2 c 3 /he, where h is Planck's constant. c Ig § 75 RADIATION DAMPING 207 We average this expression over a period of the motion. Because of the slowness of the changes in M, it is sufficient to average on the right only over r~ 3 ; this average value is computed in precisely the same way as the average of r~ 4 was found in problem 1 of § 70. As a result we find for the average loss of angular momentum per unit time the following expression : dM = 2<x(2»W 2 / e x e 2 \ 2 dt 3c 3 M 2 \mi m 2 ) [as in equation (1), we omit the average sign]. Dividing (1) by (2), we get the differential equation d\*\ _ »«? dM~ 2M which, on integration, gives : W-J^U-^ + Mm. (3) 1 ' 2M 2 V Ml) Mo The constant of integration is chosen so that for M = M , we have € = £ Q , where M and <? are the initial angular momentum and energy of the particles. The "falling in" of the particles toward one another corresponds to M->0. From (3) we see that then S— > — oo. We note that the product \&\M 2 tends toward na 2 j2, and from formula (70.3) it is clear that the eccentricity e->0, i.e. as the particles approach one another, the orbit approaches a circle. Sub- stituting (3) in (2), we determine the derivative dt/dM expressed as a function of M, after which integration with respect to M between the limits M and gives the time of fall : , M =-^te-*yw + v2-sps)-». fal1 <zV2\* \/i 3 \ m i m2 / 2. Find the Lagrangian for a system of two identical charged particles, correct to terms of fourth orderf (Ya. A. Smorodinskii and V. N. Golubenkov, 1956). Solution: The computation is conveniently done by a scheme which is somewhat different from the one used in § 65. We start from the expression for the Lagrangian of the particles and the field produced by them, a eY--— -V^-H-curIA, V c dt ) rts, we get : 8 i ,/dP-HVK=-^|w+AxH}.*- i yjE-A^-lJ(|j-A- e *)rfK For a system which does not emit dipole radiation, the integral over the infinitely distant surface gives no contribution to the terms of order 1/c 4 . The term with the total time derivative can be dropped from the Lagrangian. Thus the required fourth order terms in the Lagrangian are contained in the expression Writing E 2 -H and carrying out an integration by parts, we get : i-ilgi-A-^^-2^7 1 "?- Continuing the expansion which was done in § 65, we find the terms of fourth order in the potentials (j> and A/c) of the field produced by charge 1 at the position of charge 2 : e a 4 /? 3 1 e d 2 « 2 > = 24?^' ^0 = 2? ST.**** f See the footnote on p. 166. The third order terms in the Lagrangian drop out automatically: the terms of this order in the field produced by the particles are determined by the time derivative of the dipole moment [see (75.3)], which is conserved in the present case. 208 RADIATION OF ELECTROMAGNETIC WAVES § 76 By the transformation (18.3) with the appropriate function/, we can bring these potentials to the equivalent form rfi<2) = f Ia 1 (2) = ^ s d 2 id 3 ^ ( * Vl)+ 12^ ( ™ 3) 0) (the differentiation d/dt is done for a fixed position of the field point, i.e. of charge 2; the differentia- tion V is with respect to the coordinates of the field point). The second-order terms in the Lagrangian now give the expression! U 4> = Yc [Al(2) ' ** +A °M " Vl l+ i^i (»S+»D- (2) After performing some of the differentiations in (1), we can represent Ai(2) as c Al(2) = 8?f' '.=■ ^P*i -*<«•» Jl (where n is a unit vector in the direction from point 1 to point 2). Before making any further calculations, it is convenient to eliminate from L< 4) those terms which contain time derivatives of the velocity which are higher than first order; for this purpose we note that where c -A 1 (2).v 2 =^v 2 .|i= = ^g(v 2 .F l) -(v 2 -V)(v 2 -F 1 )-F 1 .v 2 }, j t (v 2 • F x ) = j f (v a • F 2 ) +(v a ■ V)(v 2 • F x ) is the total time derivative (differentiation with respect to both ends of the vector R!) and can be dropped from the Lagrangian. The accelerations are eliminated from the resulting expression by using the equation of motion of the first approximation: rm x = —e 2 n/R 2 , mi? 2 = e 2 n/R 2 . After a rather long computation, we finally get : £ (4) = g^ {[-«+2( Vl • v 2 ) 2 -3(n -vx) 2 (n • v 2 ) 2 +(n -vO^-Kn • v 2 ) 2 y ?] + From the symmetry in the two identical particles, it was clear beforehand that Vi == — v 2 in the system of reference in which their center of inertia is at rest. Then the fourth-order terms in the Lagrangian are: where v = v 2 — Vi. § 76. Radiation damping in the relativistic case We derive the relativistic expression for the radiation damping (for a single charge), which is applicable also to motion with velocity comparable to that of light. This force is now a four-vector g\ which must be included in the equation of motion of the charge, written in four-dimensional form : du l e .. mc— = -F k u k + g\ (76.1) as c f Here we omit the infinite terms associated with the action on the particles of their "self" fields. This operation corresponds to a "renormalization" of the masses appearing in the Lagrangian (see the footnote on p. 90).{ § 76 RADIATION DAMPING IN THE RELATIVISTIC CASE 209 To determine g { we note that for v < c, its three space components must go over into the components of the vector f/c (75.8). It is easy to see that the vector (2e 2 /3c)/(dV/tffc 2 ) has this property. However, it does not satisfy the identity g l u t = 0, which is valid for any force four-vector. In order to satisfy this condition, we must add to the expression given, a certain auxiliary four- vector, made up from the four- velocity u l and its derivatives. The three space components of this vector must become zero in the limiting case v = 0, in order not to change the correct values of f which are already given by (2e 2 /3c)j(d 2 u l /ds 2 ). The four-vector u l has this property, and therefore the required auxiliary term has the form au\ The scalar a must be chosen so that we satisfy the auxiliary relation g% = 0. As a result we find , 2e 2 /d 2 u l . k d 2 u k \ '^U^-?} (?6 - 2) In accordance with the equations of motion, this expression can be written in another form, by expressing d 2 u l /ds 2 directly in terms of the field tensor of the external field acting on the particle : du l e r ik ~r = — 2 b U k> ds mc d 2 u l e 8F ik . e 2 .. , 7T = — 2 TT u k u + —n F Fu^- ds 1 mc 2 dx l m 2 c* In making substitutions, we must keep in mind that the product of the tensor dF lk /dx l , which is antisymmetric in the indices /, k, and the symmetric tensor u t u k gives identically zero. So, 2e 3 8F ik 2e 4 2e 4 ^nyr »kU l - ^-r- 5 F il F kl u k + —j-3 (F kl u l )(F km u m ). (76.3) 3mc ox 3m c 3m c The integral of the four-force g l over the world line of the motion of a charge, passing through a given field, must coincide (except for opposite sign) with the total four-momentum AP ' of the radiation from the charge [just as the average value of the work of the force f in the nonrelativistic case coincides with the intensity of dipole radiation; see (75.6)]. It is easy to check that this is actually so. The first term in (76.2) goes to zero on performing the integration, since at infinity the particle has no acceleration, i.e. (du l /ds) = 0. We integrate the second term by parts and get : f , i 2 ? 2 f * d 2 u k J 2e 2 r (du k \(du k \ J which coincides exactly with (73.4). When the velocity of the particle approaches the velocity of light, those terms in the space components of the four- vector (76.3) increase most rapidly which come from the third derivatives of the components of the four-velocity. Therefore, keeping only these terms in (76.3) and using the relation (9.18) between the space components of the four-vector g l and the three-dimensional force f, we find for the latter: Consequently, in this case the force f is opposite to the velocity of the particle ; choosing the 210 RADIATION OF ELECTROMAGNETIC WAVES § 76 latter as the Xaxis, and writing out the four-dimensional expressions, we obtain: 2e* {E y -H z fHE z + H y f fx ~~ 3m V ~~? (76 ' 4) 2 (where we have set v = c everywhere except in the denominator). We see that for an ultrarelativistic particle, the radiation damping is proportional to the square of its energy. Let us call attention to the following interesting situation. Earlier we pointed out that the expression obtained for the radiation damping is applicable only to fields which (in the reference system K , in which the particle is at rest) are small compared with m 2 c 4 /e 3 . Let F be the order of magnitude of the external field in the reference system K, in which the particle moves with velocity v. Then in the K frame, the field has the order of magnitude F/\fl—v 2 /c 2 (see the transformation formulas in § 24). Therefore F must satisfy the condition =-«!. (76.5) v 1 - 2 c l At the same time, the ratio of the damping force (76.4) to the external force (~ eF) is of the order of e 3 F 1- and we see that, even though the condition (76.5) is satisfied, it may happen (for sufficiently high energy of the particle) that the damping force is large compared with the ordinary Lorentz force acting on the particle in the electromagnetic field. f Thus for an ultrarelativistic particle we can have the case where the radiation damping is the main force acting on the particle. In this case the loss of (kinetic) energy of the particle per unit length of path can be equated to the damping force f x alone; keeping in mind that the latter is proportional to the square of the energy of the particle, we write ax where we denote by k(x) the coefficient, depending on the x coordinate and expressed in terms of the transverse components of the field in accordance with (76.4). Integrating this differential equation, we find 1 1 f = + k(x)dx, t We should emphasize that this result does not in any way contradict the derivation given earlier of the relativistic expression for the four-force g\ in which it was assumed to be "small" compared with the four- force (e/c)F ik u k . It is sufficient to satisfy the requirement that the components of one vector be small com- pared to those of another in just one frame of reference; by virtue of relativistic invariance, the four- dimensional formulas obtained on the basis of such an assumption will be valid in any other reference frame. § 77 SPECTRAL RESOLUTION OF THE RADIATION IN THE ULTRARELATIVISTIC CASE 211 where S Q represents the initial energy of the particle (its energy for x -*■ - oo). In particular, the final energy &± of the particle (after passage of the particle through the field) is given by the formula + 00 — = h k(x) dx. We see that for $ -»• oo, the final i ^ approaches a constant limit independent of <f (I. Pomeranchuk, 1939). In other words, after passing through the field, the energy of the particle cannot exceed the energy ^ crit , defined by the equation + 00 — - = k(x)dx, ©crit J — oo or, substituting the expression for k(x), + oo 'crit PROBLEMS 1 . Calculate the limiting energy which a particle can have after passing through the field of a magnetic dipole nt ; the vector m and the direction of motion lie in a plane. Solution: We choose the plane passing through the vector nt and the direction of motion as the XZ plane, where the particle moves parallel to the Xaxis at a distance q from it. For the transverse components of the field of the magnetic dipole we have (see 44.4) : H y =0, 3(m -r) 2 — tn 2 r 2 nt . „ 2 h z = -± — ^ = ( -^ppp ^ 3 ^ cos *+ x s,n ^)e-(e 2 +^ 2 ) cos ^ (4 is the angle between nt and the Z axis). Substituting in (76.6) and performing the integration, we obtain <T crit 64m 2 c 4 £ 5 \mc 2 J 2. Write the three-dimensional expression for the damping force in the relativistic case. Solution: Calculating the space components of the four- vector (76.3), we find 3mc 3 \ c 2 1 }\dt J c \dt + 3^{ EXH+ ^ HX(HXV)+ c E(V - E »- 3m 2 c 2e 4 f/ 1 \ 2 1 (-39 § 77. Spectral resolution of the radiation in the ultrarelativistic case Earlier (in § 73) it was shown that the radiation from an ultrarelativistic particle is directed mainly in the forward direction, along the velocity of the particle : it is contained almost 212 RADIATION OF ELECTROMAGNETIC WAVES § 77 entirely within the small range of angles -J- »»-■"-? around the direction of v. In evaluating the spectral resolution of the radiation, the relation between the magnitude of the angular range A0 and the angle of deflection a of the particle in passing through the external electromagnetic field is essential. The angle a can be calculated as follows. The change in the transverse (to the direction of motion) momentum of the particle is of the order of the product of the transverse force eFf and the time of passage through the field, t ~ a/v s a/c (where a is the distance within which the field is significantly different from 0). The ratio of this quantity to the momentum mv mc P = y/l-V 2 lc 2 y/l-v 2 /c 2 determines the order of magnitude of the small angle a: eFa Dividing by A0, we find: mc a eFa Wh 2- (77.1) Ad mc v ' We call attention to the fact that it does not depend on the velocity of the particle, and is completely determined by the properties of the external field itself. We assume first that eFa > mc 2 , (77.2) that is, the total deflection of the particle is large compared with A9. Then we can say that radiation in a given direction occurs mainly from that portion of the trajectory in which the velocity of the particle is almost parallel to that direction (subtending with it an angle in the interval A0) and the length of this segment is small compared with a. The field F can be considered constant within this segment, and since a small segment of a curve can be con- sidered as an arc of a circle, we can apply the results obtained in § 74 for radiation during uniform motion in a circle (replacing HbyF). In particular, we may state that the main part of the radiation is concentrated in the frequency range eF w 7 I2T ( 77 - 3 > mc H) [see (74.16)]. In the opposite limiting case, eFa < mc 2 , (77 A) the total angle of deflection of the particle is small compared with A6. In this case the radiation is directed mainly into the narrow angular range Ad around the direction of motion, while radiation arrives at a given point from the whole trajectory. f If we choose the Xaxis along the direction of motion of the particle, then (eF) 2 is the sum of the squares of the y and z components of the Lorentz force, eE+e\/c x H, in which we can here set v s c: F 2 = (E y -H z ) 2 +(E*+H y ) 2 . §77 SPECTRAL RESOLUTION OF THE RADIATION IN THE ULTRARELATIVISTIC CASE 213 To compute the spectral distribution of the intensity, it is convenient to start in this case from the Lienard-Wiechert expressions (73.8) for the field in the wave zone. Let us compute the Fourier component 2tt J E m = — Ee i<ot dt. The expression on the right of formula (73.8) is a function of the retarded time t ', which is determined by the condition t' = t-R(t')/c. At large distances from a particle which is moving with an almost constant velocity v, we have : *'£t-^ + -n-r(t')£t-^ + -ii-Yl' c c c c (r = r(t) ^ yt is the radius vector of the particle), or , A n ' y \ R o We replace the t integration by an integration over t ', by setting dt = dt' M -— V and obtain: E co = ~2 C R 00 /f'l-vV / nX {H) >< »«')} ^"('-"^ <"'• We treat the velocity v as constant; only the acceleration w(r') is variable. Introducing the notation ca' = o)(l-—\ (77.5) and the corresponding frequency component of the acceleration, we write E^ in the form E » = ?k°G) n>< {(H) x,, 4 Finally from (66.9) we get for the energy radiated into solid angle do, with frequency in dco : 2 , dco do — . (77.6) »°> ^)_„3 2nc S) | nx {(H) xw »< An estimate of the order of magnitude of the frequencies in which the radiation is mainly concentrated in the case of (77.4) is easily made by noting that the Fourier component w^ is significantly different from zero only if the time l/co', or 1 •(-5) is of the same order as the time a/v ~ a/c during which the acceleration of the particle 214 RADIATION OF ELECTROMAGNETIC WAVES § 77 changes significantly. Therefore we find: <o~—A-^. (77.7) The energy dependence of the frequency is the same as in (77.3), but the coefficient is different. In the treatment of both cases (77.2) and (77.4) it was assumed that the total loss of energy by the particle during its passage through the field was relatively small. We shall now show that the first of these cases also covers the problem of the radiation by an ultrarelativistic particle, whose total loss of energy is comparable with its initial energy. The total loss of energy by the particle in the field can be determined from the work of the Lorentz frictional force. The work done by the force (76.4) over the path ~ a is of order e 4 F 2 a af~- H) In order for this to be comparable with the total energy of the particle, the field must exist at distances m 3 c 6 V 1 -?- e 4 F 2 But then condition (77.2) is satisfied automatically j'-i 2 2 aeF 3— l--2> mc e 3 F V <r since the field ^must necessarily satisfy condition (76.5) F J 2 < since otherwise we could not even apply ordinary electrodynamics. PROBLEMS 1. Determine the spectral distribution of the total (over all directions) radiation intensity for the condition (77.2). Solution: For each element of length of the trajectory, the radiation is determined by (74.11), where we must replace H by the value of the transverse force F at the given point and, in addition, we must go over from a discrete to a continuous frequency spectrum. This transformation is accom- plished by formally multiplying by dn and the replacement r J l dtl J T d( ° I n dn = I n -j- aco = I n — . dco co § 78 SCATTERING BY FREE CHARGES 215 Next, integrating over all time, we obtain the spectral distribution of the total radiation in the following form: + 00 2e 2 co d£< n = —dco—7=- ( 1 7, V nc + 00 oo dt, where O(w) is the Airy function of the argument rmc ■ — ,i4 [ eF \ c 2/3 The integrand depends on the integration variable t implicitly through the quantity u (F and with it u, varies along the trajectory of the particle; for a given motion this variation can be considered as a time dependence). 2. Determine the spectral distribution of the total (over all directions) radiated energy for the condition (77.4). Solution: Keeping in mind that the main role is played by the radiation at small angles to the direction of motion, we write: (.-'cos,)^-^)^-^). We replace the integration over angles do = sin 6d9d^^ Bddd^ in (77.6) by an integration over d<f> dco'\co. In writing out the square of the vector triple product in (77.6) it must be remembered that in the ultrarelativistic case the longitudinal component of the acceleration is small compared with the transverse component [in the ratio 1 — (v 2 lc 2 )], and that in the present case we can, to sufficient accuracy, consider w and v to be mutually perpendicular. As a result, we find for the spectral distribution of the total radiation the following formula : e 2 cod(o f Iwp,-! 2 I" co / v 2 \ co 2 / v< 2nc 3 J co' 2 I a>'\ c 2 ) ± 2co' 2 \ c< f(-S) dco'. § 78. Scattering by free charges If an electromagnetic wave falls on a system of charges, then under its action the charges are set in motion. This motion in turn produces radiation in all directions; there occurs, we say, a scattering of the original wave. The scattering is most conveniently characterized by the ratio of the amount of energy emitted by the scattering system in a given direction per unit time, to the energy flux density of the incident radiation. This ratio clearly has dimensions of area, and is called the effective scattering cross-section (or simply the cross-section). Let dl be the energy radiated by the system into solid angle do per second for an incident wave with Poynting vector S. Then the effective cross-section for scattering (into the solid angle do) is dl da=-= (78.1) (the dash over a symbol means a time average). The integral a of da over all directions is the total scattering cross-section. Let us consider the scattering produced by a free charge at rest. Suppose there is incident on this charge a plane monochromatic linearly polarized wave. Its electric field can be 216 RADIATION OF ELECTROMAGNETIC WAVES § 78 written in the form E = E cos (k • r — cot + a). We shall assume that the velocity acquired by the charge under the influence of the incident wave is small compared with the velocity of light (which is usually the case). Then we can consider the force acting on the charge to be eE, while the force (e/c)v x H due to the mag- netic field can be neglected. In this case we can also neglect the effect of the displacement of the charge during its vibrations under the influence of the field. If the charge carries out vibrations around the coordinate origin, then we can assume that the field which acts on the charge at all times is the same as that at the origin, that is, E = E cos (col — a). Since the equation of motion of the charge is mi = eE and its dipole moment d = ex, then .. e 2 d = - E. (78.2) m To calculate the scattered radiation, we use formula (67.7) for dipole radiation (this is justified, since the velocity acquired by the charge is assumed to be small). We also note that the frequency of the wave radiated by the charge (i.e., scattered by it) is clearly the same as the frequency of the incident wave. Substituting (78.2) in (67.7), we find e A dl = =-^ (E x n) 2 do. Anm z c i On the other hand, the Poynting vector of the incident wave is S = ^E 2 . (78.3) From this we find, for the cross-section for scattering into the solid angle do, da = ( -% ) sin 2 9 do, (78.4) where 6 is the angle between the direction of scattering (the vector n), and the direction of the electric field E of the incident wave. We see that the effective scattering cross-section of a free charge is independent of frequency. We determine the total cross-section a. To do this, we choose the polar axis along E. Then do = sin Odd d<j>; substituting this and integrating with respect to from to it, and over from to 2n, we find *-*(Af. (78-5) 3 \mc z J (This is the Thomson formula?) Finally, we calculate the differential cross-section da in the case where the incident wave is unpolarized (ordinary light). To do this we must average (78.4) over all directions of the vector E in a plane perpendicular to the direction of propagation of the incident wave (direction of the wave vector k). Denoting by e the unit vector along the direction of E, we write: (n-e) 2 = l-n a n p e a e p . § 78 SCATTERING BY FREE CHARGES 217 The averaging is done using the formula! e, and gives sin 2 9 = \ (l+^jW) = * (1+cos 2 0) 2\ k 2 / 2 where is the angle between the directions of the incident and scattered waves (the scattering angle). Thus the effective cross-section for scattering of an unpolarized wave by a free charge is 1 / e 2 \ 2 do = ~[ — 5) (1 + cos 2 0) do, (78.7) 2 \mc / The occurrence of scattering leads, in particular, to the appearance of a certain force acting on the scattering particle. One can verify this by the following considerations. On the average, in unit time, the wave incident on the particle loses energy c Wc, where W is the average energy density, and a is the total effective scattering cross-section. Since the momen- tum of the field is equal to its energy divided by the velocity of light, the incident wave loses momentum equal in magnitude to Wa. On the other hand, in a system of reference in which the charge carries out only small vibrations under the action of the force eE, and its velocity v is small, the total flux of momentum in the scattered wave is zero, to terms of higher order in v/c (in § 73 it was shown that in a reference system in which v = 0, radiation of momentum by the particle does not occur). Therefore all the momentum lost by the incident wave is "absorbed" by the scattering particle. The average force f acting on the particle is equal to the average momentum absorbed per unit time, i.e. f = aWn (78.8) (n is a unit vector in the direction of propagation of the incident wave). We note that the average force appears as a second order quantity in the field of the incident wave, while the "instantaneous" force (the main part of which is eE) is of first order in the field. Formula (78.8) can also be obtained directly by averaging the damping force (75.10). The first term, proportional to E, goes to zero on averaging, as does the average of the main part of the force, eE. The second term gives - 2e 4 — 2 Sn / e 2 \ 2 Y 2 f= 3^ Eno = yUW 4* n °' which, using (78.5), coincides with (78.8). PROBLEMS 1. Determine the effective cross-section for scattering of an elliptically polarized wave by a free charge. Solution: The field of the wave has the form E = A cos (cot+ a)+B sin (cot+ a), where A and B are mutually perpendicular vectors (see § 48). By a derivation similar to the one in the text, we find \mc 2 J (Axn) 2 +(Bxn) 2 dc = [ — \ ^^ do. t In fact, e a e e is a symmetric tensor with trace equal to 1 , which gives zero when multiplied by k a , because e and k are perpendicular. The expression given here satisfies these conditions. 218 RADIATION OF ELECTROMAGNETIC WAVES § 78 2. Determine the effective cross-section for scattering of a linearly polarized wave by a charge carrying out small vibrations under the influence of an elastic force (oscillator). Solution: The equation of motion of the charge in the incident field E = E cos (cot+<x) is r+colr = — E cos (cot+x), m where co is the frequency of its free vibrations. For the forced vibrations, we then have eE cos (cot+ix) T = . m{al — co 2 ) Calculating d from this, we find / e 2 \ 2 co* da = — - — =-2 sin2 edo \mc 2 J {col — co 2 ) 2 (9 is the angle between E and n). 3. Determine the total effective cross-section for scattering of light by an electric dipole which, mechanically, is a rotator. The frequency co of the wave is assumed to be large compared with the frequency Q of free rotation of the rotator. Solution: Because of the condition co > Q , we can neglect the free rotation of the rotator, and consider only the forced rotation under the action of the moment of the forces d x E exerted on it by the scattered wave. The equation for this motion is: Jq = dxE, where J is the moment of inertia of the rotator and Q is the angular velocity of rotation. The change in the dipole moment vector, as it rotates without changing its absolute value, is given by the formula d = H x d. From these two equations, we find (omitting the quadratic term in the small quantity Q) : d = -Ud x E) x d = j [EJ 2 -(E • d)d]. Assuming that all orientations of the dipole in space are equally probable, and averaging d 2 over them, we find for the total effective cross-section, _ 16ttJ 4 a ~ 9M 2 ' 4. Determine the degree of depolarization in the scattering of ordinary light by a free charge. Solution: From symmetry considerations, it is clear that the two incoherent polarized com- ponents of the scattered light (see § 50) will be linearly polarized: one in the plane of scattering (the plane passing through the incident and scattered waves) and the other perpendicular to this plane. The intensities of these components are determined by the components of the field of the incident wave in the plane of scattering (E N ) and perpendicular to it (E ± ), and, according to (78.3), are proportional respectively to (E, | xn) 2 -£:^cos 2 and (E x xn) 2 =E 2 ± (where is the angle of scattering). Since for the ordinary incident light, E^ =E\, the degree of depolarization [see the definition in (50.9)] is : q = cos 2 0. 5. Determine the frequency co' of the light scattered by a moving charge. Solution: In a system of coordinates in which the charge is at rest, the frequency of the light does not change on scattering (co = co'). This relation can be written in invariant form as k'iU' 1 =k i u i , where u l is the four-velocity of the charge. From this we find without difficulty /M--cos 0' ) = co (l-- cos d\ where 9 and 9' are the angles made by the incident and scattered waves with the direction of motion (v is the velocity of the charge). 6. Determine the angular distribution of the scattering of a linearly polarized wave by a charge moving with velocity v in the direction of propagation of the wave. § 78 SCATTERING BY FREE CHARGES 219 Solution: The velocity of the particle is perpendicular to the fields E and H of the incident wave, and is therefore also perpendicular to the acceleration w given to the particle. The scattered intensity is given by (73.14), where the acceleration w of the particle must be expressed in terms of the fields E and H of the incident wave by the formulas obtained in the problem in § 17. Dividing the intensity dl by the Poynting vector of the incident wave, we get the following expression for the scattering cross-section : da \mc 2 / v ' 1 — sin 9 cos <f> c 1 — - sin 9 cos (f> 1 — ( 1 — \ ] cos 2 do, where 9 and <f> are the polar angle and azimuth of the direction n relative to a system of coordinates with Z axis along E, and X axis along v (cos (n, E) = cos 9; cos (n, v) = sin 9 cos <f>). 7. Calculate the motion of a charge under the action of the average force exerted upon it by the wave scattered by it. Solution: The force (78.8), and therefore the velocity of the motion under consideration, is along the direction of propagation of the incident wave (X axis). In the auxiliary reference system K , in which the particle is at rest ( we recall that we are dealing with the motion averaged over the period of the small vibrations), the force acting on it is aW , and the acceleration acquired by it under the action of this force is a — Wo = — W m (the index zero refers to the reference system K ). The transformation to the original reference system K (in which the charge moves with velocity v) is given by the formulas obtained in the problem of § 7 and by formula (47.7), and gives : - 1- V - d v 1 dv Wo c- dt I „2 A v 2 \ 312 dt m v J^ H) Integrating this expression, we find c 1 2 3' which determines the velocity v = dx/dt as an implicit function of the time (the integration con- stant has been chosen so that v — at t = 0). 8. Determine the average force exerted on a charge moving in an electromagnetic field consisting of a superposition of waves in all directions with an isotropic distribution of directions of propagation. Solution: We write the equation of motion of the charge in four-dimensional form, du l , mc — r = g\ ds To determine the four- vector g l , we note that in a system of reference in which the charge is at rest at a given moment, in the presence of a single wave propagating along a definite direction (say, along the X axis), the equation of motion is (v x = v) dv r*/ m — = aW dt (we omit the average sign throughout). This means that the X-component of the vector g l must become (W/c)a. The four-vector (p\c)T iH u k has this property, where T ik is the energy-momentum tensor of the wave, and «' is the four- velocity of the charge. In addition, g l must satisfy the condition g l Ui = 0. This can be achieved by adding to the previous expression a four-vector of the form a.u\ 220 RADIATION OF ELECTROMAGNETIC WAVES § 79 where a is a scalar. Determining a suitably, we obtain mc d ^ = - (T ik u k -u l u k «, T kl ). (1) as c In an electromagnetic field of isotropic radiation, the Poynting vector vanishes because of symmetry, and the stress tensor a aB must have the form const • 5 a p. Noting also that we must have T\ = 0, so that a a p = T° =W, we find W ' x °afi ^ O a p. Substituting these expressions in (1), we find for the force acting on the charge: d I mv \ 4Wav Jt 3c (-5)' This force acts in the direction opposite to the motion of the charge, i.e. the charge experiences a retardation. We note that for v<^c, the retarding force is proportional to the velocity of the charge : dv AWcv m — — — . dt 3c 9. Determine the effective cross-section for scattering of a linearly polarized wave by an oscillator, taking into account the radiation damping. Solution: We write the equation of motion of the charge in the incident field in the form e 2e 2 f + co 2 r = - Eoe- tot + =— s r. In the damping force, we can substitute approximately r = — coir; then we find -imt m where y — (2e 2 /3mc 3 )co%. From this we obtain r +yr + <»ro = — E e~ m r = -E, e „ e -lat m co Q — co" — i(jyy The effective cross-section is 3 \mc z ) § 79. Scattering of low-frequency waves The scattering of a wave by a system of charges differs from the scattering by a single charge (at rest), first of all in the fact that because of the presence of internal motion of the charges of the system, the frequency of the scattered radiation can be different from the frequency of the incident wave. Namely, in the spectral resolution of the scattered wave there appear, in addition to the frequency co of the incident wave, frequencies co' differing from co by one of the internal frequencies of motion of the scattering system. The scattering with changed frequency is called incoherent (or combinational), in contrast to the coherent scattering without change in frequency. Assuming that the field of the incident wave is weak, we can represent the current density in the form j = j + j', where j is the current density in the absence of the external field, and j' is the change in the current under the action of the incident wave. Correspondingly, the vector potential (and other quantities) of the field of the system also has the form A = A + A', where A and A' are determined by the currents j and j'. Clearly, A' describes the wave scattered by the system. Let us consider the scattering of a wave whose frequency co is small compared with all the § 80 SCATTERING OF HIGH-FREQUENCY WAVES 221 internal frequencies of the system. The scattering will consist of an incoherent as well as a coherent part, but we shall here consider only the coherent scattering. In calculating the field of the scattered wave, for sufficiently low frequency co, we can use the expansion of the retarded potentials which was presented in §§67 and 71, even if the velocities of the particles of the system are not small compared with the velocity of light. Namely, for the validity of the expansion of the integral cR J c c it is necessary only that the time r -n/c ~ a/c be small compared with the time l/a>; for sufficiently low frequencies (co <£ c/a), this condition is fulfilled independently of the velocities of the particles of the system. The first terms in the expansion give H' = -y-— {H' x n + (m' x n) x n}, c R where d', m' are the parts of the dipole and magnetic moments of the system which are produced by the radiation falling on the system. The succeeding terms contain higher time derivatives than the second, and we drop them. The component H^, of the spectral resolution of the field of the scattered wave, with frequency equal to that of the incident wave, is given by this same formula, when we sub- stitute for all quantities their Fourier components: d^ = — co 2 A' m , m' w = — co 2 m' m . Then we obtain co 2 u '<o = -T^r (nxd;+nx(m;xn)}. (79.1) C K The later terms in the expansion of the field would give quantities proportional to higher powers of the small frequency. If the velocities of all the particles of the system are small (v <| c), then in (79.1) we can neglect the second term in comparison to the first, since the magnetic moment contains the ratio v/c. Then H;, = -2— fi> 2 nxd;. (79.2) C K If the total charge of the system is zero, then for co -*■ 0, d^,, and m^ approach constant limits (if the sum of the charges were different from zero, then for co = 0, i.e. for a constant field, the system would begin to move as a whole). Therefore for low frequencies (co <v/a) we can consider d^, and m^ as independent of frequency, so that the field of the scattered wave is proportional to the square of the frequency. Its intensity is consequently propor- tional to co 4 . Thus for the scattering of a low-frequency wave, the effective cross-section for (coherent) scattering is proportional to the fourth power of the frequency of the incident radiation.f § 80. Scattering of high-frequency waves We consider the scattering of a wave by a system of charges in the opposite limit, when the frequency co of the wave is large compared with the fundamental internal frequencies of the f This also applies to the scattering of light by ions as well as by neutral atoms. Because of the large mass of the nucleus, the scattering resulting from the motion of the ion as a whole can be neglected. 222 RADIATION OF ELECTROMAGNETIC WAVES § 80 system. The latter have the order of magnitude co ~ v/a, so that co must satisfy the condition co>co ~ V . (80.1) a In addition, we assume that the velocities of the charges of the system are small (v < c). According to condition (80.1), the periods of the motion of the charges of the system are large compared with the period of the wave. Therefore during a time interval of the order of the period of the wave, the motion of the charges of the system can be considered uniform. This means that in considering the scattering of short waves, we need not take into account the interaction of the charges of the system with each other, that is, we can consider them as free. Thus in calculating the velocity v\ acquired by a charge in the field of the incident wave, we can consider each of the charges in the system separately, and write for it an equation of motion of the form m a *- = eV = eE e-« cot - k - r \ at where k = (co/c)n is the wave of the incident wave. The radius vector of the charge is, of course, a function of the time. In the exponent on the right side of this equation the time rate of change of the first term is large compared with that of the second (the first is co, while the second is of order kv ~ v(co/c) < co). Therefore in integrating the equation of motion, we can consider the term r on the right side as constant. Then v'=--^-E <r , ' (w, - k - r) . (80.2) icom For the vector potential of the scattered wave (at large distances from the system), we have from the general formula (66.2): A ' " cl j j ''- 7 ^ dV = i S <eA " T * tJ f ' where the sum goes over all the charges of the system; n' is a unit vector in the direction of scattering. Substituting (80.2), we find A' = - -4- e~ i(0 ('" T°)e S - «-*•', (80.3) icR co ^ m where q = k'-k is the difference between the wave vector k = (co/c)n of the incident wave, and the wave vector k' = (co/c)n r of the scattered wave.j The value of the sum in (80.3) must be taken at the time t' = t-(R /c) (for brevity as usual, we omit the index t ' on r); the change of r in the time r • ri/c can be neglected in view of our assumption that the velocities of the particles are small. The absolute value of the vector q is « = 2-sin|, (80.4) where is the scattering angle. For scattering by an atom (or molecule), we can neglect the terms in the sum in (80.3) which come from the nuclei, because their masses are large compared with the electron mass. t Strictly speaking the wave vector k' = co'n'/c, where the frequency to" of the scattered wave may differ from co. However, in the present case of high frequencies the difference a/ -to ~&>o can be neglected. § 80 SCATTERING OF HIGH-FREQUENCY WAVES 223 Later we shall be looking at just this case, so that we remove the factor e 2 \m from the sum- mation sign, and understand by e and m the charge and mass of the electron. For the field H' of the scattered wave we find from (66.3): H'-^e-K-?)?^ e --. (80.5) c R m The energy flux into an element of solid angle in the direction n' is clH '12 Rldo = —^- 2 {n'xK ) 7 2 do. Sn u 87uc 3 m 2 Dividing this by the energy flux c|E J 2 /8n: of the incident wave, and introducing the angle 9 between the direction of the field E of the incident wave and the direction of scattering, we finally obtain the effective scattering cross-section in the form do = ( — , \mc £>-<V. sin 2 9 do. (80.6) The dash means a time average, i.e. an average over the motion of the charges of the system; it appears because the scattering is observed over a time interval large compared with the periods of motion of the charges of the system. For the wavelength of the incident radiation, there follows from the condition (80.1) the inequality X <^ ac/v. As for the relative values of X and a, both the limiting cases X > a and X <4 a are possible. In both these cases the general formula (80.6) simplifies considerably. In the case of X > a, in the expression (80.6) q • r <^ 1, since q ~ l/X, and r is of order of a. Replacing e~ ,qr by unity in accordance with this, we have: /Ze 2 \ 2 da = l — 2 ) sin2 e do (80.7) that is, the scattering is proportional to the square of the atomic number Z. We now go over to the case of X <4 a. In the square of the sum which appears in (80.6), in addition to the square modulus of each term, there appear products of the forme - ' q " (ri ~ r2) . In averaging over the motion of the charges, i.e. over their mutual separations, r x — r 2 takes on values in an interval of order a. Since q ~ l/X, X <£ a, the exponential factor e -iq- (ri-r 2 ) j s a ra pj^iy oscillating function in this interval, and its average value vanishes. Thus for X <^ a, the effective scattering cross-section is da = z(- 6 -A sin 2 9 do, (80.8) that is, the scattering is proportional to the first power of the atomic number. We note that this formula is not applicable for small angles of scattering (9 ~ X/a), since in this case q ~ 9/X~ l/a and the exponent q -r is not large compared to unity. To determine the effective coherent scattering cross-section, we must separate out that part of the field of the scattered wave which has the frequency co. The expression (80.5) depends on the time through the factor e~ i(0t , and also involves the time in the sum Se" ,,r . This latter dependence leads to the result that in the field of the scattered wave there are contained, along with the frequency co, other (though close to co) frequencies. That part of the field which has the frequency co (i.e. depends on the time only through the factor e~ l0}t ), is obtained if we average the sum 2 e~ iqr over time. In accordance with this, the expression for the effective coherent scattering cross-section da cob , differs from the total 224 RADIATION OF ELECTROMAGNETIC WAVES § 80 cross-section do in that it contains, in place of the average value of the square modulus of the sum, the square modulus of the average value of the sum, da rn u = I — 2 jnc X>~' , " r sinO do. (80.9) It is useful to note that this average value of the sum is (except for a factor) just the space Fourier component of the average distribution o(r) of the electric charge density in the atom: e ly^r = I Q (r) e -^'dV = Q q . (80.10) In case I > a, we can again replace e' q " r by unity, so that 2 \ 2 da coh = [Z zr72 ) sm 2 do. (80. 1 1) 7 I =>"i 2 mc Comparing this with the total effective cross-section (80.7), we see that do coh = da, that is, all the scattering is coherent. If X <^ a, then when we average in (80.9) all the terms of the sum (being rapidly oscillating functions of the time) vanish, so that da coh = 0. Thus in this case the scattering is completely incoherent. CHAPTER 10 PARTICLE IN A GRAVITATIONAL FIELD § 81. Gravitational fields in nonrelativistic mechanics Gravitational fields, or fields of gravity, have the basic property that all bodies move in them in the same manner, independently of mass, provided the initial conditions are the same. For example, the laws of free fall in the gravity field of the earth are the same for all bodies ; whatever their mass, all acquire one and the same acceleration. This property of gravitational fields provides the possibility of establishing an analogy between the motion of a body in a gravitational field and the motion of a body not located in any external field, but which is considered from the point of view of a noninertial system of reference. Namely, in an inertial reference system, the free motion of all bodies is uniform and rectilinear, and if, say, at the initial time their velocities are the same, they will be the same for all times. Clearly, therefore, if we consider this motion in a given noninertial system, then relative to this system all the bodies will move in the same way. Thus the properties of the motion in a noninertial system are the same as those in an inertial system in the presence of a gravitational field. In other words, a noninertial reference system is equivalent to a certain gravitational field. This is called the principle of equivalence. Let us consider, for example, motion in a uniformly accelerated reference system. A body of arbitrary mass, freely moving in such a system of reference, clearly has relative to this system a constant acceleration, equal and opposite to the acceleration of the system itself. The same applies to motion in a uniform constant gravitational field, e.g. the field of gravity of the earth (over small regions, where the field can be considered uniform). Thus a uni- formly accelerated system of reference is equivalent to a constant, uniform external field. A somewhat more general case is a nonuniformly accelerated linear motion of the reference system — it is clearly equivalent to a uniform but variable gravitational field. However, the fields to which noninertial reference systems are equivalent are not com- pletely identical with "actual" gravitational fields which occur also in inertial frames. For there is a very essential difference with respect to their behavior at infinity. At infinite distances from the bodies producing the field, "actual" gravitational fields always go to zero. Contrary to this, the fields to which noninertial frames are equivalent increase without limit at infinity, or, in any event, remain finite in value. Thus, for example, the centrifugal force which appears in a rotating reference system increases without limit as we move away from the axis of rotation ; the field to which a reference system in accelerated linear motion is equivalent is the same over all space and also at infinity. 225 226 PARTICLE IN A GRAVITATIONAL FIELD § 82 The fields to which noninertial systems are equivalent vanish as soon as we transform to an inertial system. In contrast to this, "actual" gravitational fields (existing also in an inertial reference frame) cannot be eliminated by any choice- of reference system. This is already clear from what has been said above concerning the difference in conditions at infinity between "actual" gravitational fields and fields to which noninertial systems are equivalent; since the latter do not approach zero at infinity, it is clear that it is impossible, by any choice of reference frame, to eliminate an "actual" field, since it vanishes at infinity. All that can be done by a suitable choice of reference system is to eliminate the gravita- tional field in a given region of space, sufficiently small so that the field can be considered uniform over it. This can be done by choosing a system in accelerated motion, the accelera- tion of which is equal to that which would be acquired by a particle placed in the region of the field which we are considering. The motion of a particle in a gravitational field is determined, in nonrelativistic mechanics, by a Lagrangian having (in an inertial reference frame) the form 2 L = n ^- m< ^ i (81.1) where is a certain function of the coordinates and time which characterizes the field and is called the gravitational potential.^ Correspondingly, the equation of motion of the particle is v=-grad</>. (81.2) It does not contain the mass or any other constant characterizing the properties of the particle; this is the mathematical expression of the basic property of gravitational fields. § 82. The gravitational field in relativistic mechanics The fundamental property of gravitational fields that all bodies move in them in the same way, remains valid also in relativistic mechanics. Consequently there remains also the analogy between gravitational fields and noninertial reference systems. Therefore in studying the properties of gravitational fields in relativistic mechanics, we naturally also start from this analogy. In an inertial reference system, in cartesian coordinates, the interval ds is given by the relation: ds 2 = c 2 dt 2 -dx 2 -dy 2 -dz 2 . Upon transforming to any other inertial reference system (i.e. under Lorentz transforma- tion), the interval, as we know, retains the same form. However, if we transform to a non- inertial system of reference, ds 2 will no longer be a sum of squares of the four coordinate differentials. So, for example, when we transform to a uniformly rotating system of coordinates, x = x' cosQt-y' sinfif, y = x' sinQt+y' cosQt, z = z' (Q is the angular velocity of the rotation, directed along the Z axis), the interval takes on the t In what follows we shall seldom have to use the electromagnetic potential <l>, so that the designation of the gravitational potential by the same symbol cannot lead to misunderstanding. § 82 THE GRAVITATIONAL FIELD IN RELATIVISTIC MECHANICS 227 form ds 2 = [c 2 -Q 2 (x' 2 + y' 2 J]dt 2 -dx' 2 -dy' 2 -dz' 2 + 2Qy' dx' dt-2Qx' dy' dt. No matter what the law of transformation of the time coordinate, this expression cannot be represented as a sum of squares of the coordinate differentials. Thus in a noninertial system of reference the square of an interval appears as a quadratic form of general type in the coordinate differentials, that is, it has the form ds 2 = g ik dx i dx\ (82.1) where the g ik are certain functions of the space coordinates x 1 , x 2 , x 3 and the time co- ordinate x°. Thus, when we use a noninertial system, the four-dimensional coordinate system x°, x 1 , x 2 , x 3 is curvilinear. The quantities g ik , determining all the geometric properties in each curvilinear system of coordinates, represent, we say, the space-time metric. The quantities g ik can clearly always be considered symmetric in the indices i and k {g ki = g ik ), since they are determined from the symmetric form (82.1), where g ik and g ki enter as factors of one and the same product dx l dx k . In the general case, there are ten different quantities g ik — four with equal, and 4-3/2 = 6 with different indices. In an inertial reference system, when we use cartesian space coordinates x 1 ' 2 ' 3 = x,y, z, and the time, x = ct, the quantities g ik are 9oo = h 011=922 = 933=-^ 0» = O for i#fc. (82.2) We call a four-dimensional system of coordinates with these values of g ik galilean. In the previous section it was shown that a noninertial system of reference is equivalent to a certain field of force. We now see that in relativistic mechanics, these fields are deter- mined by the quantities g ik . The same applies also to "actual" gravitational fields. Any gravitational field is just a change in the metric of space-time, as determined by the quantities g ik . This important fact means that the geometrical properties of space-time (its metric) are determined by physical phenomena, and are not fixed properties of space and time. The theory of gravitational fields, constructed on the basis of the theory of relativity, is called the general theory of relativity. It was established by Einstein (and finally formulated by him in 1916), and represents probably the most beautiful of all existing physical theories. It is remarkable that it was developed by Einstein in a purely deductive manner and only later was substantiated by astronomical observations. As in nonrelativistic mechanics, there is a fundamental difference between "actual" gravitational fields and fields to which noninertial reference systems are equivalent. Upon transforming to a noninertial reference system, the quadratic form (82.1), i.e. the quantities g ik , are obtained from their galilean values (82.2) by a simple transformation of coordinates, and can be reduced over all space to their galilean values by the inverse coordinate trans- formation. That such forms for g ik are very special is clear from the fact that it is impossible by a mere transformation of the four coordinates to bring the ten quantities g ik to a pre- assigned form. An "actual" gravitational field cannot be eliminated by any transformation of co- ordinates. In other words, in the presence of a gravitational field space-time is such that the quantities g ik determining its metric cannot, by any coordinate transformation, be brought to their galilean values over all space. Such a space-time is said to be curved, in contrast to flat space-time, where such a reduction is possible. 228 PARTICLE IN A GRAVITATIONAL FIELD § 82 By an appropriate choice of coordinates, we can, however, bring the quantities g ik to galilean form at any individual point of the non-galilean space-time : this amounts to the reduction to diagonal form of a quadratic form with constant coefficients (the values of g ik at the given point). Such a coordinate system is said to be galilean for the given pointy We note that, after reduction to diagonal form at a given point, the matrix of the quanti- ties g ik has one positive and three negative principal values. { From this it follows, in par- ticular, that the determinant g, formed from the quantities g ik , is always negative for a real space-time : g < 0. (82.3) A change in the metric of space-time also means a change in the purely spatial metric. To a galilean g ik in flat space-time, there corresponds a euclidean geometry of space. In a gravitational field, the geometry of space becomes non-euclidean. This applies both to "true" gravitational fields, in which space-time is "curved", as well as to fields resulting from the fact that the reference system is non-inertial, which leave the space-time flat. The problem of spatial geometry in a gravitational field will be considered in more detail in § 84. It is useful to give here a simple argument which shows pictorially that space will become non-euclidean when we change to a non-inertial system of reference. Let us con- sider two reference frames, of which one (K) is inertial, while the other (K') rotates uniformly with respect to K around their common z axis. A circle in the x, y plane of the K system (with its center at the origin) can also be regarded as a circle in the x', y' plane of the K' system. Measuring the length of the circle and its diameter with a yardstick in the K system, we obtain values whose ratio is equal to n, in accordance with the euclidean character of the geometry in the inertial reference system. Now let the measurement be carried out with a yardstick at rest relative to K'. Observing this process from the K system, we find that the yardstick laid along the circumference suffers a Lorentz contraction, whereas the yardstick placed radially is not changed. It is therefore clear that the ratio of the circumference to the diameter, obtained from such a measurement, will be greater than n. In the general case of an arbitrary, varying gravitational field, the metric of space is not only non-euclidean, but also varies with the time. This means that the relations between different geometrical distances change with time. As a result, the relative position of "test bodies" introduced into the field cannot remain unchanged in any coordinate system. § Thus if the particles are placed around the circumference of a circle and along a diameter, since the ratio of the circumference to the diameter is not equal to n and changes with time, it is clear that if the separations of the particles along the diameter remain unchanged the separa- tions around the circumference must change, and conversely. Thus in the general theory of relativity it is impossible in general to have a system of bodies which are fixed relative to one another. This result essentially changes the very concept of a system of reference in the general theory of relativity, as compared to its meaning in the special theory. In the latter we meant f To avoid misunderstanding, we state immediately that the choice of such a coordinate system does not mean that the gravitational field has been eliminated over the corresponding infinitesimal volume of four- space. Such an elimination is also always possible, by virtue of the principle of equivalence, and has a greater significance (see § 87). % This set of signs is called the signature of the matrix. § Strictly speaking, the number of particles should be greater than four. Since we can construct a tetra- hedron from any six line segments, we can always, by a suitable definition of the reference system, make a system of four particles form an invariant tetrahedron. A fortiori, we can fix the particles relative to one another in systems of three or two particles. § 83 CURVILINEAR COORDINATES 229 by a reference system a set of bodies at rest relative to one another in unchanging relative positions. Such systems of bodies do not exist in the presence of a variable gravitational field, and for the exact determination of the position of a particle in space we must, strictly speak- ing, have an infinite number of bodies which fill all the space like some sort of "medium". Such a system of bodies with arbitrarily running clocks fixed on them constitutes a reference system in the general theory of relativity. In connection with the arbitrariness of the choice of a reference system, the laws of nature must be written in the general theory of relativity in a form which is appropriate to any four- dimensional system of coordinates (or, as one says, in "covariant" form). This, of course, does not imply the physical equivalence of all these reference systems (like the physical equivalence of all inertial reference systems in the special theory). On the contrary, the specific appearances of physical phenomena, including the properties of the motion of bodies, become different in all systems of reference. § 83. Curvilinear coordinates Since, in studying gravitational fields we are confronted with the necessity of considering phenomena in an arbitrary reference frame, it is necessary to develop four-dimensional geometry in arbitrary curvilinear coordinates. Sections 83, 85 and 86 are devoted to this. Let us consider the transformation from one coordinate system, x°, x l , x 2 , x 3 , to another X = J \X 9 X ? X , X )) where the /' are certain functions. When we transform the coordinates, their differentials transform according to the relation d^ = Tiidx' k . (83.1) dx k Every aggregate of four quantities A { (i = 0, 1,2, 3), which under a transformation of coordinates transform like the coordinate differentials, is called a contravariant four- vector: A* = Pr k A*. (83.2) dx k Let be some scalar. Under a coordinate transformation, the four quantities d(j)/dx l transform according to the formula dx { dx' k dx 1 ' which is different from formula (83.2). Every aggregate of four quantities A t which, under a coordinate transformation, transform like the derivatives of a scalar, is called a covariant four- vector : A,J-£Ai (83.4) Because two types of vectors appear in curvilinear coordinates, there are three types of tensors of the second rank. We call a contravariant tensor of the second rank, A ik , an aggregate of sixteen quantities which transform like the products of the components of two contra- 230 PARTICLE IN A GRAVITATIONAL FIELD § 83 variant vectors, i.e. according to the law Aik dx l dx k Al , A ~B?*r A ■ (83 ' 5) A covariant tensor of rank two, transforms according to the formula _dx' l dx' m , Aik ~~dx~ i ~d^ Alm ' (83 - 6) and a mixed tensor transforms as follows : . dx l dx ,m ,, The definitions given here are the natural generalization of the definitions of four- vectors and four-tensors in galilean coordinates (§ 6), according to which the differentials dx l constitute a contravariant four-vector and the derivatives dcfr/dx 1 form a covariant four- vector.| The rules for forming four-tensors by multiplication or contraction of products of other four-tensors remain the same in curvilinear coordinates as they were in galilean coordinates. For example, it is easy to see that, by virtue of the transformation laws (83.2) and (83.4), the scalar product of two four-vectors A i B i is invariant: dx l dx' m , dx' m , A ' B ' = I^7 AB - = ^ A " B '^ A " B '- The unit four-tensor Si is defined the same as before in curvilinear coordinates: its components are again 5 k = for i # k, and are equal to 1 for / = k. If A k is a four-vector, then multiplying by 5 l k we get : A% = A\ i.e. another four- vector; this proves that S k is a tensor. The square of the line element in curvilinear coordinates is a quadratic form in the differentials dx i : ds 2 = g ik dx l dx k , (83.8) where the g ik are functions of the coordinates; g ik is symmetric in the indices / and k: 9 ik = 9ki- (83.9) Since the (contracted) product of g ik and the contravariant tensor dx f dx k is a scalar, the g ik form a covariant tensor; it is called the metric tensor. Two tensors A ik and B ik are said to be reciprocal to each other if A ik B kl = S l , In particular the contravariant metric tensor is the tensor g ik reciprocal to the tensor g ik , that is, 9ik9 kl = Z\. (83.10) The same physical quantity can be represented in contra- or co-variant components. It is obvious that the only quantities that can determine the connection between the different t Nevertheless, while in a galilean system the coordinates x l themselves (and not just their differentials) also form a four-vector, this is, of course, not the case in curvilinear coordinates. § 83 CURVILINEAR COORDINATES 231 forms are the components of the metric tensor. This connection is given by the formulas: A l = g ik A k , A t = gtk A k . (83.11) In a galilean coordinate system the metric tensor has components : 10 (0) (O)* l° - 1 ° °| (83.12) dm y « o 0-1 ' \0 -l y Then formula (83.11) gives the familiar relation A = A , A 1 ' 2 > 3 = -A u 2> 3 , etc.f These remarks also apply to tensors. The transition between the different forms of a given physical tensor is accomplished by using the metric tensor according to the formulas: A\ = g il A lk , A ik = g il g km A lm , etc. In § 6 we defined (in galilean coordinates) the completely antisymmetric unit pseudo- tensor e iklm . Let us transform it to an arbitrary system of coordinates, and now denote it by E iklm . We keep the notation e iklm for the quantities defined as before by e 0123 = 1 (or ^oi23 = -!)• Let the x' 1 be galilean, and the x v be arbitrary curvilinear coordinates. According to the general rules for transformation of tensors, we have : 6x' p dx' r dx' s dx n or piklm _ j Aklm where / is the determinant formed from the derivatives dx^dx'", i.e. it is just the Jacobian of the transformation from the galilean to the curvilinear coordinates: d(x°, x\ x 2 , x 3 ) J ~d(x'°,x'\x' 2 ,x' 3 ) This Jacobian can be expressed in terms of the determinant of the metric tensor g ik (in the system x l ). To do this we write the formula for the transformation of the metric tensor: ik _ _^L d* ( )*m y dx' 1 dx' m y ' and equate the determinants of the two sides of this equation. The determinant of the re- ciprocal tensor \g ik \ = \\g. The determinant |# (0)/m | = -1. Thus we have \\g = -/ 2 , and so/= l/yj-g. Thus, in curvilinear coordinates the antisymmetric unit tensor of rank four must be defined as E iklm = _L e ttlm m ( 83 13 ) v-fir t Whenever, in giving analogies, we use galilean coordinate systems, one should realize that such a system can be selected only in a flat space. In the case of a curved four-space, one should speak of a co- ordinate system that is galilean over a given infinitesimal element of four-volume, which can always be found. None of the derivations are affected by this change. 232 PARTICLE IN A GRAVITATIONAL FIELD § 83 The indices of this tensor are lowered by using the formula e prst gi P gi ir gi s g mt = -ge iklm , so that its covariant components are Eikim = \/-ge iklm . (83.14) In a galilean coordinate system x' 1 the integral of a scalar with respect to dQ' = dx'° dx' 1 dx' 2 dx' 3 is also a scalar, i.e. the element dQ' behaves like a scalar in the integration (§ 6). On transforming to curvilinear coordinates *', the element of integra- tion dQ.' goes over into dQ' -> - dQ = yj~^g dQ. Thus, in curvilinear coordinates, when integrating over a four- volume the quantity V -gdQ behaves like an invariant, f All the remarks at the end of § 6 concerning elements of integration over hypersurf aces, surfaces and lines remain valid for curvilinear coordinates, with the one difference that the definition of dual tensors changes. The element of "area" of the hypersurface spanned by three infinitesimal displacements is the contravariant antisymmetric tensor dS ikl ; the vector dual to it is gotten by multiplying by the tensor V ' -g e iklm , so it is equal to 4^~gdS i = -& iklm dS klm J-g. (83.15) Similarly, if df lk is the element of (two-dimensional) surface spanned by two infinitesimal displacements, the dual tensor is defined asf ^J-gdft k = y'-ge mm df lm . (83.16) We keep the designations dS t and df£ as before for ie iklm dS klm and \e iklm df lm (and not their products by V -g)\ the rules (6.14-19) for transforming the various integrals into one another remain the same, since their derivation was formal in character and not related to the tensor properties of the different quantities. Of particular importance is the rule for transforming the integral over a hypersurface into an integral over a four-volume (Gauss' theorem), which is accomplished by the substitution : dSi-tdQ—.. (83.17) f If ^ is a scalar, the quantity V —g <f>, which gives an invariant when integrated over dQ, is called a scalar density. Similarly, we speak of vector and tensor densities V — g A\ V — g A ik , etc. These quantities give a vector or tensor on multiplication by the four-volume element dQ. (the integral J A 1 V —g dQ over a finite region cannot, generally speaking, be a vector, since the laws of transformation of the vector A 1 are different at different points). t It is understood that the elements dS klm and df ik are constructed on the infinitesimal displacements dx\ dx' 1 , dx" 1 in the same way as in § 6, no matter what the geometrical significance of the coordinates x l . Then the formal significance of the elements dS t and dff k is the same as before. In particular, as before dS = dxx dx 2 dx 3 = dV. We keep the earlier definition of dV for the product of differentials of the three space coordinates; we must, however, remember that the element of geometrical spatial volume is given in curvilinear coordinates not by dV, but by Vy dV, where y is the determinant of the spatial metric tensor (which will be defined in the next section). § 84 DISTANCES AND TIME INTERVALS 233 § 84. Distances and time intervals We have already said that in the general theory of relativity the choice of a coordinate system is not limited in any way; the triplet of space coordinates x 1 , x 2 , x 3 , can be any quantities defining the position of bodies in space, and the time coordinate x° can be defined by an arbitrarily running clock. The question arises of how, in terms of the values of the quantities jc 1 , x 2 , x 3 , x°, we can determine actual distances and time intervals. First we find the relation of the proper time, which from now on we shall denote by x, to the coordinate x°. To do this we consider two infmitesimally separated events, occurring at one and the same point in space. Then the interval ds between the two events is, as we know, just cdx, where dx is the (proper) time interval between the two events. Setting dx 1 = dx 2 = dx 3 = in the general expression ds 2 = g ik dx 1 dx*, we consequently find ds 2 = c 2 dx 2 = g 00 (dx ) 2 , from which dx = -^ dx°, (84.1) c or else, for the time between any two events occurring at the same point in space, x = \[sIJ^odx°. (84.2) This relation determines the actual time interval (or as it is also called, the proper time for the given point in space) for a change of the coordinate x°. We note in passing that the quantity g 00 , as we see from these formulas, is positive: 0oo > 0. (84.3) It is necessary to emphasize the difference between the meaning of (84.3) and the meaning of the signature [the signs of three principal values of the tensor g ik (§ 82)]. A tensor g ik which does not satisfy the second of these conditions cannot correspond to any real gravita- tional field, i.e. cannot be the metric of a real space-time. Nonfulfilment of the condition (84.3) would mean only that the corresponding system of reference cannot be realized with real bodies; if the condition on the principal values is fulfilled, then a suitable transforma- tion of the coordinates can make g 00 positive (an example of such a system is given by the rotating system of coordinates, see § 89). We now determine the element dl of spatial distance. In the special theory of relativity we can define dl as the interval between two infinitesimally separated events occurring at one and the same time. In the general theory of relativity, it is usually impossible to do this, i.e. it is impossible to determine dl by simply setting dx° = in ds. This is related to the fact that in a gravitational field the proper time at different points in space has a different dependence on the coordinate x°. To find dl, we now proceed as follows. Suppose a light signal is directed from some point B in space (with coordinates x a + dx a ) to a point A infinitely near to it (and having coordinates x a ) and then back over the same path. Obviously, the time (as observed from the one point B) required for this, when multiplied by c, is twice the distance between the two points. Let us write the interval, separating the space and time coordinates : ds 2 = g aP dx a dx p + 2g 0a dx° dx* + g 00 (dx ) 2 (84.4) 234 PARTICLE IN A GRAVITATIONAL FIELD § 84 where it is understood that we sum over repeated Greek indices from 1 to 3. The interval between the events corresponding to the departure and arrival of the signal from one point to the other is equal to zero. Solving the equation ds 2 = with respect to dx°, we find two roots : d x °W -= _ {-g 0a dx ct -^(g 0a g op -g ap g 00 )dx ct dx' 3 }, 9oo dx ow =~ {-g 0a dx a + ^(gZgop-g a f l goo)dx^dx p }, ifOO (84.5) corresponding to the propagation of the signal in the two directions between A and B. If x° is the moment of arrival of the signal at A, the times when it left B and when it will return to B are, respectively, x° + dx ( 1) and x° + dx ( 2) . In the schematic diagram of Fig. 18 the solid lines are the world lines corresponding to the given coordinates x a and x a + dx a , while the dashed lines are the world lines of the signals.f It is clear that the total interval of "time" between the departure of the signal and its return to the original point is equal to dx ^ - dx ^ = — V(do«0o,r - 9 a p9oo) dx a dx*. 9oo < \ (2) x° + dx„ (D x° + dx n A B Fig. 18 The corresponding interval of proper time is obtained, according to (84.1), by multiplying by v goo/c> and the distance dl between the two points by multiplying once more by c/2. As a result, we obtain dl 2 = (-g aP +^^)dx^ x P. \ 000 / This is the required expression, defining the distance in terms of the space coordinate elements. We rewrite it in the form where dl 2 = y a p dx a dx p , v -(-a I 9oa9 °p\ y a p-\ g a p+— j \ 9oo / (84.6) (84.7) t In Fig. 18, it is assumed that dx^ > 0, dxp < 0, but this is not necessary: dx ow and dx°™ may have the same sign. The fact that in this case the value *°(A) at the moment of arrival of the signal at A might be less than the value x°(B) at the moment of its departure from B contains no contradiction, since the rates of clocks at different points in space are not assumed to be synchronized in any way. § 84 DISTANCES AND TIME INTERVALS 235 is the three-dimensional metric tensor, determining the metric, i.e., the geometric properties of the space. The relations (84.7) give the connection between the metric of real space and the metric of the four-dimensional space-time.f However, we must remember that the g ik generally depend on x°, so that the space metric (84.6) also changes with time. For this reason, it is meaningless to integrate dl; such an integral would depend on the world line chosen between the two given space points. Thus, generally speaking, in the general theory of relativity the concept of a definite distance between bodies loses its meaning, remaining valid only for infinitesimal distances. The only case where the distance can be defined also over a finite domain is that in which the g ik do not depend on the time, so that the integral J dl along a space curve has a definite meaning. It is worth noting that the tensor -y aP is the reciprocal of the contravariant three-dimen- sional tensor g*K In fact, from g ik g kl = 5\, we have, in particular, 9 aP g, y + 9 a0 9oy = F„ 9 aP g,o + 9 a °goo = 0, 9 op 9 P o + 9°°9oo = 1. (84.8) Determining g a0 from the second equation and substituting in the first, we obtain: _ n aP _ Six 9 7py — °r This result can be formulated differently, by the statement that the quantities -g* p form the contravariant three-dimensional metric tensor corresponding to the metric (84.6): y*P = -g*P. (84.9) We also state that the determinants g and y, formed respectively from the quantities g ik and y aP , are related to one another by -9 = 9oo7- ( 84 - 10 > In some of the later applications it will be convenient to introduce the three-dimensional vector g, whose covariant components are defined as g m =-—. ( 84 - n ) Considering g as a vector in the space with metric (84.6), we must define its contravariant components as g* = y ap g p . Using (84.9) and the second of equations (84.8), it is easy to see that g'-fg^-g '. (84.12) We also note the formula g 00 = — -g.ff, ( 84 - 13 > goo which follows from the third of equations (84.8). f The quadratic form (84.6) must clearly be positive definite. For this, its coefficients must, as we know from the theory of forms, satisfy the conditions 7n 7i2 7i3 721 722 733 >0. 731 732 733 Expressing y ik in terms of g tk , it is easy to show that these conditions take the form 000 001 002 <0, 010 011 012 >0, 0<O. 020 021 022 These conditions, together with the condition (84.3), must be satisfied by the components of the metric tensor in every system of reference which can be realized with the aid of real bodies. 7n >o, r 1 yi2 >o, 721 722 1 000 001 1010 01ll 236 PARTICLE IN A GRAVITATIONAL FIELD § 85 We now turn to the definition of the concept of simultaneity in the general theory of relativity. In other words, we discuss the question of the possibility of synchronizing clocks located at different points in space, i.e. the setting up of a correspondence between the readings of these clocks. Such a synchronization must obviously be achieved by means of an exchange of light signals between the two points. We again consider the process of propagation of signals between two infinitely near points A and B, as shown in Fig. 18. We should regard as simul- taneous with the moment x° at the point A that reading of the clock at point B which is half- way between the moments of departure and return of the signal to that point, i.e. the moment x° + Ax° = x° + ±(dx°( 2) + dx 0il) ). Substituting (84.5), we thus find that the difference in the values of the "time" x° for two simultaneous events occurring at infinitely near points is given by o _ Qoa.dx Ax° = -^—^g a dx a . (84.14) 000 This relation enables us to synchronize clocks in any infinitesimal region of space. Carry- ing out a similar synchronization from the point A, we can synchronize clocks, i.e. we can define simultaneity of events, along any open curve.f However, synchronization of clocks along a closed contour turns out to be impossible in general. In fact, starting out along the contour and returning to the initial point, we would obtain for Ax° a value different from zero. Thus it is, a fortiori, impossible to synchronize clocks over all space. The exceptional cases are those reference systems in which all the components g 0a are equal to zero. J It should be emphasized that the impossibility of synchronization of all clocks is a property of the arbitrary reference system, and not of the space-time itself. In any gravitational field, it is always possible (in infinitely many ways) to choose the reference system so that the three quantities g 0a become identically equal to zero, and thus make possible a complete synchro- nization of clocks (see § 90). Even in the special theory of relativity, proper time elapses differently for clocks moving relative to one another. In the general theory of relativity, proper time elapses differently even at different points of space in the same reference system. This means that the interval of proper time between two events occurring at some point in space, and the interval of time between two events simultaneous with these at another point in space, are in general different from one another. § 85. Covariant differentiation In galilean coordinates§ the differentials dA { of a vector A t form a vector, and the derivatives dAJdx* of the components of a vector with respect to the coordinates form a tensor. In curvilinear coordinates this is not so; dA t is not a vector, and dAJdx* is not a f Multiplying (84.14) by g 0Q and bringing both terms to one side, we can state the condition for syn- chronization in the form dx = g i dx i = : the "covariant differential" dx between two infinitely near simultaneous events must be equal to zero. } We should also assign to this class those cases where the g 0a can be made equal to zero by a simple transformation of the time coordinate, which does not involve any choice of the system of objects serving for the definition of the space coordinates. § In general, whenever the quantities g ik are constant. § 85 COVARIANT DIFFERENTIATION 237 tensor. This is due to the fact that dA t is the difference of vectors located at different (in- finitesimally separated) points of space; at different points in space vectors transform differently, since the coefficients in the transformation formulas (83.2), (83.4) are functions of the coordinates. It is also easy to verify these statements directly. To do this we determine the transforma- tion formulas for the differentials dA t in curvilinear coordinates. A covariant vector is transformed according to the formula _dx' k , therefore dx' k , , , fix* 3x' k JA , A , d 2 x' k , t dAi = J x - i dA' k + A' k d^=-^ dA' k + A' k ^ dxK Thus dA t does not transform at all like a vector (the same also applies, of course, to the differential of a contravariant vector). Only if the second derivatives d 2 x' k /dx l dx l = 0, i.e. if the x' k are linear functions of the x*, do the transformation formulas have the form dx' k dA^—tdAl that is, dA t transforms like a vector. We now undertake the definition of a tensor which in curvilinear coordinates plays the same role as dAJdx*- in galilean coordinates. In other words, we must transform dA i /dx k from galilean to curvilinear coordinates. In curvilinear coordinates, in order to obtain a differential of a vector which behaves like a vector, it is necessary that the two vectors to be subtracted from each other be located at the same point in space. In other words, we must somehow "translate" one of the vectors (which are separated infmitesimally from each other) to the point where the second is located, after which we determine the difference of two vectors which now refer to one and the same point in space. The operation of translation itself must be defined so that in galilean coordinates the difference shall coincide with the ordinary differential dA t . Since dA t is just the difference of the components of two infmitesimally separated vectors, this means that when we use galilean coordinates the components of the vector should not change as a result of the translation operation. But such a translation is precisely the translation of a vector parallel to itself. Under a parallel translation of a vector, its components in galilean co- ordinates do not change. If, on the other hand, we use curvilinear coordinates, then in general the components of the vector will change under such a translation. Therefore in curvilinear coordinates, the difference in the components of the two vectors after translating one of them to the point where the other is located will not coincide with their difference before the translation (i.e. with the differential dA t ). Thus to compare two infmitesimally separated vectors we must subject one of them to a parallel translation to the point where the second is located. Let us consider an arbitrary contravariant vector ; if its value at the point x l is A 1 , then at the neighboring point x l + dx l it is equal to A l + dA l . We subject the vector A 1 to an infinitesimal parallel displacement to the point x l +dx l ; the change in the vector which results from this we denote by dA\ Then the difference DA 1 between the two vectors which are now located at the same point is DA i = dA i -dA i . (85.1) 238 PARTICLE IN A GRAVITATIONAL FIELD § 85 The change 5 A 1 in the components of a vector under an infinitesimal parallel displacement depends on the values of the components themselves, where the dependence must clearly be linear. This follows directly from the fact that the sum of two vectors must transform accord- ing to the same law as each of the constituents. Thus 5 A 1 has the form SA l = -r kl A k dx l , (85.2) where the T l kl are certain functions of the coordinates. Their form depends, of course, on the coordinate system; for a galilean coordinate system T l kl = 0. From this it is already clear that the quantities P kl do not form a tensor, since a tensor which is equal to zero in one coordinate system is equal to zero in every other one. In a curvilinear space it is, of course, impossible to make all the T l kl vanish over all of space. But we can choose a coordinate system for which the r l kl become zero over a given infinitesimal region (see the end of this sectionf). The quantities rj, are called Christoffel symbols. In addition to the quantities P kl we shall later also use quantities T, kl J defined as follows : If, kl = Qim^U- (85.3) Conversely, ni = 9 im r m>kl . (85.4) It is also easy to relate the change in the components of a co variant vector under a parallel displacement to the Christoffel symbols. To do this we note that under a parallel displace- ment, a scalar is unchanged. In particular, the scalar product of two vectors does not change under a parallel displacement. Let A t and B l be any covariant and contravariant vectors. Then from <5(^ £ -5') = 0, we have B l SAi = -A t 5B l = Ti^Ai dx l or, changing the indices, B l SAi = TlAkB 1 dx l . From this, in view of the arbitrariness of the B\ SA^T^dx 1 , (85.5) which determines the change in a covariant vector under a parallel displacement. Substituting (85.2) and dA l = (dA l /dx l ) dx l in (85.1), we have DA 1 = ( — . +r kl A k dx l . (85.6) Jx l Similarly, we find for a covariant vector DA i = ( d ^i-T k l A k )dx l . (85.7) \dx l The expressions in parentheses in (85.6) and (85.7) are tensors, since when multiplied by the vector dx l they give a vector. Clearly, these are the tensors which give the desired generalization of the concept of a derivative to curvilinear coordinates. These tensors are called the covariant derivatives of the vectors A 1 and A t respectively. We shall denote them by A 1 . k and A i; k . Thus, DA i = A i . l dx l \ DA^A^dx 1 , (85.8) f This is precisely the coordinate system which we have in mind in arguments where we, for brevity's sake, speak of a "galilean" system; still all the proofs remain applicable not only to flat, but also to curved space. % In place of r^ and r f , w , the symbols < . [ and .are sometimes used. § 85 COVARIANT DIFFERENTIATION 239 while the co variant derivatives themselves are : A^^+TlA*, (85.9) A tll = d ^-I* a A k . (85.10) In galilean coordinates, P kl = 0, and covariant differentiation reduces to ordinary differen- tiation. It is also easy to calculate the covariant derivative of a tensor. To do this we must deter- mine the change in the tensor under an infinitesimal parallel displacement. For example, let us consider any contravariant tensor, expressible as a product of two contravariant vectors A l B k . Under parallel displacement, 5{A l B k ) = A i 8B k + B k dA i = -A i T k lm B l dx m -B k Y\ m A l dx m . By virtue of the linearity of this transformation we must also have, for an arbitrary tensor A ik dA ik = -{A im T k ml + A mk r m ddx l . (85.11) Substituting this in DA ik = dA ik -SA ik = A ik ;l dx l , we get the covariant derivative of the tensor A ,k in the form A% = 8 ^T+T i ml A mk +r k l A i '". (85.12) ox In completely similar fashion we obtain the covariant derivative of the mixed tensor A l k and the covariant tensor A ik in the form 4;/ = ^-rr^+r'A m , (85.13) A ik;l = °^ -TJ}A mk -TZA im . (85.14) One can similarly determine the covariant derivative of a tensor of arbitrary rank. In doing this one finds the following rule of covariant differentiation. To obtain the covariant derivative of the tensor A ; ; ; with respect to x\ we add to the ordinary derivative dA ; ; ; /dx l for each covariant index i(A\ \ ;) a term -T k u A; k ; , and for each contravariant index i(A\ \ ;) a term +rj^4; fc ;. One can easily verify that the covariant derivative of a product is found by the same rule as for ordinary differentiation of products. In doing this we must consider the covariant derivative of a scalar as an ordinary derivative, that is, as the covariant vector (f> k = dcj)/dx k , in accordance with the fact that for a scalar d(p = 0, and therefore D(f> = d(f). For example, the covariant derivative of the product A i B k is {A i B k ). l = A i , l B k + A i B k . l . If in a covariant derivative we raise the index signifying the differentiation, we obtain the so-called contravariant derivative. Thus, AS k = g kl A i;l , A i -' k = g kl A\ tl . We prove that the Christoffel symbols Y kl are symmetric in the subscripts. Since the covariant derivative of a vector A i;k is a tensor, the difference A ixk —A kxi is also a tensor. 240 PARTICLE IN A GRAVITATIONAL FIELD § 85 Let the vector A t be the gradient of a scalar, that is, A t = d^/dx 1 . Since dAJdx* = d 2 (f)/dx k dx i = dAJdx 1 , with the help of (85.10) we have Ah-, i A i; k — (T ik — r ki ) x dx l In a galilean coordinate system the covariant derivative reduces to the ordinary derivative, and therefore the left side of our equation becomes zero. But since A k . i-A i; k is a tensor, then being zero in one system it must also be zero in any coordinate system. Thus we find that rL = r\ k . (85.15) Clearly, also, r,-, H = r f>tt . (85.16) In general, there are altogether forty different quantities P kl ; for each of the four values of the index i there are ten different pairs of values of the indices k and / (counting pairs obtained by interchanging k and / as the same). In concluding this section we present the formulas for transforming the Christoffel symbols from one coordinate system to another. These formulas can be obtained by com- paring the laws of transformation of the two sides of the equations defining the covariant derivatives, and requiring that these laws be the same for both sides. A simple calculation gives lu-K P dx , m dxk dxl + dxkdxl ^. (85.17) From this formula it is clear that the quantity P kl behaves like a tensor only under linear transformations [for which the second term in (85.17) drops out]. Formula (85.17) enables us to prove easily the assertion made above that it is always possible to choose a coordinate system in which all the P kl become zero at a previously assigned point (such a system is said to be locally-geodesic (see § 87).f In fact, let the given point be chosen as the origin of coordinates, and let the values of the T l kl at it be initially (in the coordinates x { ) equal to (P k i) . In the neighborhood of this point, we now make the transformation x'^x'+KrDoxV. (85.18) Then / d 2 x' m dx* \ \dx k dx l dx' m ) = (FL)o and according to (85.17), all the T'™ become equal to zero. We note that for the transformation (85.18). fa?).-*" so that it does not change the value of any tensor (including the tensor g ik ) at the given point, so that we can make the Christoffel symbols vanish at the same time as we bring the g ik to galilean form. t It can also be shown that, by a suitable choice of the coordinate system, one can make all the !"£, go to zero not just at a point but all along a given curve. § 86 THE RELATION OF THE CHRISTOFFEL SYMBOLS TO THE METRIC TENSOR 241 § 86. The relation of the Christoffel symbols to the metric tensor Let us show that the covariant derivative of the metric tensor g ik is zero. To do this we note that the relation DA t = g ik DA k is valid for the vector DA h as for any vector. On the other hand, A t = g ik A k , so that DA V = D(g ik A k ) = g ik DA k +A k Dg ik . Comparing with DA t = g ik DA k , and remembering that the vector A k is arbitrary, Dg ik = 0. Therefore the covariant derivative 9 ik ..i = 0. (86.1) Thus g ik may be considered as a constant during covariant differentiation. The equation g ik . , = can be used to express the Christoffel symbols P kl in terms of the metric tensor g ik . To do this we write in accordance with the general definition (85.14): _°9ik_ r m —n r m — dik r — r — o dik;l— <, i 9mk l il dim 1 kl — * i l k,il L i,kl ~ u - Thus the derivatives of g ik are expressed in terms of the Christoffel symbols. f We write the values of the derivatives of g ik , permuting the indices /, k, I: <Hhk _ r r dx l - lfc >» + 1 «- fc " M'_r +r dx k ~ Ukl+ ^ dQki _ r r ~ p. i ~ l l.ki l k, li- Taking half the sum of these equations, we find (remembering that T t kl = T; lk ) 1 /dgtk dffi. 2 V dx l dx From this we have for the symbols T[ t = g im T mkh r IH ==-(^? + ^-^Y (86.2) Tkl ~~2 g Ve7 + ~d?-dx»>)' (86,3) These formulas give the required expressions for the Christoffel symbols in terms of the metric tensor. We now derive an expression for the contracted Christoffel symbol T l ki which will be important later on. To do this we calculate the differential dg of the determinant g made up from the components of the tensor g ik ; dg can be obtained by taking the differential of each component of the tensor g ik and multiplying it by its coefficient in the determinant, i.e. by the corresponding minor. On the other hand, the components of the tensor g ik reciprocal to g ik are equal to the minors of the determinant of the g ik , divided by the determinant. f Choosing a locally-geodesic system of coordinates therefore means that at the given point all the first derivatives of the components of the metric tensor vanish. 242 PARTICLE IN A GRAVITATIONAL FIELD § 86 Therefore the minors of the determinant g are equal to gg ik . Thus, dg = gg ik dg ik = -gg ik dg ik (86.4) (since g ik g ik = 8\ = 4, g ik dg ik = -g ik dg ik ). From (86.3), we have r» = l n im ( d9mk d9mi d9ki " ki 2 9 [dx 1 + dx k dx" Changing the positions of the indices m and i in the third and first terms in parentheses, we see that these two terms cancel each other, so that 1 n im d 9im 2 9 dx k ' Hi = ^ flf or, according to (86.4), 1 dg d In V — g 2g dx k ~ dx ; It is useful to note also the expression for the quantity g kl T l kl ; we have r « = ^ = — lur^- (86-5) u ~2 0g \dl + ~d^~^r)- 9 9 \d7~i With the help of (86.4) this can be transformed to *»!<„-- * *^9. (86.6) yj-g dx k For various calculations it is important to remember that the derivatives of the contra- variant tensor g lk are related to the derivatives of g ik by the relations 9il dx^-~ 9 d?> (86J) (which are obtained by differentiating the equality g u g lk = <5*). Finally we point out that the derivatives of g lk can also be expressed in terms of the quantities T kl . Namely, from the identity g ik . , = it follows directly that da ik j^ = -T l ml g mk -T k ml g im . (86.8) With the aid of the formulas which we have obtained we can put the expression for A 1 , j, the generalized divergence of a vector in curvilinear coordinates, in convenient form. Using (86.5), we have A --'-dx t+lliA ~ dx i+A dx 1 or, finally, 1 diyf^gA 1 ) , arn ^ A\ , = -7= — — — -. (86.9 ) V-flf dx* We can derive an analogous expression for the divergence of an antisymmetric tensor A ik . From (85.12), we have 8A ik Aik ,pi Amk.j^k Aim A ik — p, k + A mk A + l mk^ 1 • § 87 MOTION OF A PARTICLE IN A GRAVITATIONAL FIELD 243 But, since A mk = -A km , ri A mk — _F' A km = 1 mk A — 1 km^ 1 v ' Substituting the expression (86.5) for r£, fc , we obtain A'x.^-L, 8 ^- 9 ^. (86.10) y/-g d* k Now suppose A ik is a symmetric tensor; we calculate the expression A k . k for its mixed components. We have a" - sa Kt*a< t'a"- l d< - A ^- g) PA* i4 «"-a? +r " 4 '- r " 4 '-.7^ Bx " " " The last term here is equal to 1 ((Hki ddki _ d(kk\ A ki 2\dx k dx f dx l J Because of the symmetry of the tensor A kl , two of the terms in parentheses cancel each other, leaving k _ * d(V-g^ ) _ldga Akl yfZTg dx k 2 dx l A* = -J= ^Zf^ _ i ^ ^« (86.11) l i;k In cartesian coordinates, dA i Jdx k -dAJdx i is an antisymmetric tensor. In curvilinear coordinates this tensor is A iik —A kii . However, with the help of the expression for A i;k and since T l kl = T\ k , we have A. k -A k . = d A-—-. (86-12) A:fe Akil dx k dx l v } Finally, we transform to curvilinear coordinates the sum d 2 (f)ldx i dx l of the second derivatives of a scalar <£. It is clear that in curvilinear coordinates this sum goes over into 4>\ \. But (f>. i = d4>ldx\ since covariant differentiation of a scalar reduces to ordinary dif- ferentiation. Raising the index /, we have * 9 dx k ' and using formula (86.9), we find It is important to note that Gauss' theorem (83.17) for the transformation of the integral of a vector over a hypersurface into an integral over a four- volume can, in view of (86.9), be written as j> A^^gdSi = j AKty/^gdQ. (86.14) § 87. Motion of a particle in a gravitational field The motion of a free material particle is determined in the special theory of relativity from the principle of least action, 5S= -mcd f ds = 0, (87.1) 244 PARTICLE IN A GRAVITATIONAL FIELD § 87 according to which the particle moves so that its world line is an extremal between a given pair of world points, in our case a straight line (in ordinary three-dimensional space this corresponds to uniform rectilinear motion). The motion of a particle in a gravitational field is determined by the principle of least action in this same form (87.1), since the gravitational field is nothing but a change in the metric of space-time, manifesting itself only in a change in the expression for ds in terms of the dx\ Thus, in a gravitational field the particle moves so that its world point moves along an extremal or, as it is called, a geodesic line in the four-space x°, x 1 , x 2 , x 3 ; however, since in the presence of the gravitational field space-time is not galilean, this line is not a "straight line", and the real spatial motion of the particle is neither uniform nor rectilinear. Instead of starting once again directly from the principle of least action (see the problem at the end of this section), it is simpler to obtain the equations of motion of a particle in a gravitational field by an appropriate generalization of the differential equations for the free motion of a particle in the special theory of relativity, i.e. in a galilean four-dimensional coordinate system. These equations are du'/ds = or du l = 0, where u l = dx l /ds is the four- velocity. Clearly, in curvilinear coordinates this equation is generalized to the equation Du l = 0. (87.2) From the expression (85.6) for the covariant differential of a vector, we have du i + T i ia u k dx l =$. Dividing this equation by ds, we have d 2 x i . dx k dx l This is the required equation of motion. We see that the motion of a particle in a gravita- tional field is determined by the quantities T l kl . The derivative d 2 x i /ds 2 is the four-accelera- tion of the particle. Therefore we may call the quantity —mT l kl u k u l the "four-force", acting on the particle in the gravitational field. Here, the tensor g ik plays the role of the "potential" of the gravitational field — its derivatives determine the field "intensity" T kl .^ In § 85 it was shown that by a suitable choice of the coordinate system one can always make all the T l kl zero at an arbitrary point of space-time. We now see that the choice of such a locally-inertial system of reference means the elimination of the gravitational field in the given infinitesimal element of space-time, and the possibility of making such a choice is an expression of the principle of equivalence in the relativistic theory of gravitation. As before, we define the four-momentum of a particle in a gravitational field as p l = mcu\ (87.4) Its square is PiP 1 = m 2 c 2 . (87.5) t We also give the form of the equations of motion expressed in terms of covariant components of the four-acceleration. From the condition Du t = 0, we find —r — r fc , u u k u l = 0. ds Substituting for T k , u from (86.2), two of the terms cancel and we are left with ds 2 dx* § 87 MOTION OF A PARTICLE IN A GRAVITATIONAL FIELD 245 Substituting — dS/dx* for p h we find the Hamilton-Jacobi equation for a particle in a gravitational field: •*IS-" v - a (87 - 6) The equation of a geodesic in the form (87.3) is not applicable to the propagation of a light signal, since along the world line of the propagation of a light ray the interval ds, as we know, is zero, so that all the terms in equation (87.3) become infinite. To get the equa- tions of motion in the form needed for this case, we use the fact that the direction of pro- pagation of a light ray in geometrical optics is determined by the wave vector tangent to the ray. We can therefore write the four-dimensional wave vector in the form k 1 = dx l /dA, where X is some parameter varying along the ray. In the special theory of relativity, in the propagation of light in vacuum the wave vector does not vary along the path, that is, dk l = (see § 53). In a gravitational field this equation clearly goes over into Dk l = or dk* — +r i kl k k k l = (87.7) (these equations also determine the parameter X). The absolute square of the wave four-vector (see § 48) is zero, that is, fc,fc f = 0. (87.8) Substituting dif/Jdx 1 in place of k t (^ is the eikonal), we find the eikonal equation in a gravita- tional field In the limiting case of small velocities, the relativistic equations of motion of a particle in a gravitational field must go over into the corresponding non-relativistic equations. In this we must keep in mind that the assumption of small velocity implies the requirement that the gravitational field itself be weak; if this were not so a particle located in it would acquire a high velocity. Let us examine how, in this limiting case, the metric tensor g ik determining the field is related to the nonrelativistic potential (j> of the gravitational field. In nonrelativistic mechanics the motion of a particle in a gravitational field is determined by the Lagrangian (81.1). We now write it in the form TtlV L = -mc 2 +—- - m& (87. 10) adding the constant — mc 2 .\ This must be done so that the nonrelativistic Lagrangian in the absence of the field, L = —mc 2 + mv 2 /2, shall be the same exactly as that to which the corresponding relativistic function L = —mc 2 y/\—v 2 /c 2 reduces in the limit as v/c -*■ 0. Consequently, the nonrelativistic action function S for a particle in a gravitational field has the form S = Ldt = -mc (c- ~ + -J dt. t The potential <f> is, of course, denned only to within an arbitrary additive constant. We assume through- out that one makes the natural choice of this constant so that the potential vanishes far from the bodies producing the field. 246 PARTICLE IN A GRAVITATIONAL FIELD § 87 Comparing this with the expression S = —mc$ds, we see that in the limiting case under consideration ds = [ c — — + — ] dt. K*9 Squaring and dropping terms which vanish for c -> oo, we find ds 2 = (c 2 + 2(l))dt 2 -dr 2 . (87.11) where we have used the fact that v dt = dr. Thus in the limiting case the component g 00 of the metric tensor is 26 c As for the other components, from (87.11) it would follow that g aP = S aP , g 0<x = 0- Actually, however, the corrections to them are, generally speaking, of the same order of magnitude as the corrections to g 00 (for more detail, see § 106). The impossibility of deter- mining these corrections by the method given above is related to the fact that the corrections to the g aP , though of the same order of magnitude as the correction to g 00 , would give rise to terms in the Lagrangian of a higher order of smallness (because in the expression for ds 2 the components g ap are not multiplied by c 2 , while this is the case for g 00 ). PROBLEM Derive the equation of motion (87.3) from the principle of least action (87.1). Solution: We have: Sds 2 — Ids Sds = S(g ik dx 1 dx k ) = dx 1 dx k -^ Sx l +2g ik dx l dSx k . ox 1 Therefore . . , dx l dx k dg ik dx l dSx k . = — mc C fl dx l dx k dg ik d ( dx K \ 1 J 2i IT* "a? dx ds ( gi * -ds) dx ) ds (in integrating by parts, we use the fact that Sx k = at the limits). In the second term in the integral, we replace the index k by the index /. We then find, by equating to zero the coefficient of the arbitrary variation Sx l : 1 !,««* d0ik d (a ill - l ,/u« dgik -a ^ -u'u k d9il - Noting that the third term can be written as - l u l u k ( d ^ + d9 ^\ 2 U \dx k+ dx 1 )' and introducing the Christoffel symbols r ; , ifc in accordance with (86.2), we have: dui i t- i k n gu-r+T Uik u i u K = 0. ds Equation (87.3) is obtained from this by raising the index /. § 88 THE CONSTANT GRAVITATIONAL FIELD 247 § 88. The constant gravitational field A gravitational field is said to be constant if one can choose a system of reference in which all the components of the metric tensor are independent of the time coordinate x° ; the latter is then called the world time. The choice of a world time is not completely unique. Thus, if we add to x° an arbitrary function of the space coordinates, the g ik will still not contain x°; this transformation corresponds to the arbitrariness in the choice of the time origin at each point in space.f In addition, of course, the world time can be multiplied by an arbitrary constant, i.e. the units for measuring it are arbitrary. Strictly speaking, only the field produced by a single body can be constant. In a system of several bodies, their mutual gravitational attraction will give rise to motion, as a result of which the field produced by them cannot be constant. If the body producing the field is fixed (in the reference system in which the g ik do not depend on x°), then both directions of time are equivalent. For a suitable choice of the time origin at all the points in space, the interval ds should in this case not be changed when we change the sign of x°, and therefore all the components g 0a of the metric tensor must be identically equal to zero. Such constant gravitational fields are said to be static. However, for the field produced by a body to be constant, it is not necessary for the body to be at rest. Thus the field of an axially symmetric body rotating uniformly about its axis will also be constant. However in this case the two time directions are no longer equivalent by any means — if the sign of the time is changed, the sign of the angular velocity is changed. Therefore in such constant gravitational fields (we shall call them stationary fields) the components g 0x of the metric tensor are in general different from zero. The meaning of the world time in a constant gravitational field is that an interval of world time between events at a certain point in space coincides with the interval of world time between any other two events at any other point in space, if these events are respectively simultaneous (in the sense explained in § 84) with the first pair of events. But to the same interval of world time x° there correspond, at different points of space, different intervals of proper time t. The relation between world time and proper time, formula (84.1), can now be written in the form t = -V-0oo*°. (88.1) applicable to any finite time interval. If the gravitational field is weak, then we may use the approximate expression (87.12), and (88.1) gives -?(■♦» (88.2) f It is easy to see that under such a transformation the spatial metric, as expected, does not change. In fact, under the substitution x°-^x°+f(x 1 ,x 2 ,x 3 ) with an arbitrary function /(x 1 , x 2 , x 3 ), the components g tk change to ffaff -* g«0 +goof. af, B +9oaf. +ffopf, «, doa-^ffOa+goof.a, goQ-^doO, where /, « = dfjdx a . This obviously does not change the tensor (84.7). 248 PARTICLE IN A GRAVITATIONAL FIELD § 88 Thus proper time elapses the more slowly the smaller the gravitational potential at a given point in space, i.e., the larger its absolute value (later, in § 96, it will be shown that the poten- tial (f> is negative). If one of two identical clocks is placed in a gravitational field for some time, the clock which has been in the field will thereafter appear to be slow. As was already indicated above, in a static gravitational field the components g 0a of the metric tensor are zero. According to the results of § 84, this means that in such a field synchronization of clocks is possible over all space. We note also that the element of spatial distance in a static field is simply: dl 2 = -g aP dx a dx p . (88.3) In a stationary field the g 0a are different from zero and the synchronization of clocks over all space is impossible. Since the g ik do not depend on x°, formula (84.14) for the dif- ference between the values of world time for two simultaneous events occurring at different points in space can be written in the form Ax°=-( 9 -^ (88.4) J 000 for any two points on the line along which the synchronization of clocks is carried out. In the synchronization of clocks along a closed contour, the difference in the value of the world time which would be recorded upon returning to the starting point is equal to the integral Ax =-& 9 -^— (88.5) J 9oo taken along the closed contour, f Let us consider the propagation of a light ray in a constant gravitational field. We have seen in § 53 that the frequency of the light is the time derivative of the eikonal \j/ (with opposite sign). The frequency expressed in terms of the world time x°/c is therefore co — —c(d\l//dx°). Since the eikonal equation (87.9) in a constant field does not contain x° explicitly, the frequency co remains constant during the propagation of the light ray. The frequency measured in terms of the proper time is a> = —(dij//dx); this frequency is different at different points of space. From the relation # _ _# dx° _ _# _^_ Jx"dx~° dx~ dx° V^o' we have co = -p=. (88.6) vtfoo In a weak gravitational field we obtain from this, approximately, a> = a> (l-^). (88.7) We see that the light frequency increases with increasing absolute value of the potential of the gravitational field, i.e. as we approach the bodies producing the field; conversely, as the light recedes from these bodies the frequency decreases. If a ray of light, emitted at a point f The integral (88.5) is identically zero if the sum g 0a dx a /g 00 is an exact differential of some function of the space coordinates. However, such a case would simply mean that we are actually dealing with a static field, and that all the g 0a could be made equal to zero by a transformation of the form x -+x°+f(x a ). § 88 THE CONSTANT GRAVITATIONAL FIELD 249 where the gravitational potential is 4>u nas ( at that point) the frequency co, then upon arriving at a point where the potential is <£ 2 , it will have a frequency (measured in units of the proper time at that point) equal to co i0i c 2 ;H)-0+^> A line spectrum emitted by some atoms located, for example, on the sun, looks the same there as the spectrum emitted by the same atoms located on the earth would appear on it. If, however, we observe on the earth the spectrum emitted by the atoms located on the sun, then, as follows from what has been said above, its lines appear to be shifted with respect to the lines of the same spectrum emitted on the earth. Namely, each line with frequency co will be shifted through the interval Aco given by the formula Aco = ^^co, (88.8) where $ x and <f) 2 are the potentials of the gravitational field at the points of emission and observation of the spectrum respectively. If we observe on the earth a spectrum emitted on the sun or the stars, then I0J > |0 2 |» and from (88.8) it follows that Aco < 0, i.e. the shift occurs in the direction of lower frequency. The phenomenon we have described is called the "red shift". The occurrence of this phenomenon can be explained directly on the basis of what has been said above about world time. Because the field is constant, the interval of world time during which a certain vibration in the light wave propagates from one given point of space to another is independent of jc°. Therefore it is clear that the number of vibrations occurring in a unit interval of world time will be the same at all points along the ray. But to one and the same interval of world time there corresponds a larger and larger interval of proper time, the further away we are from the bodies producing the field. Consequently, the frequency, i.e. the number of vibrations per unit proper time, will decrease as the light recedes from these masses. During the motion of a particle in a constant field, its energy, defined as dS_ ~ C cV>' the derivative of the action with respect to the world time, is conserved; this follows, for example, from the fact that x° does not appear explicitly in the Hamilton- Jacobi equation. The energy defined in this way is the time component of the covariant four-vector of momentum p k = mcu k = mcg ki u l . In a static field, ds 2 = g 00 (dx ) 2 — dl 2 , and we have for the energy, which we here denote by i Q , i dx° , dx° <f = mc z g 00 -—- = mc z g 00 ds aOO Jg 00 (dx ) 2 -dl 2 We introduce the velocity dl cdl dx \/g odx° of the particle, measured in terms of the proper time, that is, by an observer located at the 250 PARTICLE IN A GRAVITATIONAL FIELD given point. Then we obtain for the energy o — 7 1 - 1- (88.9) This is the quantity which is conserved during the motion of the particle. It is easy to show that the expression (88.9) remains valid also for a stationary field, if only the velocity v is measured in terms of the proper time, as determined by clocks syn- chronized along the trajectory of the particle. If the particle departs from point A at the moment of world time x° and arrives at the infinitesimally distant point B at the moment x°+dx°, then to determine the velocity we must now take, not the time interval (x° + dx )— x° = dx°, but rather the difference between x° + dx° and the moment x° — (goJffoo)dx a which is simultaneous at the point B with the moment x° at the point A : (x° + dx )- (x°- ^ dx a ) = Jx°+ — dx\ \ 9oo J 9oo Multiplying by v g 00 /c, we obtain the corresponding interval of proper time, so that the velocity is c dx x v* = -r- , (88.10) yjh(dx —g a dx a ) where we have introduced the notation h = g o, g*=-^ (88.11) 000 for the three-dimensional vector g (which was already mentioned in § 84) and for the three- dimensional scalar g 00 . The covariant components of the velocity v form a three-dimen- sional vector in the space with metric y ap , and correspondingly the square of this vector is to be taken asf v^ltpv*, v 2 = v a v*. (88.12) We note that with such a definition, the interval ds is expressed in terms of the velocity in the usual fashion: ds 2 = #oo (d* ) 2 + 2#oa dx° dx a + g ap dx a dx p = h(dx°-g a dx a ) 2 -dl 2 = h(dx°-g a dx a ) 2 (\ - ^A (88.13) The components of the four- velocity j dx l ds f In our further work we shall repeatedly introduce, in addition to four-vectors and four-tensors, three- dimensional vectors and tensors denned in the space with metric y a/> ; in particular the vectors g and v, which we have already used, are of this type. Just as in four dimensions the tensor operations (in particular, raising and lowering of indices) are done using the metric tensor g ik , so, in three dimensions these are done using the tensor y aB . To avoid misunderstandings that may arise, we shall denote three-dimensional quantities by symbols other than those used for four-dimensional quantities. § 88 THE CONSTANT GRAVITATIONAL FIELD 251 are «■-— f=>. «°= J== + -^L=. (88.14) The energy is S Q = mc 2 g 0i u l = mc 2 h{u —g gL u a ), and after substituting (88.14), takes the form (88.9). In the limiting case of a weak gravitational field and low velocities, by substituting g 00 = 1 + (20/ c 2 ) in (88.9), we get approximately: 2 mv <f = mc 2 +-— + m<£, (88.15) where w<£ is the potential energy of the particle in the gravitational field, which is in agree- ment with the Lagrangian (87.10). PROBLEMS 1. Determine the force acting on a particle in a constant gravitational field. Solution: For the components of r fci which we need, we find the following expressions : 1 00 ry "; > Th = ^{gt B -g B a )-\g e h' a , r a 0y = x a ey + -[ge(g : y a -g%+gy(g : i J a -g%)]+ ^ 9ng?h' a . In these expressions all the tensor operations (covariant differentiation, raising and lowering of indices) are carried out in the three-dimensional space with metric y a0 , on the three-dimensional vector g a and the three-dimensional scalar h (88.11); X% y is the three-dimensional Christoffel symbol, constructed from the components of the tensor y ae in just the same way as r£, is con- structed from the components of g lk ; in the computations we use (84.9-12). Substituting (1) in the equation of motion du a — = -r(w yo 2 -2r^*/v-r« v wV' and using the expression (88.14) for the components of the four-velocity, we find after some simple transformations : The force f acting on the particle is the derivative of its momentum p with respect to the (syn- chronized) proper time, as defined by the three-dimensional covariant differential: L v 2 Dp a I. v 2 d mv a fl V^ 252 PARTICLE IN A GRAVITATIONAL FIELD § 88 From (2) we therefore have (for convenience we lower the index a): \-— I \ ■*" */ C J or, in the usual three-dimensional notation,! f= ^f i-Vln VA + Vfl-x(curlg)i. (3) 7-1 We note that if the body is at rest, then the force acting on it [the first term in (3)] has a potential. For low velocities of motion the second term in (3) has the form mcVh\ x (curl g) analogous to the Coriolis force which would appear (in the absence of the field) in a coordinate system rotating with angular velocity Q = - VAcurlg. 2. Derive Fermat's principle for the propagation of a ray in a constant gravitational field. Solution: Fermat's principle (§ 53) states: S$ k a dx a = 0, where the integral is taken along the ray, and the integral must be expressed in terms of the frequency co (which is constant along the ray) and the coordinate differentials. Noting that k = — dif//dx ^ (g> /c), we write: — = k = g 0i k l = g 00 k° +g 0a k a = h(k° —g a k a ). Substituting this in the relation k t k l = g^Wk* = 0, written in the form h(k° -g* k a f - y aB k a k B = 0, t In three-dimensional curvilinear coordinates, the unit antisymmetric tensor is defined as Vy where e 12 3 = e 123 = 1, and the sign changes under transposition of indices [compare (83.13-14)]. Accord- ingly the vector c = axb, defined as the vector dual to the antisymmetric tensor c By = a B b y —a y b B , has components : Conversely, c a = %Vy e aBy c"" = Vy e aBy a B b\ c a = -i- e ae *c By = ^- e aBr a B b y 2Vy Vy c aB = V~y e aBy c\ c aB = 4= e a ^c y . ■Vy In particular, curl a should be understood in this same sense as the vector dual to the tensor a0:«—a a ;e = (^a B /dx a )—(da a /dx B ), so that its contravariant components are (curlar = ^-^^-^Y 2Vy ydx" dx*J In this same connection we repeat that for the three-dimensional divergence of a vector [see (86.9)]: divsi=~-^- a {Vya a ). To avoid misunderstandings when comparing with formulas frequently used for the three-dimensional vector operations in orthogonal curvilinear coordinates (see, for example, Electrodynamics of Continuous Media, appendix), we point out that in these formulas the components of the vectors are understood to be the quantities Vg^.A\= Va[A*), Vg~ZA 2 , Vg~^ A 3 . § 89 ROTATION 253 we obtain : 1 /m„\ 2 y a0 k a k e = O. 1/cD^ 2 (tJ h Noting that the vector k a must have the direction of the vector dx a , we then find: COp dx a cVh dl where dl (84.6) is the element of spatial distance along the ray. In order to obtain the expression for k a , we write k« = g a % = g a0 k +g ae k ff = -g« — -y ae k , c so that (*-^'M(^+*} Finally, multiplying by dx a , we obtain Fermat's principle in the form (dropping the constant factor c»o/c): In a static field, we have simply: •Kfi*-")'" \ S I -p = 0. Vh We call attention to the fact that in a gravitational field the ray does not propagate along the shortest line in space, since the latter would be defined by the equation S J" dl = 0. § 89. Rotation As a special case of a stationary gravitational field, let us consider a uniformly rotating reference system. To calculate the interval ds we carry out the transformation from a system at rest (inertial system) to the uniformly rotating one. In the coordinates r' , $', z', t of the system at rest (we use cylindrical coordinates r', $', z'), the interval has the form ds 2 = c 2 dt 2 -dr' 2 -r' 2 d4>' 2 -dz' 2 . (89.1) Let the cylindrical coordinates in the rotating system be r, §, z. If the axis of rotation coincides with the axes Z and Z', then we have r' = r, z' = z, $' = <f) + £lt, where Q is the angular velocity of rotation. Substituting in (89.1), we find the required expression for ds 2 in the rotating system of reference : ds 2 = (c 2 -QV) dt 2 -2Qr 2 d(f) dt-dz 2 -r 2 d(f> 2 -dr 2 . (89.2) It is necessary to note that the rotating system of reference can be used only out to distances equal to c/Q. In fact, from (89.2) we see that for r > c/Cl, g 00 becomes negative, which is not admissible. The inapplicability of the rotating reference system at large distances is related to the fact that there the velocity would become greater than the velocity of light, and therefore such a system cannot be made up from real bodies. As in every stationary field, clocks on the rotating body cannot be uniquely synchronized at all points. Proceeding with the synchronization along any closed curve, we find, upon returning to the starting point, a time differing from the initial value by an amount [see 254 PARTICLE IN A GRAVITATIONAL FIELD § 90 (88.5)] 9o* j a IX fir2 # C J #00 c J QV 2„2 or, assuming that Qr/c <^ 1 (i.e. that the velocity of the rotation is small compared with the velocity of light), Q f „ 2Q At= -j\ r 2 dcj) =+-yS, (89.3) where 5 is the projected area of the contour on a plane perpendicular to the axis of rotation (the sign + or — holding according as we traverse the contour in, or opposite to, the direc- tion of rotation). Let us assume that a ray of light propagates along a certain closed contour. Let us cal- culate to terms of order v/c the time t that elapses between the starting out of the light ray and its return to the initial point. The velocity of light, by definition, is always equal to c, if the times are synchronized along the given closed curve and if at each point we use the proper time. Since the difference between proper and world time is of order v 2 /c 2 , then in calculating the required time interval t to terms of order v/c this difference can be neglected. Therefore we have L 2Q „ c c where L is the length of the contour. Corresponding to this, the velocity of light, measured as the ratio L/t, appears equal to c±2Qj. (89.4) This formula, like the first approximation for the Doppler effect, can also be easily derived in a purely classical manner. PROBLEM Calculate the element of spatial distance in a rotating coordinate system. Solution: With the help of (84.6) and (84.7), we find r 2 dd> 2 dl 2 = dr 2 +dz 2 +- * c 2 which determines the spatial geometry in the rotating reference system. We note that the ratio of the circumference of a circle in the plane z — constant (with center on the axis of rotation) to its radius r is 2n/Vl-Q. 2 r 2 /c 2 , i.e. larger than 2n. § 90. The equations of electrodynamics in the presence of a gravitational field The electromagnetic field equations of the special theory of relativity can be easily generalized so that they are applicable in an arbitrary four-dimensional curvilinear system of coordinates, i.e., in the presence of a gravitational field. § 90 EQUATIONS OF ELECTRODYNAMICS IN A GRAVITATIONAL FIELD 255 The electromagnetic field tensor in the special theory of relativity is defined as F ik = (dA k /dx i ) — (dA i Jdx k ). Clearly it must now be defined correspondingly as Fik = A k;i —A i;k . But because of (86.12), and therefore the relation of F ik to the potential A t does not change. Consequently the first pair of Maxwell equations (26.5) also does not change its formf In order to write the second pair of Maxwell equations, we must first determine the current four-vector in curvilinear coordinates. We do this in a fashion completely analogous to that which we followed in § 28. The spatial volume element, constructed on the space coordinate elements dx 1 , dx 2 , and dx 3 , is Vy dV, where y is the determinant of the spatial metric tensor (84.7) and dV = dx 1 dx 2 dx 3 (see the footnote on p. 232). We introduce the charge density q according to the definition de = g^Jy dV, where de is the charge located within the volume element vy dV. Multiplying this equation on both sides by dx 1 , we have: /- q I dx l de dx 1 = q dx 1 Vy dx 1 dx 2 dx 3 = ■ , — v — g d£l ~—q V0oo dx [where we have used the formula —g = yg 00 (84.10)]. The product V —gdQ. is the in- variant element of four-volume, so that the current four- vector is defined by the expression oc dx 1 (90.3) \f g 00 dx° (the quantities dx l jdx° are the rates of change of the coordinates with the "time" x°, and do not constitute a four- vector). The component y° of the current four- vector, multiplied by V g 00 /c, is the spatial density of charge. For point charges the density g is expressed as a sum of (5-functions, as in formula (28.1). We must, however, correct the definition of these functions for the case of curvilinear co- ordinates. By <5(r) we shall again mean the product Six 1 ) d(x 2 ) S(x 3 ), regardless of the geometrical meaning of the coordinates jc 1 , x 2 , x 3 ; then the integral over dV(and not over yJydV) is unity: j" <5(r) dV= 1. With this same definition of the ^-functions, the charge density is Q Q = H~T d ( r -r a X a Vy and the current four-vector is ^E-^O-O™ (90.4) Conservation of charge is expressed by the equation of continuity, which differs from f It is easily seen that the equation can also be written in the form F lk;l + F li ; k + F kl ;i=0, from which its covariance is obvious. 256 PARTICLE IN A GRAVITATIONAL FIELD § 90 (29.4) only in replacement of the ordinary derivatives by covariant derivatives: /; i = -7=1= ■/-. (y/^0 f) = (90.5) V -g ox [using formula (86.9)]. The second pair of Maxwell equations (30.2) is generalized similarly; replacing the ordinary derivatives by covariant derivatives, we find : 1 d i Ait F ik ;k = -= — (y/-g F ik ) = - -/ (90.6) yj-gdx k c [using formula (86.10)]. Finally the equations of motion of a charged particle in gravitational and electromagnetic fields is obtained by replacing the four-acceleration du l /ds in (23.4) by Du l /ds: Du l /du l ■ , A e ., mc — = mc(— +T l kl u k u l ) = - F lk u k . (90.7) ds \ds ) c PROBLEM Write the Maxwell equations in a given gravitational field in three-dimensional form (in the three-dimensional space with metric y aP ), introducing the three-vectors E, D and the antisymmetric three-tensors B ae and H aB according to the definitions : E a = Fo a , Bag = F a p, D a = -V^o F°\ H aB = V^o F aB . (1) Solution: The quantities introduced above are not independent. Writing out the equations F 0a = goi 9a m F lm , F" = g a V m Fi m , and introducing the three-dimensional metric tensor y aB = —g aB +hg a g B [with g and h from (88.11)], and using formulas (84.9) and 84.12), we get: F H aP D* = -^+g>H aft B«e = - 7 =+g< 1 E«-g"Ee. (2) Vh Vh We introduce the vectors B and H, dual to the tensors B aB and H aB , in accordance with the definition : B"=-^ 7 ,e°»B tr , Ha=-\y/ye a ,yH» (3) (see the footnote on p. 252; the minus sign is introduced so that in galilean coordinates the vector: H and B coincide with the ordinary magnetic field intensity). Then (2) can be written in the forms D = -^L+Hxg, B = ^+gxE. (4) Vh Vh Introducing definition (1) in (90.2), we get the equations: dB aB dBya dB Br _ ~dx T + 'dx T ~dx"~ ' dBas , dEa _ dE B _ Ix* ~dx° ~ ~dx« ~ ' or, changing to the dual quantities (3): 1 r) divB = 0, curlE= --(VyB) (5) cy y ot § 90 EQUATIONS OF ELECTRODYNAMICS IN A GRAVITATIONAL FIELD 257 (x° = ct ; the definitions of the operations div and curl are given in the footnote on p. 252). Similarly we find from (90.6) the equations 1 /} _ IP chc. a _ (Vy H«")+ -j= — (V~y D«) = -4ng -—, Vy dx B Vy dx° dx or, in three-dimensional notation : div D = Aizq, curl H = -^= - (Vy D) + — s, (6) cVy dt c where s is the vector with components s a = q dx a \dt. We also write the continuity equation (90.5) in three-dimensional form: 4"--(Vye)+divs-0. (7) Vy dt The reader should note the analogy (purely formal, of course) of equations (5) and (6) to the Maxwell equations for the electromagnetic field in material media. In particular, in a static gravita- tional field the quantity Vy drops out of the terms containing time derivatives, and relation (4) reduces to D = E/Vh, B = H/Vh. We may say that with respect to its effect on the electromagnetic field a static gravitational field plays the role of a medium with electric and magnetic permeabilities e = fi = 1/V/C CHAPTER 11 THE GRAVITATIONAL FIELD EQUATIONS § 91. The curvature tensor Let us go back once more to the concept of parallel displacement of a vector. As we said in § 85, in the general case of a curved four-space, the infinitesimal parallel displacement of a vector is defined as a displacement in which the components of the vector are not changed in a system of coordinates which is galilean in the given infinitesimal volume element. If x 1 = x\s) is the parametric equation of a certain curve (s is the arc length measured from some point), then the vector u l = dx l \ds is a unit vector tangent to the curve. If the curve we are considering is a geodesic, then along it Du l = 0. This means that if the vector u l is subjected to a parallel displacement from a point x' on a geodesic curve to the point x l +dx l on the same curve, then it coincides with the vector u l + du l tangent to the curve at the point x l + dx l . Thus when the tangent to a geodesic moves along the curve, it is displaced parallel to itself. On the other hand, during the parallel displacement of two vectors, the "angle" between them clearly remains unchanged. Therefore we may say that during the parallel displace- ment of any vector along a geodesic curve, the angle between the vector and the tangent to the geodesic remains unchanged. In other words, during the parallel displacement of a vector, its component along the geodesic must be the same at all points of the path. Now the very important result appears that in a curved space the parallel displacement of a vector from one given point to another gives different results if the displacement is carried out over different paths. In particular, it follows from this that if we displace a vector parallel to itself along some closed contour, then upon returning to the starting point, it will not coincide with its original value. In order to make this clear, let us consider a curved two-dimensional space, i.e., any curved surface. Figure 19 shows a portion of such a surface, bounded by three geodesic curves. Let us subject the vector 1 to a parallel displacement along the contour made up of Fig. 19. 258 § 91 THE CURVATURE TENSOR 259 these three curves. In moving along the line AB, the vector 1, always retaining its angle with the curve unchanged, goes over into the vector 2. In the same way, on moving along BC it goes over into 3. Finally, on moving from C to A along the curve CA, maintaining a constant angle with this curve, the vector under consideration goes over into 1', not co- inciding with the vector 1. We derive the general formula for the change in a vector after parallel displacement around any infinitesimal closed contour. This change AA k can clearly be written in the form § SA k , where the integral is taken over the given contour. Substituting in place of 8A k the expression (85.5), we have AA^jrlAidx 1 (91.1) (the vector A x which appears in the integrand changes as we move along the contour). For the further transformation of this integral, we must note the following. The values of the vector A t at points inside the contour are not unique; they depend on the path along which we approach the particular point. However, as we shall see from the result obtained below, this non-uniqueness is related to terms of second order. We may therefore, with the first-order accuracy which is sufficient for the transformation, regard the components of the vector Ai at points inside the infinitesimal contour as being uniquely determined by their values on the contour itself by the formulas 5A t = TlA„dx l , i.e., by the derivatives dAj ~dx l — F" A (91.2) Now applying Stokes' theorem (6.19) to the integral (91.1) and considering that the area enclosed by the contour has the infinitesimal value A/' m , we get: AA,= c{T km Ad cirUAd dx l dx r A/' dT km dx l dKi . w dA i *' A -4-F 1 dx m ' km dx l ■ V 1 dA t dx m Af lm Substituting the values of the derivatives (91.2), we get finally: AA k = $Ri lm A i Af lm , where JRL, is a tensor of the fourth rank: Rklm — arL art, dx l dx n iri r n —T l F" "t" 1 nl L km l nm L kh (91.3) (91.4) That R[ lm is a tensor is clear from the fact that in (91.3) the left side is a vector— the dif- ference AA k between the values of vectors at one and the same point. The tensor R l klm is called the curvature tensor or the Riemann tensor. It is easy to obtain a similar formula for a contravariant vector A k . To do this we note, since under parallel displacement a scalar does not change, that A(A k B k ) = 0, where B k is any covariant vector. With the help of (91.3), we then have A(A k B k ) = A k AB k + B k AA k = ±A k B i R l klm Af lm +B k AA k = = B k (AA k + lrA i R k lm Af ,m ) = 0, or, in view of the arbitrariness of the vector B k , AA k = -iR^A'Af 1 ™. (91.5) 260 THE GRAVITATIONAL FIELD EQUATIONS § 92 If we twice differentiate a vector A t covariantly with respect to x k and x l , then the result generally depends on the order of differentiation, contrary to the situation for ordinary differentiation. It turns out that the difference A i . kil —A ii i ik is given by the same curvature tensor which we introduced above. Namely, one finds the formula ^i;k;i-A i;l;k = A m RJ n kh (91.6) which is easily verified by direct calculation in the locally-geodesic coordinate system Similarly, for a contravariant vector,! A i . k . l -A i . l . k =-A m R i mU . (91.7) Finally, it is easy to obtain similar formulas for the second derivatives of tensors [this is done most easily by considering, for example, a tensor of the form A t B k , and using formulas (91.6) and (91.7); because of the linearity, the formulas thus obtained must be valid for an arbitrary tensor A ik ]. Thus A ik; l; m ~ A ik; m; I = A in R klm + ^nk^ilm- (91.8) Clearly, in a flat space the curvature tensor is zero, for, in a flat space, we can choose coordinates such that over all the space all the r kt = 0, and therefore also R klm = 0. Because of the tensor character of R klm it is then equal to zero also in any other coordinate system. This is related to the fact that in a flat space parallel displacement is a single- valued operation, so that in making a circuit of a closed contour a vector does not change. The converse theorem is also valid: if R[ lm = 0, then the space is flat. Namely, in any space we can choose a coordinate system which is galilean over a given infinitesimal region. I f R kim = 0, then parallel displacement is a unique operation, and then by a parallel dis- placement of the galilean system from the given infinitesimal region to all the rest of the space, we can construct a galilean system over the whole space, which proves that the space is Euclidean. Thus the vanishing or nonvanishing of the curvature tensor is a criterion which enables us to determine whether a space is flat or curved. We note that although in a curved space we can also choose a coordinate system which will be locally geodesic at a given point, at the same time the curvature tensor at this same point does not go to zero (since the derivatives of the r l kl do not become zero along with the r kl ). § 92. Properties of the curvature tensor From the expression (91.4) it follows immediately that the curvature tensor is anti- symmetric in the indices / and m: R klm = — R kmi- (92.1) Furthermore, one can easily verify that the following identity is valid: Rl klm + R mkl + R \mk = 0- (92.2) In addition to the mixed curvature tensor R klm , one also uses the covariant curvature t Formula (91.7) can also be obtained directly from (91.6) by raising the index i and using the symmetry properties of the tensor R iklm (§92). § 92 PROPERTIES OF THE CURVATURE TENSOR 261 tensorf RikM^gMm- (92.3) By means of simple transformations it is easy to obtain the following expressions for Riklm'' D 1 / d dim , d g kl d g iX d g km \ _ ( Riklm== 2Wd? + d^d^'dx k dx m dx'dx 1 ) 9np{kl im km il) ' K } (for actual calculations the last term is more conveniently written as g np (T„ tkl T Ptim - -*■ n, km *■ p, il))' From this expression one sees immediately the following symmetry properties: Riklm = —Rkilm = —Rikml (92.5) Rum = Rlmik, (92.6) i.e. the tensor is antisymmetric in each of the index pairs i, k and /, m, and is symmetric under the interchange of the two pairs with one another. From these formulas it follows, in particular, that all components R iklm , in which i = k or / = m are zero. For R iklm as for R[ lm , the identity (92.2) is valid: Riki m +Rimki + Ru m k = 0. (92.7) Furthermore, from the relations (92.5)-(92.6) it follows that if we cyclically permute any three indices in R iklm and add the three components obtained, then the result will be zero. Finally, we also prove the Bianchi identity : Rlkl;m + R?mk;l + Rilm;k = 0. (92.8) It is most conveniently verified by using a locally-geodesic coordinate system. Because of its tensor character, the relation (92.8) will then be valid in any other system. Differentiating (91.4) and then substituting in it P kl = 0, we find for the point under consideration _dR? kl _ d 2 n d 2 r? k K ikl;m dx m ^m^k famfaV With the aid of this expression it is easy to verify that (92.8) actually holds. From the curvature tensor we can, by contraction, construct a tensor of the second rank. This contraction can be carried out in only one way: contraction of the tensor R iklm on the indices i and k or / and m gives zero because of the antisymmetry in these indices, while contraction on any other pair always gives the same result, except for sign. We define the tensor R ik (the Ricci tensor) asf R ik = 9 m Riimk — Riik- (92.9) According to (91.4), we have: dx l dx k This tensor is clearly symmetric: ** = *«. (92.11) t In this connection it would be more correct to use the notation iV^m which clearly shows the position of the index which has been raised. % In the literature one also finds another definition of the tensor R ik , using contraction of R iMm on the first and last indices. This definition differs in sign from the one used here. Rik = -^--~ k +T l ik TZ-Tr l T[ m . (92.10) 262 THE GRAVITATIONAL FIELD EQUATIONS § 92 Finally, contracting R ik , we obtain the invariant R = g ik R ik = g il 9 km Rm m , (92.12) which is called the scalar curvature of the space. The components of the tensor R ik satisfy a differential identity obtained by contracting the Bianchi identity (92.8) on the pairs of indices ik and In: R ^ = 2^ (92.13) Because of the relations (92.5-7) not all the components of the curvature tensor are independent. Let us determine the number of independent components. The definition of the curvature tensor as given by the formulas written above applies to a space of an arbitrary number of dimensions. Let us first consider the case of two dimen- sions, i.e. an ordinary surface; in this case (to distinguish them from four-dimensional quantities) we denote the curvature tensor by P abci and the metric tensor by y ab , where the indices a, b, . . . run through the values 1, 2. Since in each of the pairs ab and cd the two indices must have different values, it is obvious that all the non-vanishing components of the curvature tensor coincide or differ in sign. Thus in this case there is only one independent component, for example P 1212 . It is easily found that the scalar curvature is p = "~^T~' y - M = yiiV22-(yi2) • (92.14) The quantity P/2 coincides with the Gaussian curvature K of the surface: P 1 o - " - (92.15) 2 QiQ 2 v ' where the q u q 2 are the principal radii of curvature of the surface at the particular point (remember that q ± and q 2 are assumed to have the same sign if the corresponding centers of curvature are on one side of the surface, and opposite signs if the centers of curvature lie on opposite sides of the surface; in the first case K> 0, while in the second K< O.f Next we consider the curvature tensor in three-dimensional space; we denote it by P ap d and the metric tensor by y ap , where the indices a, p run through values 1, 2, 3. The index pairs ccp and yd run through three essentially different sets of values: 23, 31, and 12 (per- mutation of indices in a pair merely changes the sign of the tensor component). Since the tensor P aPy5 is symmetric under interchange of these pairs, there are all together 3-2/2 independent components with different pairs of indices, and three components with identical pairs. The identity (92.7) adds no new restrictions. Thus, in three-dimensional space the curvature tensor has six independent components. The symmetric tensor P ap has the same number. Thus, from the linear relations P aB = g yd P yaSf} all the components of the tensor P afty5 can be expressed in terms of P ap and the metric tensor y ap (see problem 1). If we choose a system of coordinates that is cartesian at the particular point, then by a t Formula (92.15) is easy to get by writing the equation of the surface in the vicinity of the given point (x = y = 0) in the form z = ix 2 ^) + (y 2 /2e 2 ). Then the square of the line element on it is dl* = (l+ ^) dx 2 + (l + £) dy*+2^-dxdy. Calculation of P 12 \2 at the point x = y = using formula (92.4) (in which only terms with second derivatives of the y aB are needed) leads to (92.15). § 92 PROPERTIES OF THE CURVATURE TENSOR 263 suitable rotation we can bring the tensor P ap to principal axes.f Thus the curvature tensor of a three-dimensional space at a given point is determined by three quantities. $ Finally we go to four-dimensional space. The pairs of indices ik and Im in this case run through six different sets of values: 01, 02, 03, 23, 31, 12. Thus there are six components of R iklm with identical, and 6-5/2 with different, pairs of indices. The latter, however, are still not independent of one another; the three components for which all four indices are different are related, because of (92.7), by the identity: ^0123 + ^0312 + ^0231 =0- (92.16) Thus, in four-space the curvature tensor has a total of twenty independent components. By choosing a coordinate system that is galilean at the given point and considering the transformations that rotate this system (so that the g ik at the point are not changed), one can achieve the vanishing of six of the components of the curvature tensor (since there are six independent rotations of a four-dimensional coordinate system). Thus, in the general case the curvature of four-space is determined at each point by fourteen quantities. If R. k = 0, § then the curvature tensor has a total of ten independent components in an arbitrary coordinate system. By a suitable transformation we can then bring the tensor Rikim ( at tne given point of four-space) to a "canonical" form, in which its components are expressed in general in terms of four independent quantities ; in special cases this number may be even smaller. (The classification of the possible canonical types for the tensor R iklm was found by A. Z. Petrov, 1950; see problem 3.) If, however, R ik # 0, then the same classification can be used for the curvature tensor after one has subtracted from it a particular part that is expressible in terms of the com- ponents R ik . Namely, we construct the tensor^f Ciklm = Riklm~iRil9km + iKim9kl + $Rkl9im — %Rkm9u + iR(9il9km~9lm9kl)- (92.17) It is easy to see that this tensor has all the symmetry properties of the tensor R ikUn , but vanishes when contracted on a pair of indices (// or km). PROBLEMS 1. Express the curvature tensor P aei6 of three-dimensional space in terms of the second-rank tensor P aB . Solution: We look for P aByd in the form P<x0y« = Aay7ed — A a6 7/Sy + A 06 lay Afiyjad, t For the actual determination of the principal values of the tensor P a0 there is no need to transform to a coordinate system that is cartesian at the given point. These values can be found by determining the roots X of the equation \P ae ~Xy ae \=0. t Knowledge of the tensor P aevd enables us to determine the Gaussian curvature K of an arbitrary surface in the space. Here we note only that if the x 1 , x 2 , x 2 are an orthogonal coordinate system, then 7iiy 2 2— (712) 2 is the Gaussian curvature for the "plane" perpendicular (at the given point) to the x 3 axis; by a "plane" we mean a surface formed by geodesic lines. § We shall see later (§ 95) that the curvature tensor for the gravitational field in vacuum has this property. If This complicated expression can be written more compactly in the form : Ciklm = Riklm — Rlliffklm-'r Rmli0kll + iRgilldklm, where the square brackets imply antisymmetrization over the indices contained in them : Auk-} = KAik—Akt). 264 THE GRAVITATIONAL FIELD EQUATIONS § 92 which satisfies the symmetry conditions; here A aB is some symmetric tensor whose relation to P a0 is determined by contracting the expression we have written on the indices a and y. We thus find: PaB=Ay a0 +A a0 , A a0 =P ae —iPy a0 , and finally, p PaBYd =PccYYea—P a dy0v+PBSyccY—P0yy a d+ j (yadypY — yaYVBi)' 2. Calculate the components of the tensors R mm and R ik for a metric in which a ik = for / i= k (B. K. Harrison, 1960). Solution: We represent the nonzero components of the metric tensor in the form Qa = e t e 2F <, e = 1, e a = — 1. The calculation according to formula (92.4) gives the following expressions for the nonzero com- ponents of the curvature tensor: Rim = e l e 2F ' [F,. k F ki +F Uk F u t -F,. t F u k -F u ,. J, i^k^l; Ruu = e 1 e^{F Ui F l , i -Fl i -F ul , i )+e l e^{F l , l F ul -Fl l -F ul , l )- (no summation over repeated indices!). The subscripts preceded by a comma denote ordinary differentiation with respect to the corresponding coordinate. Contracting the curvature tensor on two indices, we obtain : i? Jfc = S (F,.*F fc . ( +F f>fc F u -F, i ,F, ifc -F, 1 ,. fc ),i#*; Ru= S[F M F i . J -Ff. J -F / . J , i + ei ^e 2 ^-^(F J>! F M -F i 2 >i -F M>I -F <>I £ F MlI )]. 3. Consider the possible types of canonical forms of the curvature tensor when R ik = 0. Solution: We shall assume that the metric at the given point in four-dimensional space has been brought to galilean form. We write the set of twenty independent components of the tensor R iklm as a collection of three three-dimensional tensors defined as follows: AaB — RoccO0> CaB = i^ayi^PAu RydAn, BaB = 2 e «Yi RoBYd (1) (e aPY is the unit antisymmetric tensor; since the three-dimensional metric is cartesian, there is no need to deal with the difference between upper and lower indices in the summation). The tensors A aB and C ae are symmetric by definition; the tensor B ae is asymmetric, while its trace is zero because of (92.16). According to the definitions (1) we have, for example, -"11 == -Roi23» B12 — -ft()131> Bx3 = i?0112> di = /?2323> • • • It is easy to see that the conditions R km =g il R mm = are equivalent to the following relations between the components of the tensors (1): Aaa = 0, BaB = Bff a , AaB = — CaB- (2) We also introduce the symmetric complex tensor D a0 = i(AaB + 2iB a g-Ca ) = A a B + iB aB . (3) This combining of the two real three-dimensional tensors A aB and B a0 into one complex tensor corresponds precisely to the combination (in § 25) of the two vectors E and H into the complex vector F, while the resulting relation between D aB and the four-tensor R mm corresponds to the relation between F and the four-tensor F lk . It then follows that four-dimensional transformations of the tensor R iklm are equivalent to three-dimensional complex rotations carried out on the tensor D ae . With respect to these rotations one can define eigenvalues k = k'+ik" and eigenvectors n a (complex, in general) as solutions of the system of equations D a gn = kn a . (4) § 92 PROPERTIES OF THE CURVATURE TENSOR 265 The quantities k are the invariants of the curvature tensor. Since the trace #«« = 0, the sum of the roots of equation (4) is zero : A U) +A (2) +A (3) =0> Depending on the number of independent eigenvectors n a , we arrive at the following classification of possible cases of reduction of the curvature tensor to the canonical Petrov types I-III. (I) There are three independent eigenvectors. Then their squares n a n a are different from zero and by a suitable rotation we can bring the tensor D a0 , and with it A a0 and B ttP , to diagonal form: nay \ IX 1 ** \ A ae =i x*y , *„,= V* (I) \ -A^'- W \ -X™"- W In this case the curvature tensor has four independent invariants.t The complex invariants X a \ A (2) are expressed algebraically in terms of the complex scalars h = ~ (Rm m R iklm -iRm m R iklm ), / a = 1 (R iklm R**vr Rpr tic + i R mm R^4 pT ac )t where the asterisk over a symbol denotes the dual tensor: * Riklm = ^Eucpr R Pr im- Calculating h and I 2 using (I), we obtain: h = K^ (1)2 + ^ (2)2 + * <3>2 )> 7 2 = P (1) ^ 2) U (1) + A< 2 >). (5) These formulas enable us to calculate X a \ A (2) starting from the values of R mm in any reference system. (II) There are two independent eigenvectors. The square of one of them is then equal to zero, so that it cannot be chosen as the direction of one of the coordinate axes. One can, however, take it to lie in the x 1 , x 2 plane; then n 2 = in u n 3 = 0. The corresponding equations (4) give: #ii+i D 12 = X, D 22 -i D 12 = X, so that D n = X— in, D 22 = X+in, D 12 = n. The complex quantity X = X'+iX" is a scalar and cannot be changed. But the quantity n can be given any nonzero value by a suitable complex rotation; we can therefore, without loss of generality, assume it to be real. As a result we get the following canonical type for the real tensors A ae and Bag'. IX' n \ IX"-n \ A aB = lfi X' 1, B ap = l o r+ M . (II) \0 -2X'I \ -2k' J In this case there are just two invariants k' and k". Then, in accordance with (5), h = k 2 , h = k 3 , so that I\ = 11. (III) There is just one eigenvector, and its square is zero. All the eigenvalues k are then identical and consequently equal to zero. The solutions of equations (4) can be brought to the form #n = #22 = #12 = 0, Di3 = M, #23 = m, so that /0 A /0 0\ A aB = (o 0), B a , = lo A (III) \ji 0/ \0 n 0/ In this case the curvature tensor has no invariants at all and we have a peculiar situation : the four- space is curved, but there are no invariants which could be used as a measure of its curvature. (The same situation occurs in the degenerate case (II) when k' = k" = 0; this case is called type N.) t The degenerate case when k ay = A (2) ', k a) " = A (2) " is called Type D in the literature. 266 THE GRAVITATIONAL FIELD EQUATIONS § 93 § 93. The action function for the gravitational field To arrive at the equations determining the gravitational field, it is necessary first to determine the action S g for this field. The required equations can then be obtained by varying the sum of the actions of field plus material particles. Just as for the_electromagnetic field, the action S g must be expressed in terms of a scalar integral j" (?V -g dQ, taken over all space and over the time coordinate x° between two given values. To determine this scalar we shall start from the fact that the equations of the gravitational field must contain derivatives of the "potentials" no higher than the second (just as is the case for the electromagnetic field). Since the field equations are obtained by varying the action, it is necessary that the integrand G contain derivatives of g ik no higher than first order; thus G must contain only the tensor g ik and the quantities rj However, it is impossible to construct an invariant from the quantities g ik and r£, alone. This is immediately clear from the fact that by a suitable choice of coordinate system we can always make all the quantities r kl zero at a given point. There is, however, the scalar R (the curvature of the four-space), which though it contains in addition to the g ik and its first derivatives also the second derivatives ofj^, is linear in the second derivatives. Because of this linearity, the invariant integral J" R^J-g dQ can be transformed by means of Gauss' theorem to the integral of an expression not containing the second derivatives. Namely, j" R\l -g dQ can be presented in the form j R^g dCl = j Gj—g dCl + j *£^5 dn , where G contains only the tensor g ik and its first derivatives, and the integrand of the second integral has the form of a divergence of a certain quantity w l (the detailed calculation is given at the end of this section). According to Gauss' theorem, this second integral can be transformed into an integral over a hypersurface surrounding the four-volume over which the integration is carried out in the other two integrals. When we vary the action, the variation of the second term on the right vanishes, since in the principle of least action, the variations of the field at the limits of the region of integration are zero. Consequently, we may write <5 j Ry/-gdQ = d j GyJ-g dQ. The left side is a scalar; therefore the expression on the right is also a scalar (the quantity G itself is, of course, not a scalar). The quantity G satisfies the condition imposed above, since it contains only the g ik and its derivatives. Thus we may write "•--isW^^'-isW*^* 1 ' (931) where k is a new universal constant. Just as was done for the action of the electromagnetic field in § 27, we can see that the constant k must be positive (see the end of this section). The constant k is called the gravitational constant. The dimensions of A: follow from (93. 1). The action has dimensions gm-cm 2 -sec _1 ; all the coordinates have the dimensions cm, the g ik are dimensionless, and so R has dimensions cm -2 . As a result, we find that k has the dimensions cm 3 -gm -1 -sec~ 2 . Its numerical value is k = 6.67 x 10" 8 cm 3 -gm _1 -sec~ 2 . (93.2) § 93 THE ACTION FUNCTION FOR THE GRAVITATIONAL FIELD 267 We note that we could have set k equal to unity (or any other dimensionless constant). However, this would fix the unit of mass.f Finally, let us calculate the quantity G of (93.1). From the expression (92.10) for R ik , we have / ~ ~ / ik r» / I ik^ik „ik°* H , n ikr-l rm _ n ik T m T l I J -g R = j -g g lk R ik = \/ -g Y M~ 9 fa* 9 ik lm 9 1 " 1 * M J- In the first two terms on the right, we have Dropping the total derivatives, we find V^ g = rr m ^ (V^ rt-ft £i (V^ ^ fc )-(nrL-r:,r»'V^. With the aid of formulas (86.5)-(86.8), we find that the first two terms on the right are equal to y/-g multiplied by = 2g v (r™r fcm — r lfe rj^). Finally, we have G = g i Xr7 l r km -r\ k rr m )- (93-3) The components of the metric tensor are the quantities which determine the gravitational field. Therefore in the principle of least action for the gravitational field it is the quantities g ik which are subjected to variation. However, it is necessary here to make the following fundamental reservation. Namely, we cannot claim now that in an actually realizable field the action integral has a minimum (and not just an extremum) with respect to all possible variations of the g ik . This is related to the fact that not every change in the g ik is associated with a change in the space-time metric, i.e. with a real change in the gravitational field. The components g ik also change under a simple transformation of coordinates connected merely with the shift from one system to another in one and the same space-time. Each such coordinate transformation is generally an aggregate of four independent transformations. In order to exclude such changes in g ik which are not associated with a change in the metric, we can impose four auxiliary conditions and require the fulfillment of these conditions under the variation. Thus, when the principle of least action is applied to a gravitational field, we t If one sets k = c 2 , the mass is measured in cm, where 1 cm - 1.35 x 10 28 gm. Sometimes one uses in place of k the quantity x = !?* = 1.86 x 10- 27 cm gm- 1 , c 2 which is called the Einstein gravitational constant. 268 THE GRAVITATIONAL FIELD EQUATIONS § 94 can assert only that we can impose auxiliary conditions on the g ik , such that when they are fulfilled the action has a minimum with respect to variations of the g ik .\ Keeping these remarks in mind, we now show that the gravitational constant must be positive. As the four auxiliary conditions mentioned, we use the vanishing of the three components g 0a , and the constancy of the determinant \g afi \ made up from the components 0o« = 0, \g ap \ = const; from the last of these conditions we have n *» d 9*p d i | n 9 ^ = d?\9 aP \-0. We are here interested in those terms in the integrand of the expression for the action which contain derivatives of g ik with respect to x° (cf. p. 68). A simple calculation using (93.3) shows that these terms in G are _ t n *P n r* n oo dffay ddps 4 9 9 9 dx° 8x°- It is easy to see that this quantity is essentially negative. Namely, choosing a spatial system of coordinates which is cartesian at a given point at a given moment of time (so that 9*p = sf" = -<5«/})> we obtain: _ i „oo f d San\ 2 4 9 \dx )' and, since g°° = l/g 00 > 0, the sign of the quantity is obvious. By a sufficiently rapid change of the components g afi with the time x° (within the time interval between the limits of integration of x°) the quantity G can consequently be made as large as one likes. If the constant k were negative, the action would then decrease without limit (taking on negative values of arbitrarily large absolute magnitude), that is, there could be no minimum. § 94. The energy-momentum tensor In § 32 the general rule was given for calculating the energy-momentum tensor of any physical system whose action is given in the form of an integral (32.1) over four-space. In curvilinear coordinates this integral must be written in the form (94.1) (in galilean coordinates g = - 1, and S goes over into $AdVdt). The integration extends over all the three-dimensional space and over the time between two given moments, i.e., over the infinite region of four-space contained between two hypersurfaces. As already discussed in § 32, the energy-momentum tensor, calculated from the formula (32.5), is generally not symmetric, as it should be. In order to symmetrize it, we had to add t We must emphasize, however, that everything we have said has no effect on the derivation of the field equations from the principle of least action (§ 95). These equations are already obtained as a result of the requirement that the action be an extremum (i.e., vanishing of the first derivative), and not necessarily a minimum. Therefore in deriving them we can vary all of the g ik independently. § 94 THE ENERGY-MOMENTUM TENSOR 269 to (32.5) suitable terms of the form (d/8x l )\l/ m , where ^ m = -il/ ilk . We shall now give another method of calculating the energy-momentum tensor which has the advantage of leading at once to the correct expression. In (94.1) we carry out a transformation from the coordinates x l to the coordinates x H = x l +C, where the §' are small quantities. Under this transformation the g vk are trans- formed according to the formulas : g (x)-g (x) dxl dxm -g \b l+ dxl J \d m + dxm J ~ a * (x i ) + a im<!? + a?? ~g {x)+g dxm +g dxl . Here the tensor g' ik is a function of the x' 1 , while the tensor g ik is a function of the original coordinates x l . In order to represent all terms as functions of one and the same variables, we expand g' ik (x l + £ l ) in powers of £'. Furthermore, if we neglect terms of higher order in £', we can in all terms containing £', replace g nk by g ik . Thus we find r)a ik dl k d£ l n >ik, 1} _ n ik( x l\_fl V JL_ ± JU, 1^_ ■ fl « li. g (x)-g (x) £ dxl +g ^+g ^ It is easy to verify by direct trial that the last three terms on the right can be written as a sum £ i; k + £ fe; ' of contravariant derivatives of the £'. Thus we finally obtain the transforma- tion of the g ik in the form g ' ik = g ik + 5g ik , dg ik = £ i; * + f* ;l . (94.2) For the co variant components, we have : G'ik = dik + $9ik, $9ik = ~ Zi; k-£k;i (94.3) (so that, to terms of first order we satisfy the condition g iX g' kl = <5*).f Since the action S is a scalar, it does not change under a transformation of coordinates. On the other hand, the change SS in the action under a transformation of coordinates can be written in the following form. As in § 32, let q denote the quantities defining the physical system to which the action S applies. Under coordinate transformation the quantities q change by bq. In calculating SS we need not write terms containing the changes in q. All such terms must cancel each other by virtue of the "equations of motion" of the physical system, since these equations are obtained by equating to zero the variation of S with respect to the quantities q. Therefore it is sufficient to write the terms associated with changes in the g ik . Using Gauss' theorem, and setting Sg lk = at the integration limits, we find SS in the formj f We note that the equations £i;fc + {k;i = determine the infinitesimal coordinate transformations that do not change the metric. In the literature these are often called the Killing equations. t It is necessary to emphasize that the notation of differentiation with respect to the components of the symmetric tensor g ik , which we introduce here, has in a certain sense a symbolic character. Namely, the derivative dF/dg (k (F is some function of the g ik ) actually has a meaning only as the expression of the fact that dF= (dF/dg (k )dg ik . But in the sum (dF/dg tk )dg ik , the terms with differentials dg ik , of components with i # k, appear twice. Therefore in differentiating the actual expression for F with respect to any definite component g ik with i ^ k, we would obtain a value which is twice as large as that which we denote by 3Fldg lk . This remark must be kept in mind if we assign definite values to the indices /, k, in formulas in which the derivatives with respect to g tk appear. 270 THE GRAVITATIONAL FIELD EQUATIONS § 94 Iff^A aV- dx l „ a^* a a7 fir A dg'° dfi <V' fc dfi. Here we introduce the notation -V-0T tt = d\J — g A d d\J — gA dg ik dx l a d^" Then 55 takes the formf dS = 2c J T^J-Q da= ~h\ Tikd 9^~V dQ. (94.4) (94.5) (note that g ik dg lk = -g lk 5g ik , and therefore T ik dg ik = -T ik dg ik ). Substituting for Sg ik the expression (94.2), we have, making use of the symmetry of the tensor T ik , 5S = t\ T i^ i '' k + ^' i ^ Zr d <M = lj T ik e' k yl^g da. Furthermore, we transform this expression in the following way : SS = c / (^ W-^ dn ~ \ j Tukt'yf^g d^ (94.6) Using (86.9), the first integral can be written in the form d and transformed into an integral over a hypersurface. Since the £' vanish at the limits of integration, this integral drops out. Thus, equating SS to zero, we find Ij^M-gmdn, SS = - 1 c JT k iik ?y/-gdQ = 0. Because of the arbitrariness of the <f it then follows that T k ;k = 0. (94.7) Comparing this with equation (32.4) dT ik jd^ = 0, valid in galilean coordinates, we see that the tensor T ik , defined by formula (94.4), must be identical with the energy-momentum tensor — at least to within a constant factor. It is easy to verify, carrying out, for example, the calculation from formula (94.4) for the electromagnetic field ( A --is '•*•-- is ™-« v "> that this factor is equal to unity. t In the case we are considering, the ten quantities Sg ik are not independent, since they are the result of a transformation of the coordinates, of which there are only four. Therefore from the vanishing of SS it does not follow that T ik = 0\ § 94 THE ENERGY-MOMENTUM TENSOR 271 Thus, formula (94.4) enables us to calculate the energy-momentum tensor by dif- ferentiating the function A with respect to the components of the metric tensor (and their derivatives). The tensor T ik obtained in this way is symmetric. Formula (94.4) is convenient for calculating the energy-momentum tensor not only in the case of the presence of a gravita- tional field, but also in its absence, in which case the metric tensor has no independent significance and the transition to curvilinear coordinates occurs formally as an intermediate step in the calculation of T ik . The expression (33.1) for the energy-momentum tensor of the electromagnetic field must be written in curvilinear coordinates in the form T ik = ^ (-F n F k l + \ F lm F lm g ik \ (94.8) For a macroscopic body the energy-momentum tensor is T ik = (p + s)u i u k -pg ik . (94.9) We note that the quantity T 00 is always positive :f r 00 > 0. (94.10) (No general statement can be made about the mixed component T%.) PROBLEM Consider the possible cases of reduction to canonical form of a symmetric tensor of second rank in a pseudo-euclidean space. Solution: The reduction of a symmetric tensor A ik to principal axes means that we find "eigen- vectors" n l for which A ik n k = Xn { . (1) The corresponding principal (or "proper") values X are obtained from the condition for consistency of equation (1), i.e. as the roots of the fourth degree equation \A ik -Xg ik \=0, (2) and are invariants of the tensor. Both the quantities X and the eigenvectors corresponding to them may be complex. (The components of the tensor A ik itself are of course assumed to be real.) From equation (1) it is easily shown in the usual fashion that two vectors w t (1) and « ( (2) which correspond to different principal values X w and A (2) are "mutually perpendicular" : „w n wi = 0, (3) In particular, if equation (2) has complex-conjugate roots X and X*, to which there correspond the complex-conjugate vectors n t and n*, then we must have n i n t *=0. (4) The tensor A ik is expressed in terms of its principal values and the corresponding eigenvectors by the formula 4 = s^-^ (5) (so long as none of the quantities n t n 1 is equal to zero — cf. below). t We have T 00 = eu%+p(ul—g o)- The first term is always positive. In the second term we write g odx°+g 0a dx a u =goou +g 0a u a =' ds and obtain after a simple transformation g op(dl/ds) 2 , where dl is the element of spatial distance (84.6); from this it is clear that the second term of T Q0 is also positive. The same result can also be shown for the tensor (94.8). 272 THE GRAVITATIONAL FIELD EQUATIONS § 95 Depending on the character of the roots of equation (2), the following three situations may occur. (a) All four eigenvalues X are real. Then the vectors n t are also real, and since they are mutually perpendicular, three of them must have spacelike directions and one a timelike direction (and are normalized by the conditions n t n l = — 1 and itxn 1 = 1, respectively). Choosing the directions of the coordinates along these vectors, we bring the tensor A ik to the form 'F 4 = l o o -x* o ' (6) -X^j (b) Equation (2) has two real roots (7 (2) , A (3) ) and two complex-conjugate roots (X'±iX"). We write the complex-conjugate vectors n u nf, corresponding to the last two roots in the form a t ±ibi; since they are defined only to within an arbitrary complex factor, we can normalize them by the condition riiii 1 = n*n l * = 1. Also using equation (4), we find a i a l +b i b i =0 t a { b l = 0, a l a i -b l b i ^\, so that «i« i= =J, bib l = —^, i.e. one of these vectors must be spacelike and the other timelike, f Choosing the coordinate axes along the vectors a 1 , b\ n (2H , n i3)i , we bring the tensor to the form: X' X" , X" ~X' . 4 = ' o -*» o I (7) -a< 3 \ (c) If the square of one of the vectors n l is equal to zero {riin 1 = 0), then this vector cannot be chosen as the direction of a coordinate axis. We can however choose one of the planes x°, x" so that the vector n l lies in it. Suppose this is the jt°, x 1 plane; then it follows from Wjw' =0 that n° = n 1 , and from equation (1) we have ,4oo+^oi = X, A^+A^ = -X, so that A X1 = — X+n, Aoo = X+n, Aox = — n, where n is a quantity which is not invariant but changes under rotations in the x°, x 1 plane; it can always be made real by a suitable rotation. Choosing the axes x 2 , x 3 along the other two (spacelike) vectors « (2)i , « (3)i , we bring the tensor A ik to the form 'X+fi -n — fi —X+m 0. ^ fc = l o -*» o I (8) -x&> This case corresponds to the situation when two of the roots (A< 0) , A (1) ) of equation (2) are equal. We note that for the physical energy-momentum tensor T lk of matter moving with velocities less than the velocity of light only case (a) can occur; this is related to the fact that there must always exist a reference system in which the flux of the energy of the matter, i.e. the components T aQ are equal to zero. For the energy-momentum tensor of electromagnetic waves we have case (c) with X& = A (3) = X = (cf. p. 82); it can be shown that if this were not the case there would exist a reference frame in which the energy flux would exceed the value c times the energy density. § 95. The gravitational field equations We can now proceed to the derivation of the equations of the gravitational field. These equations are obtained from the principle of least action 5(S m +S g ) = 0, where S g and S m are the actions of the gravitational field and matter respectively. We now subject the gravitational field, that is, the quantities g ik , to variation. t Since only one of the vectors can have a timelike direction, it then follows that equation (2) cannot have two pairs of complex-conjugate roots. § 95 THE GRAVITATIONAL FIELD EQUATIONS 273 Calculating the variation 5S g , we have 5 j Ry/^j dQ = dj g ik R ik y/~g dQ = J* {R^yf^i 5g ik + R ik g ik 8y/~g + g ik yr^g 5R ik }dQ. From formula (86.4), we have &\l-g = -— t=<5#= - -V^7 g ik 8g ik ; 2 v -g z substituting this, we find <5 j R^~gdQ = J {R ik -^g ik RW k sJ~g dQ + j g ik 5R ik y/~^g dQ. (95.1) For the calculation of 5R ik we note that although the quantities r£, do not constitute a tensor, their variations 3P kl do form a tensor, for T k t A k dx l is the change in a vector under parallel displacement [see (85.5)] from some point P to an infinitesimally separated point P'. Therefore dT k u A k dx l is the difference between the two vectors, obtained as the result of two parallel displacements (one with the unvaried, the other with the varied T l kl ) from the point P to one and the same point P '. The difference between two vectors at the same point is a vector, and therefore 5T kl is a tensor. Let us use a locally geodesic system of coordinates. Then at that point all the T kl = 0. With the help of expression (92. 10) for the R ik , we have (remembering that the first derivatives of the g lk are now equal to zero) a*XR - n ik I 9 ST 1 —fip\- a ik — 5P - a il — 8T k - — g3R *~ 9 \dx~i 5Tik -dx kdTil r 9 dx l6Tik 9 dx ldrik ~dx 1 ' where W l = g ik dP ik -g il 5r k k . Since w l is a vector, we may write the relation we have obtained, in an arbitrary coordinate system, in the form g ik ZR ik = [replacing dw l /dx l by w l . , and using (86.9)]. Consequently the second integral on the right side of (95.1) is equal to j g ik dR ik sf^gdQ = j d(y/-g w l ) dQ, 8x l and by Gauss' theorem can be transformed into an integral of w l over the hypersurface surrounding the whole four- volume. Since the variations of the field are zero at the integra- tion limits, this term drops out. Thus, the variation 5S g is equal tof J (R ik - \ 9ik^) §g ik J^g dQ. (95.2) We note that if we had started from the expression c 3 ^--issj^-' dQ. t We note here the following curious fact. If we calculate the variation S J R V —g dCl [with R llc from (92.10)], considering the r£, as independent variables and the g ik as constants, and then use expression (86.3) for the TU, we would obtain, as one easily verifies, identically zero. Conversely, one could determine the relation between the r^, and the metric tensor by requiring that the variation we have mentioned should vanish. 274 THE GRAVITATIONAL FIELD EQUATIONS for the action of the field, then we would have obtained § 95 dS= - 9 16nk c 3 C (d(G^-g) d d(G^-g)] I dg l dx l ox Sg ik dQ. Comparing this with (95.2), we find the following relation: 1 1 R ik~ ~ 9ikR — ~7= 1 yJ-g d(Gy/-g) d d{G^g) dg ik dx l dg* dx l (95.3) For the variation of the action of the matter we can write immediately from (94.5): SSm = 2c\ T * d 9 ik ^~9 dQ, (95.4) where T ik is the energy-momentum tensor of the matter (including the electromagnetic field). Gravitational interaction plays a role only for bodies with sufficiently large mass (because of the smallness of the gravitational constant), and therefore in studying the gravitational field we usually have to deal with macroscopic bodies. Corresponding to this we must usually write for T ik the expression (94.9). Thus, from the principle of least action 3S m + SS g = we find: ~ 3 * ' ' 8nk I6nk J \Rit- j 9ikR~ -^ T ik ) 5g lk yl-g dQ = 0, from which, in view of the arbitrariness of the Sg ik. or, in mixed components, 1 Snk 2 c 2 c (95.5) (95.6) These are the required equations of the gravitational field— the basic equation of the general theory of relativity. They are called the Einstein equations. Contracting (95.6) on the indices i and k, we find ink R= - T; (T = T'i). Therefore the equations of the field can also be written in the form Snk R ik = ~T c yTik-^dikTJ. (95.7) (95.8) Note that the equations of the gravitational field are nonlinear equations. Therefore for gravitational fields the principle of superposition is not valid, contrary to the case for the electromagnetic field in the special theory of relativity. It is necessary, however, to remember that actually one has usually to deal with weak gravitational fields, for which the equations of the field in first approximation are linear (see the following section). For such fields, in this approximation, the principle of superposition is valid. § 95 THE GRAVITATIONAL FIELD EQUATIONS 275 In empty space T ik = 0, and the equations of the gravitational field reduce to the equation R ik = 0. (95.9) We mention that this does not at all mean that in vacuum, spacetime is flat; for this we would need the stronger conditions R l klm = 0. The energy-momentum tensor of the electromagnetic field has the property that T\ = [see (33.2)]. From (95.7), it follows that in the presence of an electromagnetic field without any masses the scalar curvature of spacetime is zero. As we know, the divergence of the energy-momentum tensor is zero: 7t ;fc = 0; (95.10) therefore the divergence of the left side of equation (95.6) must be zero. This is actually the case because of the identity (92.13). Thus the equation (95.10) is essentially contained in the field equations (95.6). On the other hand, the equation (95.10), expressing the law of conservation of energy and momen- tum, contains the equation of motion of the physical system to which the energy-momentum tensor under consideration refers (i.e., the equations of motion of the material particles or the second pair of Maxwell equations). Thus the equations of the gravitational field also contain the equations for the matter which produces this field. Therefore the distribution and motion of the matter producing the gravitational field cannot be assigned arbitrarily. On the contrary, they must be determined (by solving the field equations under given initial conditions) at the same time as we find the field produced by the matter. We call attention to the difference in principle between the present situation and the one we had in the case of the electromagnetic field. The equations of that field (the Maxwell equations) contain only the equation of conservation of the total charge (the continuity equation), but not the equations of motion of the charges themselves. Therefore the distribu- tion and motion of the charges can be assigned arbitrarily, so long as the total charge is constant. Assignment of this charge distribution then determines, through Maxwell's equations, the electromagnetic field produced by the charges. We must, however, make it clear that for a complete determination of the distribution and motion of the matter in the case of the Einstein equations one must still add to them the equation of state of the matter, i.e. an equation relating the pressure and density. This equation must be given along with the field equations.! The four coordinates x l can be subjected to an arbitrary transformation. By means of these transformations we can arbitrarily assign four of the ten components of the tensor g ik . Therefore there are only six independent quantities g ik . Furthermore, the four components of the four-velocity u\ which appear in the energy-momentum tensor of the matter, are related to one another by u% = 1, so that only three of them are independent. Thus we have ten field equations (95.5) for ten unknowns, namely, six components of g ik , three components of u\ and the density e/c 2 of the matter (or its pressure p). For the gravitational field in vacuum there remain a total of six unknown quantities (components of g ik ) and the number of independent field equations is reduced corres- pondingly: the ten equations R ik = are connected by the four identities (92.13). t Actually the equation of state relates to one another not two but three thermodynamic quantities, for example the pressure, density and temperature of the matter. In applications in the theory of gravitation, this point is however not important, since the approximate equations of state used here actually do not depend on the temperature (as, for example, the equation p = for rarefied matter, the limiting extreme-relativistic equation p = e/3 for highly compressed matter, etc.). 276 THE GRAVITATIONAL FIELD EQUATIONS § 95 We mention some peculiarities of the structure of the Einstein equations. They are a system of second-order partial differential equations. But the equations do not contain the time derivatives of all ten components g ik . In fact it is clear from (92.4) that second derivatives with respect to the time are contained only in the components R 0a0p of the curvature tensor, where they enter in the form of the term -\g aP (the dot denotes differentiation with respect to x°); the second derivatives of the components g 0a and g 00 do not appear at all. It is therefore clear that the tensor R ik , which is obtained by contraction of the curvature tensor, and with it the equations (95.5), also contain the second derivatives with respect to the time of only the six spatial components g aP . It is also easy to see that these derivatives enter only in the ^-equation of (95.6), i.e. the equation Rl-MR-^Tl (95.11) c The J and % equations, i.e. the equations R°o-lR = S -fn, K^-fTl (95.12) contain only first-order time derivatives. One can verify this by checking that in forming the quantities R° a and R%-%R = $(R%-Rl) from R iklm by contraction, the components of the form R 0a0p actually drop out. This can be seen even more simply from the identity (92.13), by writing it in the form (Rl-tffRU = ~ (R a i-im) ia (95.13) (i = 0, 1, 2, 3). The highest time derivatives appearing on the right side of this equation are second derivatives (appearing in the quantities Rf, R). Since (95.13) is an identity, its left side must consequently contain no time derivatives of higher than second order. But one time differentiation already appears explicitly in it; therefore the expressions Rf—^SfR themselves cannot contain time derivatives of order higher than the first. Furthermore, the left sides of equations (95.12) also do not contain the first derivatives g 0a and g 00 (but only the derivatives g ap ). In fact, of all the T i>kl , only r a00 and r 000 contain these quantities, but these latter in turn appear only in the components of the curvature tensor of the form R 0a o P which, as we already know, drop out when we form the left sides of equations (95.12). If one is interested in the solution of the Einstein equations for given initial conditions (in the time), we must consider the question of the number of quantities for which the initial spatial distribution can be assigned arbitrarily. The initial conditions for a set of equations of second order must include both the quantities to be differentiated as well as their first time derivatives. But since in the present case the equations contain second derivatives of only the six g aP , not all the g ik and g ik can be arbitrarily assigned. Thus, we may assign (in addition to the velocity and density of the matter) the initial values of the functions g aP and g aP , after which the four equations (95.12) determine the admissible initial values of g 0a and g 00 ; in (95.11) the initial values of g 0a still remain arbitrary. Among the initial conditions thus assigned there are some functions whose arbitrariness is related simply to the arbitrariness in choice of the four-dimensional coordinate system. But the only thing that has real physical meaning is the number of "physically different" arbitrary functions, which cannot be reduced by any choice of coordinate system. From § 95 THE GRAVITATIONAL FIELD EQUATIONS 277 physical arguments it is easy to see that this number is eight: the initial conditions must assign the distribution of the matter density and of its three velocity components, and also of four other quantities characterizing the free gravitational field in the absence of matter (see later in § 102); for the free gravitational field in vacuum only the last four quantities should be fixed by the initial conditions. PROBLEM Write the equations for a constant gravitational field, expressing all the operations of differentia- tion with respect to the space coordinates as covariant derivatives in a space with the metric y a$ (84.7). Solution: We introduce the notation g 00 =h, g 0a = -hg a (88.11) and the three-dimensional velocity v a (89.10). In the following all operations of raising and lowering indices and of covariant differentiation are carried out in the three-dimensional space with the metric y aB , on the three- dimensional vectors g a , V and the three-dimensional scalar h. The desired equations must be invariant with respect to the transformation x a -*x a , x°-+x°+f(x a ), (1) which does not change the stationary character of the field. But under such a transformation, as is easily shown (see the footnote on p. 247), g a ~^g a -8f/8x a , while the scalar h and the tensor y aB = -g a0 +hg a g ff are unchanged. It is therefore clear that the required equations, when expressed in terms of y aB , h and g a , can contain g a only in the form of combinations of derivatives that con- stitute a three-dimensional antisymmetric tensor: _ _ 8g _ 8g a ( . which is invariant under such transformations. Taking this fact into account, we can drastically simplify the computations by setting (after computing all the derivatives appearing in R ik ) #« = and g a; }+g B -, « =0.f The Christoffel symbols are: 1 00 2 y ; <*> 1 00 -**■■ r° 1 a0 -£*■ a+^g B f a $+ •• • J x 00 = - t a 2 Jb ~2^ h '' a ' r° 1 a0 -\\ fSg a 8g e \ _ \dx e 8x a ) Th (gah - ■*■ 0Y — } a — 2(g0fy a +9rft ,')+ • • • -g0h;a)+gvK0+ The terms omitted (indicated by the dots) are quadratic in the components of g a \ these terms do drop out when we set g a = after performing the differentiations in R ik (92.10). In the calculations one uses formulas (84.9), (84.12-13); the X« y are three-dimensional Christoffel symbols constructed from the metric y a/3 . t To avoid any misunderstanding we emphasize that this simplified method for making the computations, which gives the correct field equations, would not be applicable to the calculation of arbitrary components of the R ik itself, since they are not invariant under the transformation (1). In equations (3)-(5) on the left are given those components of the Ricci tensor which are actually equal to the expressions given. These components are invariant under (1). r.T.F. 10 278 THE GRAVITATIONAL FIELD EQUATIONS § 96 The tensor T ik is calculated using formula (94.9) with the «' from (88.14) (where again we set As a result of the calculations, we obtain the following equations from (95.8): (3) c c 2 h „ „„ 1 , /- „ 8ttA: — (V /i) :a: 2 Vh c (p+e)v°v B ' e-p 2 y ' K-5) (5) Here ?" s is a three-dimensional tensor constructed from the y aB in the same way as R tk is con- structed from the g ik .-f § 96. Newton's law In the Einstein field equations we now carry out the transition to the limit of non- relativistic mechanics. As was stated in § 87, the assumption of small velocities of all particles requires also that the gravitational field be weak. The expression for the component g 00 of the metric tensor (the only one which we need) was found, for the limiting case which we are considering, in § 87 : c Further, we can use for the components of the energy-momentum tensor the expression (35.4) T* = iic 2 UiU k , where n is the mass density of the body (the sum of the rest masses of the particles in a unit volume; we drop the subscript on fi). As for the four- velocity u\ since the macroscopic motion is also considered to be slow, we must neglect all its space components and retain only the time component, that is, we must set u a = 0, u° = u = 1. Of all the components T\, there thus remains only T° = iic 2 . (96.1) The scalar T = T\ will be equal to this same value fie 2 . We write the field equations in the form (95.8): for / = k = R k t = One easily verifies that in the approximation we are considering all the other equations vanish identically. t The Einstein equations can also be written in an analogous way for the general case of a time-dependent metric. In addition to space derivatives they will also contain time derivatives of the quantities y a0 , g a , and ft. See A. L. Zel'manov, Doklady Acad. Sci., U.S.S.R. 107, 815 (1956). §96 NEWTON'S LAW 279 For the calculation of R% from the general formula (92.10), we note that terms containing derivatives of the quantities r ki are in every case quantities of the second order. Terms con- taining derivatives with respect to *° = ct are small (compared with terms with derivatives with respect to the coordinates x a ) since they contain extra powers of l/c. As a result, there remains R 00 = R% = dr a 00 jdx a . Substituting 1 af dg 00 1 # we find 00 " 2 9 dx* c o 1^ 1 AJL Thus the field equations give A<£ = AnkpL. (96.2) This is the equation of the gravitational field in nonrelativistic mechanics. It is completely analogous to the Poisson equation (36.4) for the electric potential, where here in place of the charge density we have the mass density multiplied by - k. Therefore we can immediately write the general solution of equation (96.2) by analogy with (36.8) in the form 4> «-*J"^. < 96 ' 3 ) This formula determines the potential of the gravitational field of an arbitrary mass distribu- tion in the nonrelativistic approximation. In particular, we have for the potential of the field of a single particle of mass m 4 = - ^ (96-4) iv and, consequently, the force F= -m'id^jdR), acting in this field on another particle (mass m'), is equal to F=--JT- ( 965 > This is the well-known law of attraction of Newton. The potential energy of a particle in a gravitational field is equal to its mass multiplied by the potential of the field, in analogy to the fact that the potential energy in an electric field is equal to the product of the charge and the potential of the field. Therefore, we may write, by analogy with (37.1), for the potential energy of an arbitrary mass distribution, the expression U = i f n4> dV. (96.6) For the Newtonian potential of a constant gravitational field at large distances from the masses producing it, we can give an expansion analogous to that obtained in §§ 40-41 for the electrostatic field. We choose the coordinate origin at the inertial center of the masses. Then the integral J /ir dV, which is analogous to the dipole moment of a system of charges, vanishes identically. Thus, unlike the case of the electrostatic field, in the case of the gravita- tional field we can always eliminate the "dipole terms". Consequently, the expansion of the potential has the form : M 1 _ d 2 _ J_ ^-^rM^WJx-.r^-^ (% - 7) 280 THE GRAVITATIONAL FIELD EQUATIONS § 96 - where M = j" \i dV is the total mass of the system, and the quantity D aP = j ii(3x a x p -r 2 d aP )dV (96.8) may be called the mass quadrupole moment tensor.^ It is related to the usual moment of inertia tensor J a p = j v(r 2 d a p-x a x p )dV by the obvious relation D aP = J y7 5 aP -3J aP . (96.9) The determination of the Newtonian potential from a given distribution of masses is the subject of one of the branches of mathematical physics; the exposition of the various methods for this is not the subject of the present book. Here we shall for reference purposes give only the formulas for the potential of the gravitational field produced by a homo- geneous ellipsoidal body. J Let the surface of the ellipsoid be given by the equation ~2 + 172 + ~2 = l> a>b>c. (96.10) a 1 b l c z Then the potential of the field at an arbitrary point outside the body is given by the following formula : oo tp=-nnabck [(i--^- +T ^---^-)^. (96.11) J \ a 2 +s b 2 + s c 2 + s/R s 4 . j? s = V(a 2 + s)(fc 2 + s)(c 2 + s), where £ is the positive root of the equation x 2 v 2 z 2 + 7#-. + -rr-- = l. ( 96 - 12 > a 2 + i; b 2 + £ c 2 + Q The potential of the field in the interior of the ellipsoid is given by the formula 00 q> = -n^iabck f ( 1 - -^— - -J~ £-) £ (96.13) J V a 2 + s b 2 + s c 2 + s/R s o which differs from (96.1 1) in having the lower limit replaced by zero ; we note that this expres- sion is a quadratic function of the coordinates x, y, z. The gravitational energy of the body is obtained, according to (96.6), by integrating the expression (96.13) over the volume of the ellipsoid. This integral can be done by elementary methods,§ and gives: oo 3km 2 r\lf a 2 b 2 c 2 \ 1 ds = U ~ 8 J [5\a 2 + s + b 2 + s + c 2 + s) \R S o t We here write all indices «, as subscripts, not distinguishing between co- and contravariant components, in accordance with the fact that all operations are carried out in ordinary Newtonian (Euclidean) space. t The derivation of these formulas can be found in the book of L. N. Stretenskii, Theory of the Newtonian Potential, Gostekhizdat, 1946. § The integration of the squares x?, y 2 , z 2 is most simply done by making the substitution x = ax', y = by', z = cz', which reduces the integral over the volume of the ellipsoid to an integral over the volume of the unit sphere. § 96 NEWTON'S LAW 281 (96.14) 3km 2 C \2 . ( 1 \ Ids o ( m =*!L abc n is the total mass of the body); integrating the first term by parts, we obtain finally: 00 U = _ V??L f d -l (96.15) io J r; o All the integrals appearing in formulas (96.11)-(96.14) can be expressed in terms of elliptic integrals of the first and second kind. For ellipsoids of rotation, these integrals are expressed in terms of elementary functions. In particular, the gravitational energy of an oblate ellipsoid of rotation (a = b > c) is l7= __4^L C0S - 1 C - (96.16) 5vV- 2 a and for a prolate ellipsoid of rotation (a>b = c): u= _^m^ cosh -i a . (96.17) For a sphere (a = c) both formulas give the value U = -3km 2 J5a, which, of course, can also be obtained by elementary methods.f PROBLEM Determine the equilibrium shape of a homogeneous gravitating mass of liquid which is rotating as a whole. Solution: The condition of equilibrium is the constancy on the surface of the body of the sum ot the gravitational potential and the potential of the centrifugal forces: 4>—— (x 2 +y 2 ) = const. (Q is the angular velocity; the axis of rotation is the z axis). The required shape is that of an oblate ellipsoid of rotation. To determine its parameters we substitute (96.13) in the condition of equi- librium, and eliminate z 2 by using equation (96.10); this gives: [f ds _J^_ <? f ds 1 , (x2+y2) [j (^+sfVW+s 2 *» ka2c a 2 ) (a 2 +s)(c 2 +sy' 2 \- COmt •' hom which it follows that the expression in the square brackets must vanish. Performing the in- tegration, we get the equation (a 2 +2c 2 )c , c 3c 2 CI 2 cos -1 — — _ 25 (^lY ' 3 m v /3 /f V /3 . _ ~6 \J ) m 10 / 3 * \a) ( a 2_ c 2)3/2 Wa a a 2_ c 2 2llkll (M = f ma 2 Q is the angular momentum of the body around the z axis), which determines the t The potential of the field inside a homogeneous sphere of radius a is q>= -2nkM\a z - r -). 282 THE GRAVITATIONAL FIELD EQUATIONS § 97 ratio of the semiaxes c/a for given ft or M. The dependence of the ratio c/a on M is single-valued; c/a increases monotonically with increasing M. However, it turns out that the symmetrical form which we have found is stable (with respect to small perturbations) only for not too large values of M.f The stability is lost for M = 2.89 k 1 ' 2 m 5 ' 3 n 1 ' 6 (when c/a = 0.58). With further increase of M, the equilibrium shape becomes a general ellipsoid with gradually decreasing values of b/a and c/a (from 1 and from 0.58, respectively). This shape in turn becomes unstable for M = 3.84 k 112 m 513 v' 1 ' 6 (when a:b:c = 1 :0.43:0.34). § 97. The centrally symmetric gravitational field Let us consider a gravitational field possessing central symmetry. Such a field can be produced by any centrally symmetric distribution of matter; for this, of course, not only the distribution but also the motion of the matter must be centrally symmetric, i.e. the velocity at each point must be directed along the radius. The central symmetry of the field means that the space-time metric, that is, the expression for the interval ds, must be the same for all points located at the same distance from the center. In euclidean space this distance is equal to the radius vector; in a non-euclidean space, such as we have in the presence of a gravitational field, there is no quantity which has all the properties of the euclidean radius vector (for example to be equal both to the distance from the center and to the length of the circumference divided by In). Therefore the choice of a "radius vector" is now arbitrary. If we use "spherical" space coordinates r, 9, <f>, then the most general centrally symmetric expression for ds 2 is ds 2 = h(r, i) dr 2 + k(r, t)(sin 2 9 d<f> 2 + d9 2 ) + l(r, t) dt 2 + a(r, t) dr dt, (97.1) where a, h, k, I are certain functions of the "radius vector" r and the "time" t. But because of the arbitrariness in the choice of a reference system in the general theory of relativity, we can still subject the coordinates to any transformation which does not destroy the central symmetry of ds 2 ; this means that we can transform the coordinates r and t according to the formulas r=fi(r',f), t=f 2 (r',t'), where/!, f 2 are any functions of the new coordinates r', t'. Making use of this possibility, we choose the coordinate r and the time t in such a way that, first of all, the coefficient air, t) of dr dt in the expression for ds 2 vanishes and, secondly, the coefficient k(r, t) becomes equal simply to — r 2 .% The latter condition implies that the radius vector r is defined in such a way that the circumference of a circle with center at the origin of coordinates is equal to 2nr (the element of arc of a circle in the plane 9 = n/2 is equal to dl = r d4>). It will be convenient to write the quantities h and / in exponential form, as — e x and cV respectively, where X and v are some functions of r and t. Thus we obtain the following expression for ds 2 : ds 2 = e v c 2 dt 2 -r 2 (d9 2 + sin 2 9 d(j> 2 ) - e* dr 2 . (97.2) Denoting by x°,x 1 ,x 2 ,x 3 , respectively, the coordinates ct, r, 9, <j), we have for the f References to the literature concerning this question can be found in the book by H. Lamb, Hydro- dynamics, chap. XII. % These conditions do not determine the choice of the time coordinate uniquely. It can still be subjected to an arbitrary transformation / =f(t'), not containing r. § 97 THE CENTRALLY SYMMETRIC GRAVITATIONAL FIELD 283 nonzero components of the metric tensor the expressions goo = e\ 9n = -e\ 9n= ~r\ g 33 =-r 2 sin 2 e. Clearly, g 00 = e- v , g xl =-e~\ g 22 = -r' 2 , g 33 = -r~ 2 sin" 2 0. With these values it is easy to calculate the r£, from formula (86.3). The calculation leads to the following expressions (the prime means differentiation with respect to r, while a dot on a symbol means differentiation with respect to ct): ln "2' v' ll0 ~2' T 2 33 = -sin 9 5 111 2 ' 1 22 == ^ ' r 1 - v -<r A 1 oo — ^ : r 2 - r 3 - - 1 12 "A 13 - r » r| 3 = cot 0, 1 oo - 2 • Tio — ~ > rl 3 = -rsin 2 Ge -A- (97.3) All other components (except for those which differ from the ones we have written by a transposition of the indices k and /) are zero. To get the equations of gravitation we must calculate the components of the tensor RJ; according to formula (92.10). A simple calculation leads to the following equations: 8 ^ri = -e-^ + A) + l, (97.4) 8rcfc._, Snk _, 1 _,/.. v' 2 v'-X' VX'\ 1 _„ ( v . A 2 Av c Snk nk . . 8tt/c . 1 _,/ v' 2 v'-A' vT\ , 1 _ v /v A 2 Av\ (97.6) / J" **Th--e~^.- (97.7) c . r The other components vanish identically. Using (94.9), the components of the energy momentum tensor can be expressed in terms of the energy density e of the matter, its pressure p, and the radial velocity v. The equations (97.4-7) can be integrated exactly in the very important case of a centrally symmetric field in vacuum, that is, outside of the masses producing the field. Setting the energy-momentum tensor equal to zero, we get the following equations : '- l (i+?)-h-°- (97 - 8) ^(7-^) + 4 = 0, (97.9) A = (97.10) 284 THE GRAVITATIONAL FIELD EQUATIONS § 97 [we do not write the fourth equation, that is, equation (97.5), since it follows from the other three equations]. From (97.10) we see directly that X does not depend on the time. Further, adding equations (97.8) and (97.9), we find X' + v' = 0, that is, A + v=J(t), (97.11) where f(t) is a function only of the time. But when we chose the interval ds 2 in the form (97.2), there still remained the possibility of an arbitrary transformation of the time of the form t =f{t'). Such a transformation is equivalent to adding to v an arbitrary function of the time, and with its aid we can always make/(0 in (97.11) vanish. And so, without any loss in generality, we can set A+v = 0. Note that the centrally symmetric gravitational field in vacuum is automatically static. The equation (97.9) is easily integrated and gives : e - w = 1+ ^<. (97 . 12) r Thus, at infinity (r-> oo), e~ x = e v = 1, that is, far from the gravitating bodies the metric automatically becomes galilean. The constant is easily expressed in terms of the mass of the body by requiring that at large distances, where the field is weak, Newton's law should hold.| In other words, we should have g 00 = l + (20/c 2 ), where the potential <f> has its Newtonian value (96.4) $ = —(Jcm/r) (m is the total mass of the bodies producing the field). From this it is clear that const = —(2km/c 2 ). This quantity has the dimensions of length; it is called the gravitational radius r g of the body : r,J^. (97.13) Thus we finally obtain the space-time metric in the form : ds 2 = U- r A c 2 dt 2 -r 2 (sm 2 6 d<f> 2 + d0 2 )- -^y. (97.14) r This solution of the Einstein equations was found by K. Schwarzschild (1916). It completely determines the gravitational field in vacuum produced by any centrally-symmetric distribu- tion of masses. We emphasize that this solution is valid not only for masses at rest, but also when they are moving, so long as the motion has the required symmetry (for example, a centrally-symmetric pulsation). We note that the metric (97.14) depends only on the total mass of the gravitating body, just as in the analogous problem in Newtonian theory. The spatial metric is determined by the expression for the element of spatial distance: dr 2 dl 2 = — +r 2 (sin 2 9 d(l> 2 + d0 2 ). (97.15) r The geometrical meaning of the coordinate r is determined by the fact that in the metric (97.15) the circumference of a circle with its center at the center of the field is 2nr. But the f For the field in the interior of a spherical cavity in a centrally symmetric distribution, we must have const = 0, since otherwise the metric would have a singularity at r = 0. Thus the metric inside such a cavity is automatically galilean, i.e., there is no gravitational field in the interior of the cavity (just as in Newtonian theory). § 97 THE CENTRALLY SYMMETRIC GRAVITATIONAL FIELD 285 distance between two points r t and r 2 along the same radius is given by the integral [-4^=>r,-r i . (97.16) l M Furthermore, we see that g 00 ^ 1. Combining with the formula (84.1) dx = y/g 00 dt, defining the proper time, it follows that dx < dt. (97.17) The equality sign holds only at infinity, where / coincides with the proper time. Thus at finite distances from the masses there is a "slowing down" of the time compared with the time at infinity. Finally, we present an approximate expression for ds 2 at large distances from the origin of coordinates : ds 2 = ds 2 - 2 -— (dr 2 + c 2 dt 2 ). (97.18) c r The second term represents a small correction to the galilean metric ds 2 ,. At large distances from the masses producing it, every field appears centrally symmetric. Therefore (97.18) determines the metric at large distances from any system of bodies. Certain general considerations can also be made concerning the behavior of a centrally symmetric gravitational field in the interior of the gravitating masses. From equation (97.6) we see that for r -> 0, A must also vanish at least like r 2 ; if this were not so the right side of the equation would become infinite for r -* 0, that is, T° would have a singular point at r = 0, which is physically impossible. Formally integrating (97.6) with the limiting con- dition A| P=0 = °> we obtain r A = -In |l-^ j T Q r 2 dr}. (97.19) o Since, from (94.10), T° = e~ v T 00 > 0, is it clear that A ^ 0, that is, e k > 1. (97.20) Subtracting equation (97.6) term by term from (97.4), we get: -a o , ( £ + P)( 1 + ^) e V + ao= 8 4Vs-t|)= — V^°> c 2 i.e. v' + A'$s0. But for r-+oo (far from the masses) the metric becomes galilean, i.e. v _y o, A -* 0. Therefore, from v' -1- A' ^ it follows that over all space v + A^0. (97.21) Since A $s 0, it then follows that v < 0, i.e. e v < 1. (97.22) The inequalities obtained show that the above properties (97.16) and (97.17) of the spatial metric and the behaviour of clocks in a centrally symmetric field in vacuum apply equally well to the field in the interior of the gravitating masses. 286 THE GRAVITATIONAL FIELD EQUATIONS § 97 If the gravitational field is produced by a spherical body of "radius" a, then for r > a, we have T% = 0. For points with r > a, formula (97.19) therefore gives A=-lnO-^ I TlrUr c r /«' o On the other hand, we can here apply the expression (97.14) referring to vacuum, according to which i /. 2km\ Equating the two expressions, we get the formula a m = p J T° r 2 dr, (97.23) o expressing the total mass of a body in terms of its energy-momentum tensor. PROBLEMS 1. Determine the spatial curvature in a centrally symmetric gravitational field in vacuum. Solution: The components of the spatial curvature tensor P a g y6 can be expressed in terms of the components of the tensor P ae (and the tensor y a0 ) so that we need only calculate P a0 (see problem 1 in § 92). The tensor P a0 is expressed in terms of y a0 just as R tk is expressed in terms of g ik . Using the values of y a0 from (97.15), we find from the calculations: pe = p<t> = JjL P r — — 2r 3 ' r 3 ' and PI = for a # yS. We note that P e g , P% > 0, P\ < 0, while P = P a a = 0. From the formula given in problem 1 of § 92, we find: Prere = (P r r +P"e)yrrVe6 = ~P% Yrr 7ee, Prd>r<P — P e yrr Yd><t>i Pe<t>o$ = — P r 7ee V<t>0- It then follows (see the footnote on p. 263) that for a "plane" perpendicular to the radius, the Gaussian curvature is (which means that, in a small triangle drawn on the "plane" in the neighborhood of its intersection with the radius perpendicular to it, the sum of the angles of the triangle is greater than n). As to the "planes" which pass through the centre, their Gaussian curvature K<0; this means that the sum of the angles of a small triangle in such a "plane" is less than n (however this does not refer to the triangles embracing the centre — the sum of the angles in such a triangle is greater than n). 2. Determine the form of the surface of rotation on which the geometry would be the same as on a "plane" passing through the origin in a centrally symmetric gravitational field in vacuo. Solution: The geometry on the surface of rotation z = z(r) is determined (in cylindrical co- ordinates) by the element of length: dl 2 = dr 2 +dz 2 +r 2 d<p 2 = dr 2 (\+z' 2 )+r 2 dy 2 . Comparing with the element of length (97.4) in the "plane" 6 = n/2 dr 2 dl 2 =r 2 d<p 2 +~ T , \-r g \r we find 1+V (-?)"• § 98 MOTION IN A CENTRALLY SYMMETRIC GRAVITATIONAL FIELD 287 from which z = 2Vr g (r-r g ). For r = r g this function has a singularity— a branch point. The reason for this is that the spatial metric (97.15) in contrast to the space-time metric (97.14), actually has a singularity at r = r g . The general properties of the geometry on "planes" passing through the center, which were mentioned in the preceding problem, can also be found by considering the curvature in the pictorial model given here. 3. Transform the interval (97.11) to such coordinates that its element of spatial distance has conformal— euclidean form, i.e. dl is proportional to its euclidean expression. Solution: Setting we get from (97.14) <fc 2 = ^ 1 c'dt'-fl+Z-) (dp 2 +P*d9*+p 2 sin* 6d? 2 ). The coordinates p, 6, y are called isotropic spherical coordinates; instead of them we can also introduce isotropic cartesian coordinates x, y, z. In particular, at large distances (p » r g ) we have approximately: rfja = A _ r A C 2 dt 2_ (l+ r j\ (dx 2 +dy 2 +dz 2 ). § 98. Motion in a centrally symmetric gravitational field Let us consider the motion of a body in a centrally symmetric gravitational field. As in every centrally symmetric field, the motion occurs in a single "plane" passing through the origin ; we choose this plane as the plane 9 = n/2. To determine the trajectory of the body (with mass m), we use the Hamilton- Jacobi equation: ik dS 8S 2 2 9 &?a?- mc - a Using the g ik given in the expression (97.14), we find the following equation: ( d A\ 2 ^( d l\ 2 -L( d A\ \cdtj \drj r 2 \d<i>) e-[~] -^ff I 4(p) -mV = 0, (98.1) where e v = l-'- 9 (98.2) r ,2 (m' is the mass of the body producing the field; r g = 2km' J c 1 is its gravitational radius). By the general procedure for solving the Hamilton-Jacobi equation, we look for an S in the form S= -£ t + M<f> + S r (r), (98.3) with constant energy <f and angular momentum M. Substituting this in (98.1), we find the equation c r \or J 288 THE GRAVITATIONAL FIELD EQUATIONS § 98 from which s ' = jV? e " 2v -( mV+ ?) c-v - dr -it S-mV) + »Vrr. M> V'^ c 2 (r-r g ) 2 r(r-r g )_ The trajectory is determined f by the equation 8S/8M = const, from which <f> = J — -,_,., (98.5) VH^)H) dr, (98.6) This integral reduces to an elliptic integral. For the motion of a planet in the field of attraction of the Sun, the relativistic theory leads to only an insignificant correction compared to Newton's theory, since the velocities of the planets are very small compared to the velocity of light. In the integrand in the equation (98.5) for the trajectory, this corresponds to a small value for the ratio rjr, where r g is the gravitational radius of the Sun.J To investigate the relativistic corrections to the trajectory, it is convenient to start from the expression (98.4) for the radial part of the action, before differentiation with respect to M. We make a transformation of the integration variable, writing r r(r-r g ) = r' 2 , i.e. r-^-^r', as a result of which the second term under the square root takes the form M 2 /r' 2 . In the first term we make an expansion in powers of rjr', and obtain to the required accuracy: S r = 1 2<f m + %- J + - (2m 2 m'/c + 4<Tmr 9 )- - 2 ( M 2 — 1 \ where for brevity we have dropped the prime on r' and introduced the non-relativistic energy $' (without the rest energy). The correction terms in the coefficients of the first two terms under the square root have only the not particularly interesting effect of changing the relation between the energy and momentum of the particle and changing the parameters of its Newtonian orbit (ellipse). But the change in the coefficient of 1/r 2 leads to a more fundamental effect — to a systematic (secular) shift in the perihelion of the orbit. Since the trajectory is defined by the equation (f> + (dS r /dM) = const, the change of the angle $ after one revolution of the planet in its orbit is **--m AS - where AS r is the corresponding change in S r . Expanding S r in powers of the small correction to the coefficient of 1/r 2 , we get: , m -hm 2 c 2 r 2 oAS< 0) AM oM where AS ( r 0) corresponds to the motion in the closed ellipse which is unshifted. Differentiating t See Mechanics, § 47. t For the Sun, r g = 3 km; for Earth, r g = 0.44 cm. § 98 MOTION IN A CENTRALLY SYMMETRIC GRAVITATIONAL FIELD 289 this relation with respect to M, and using the fact that - — AS< 0) = A0 (O) = 2n, dM r we find: n -,-,,-> Znm 2 c 2 rl „ 6nk 2 m 2 m' 2 The second term is the required angular displacement H of the Newtonian ellipse during one revolution, i.e. the shift in the perihelion of the orbit. Expressing it in terms of the length a of the semimajor axis and the eccentricity e of the ellipse by means of the formula M 2 n 2 , km'm we obtain :f c 2 a{\-e z ) Next we consider the path of a light ray in a centrally symmetric gravitational field. This path is determined by the eikonal equation (87.9) y dx l dx k which differs from the Hamilton-Jacobi equation only in having m set equal to zero. There- fore the trajectory of the ray can be obtained immediately from (98.5) by setting m = 0; at the same time, in place of the energy S = -(dSjdt) of the particle we must write the frequency of the light, co = -(#/df). Also introducing in place of the constant M a constant q defined by q = cMjw , we get: ^ = f =£==. (98.8) r dr If we neglect the relativistic corrections (r, -►()), this equation gives r = <?/cos <f>, i.e. a straight line passing at a distance q from the origin. To study the relativistic corrections, we proceed in the same way as in the previous case. For the radial part of the eikonal we have [see (98.4)]: Making the same transformations as were u sed to go fro m (98.4) to (98.6), we find: Expanding the integrand in powers of rjr, we have : \ = ^(o, + M?2 f dr^ = ^ <o> + r_£o cosh -i 1 Vr ¥r c J Vr 2 -e 2 c e where ^< 0) .corresponds to the classical straight ray. t Numerical values of the shifts determined from formula (98.7) for Mercury and Earth are equal, respectively, to 43.0" and 3.8" per century. Astronomical measurements give 43.1 ±0.4 and 5.0 +1.2 , in excellent agreement with theory. 290 THE GRAVITATIONAL FIELD EQUATIONS § 99 The total change in \J/ r during the propagation of the light from some very large distance R to the point r = q nearest to the center and then back to the distance R is equal to A<A r = A^°> + 2^cosh- li? . C Q The corresponding change in the polar angle $ along the ray is obtained by differentiation with respect to M = q /<dc: dM dM Q sJr 2 - q 2 ' Finally, going to the limit R -> oo, and noting that the straight ray corresponds to A0 = n, we get: A0 = 7T+?^. This means that under the influence of the field of attraction the light ray is bent: its trajectory is a curve which is concave toward the center (the ray is "attracted" toward the center), so that the angle between its two asymptotes differs from n by 2r a 4km' b<$> = ^ i = - T -\ (98.9) in other words, the ray of light, passing at a distance q from the center of the field, is deflected through an angle <50.f § 99. The synchronous reference system As we know from § 84, the condition for it to be possible to synchronize clocks at dif- ferent points in space is that the components g 0a of the metric tensor be equal to zero. If, in addition, g 00 = 1, the time coordinate x° = t is the proper time at each point in space. J A reference system satisfying the conditions 9oa = 0, g 00 = 1 (99.1) is said to be synchronous. The interval element is such a system is given by the expression ds 2 = dt 2 - y aP dx* dx p , (99.2) where the components of the spatial metric tensor are the same (except for sign) as the g aP : y«p=-g aP . (9*3) The three-dimensional tensor g aP determines the spatial metric. In the synchronous reference system the time lines are geodesies in the four-space. The four- vector u l = dx*/ds, which is tangent to the world line x 1 , x 2 , x 3 = const, has com- ponents u a = 0, u° = 1, and automatically satisfies the geodesic equations: du l , , — +r kl u k u l = r 00 = o, ds since, from the conditions (99.1), the Christoffel symbols Too and Too vanish identically. t For a ray just skirting the edge of the Sun, d<f> = 1.75". % In this section we set c = 1 . § 99 THE SYNCHRONOUS REFERENCE SYSTEM 291 It is also easy to see that these lines are normal to the hypersurfaces / = const. In fact, the four-vector normal to such a hypersurface, n t = dt/dx\ has covariant components n = 1 n =0 With the conditions (99.1), the corresponding contravanant components are also n° = 1, rf = 0, i.e., they coincide with the components of the four-vector u l which is tangent to the time lines. . Conversely, these properties can be used for the geometrical construction of a syn- chronous reference system in any space-time. For this purpose we choose as our starting surface any spacelike hypersurface, i.e., a hypersurface whose normals at each point have a time-like direction (they lie inside the light cone with its vertex at this point); all elements ot interval on such a hypersurface are spacelike. Next we construct the family of geodesic lines normal to this hypersurface. If we now choose these lines as the time coordinate lines and determine the time coordinate t as the length s of the geodesic line measured from the initial hypersurface, we obtain a synchronous reference system. It is clear that such a construction, and the selection of a synchronous reference system, is always possible in principle. Furthermore, this choice is still not unique. A metric of the form (99.2) allows any trans- formation of the space coordinates which does not affect the time, and also transformations corresponding to the arbitrariness in the choice of the initial hypersurface for the geometrical construction. . . . The transformation to the synchronous reference system can, m principle, be done analytically by using the Hamilton-Jacobi equation; the basis of this method is the fact that the trajectories of a particle in a gravitational field are just the geodesic lines. The Hamilton-Jacobi equation for a particle (whose mass we set equal to unity) in a gravitational field is y dx l dx k (where we denote the action by t). Its complete integral has the form: t=M*,x 1 ) + A(?), ( 99 - 5 > where /is a function of the four coordinates x l and the three parameters ?; the fourth constant A we treat as an arbitrary function of the three ? . With such a representation for t, the equations for the trajectory of the particle can be obtained by equating the derivatives dx/d^* to zero, i.e. M. = - b A (99.6) For each set of assigned values of the parameters £", the right sides of equations (99.6) have definite constant values, and the world line determined by these equations is one of the possible trajectories of the particle. Choosing the quantities f , which are constant along the trajectory, as new space coordinates, and the quantity t as the new time coordinate, we get the synchronous reference system; the transformation which takes us from the old coordinates to the new is given by equations (99.5-6). In fact it is guaranteed that for such a transformation the time lines will be geodesies and will be normal to the hypersurfaces t = const. The latter point is obvious from the mechanical analogy: the four-vector -dxjdx 1 which is normal to the hypersurface coincides in mechanics with the four-momentum of the particle, and therefore coincides in direction with its four-velocity u\ i.e. with the four- vector tangent to the trajectory. Finally the condition g 00 = 1 is obviously satisfied, since the 292 THE GRAVITATIONAL FIELD EQUATIONS § 99 derivative -dr/ds of the action along the trajectory is the mass of the particle, which we set equal to 1 ; therefore \dxjds\ = 1. We write the Einstein equations in the synchronous reference system, separating the operations of space and time differentiation in the equations. We introduce the notation *->=^f (99.7) for the time derivatives of the three-dimensional metric tensor; these quantities also form a three-dimensional tensor. All operations of shifting indices and covariant differentiation of tie three-dimensional tensor x aP will be done in three-dimensional space with the metric V«0-t We note that the sum < is the logarithmic derivative of the determinant y = \y af5 \ : < = f pdj ^ = ^ln(y). (99.8) For the Christoffel symbols we find the expressions: yO _ pa _ pO _ (\ 1 00 — * 00 — L 0a ~ u > rS, = K/»» r 0/} = ^, T% = k* Pv (99.9) where X* By are the three-dimensional Christoffel symbols formed from the tensor y aB . A calculation using formula (92.10) gives the following expressions for the components of the tensor K lk : 1 d « 1 8 a *oo=-^<-4^, Ro a =- 2 (Xa;l>-X P p;J, (99.10) Id 1 p = 2dt XaP+ 4 ^ ** ~ 2x " *'** + PaP ' Here P aP is the three-dimensional Ricci tensor which is expressed in terms of y ap in the same way as P ik is expressed in terms of g ik . All operations of raising indices and of covariant differentiation are carried out in the three-dimensional space with the metric y ap . We write the Einstein equations in mixed components : 1 r\ 1 R o=-~j t <~ 2 *J*? = ** k ( T o-iT), (99.11) 2dt * 4 2 K = - «r4«) = **kT° a , (99.12) Rt =-P p a - -^ (V? 4) = *nk(Tl-i8ln (99.13) 2s/ydt A characteristic feature of synchronous reference systems is that they are not stationary: the gravitational field cannot be constant in such a system. In fact, in a constant field we would have x ap = 0. But in the presence of matter the vanishing of all the x aB would con- tradict (99.11) (which has a right side different from zero). In empty space we would find t But this does not, of course, apply to operations of shifting indices in the space components of the four- tensors R ik , T ik (see the footnote on p. 250). Thus T a must be understood as before to be g ey T ya ^-g B0 T 0a , which in the present case reduces to g ev T ya and differs in sign from y ey T Ya . § 99 THE SYNCHRONOUS REFERENCE SYSTEM 293 from (99 13) that all the P aP , and with them all the components of the three-dimensional curvature tensor P aPyS , vanish, i.e. the field vanishes entirely (in a synchronous system with a euclidean spatial metric the space-time is flat). At the same time the matter filling the space cannot in general be at rest relative to the synchronous reference frame. This is obvious from the fact that particles of matter within which there are pressures generally move along lines that are not geodesies; the world line of a particle at rest is a time line, and thus is a geodesic in the synchronous reference system. An exception is the case of "dust" (p = 0). Here the particles interacting with one another will move along geodesic lines; consequently, in this case the condition for a synchronous reference system does not contradict the condition that it be comoving with the matter.t For other equations of state a similar situation can occur only in special cases when the pressure gradient vanishes in all or in certain directions. From (99,11) one can show that the determinant -g = y of the metric tensor in a syn- chronous system must necessarily go to zero in a finite length of time. To prove this we note that the expression on the right side of this equation is positive for any distribution of the matter. In fact, in a synchronous reference system we have for the energy-momentum tensor (94.9): (v + e)v 2 T -$T = & + 3p)+ yJ Y Z ~ [the components of the four-velocity are given by (88.14)]; this quantity is clearly positive. The same statement is also true for the energy-momentum tensor of the electromagnetic field (T = 0, T% is the positive energy density of the field). Thus we have from (99.11): (where the equality sign applies in empty space). Using the algebraic inequality^ kJkJ > H*3 2 we can rewrite (99.14) in the form ^<+i«) 2 <o or -(-)>-. (99.15) dtWJ'6 Suppose, for example, that at a certain time < > 0. Then as t decreases the quantity l/< decreases, with a finite (nonzero) derivative, so that it must go to zero (from positive values) in the course of a finite time. In other words, *£ goes to + oo, but since >£ = d In y/dt, this means that the determinant y goes to zero [no faster than t 6 , according to the inequality t Even in this case, in order to be able to choose a "synchronously comoving" system of reference, it is still necessary that the matter move "without rotation". In the comoving system the contravariant components of the velocity are u° = 1, W = 0. If the reference system is also synchronous, the covariant components must satisfy u = 1, u a = 0, so that its four-dimensional curl must vanish: dut du k _ But this tensor equation must then also be valid in any other reference frame. Thus, in a synchronous, but not comoving, system, we then get the condition curl v = for the three-dimensional velocity v. X Its validity can easily be seen by bringing the tensor x a0 to diagonal form (at a given instant of time). 294 THE GRAVITATIONAL FIELD EQUATIONS § 99 (99.15)]. If on the other hand x" < at the initial time, we get the same result for increasing times. This result does not, however, by any means prove that there must be a real physical singularity of the metric. A physical singularity is one that is characteristic of the space- time itself, and is not related to the character of the reference frame chosen (such a singularity should be characterized by the tending to infinity of various scalar quantities — the matter density, or the invariants of the curvature tensor). The singularity in the synchronous reference system, which we have proven to be inevitable, is in general actually fictitious, and disappears when we change to another (nonsynchronous) reference frame. Its origin is evident from simple geometrical arguments. We saw earlier that setting up a synchronous system reduces to the construction of a family of geodesic lines orthogonal to any space-like hypersurface. But the geodesic lines of an arbitrary family will, in general, intersect one another on certain enveloping hyper- surfaces — the four-dimensional analogues of the caustic surfaces of geometrical optics. We know that intersection of the coordinate lines gives rise to a singularity of the metric in the particular coordinate system. Thus there is a geometrical reason for the appearance of a singularity, associated with the specific properties of the synchronous system and therefore not physical in character. In general an arbitrary metric of four-space also permits the existence of nonintersecting families of geodesic lines. The unavoidable vanishing of the determinant in the synchronous system means that the curvature properties of a real (non- flat) space-time (which are expressed by the inequality R% ^ 0) that are permitted by the field equations exclude the possibility of existence of such families, so that the time lines in a synchronous reference system necessarily intersect one another.f We mentioned earlier that for dustlike matter the synchronous reference system can also be comoving. In this case the density of the matter goes to infinity at the caustic — simply as a result of the intersection of the world trajectories of the particles, which coincide with the time lines. It is, however, clear that this singularity of the density can be eliminated by introducing an arbitrarily small nonzero pressure of the matter, and in this sense is not physical in character. PROBLEMS 1 . Find the form of the solution of the gravitational field equations in vacuum in the vicinity of a point that is not singular, but regular in the time. Solution: Having agreed on the convention that the time under consideration is the time origin, we look for y a0 in the form: Y«0 =a a 0+tb a i, + t 2 c aP + . . . , (1) t For the analytic structure of the metric in the vicinity of a fictitious singularity in a synchronous reference system, see E. M. Lifshitz, V. V. Sudakov and I. M. Khalatnikov, JETP 40, 1847, 1961, (Soviet Phys.—JETP, 13, 1298, 1961). The general character of the metric is clear from geometrical considerations. Since the caustic hyper- surface always contains timelike intervals (the line elements of the geodesic time lines at their points of tan- gency to the caustic), it is not spacelike. Furthermore, on the caustic one of the principal values of the metric tensor y a$ vanishes, corresponding to the vanishing of the distance (<5) between two neighboring geodesies that intersect one another at their point of tangency to the caustic. The quantity 3 goes to zero as the first power of the distance (/) to the point of intersection. Thus the principal value of the metric tensor, and with it the determinant y, goes to zero like I 2 . The synchronous reference system can also be constructed so that the time lines intersect on a set of points having lower dimensionality than a hypersurface — on a two-dimensional surface that may be called the focal surface corresponding to the family of geodesies. The analytic construction of such a metric is given by V. A. Belinskil and I. M. Khalatnikov, JETP 49, 1000, 1965 (Soviet Phys.—JETP 22, 694, 1966). § 99 ' THE SYNCHRONOUS REFERENCE SYSTEM 295 where «.„ b.» c a , are functions of the space coordinates. In this same approximation the reciprocal tensor is: „,, ,. „„ yae = a ^-tb a6 ^t\b a %-c aR \ where a" is the tensor reciprocal to a.» and the raising of indices of the other tensors is done by using a aB . We also have: x a e = b aB +2tc aP , K = b' a +t(c B a -b ay b<»). The Einstein equations (99. 11-13) lead to the following relations : flo=-c+ifW=0, (2) iJ^K^-.-^^+^l-c^+K^^^+^^+^^.-KA^^l-o, (3) K = -Pi -ib>b+V>lb>-C. = (4) (6s /,« cs c-) The operations of covariant differentiation are carried out in the three-dimensional space with metric a aB ; the tensor P a , is also defined with respect to this metric^ From (4) the coefficients c, are completely determined in terms of the coefficients a aP and b aB . Then (2) gives the relation ,-v From the terms of zero order in (3) we have: b B a ., e =b., a . (6) The terms ~ / in this equation vanish identically when we use (5) and (6) and the identity pe _ xp fseg (92 1 3)1 "" Thus 'the twelve quantities «.,, 6., are related to one another by the one relation (5) and the three relations (6), so that there remain eight arbitrary functions of the three space coordinates. Of these, three are related to the possibility of arbitrary transformations of the three space co- ordinates, and one to the arbitrariness in choosing the initial hypersurface for setting up the synchronous reference system. Therefore we are left with the correct number (see the end of § 95) of four "physically different" arbitrary functions. 2. Calculate the components of the curvature tensor R iklm in the synchronous reference system. Solution: Using the Christoffel symbols (99.10) we find from (92.4): R a eyi= —Patrt + lOt'aXer — X'rXf)* RoaPy == HyXay; B XaB; y)> 18 1 Robots ~~2 8f Xafi ~~ 4 Ka v X/3 ' where P af1i is the three-dimensional curvature tensor corresponding to the three-dimensional m t^Find 'the general form of the infinitesimal transformation from one synchronous reference system to another. Solution: The transformation has the form ,_»,+ ,,(*!, x \ x 3 ), x a -+x a + ?(x\ x 2 , x 3 , t), where m and {« are small quantities. We are guaranteed that the condition goo = 1 is satisfied by keeping <p independent of t\ to maintain the condition g 0a = 0, we must satisfy the equations d£ e __ 8<p 7a "Tt ~aP from which where the/* are again small quantities (forming a three-dimensional vector f). The spatial metric tensor y a0 is replaced by [as can be easily verified using (94.3)]. _ Thus the transformation contains four arbitrary functions fa f) of the space coordinates. 296 THE GRAVITATIONAL FIELD EQUATIONS § 100 § 100. Gravitational collapse In the Schwarzschild metric (97.14), g 00 goes to zero and g tl to infinity at r = r g (on the "Schwarzschild sphere"). This could give the basis for concluding that there must be a singularity of the space-time metric and that it is therefore impossible for bodies to exist that have a "radius" (for a given mass) that is less than the gravitational radius. Actually, however, this conclusion would be wrong. This is already evident from the fact that the determinant g = -r 4 sin 2 has no singularity at r g = r, so that the condition g < (82.3) is not violated. We shall see that in fact we are dealing simply with the impossibility of establishing a suitable reference system for r < r g . To make clear the true character of the space-time metric in this domainf we make a transformation of the coordinates of the form: Cf(r)dr „ r dr cx= ±ct±\ J -^—, R = ct+\- . (100.1) Then ds 2 = — j 2 (c 2 dx 2 -f 2 dR 2 )-r 2 (d6 2 + sin 2 d<$>\ We eliminate the singularity at r = r g by choosing/(r) so thatf(r g ) = 1. If we set/(r) = \JrJr, then the new coordinate system will also be synchronous (g xt =1). First choosing the upper sign in (100.1), we have: *-"-/ (l-f)dr r Ir , 2r 3 ' 2 dr = (-?) J. \f 3 r 1/2 ' or 2/3 r = (J ( *~ CT) j ^ (100 - 2) (we set the integration constant, which depends on the time origin, equal to zero). The element of interval is : "3 1 4/3 (R-ct) ds 2 = c 2 dx ° 2 3 VM . 2/ W + sin 2 0<^ 2 ). (100.3) In these coordinates the singularity on the Schwarzschild sphere [to which there corres- ponds the equality f (R — ex) = r g ] is absent. The coordinate R is everywhere spacelike, while x is timelike. The metric (100.3) is nonstationary. As in every synchronous reference system, the time lines are geodesies. In other words, "test" particles at rest relative to the reference system are particles moving freely in the given field. To given values of r there correspond world lines R—cx = const (the sloping straight lines in Fig. 20). The world lines of particles at rest relative to the reference system are shown on this diagram as vertical lines ; moving along these lines, after a finite interval of f This was first done by D. Finkelstein (1958) using a different transformation. The particular metric (100.3) was first found in a different way by Lemaitre (1938) and in the present connection by Yu. Rylov (1961). § ioo GRAVITATIONAL COLLAPSE 297 Fig. 20. proper time the particles "fall in" to the center of the field (r = 0), which is the location of the true singularity of the metric. Let us consider the propagation of radial light signals. The equation ds = (tor 0, (j) = const) gives for the derivative dxjdR along the ray: dx dR = + fe M ) -- + /* 1/3 x V r' (100.4) the two signs corresponding to the two boundaries of the light "cone" with its vertex at the given world point. When r > r g (point a in Fig. 20) the slope of these boundaries satisfies \c dx/dR\ < 1, so that the straight line r = const (along which c dx/dR = 1) falls inside the cone. But in the region r < r g (point a') we have \c dx/dR\ > 1, so that the line r = const, the world line of a particle at rest relative to the center of the field, lies outside the cone. Both boundaries of the cone intersect the line r = at a finite distance, approach- ing it along a vertical. Since no causally related events can lie on the world line outside the light cone, it follows that in the region r < r g no particles can be at rest. Here all interactions and signals propagate in the direction toward the center, reaching it after a finite interval of time x. Similarly, choosing the lower signs in (100.1) we would obtain an "expanding reference system with a metric differing from (100.3) by a change of the sign of t. It corresponds to a space-time in which (in the region r < r g ) again rest is impossible, but all signals propagate outward from the center. The results described here can be applied to the problem of the behavior of massive bodies in the general theory of relativity.f The investigation of the relativistic conditions for equilibrium of a spherical body shows that for a body of sufficiently large mass, states of static equilibrium cannot exist.J It is t We caution against any applications to elementary particles: the entire theory presented in this book already loses its validity for dimensions ~h/mc, which is by an enormous factor (~ 10 40 ) greater than km/c . t See Statistical Physics, § 111. 298 THE GRAVITATIONAL FIELD EQUATIONS § 100 clear that such a body must contract without limit (i.e. it must undergo "gravitational collapse").^ In a reference system not attached to the body and galilean at infinity [metric (97.14)], the radius of the central body cannot be less than r r This means that according to the clocks t of a distant observer the radius of the contracting body only approaches the gravitational radius asymptotically as /-► oo. It is easy to find the limiting form of this dependence. A particle on the surface of the contracting body is at all times in the field of attraction of a constant mass m, the total mass of the body. As r -» r g the gravitational force becomes very large; but the density of the body (and with it, the pressure) remains finite. Neglecting the pressure forces for this reason, we reduce the determination of the time dependence r = r(t) of the radius of the body to a consideration of the free fall of a test particle in the field of the mass m. The function r(t) for fall in a Schwarzschild field can be found (using the Hamilton- Jacobi method) from the equation dS/d£ = const, with the action S from (98.3-4), where, for the case of purely radial motion, the angular momentum M = 0. Thus we get • /, * ct = i-- 9 L/(^) 2 -i + (100.5) (for brevity we omit the subscript on <f). This integral diverges like r g In (r-r ) for r -* r . Thus we find the asymptotic formula for the approach of r to r : r—r g = const e~ (ct/tg \ (100.6) Although the rate of contraction as observed from outside goes to zero asymptotically, the velocity of fall of the particles, as measured in their proper time, increases and approaches the velocity of light. In fact, according to the definition (88.10): „._„.,,. _»i.(*y 9 oo \dtj Taking gl t and g 00 from (97.14) and dr/dt from (100'5), we find that v 2 -> c 2 . The approach to the gravitational radius, which according to the clocks of the outside observer takes an infinite time, occupies only a finite interval of proper time (i.e. time in the reference system comoving with the body). This is already clear from the analysis given above, but one can also verify it directly by computing the proper time t as the invariant integral 1/c j" ds. Carrying out the calculation in the Schwarzschild reference system and taking dr/dt for the falling particle from (100.5), we get: -•/</©■('-?)- :V- J „ . , . dr ex V r \mc 2 J This integral converges for r-*r g . Having reached the gravitational radius (as measured by proper time), the body will con- tinue to contract, with all of its particles arriving at the center within a finite time. We do not, however, observe this process from outside the system; we have seen that no signals emerge from the Schwarzschild sphere (in the "contracting" reference system). t The essential properties of this phenomenon were first explained by J. R. Oppenheimer and H. Snyder (1939). 9Q9 c 10 Q GRAVITATIONAL COLLAPSE With respect to an external observer the contraction to the gravitational radius is accompanied by a "closing up" of the body. The time for propagation of signals sent from the body tends to infinity: for a light signal c dt = drl(\-r g lr), and the integral r [like the integral (100.5)] diverges for r - r 9 . Intervals of proper time on the surface of the body are shortened, as compared to intervals of time t for the distant observer in the ratio l-rlr: consequently, as r-+r g all processes on the body appear to be frozen with respect to the external observer. Such a "congealing" body interacts with surrounding bodies only through its static gravitational field. The question of gravitational collapse of nonspherical bodies has not been clarified much at present. One can apparently assert that for small deviations from sphericity collapse occurs (relative to the system of a distinct observer) to the same state of congealing of the body, and in the comoving reference system collapse occurs until it reaches the Schwarz- schild sphere; the ensuing fate of the body in the comoving system is, however not clear.t In conclusion we make one further remark of a methodological nature. We have seen that for the central field in vacuum the "system of the outside observer" that is mertial at infinity is not complete: there is no place in it for the world lines of particles moving inside the Schwarzschild sphere. The metric (100.3) is still applicable inside the Schwarzschild sphere but this system too is not complete in a certain sense. Consider, in this system, a particle carrying out a radial motion in the direction away from the center. As x - oo its world line goes out to infinity, while for x -+ - co it must approach asymptotically to r = r g , since in this metric, within the Schwarzschild sphere motion can occur only along the direc- tion to the center. On the other hand, emergence of the particle from r = r g to any given point r > r a occurs within a finite interval of proper time. In terms of proper time the particle must approach the Schwarzschild sphere from inside before it can begin to move outside it; but this part of the history of the particle is not kept by the particular reference system 4 PROBLEMS 1 For a particle in the field of a spherical body contracting to the gravitational radius, find the range of distances within which motion in a circular orbit is possible (S. A. Kaplan 1949). Solution: The dependence r = r(t) for a particle moving in the SchwarzschUd field w th an angular momentum M different from zero is obtained in a way analogous to (100.5), in differential form, 1 dr mc 2 // <? V_ 1 , r g M 2 M 2 r g (1) r (where m is the mass of the particle and r g the gravitational radius of the body). Equating the integrand in (1) to zero, we get the function #(r), which here replaces the potential curve of non- relativistic theory ; in Fig. 21 these curves are shown for different values of the angular momentum M. t See A. G. Doroshkevich, Ya. B. Zel'dovich and I. D. Novikov, JETP 49, 170, 1965, {Soviet Phys.~ t The construction of a reference system that is not incomplete in this way is considered in problem 5 of this section. 300 THE GRAVITATIONAL FIELD EQUATIONS § 100 e/mc 0-943 Fig. 21. The radii of the circular orbits and the corresponding energies are determined by the extrema of the curves, where the minima correspond to stable, and the maxima to unstable orbits. For M > V3 mcr g , each curve has one minimum and one maximum. As M increases from V3 mcr g to oo the coordinates of the minimum increase from 3r g to oo (and the corresponding energies from V8/9 mc 2 to mc 2 ); the coordinates of the maximum decrease from 3r g to 3r 9 /2 (while the corres- ponding energies go from V8/9 mc 2 to oo). For r < 3r g /2 there are no circular orbits. 2. For motion in this same field determine the cross-section for gravitational capture of (a) non- relativistic, (b) ultrarelativistic, particles coming from infinity (Ya. B. Zel'dovich and I. D. Novikov, 1964). Solution: (a) For a nonrelativistic velocity »«, (at infinity) the energy of the particle is £ « mc 2 . From Fig. 21 we see that the line £ = mc 2 lies above all the potential curves with angular momenta M < 2mcr g , i.e. all those with impact parameters q < 2cr g /v 00 . All particles with such values of q undergo gravitational capture: they reach the Schwarzschild sphere (asymptotically, as /-»oo) and do not emerge again to infinity. The capture cross-section is 4nr: \v„) (b) In equation (1) of problem 1 the transition to the ultrarelativistic particle (or to a light ray) is achieved by the substitution w->0. Also introducing the impact parameter e= cM/£, we get: 1 1- cdt -J l ~$ -^ + Q r g Setting the integrand equal to zero, we get the closest distance to the center, r mln , which the orbit reaches. This quantity attains its smallest value (r min = 3r g /2) for q = 3V3rJ2; for smaller values of q the particle moves to the Schwarzschild sphere. Thus we get for the capture cross-section 27 . nri 3. Find the equations of a centrally-symmetric gravitational field in matter in the comoving reference system. Solution: We make use of the two possible transformations of the coordinates r, t in the element of interval (97.1) in order to, first, make the coefficient a(r, t) of drdt vanish, and second, to make the radial velocity of the matter vanish at each point (because of the central symmetry, other com- ponents of the velocity are not present). After this is done, r and / can still be subjected to an arbitrary transformation of the form r = r(r') and / = /(/ '). (2) § 100 GRAVITATIONAL COLLAPSE We denote the radial coordinate and time selected in this way by R and r, and the ^ coefficknts h, M, by -e\ -«*, e\ respectively (where A, n and v are functions of R and r). We then have tor the line element: ,,,,, . j.,* m In the comoving reference system the components of the energy-momentum tensor are: A quite lengthy calculation leads to the following field equations: _^T 2 =— n = -e-X2v"+v ,2 +2/i"+At' 2 -/x'A'-v'A'+MV)+ + - ^-v(Av + /iv-A/i-2l-A 2 -2/i-/i 2 ), (3) 4 ^ Tl=0 = \e-W+»»'-l»'-v'M) (5) (where the prime denotes differentiation with respect to R, and the dot with respect to ct). General relations for the A, * v can be easily found if we start from the equations r, ^ = ^ wh °h are contained in the field equations. Using formula (86.11) we get the following two equations. If p is known as a function of e, equation (6) is integrated in the form: x+to—ij-g-sfiW. --*/£+/*>. (7) where the functions MR) and/ 2 (r) can be chosen arbitrarily in view of the possibility of making arbitrary transformations of the type R = R(R') and t = t(t'). 4. Find the general solution of the equations for a centrally-symmetric gravitational field in the comoving reference for the case of dustlike matter, i.e. for p = (R. Tolman, 1934).? Solution: From equations (6) we see that if p = we can set v = 0, which gives a unique choice of the time t (in other words the reference system can be chosen to be comoving and at the same time synchronous-in accordance with the general statement on p. 293). In pace of »(R, t) we introduce the function Ki?,t) = e B/2 , representing the "radius", defined so that 2nr is the circumference (of a circle with center at the origin); then the line element is ds 2 = dx 2 -e* dR 2 ~r 2 {R, T)(rf0 2 +sin 2 6 dip 2 ). Equation (5) takes the form AV = 2r', and is immediately integrated over the time, giving /o «'2 r (8) 1+/ where f(R) is an arbitrary function subject only to the condition 1 +/> 0. Substituting this expres- sion in equation (2) (substitution in (3) gives nothing new), we get: 2rr+r 2 -f=0. The first integral of this equation is t In problems 4 and 5, we set c = 1. 302 THE GRAVITATIONAL FIELD EQUATIONS § 100 where F(R) is another arbitrary function. Integrating once more, we get : to(R)-t = jVf^+Fr-~ sinh" 1 /^ for f>0, ■tM-t =yVfi*+&+— jy^^-'J^f for /<0, (10) ■ffl F^to-t) 2 ' 3 for f=0. In the first two cases the dependence r(R, r) can also be given in parametric form: F F r = y(co&hti-l\ T ~x=—j- 2 (smhn-ri) for />0. F F r = ^Yf (1 -cos rj), t -t== (»7-siiH7) for f< (10a) (where 77 is the parameter). Substituting (8) in (4) and eliminating /by using (9), we find for e: S" ke = ^ 2 - (ID Formulas (8)— (11) give the required general solution. We note that it depends essentially not on three, but only on two arbitrary functions determining the relation between /, F and t , since the coordinate R can still be subjected to an arbitrary transformation R = R(R'). This number corres- ponds exactly to the maximum possible number of "physically different" arbitrary functions for this case (see p. 276): the initial centrally-symmetric distribution of the matter is fixed by two quantities (the distributions of density and of radial velocity), while a free gravitational field with central symmetry does not exist. The overall sign in equations (10) is chosen so that the contraction of the sphere corresponds to 5750-*- —0. A complete solution for the collapse of the sphere requires specific inclusion of the initial conditions and "matching" on the boundary of the sphere with the Schwarzschild solution for empty space. But the limiting character of the metric inside the sphere follows immediately from the formulas given here. For t->t (R) the function r(R, t) goes to zero according to the law /3\ 4 / 3 r2 ~(2J F2/3 ( T °- T ) 4/3 > while the function e K goes to infinity like v 2/3 T '2 F 2/3 J ©' 1+/ (to-*) 2/3 * This means that all radial distances (in the comoving reference system) go to infinity, and all azimethal distances go to zero, while all volumes also go to zero (like t— T ).f Correspondingly, the matter density increases without limit : IF' 8nke x — — ; 3Ft (t — t) Thus, in accordance with the remarks in the text, there is a collapse of the whole distribution in to the center. J In the spatial case where the function r (R) = const (i.e. all the particles of the sphere reach the f The geometry on "planes" passing through the center is thus like that on a cone-shaped surface which is stretching in the course of time along its generators and at the same time contracting along the circles drawn on it. J It is understood that for e ->■ co the assumption of dustlike matter is not permissible from the physical point of view; one should use the ultrarelativistic equation of states = e/3. It appears that the general charac- ter of the contraction is to a large extent independent of the equation of state; see E. M. Lifshitz and I. M. Khalatnikov, JETP 39, 149, 1960, {Soviet Phys.—JETP, 12, 108, 1961). § 100 GRAVITATIONAL COLLAPSE 303 center simultaneously) the metric inside the contracting sphere has a different character. In this case F 2 ' 3 (t -t) 4/3 , (r 4F 4/3 (/+l) (to-t) 4 ' 3 , Siike i.e. as t 3(t -t) 2 ' 1 e. u S T-*r all distances, both tangential and radial, tend to zero according to the same law [~(t -t) 2 ' 3 ]; the matter density goes to infinity like (r -r)- 2 , and in the limit its distribution tends toward uniformity. „«u«« The case t = const includes, in particular, the collapse of a completely homogeneous sphere. Assuming (for example, for /> 0) F/2/ 3 ' 2 = a , f= sinh 2 * (where a is a constant), we get the metric * ds 2 = dT 2 -a 2 (r)[dR 2 +sinh 2 R(d0 2 +sm 2 d<p 2 )], where the dependence a(r) is given by the parametric equations a = 0o(cosh?/-l), T -T = ao(sinh>7-/7). The density is o / 6a ° a This solution coincides with the metric of a universe that is completely filled with homogeneous matter (§ 109). This is an entirely natural result, since a sphere cut out of a uniform distribution of matter has central symmetry. . 5. By a suitable choice of the functions F, f, r in the Tolman solution (problem 4), construct the most complete reference system for the field of a point mass. f Solution ■ When F = const * 0, we have from (1 1), e = 0, so that the solution applies to empty space, i e. it describes the field produced by a point mass (which is located at ihe center where there is a singularity of the metric). So setting F = I, f= 0, r (R) = R, we get the metric (100.3). J To achieve our purpose, we must start from a solution that contains both expanding and con- tracting" space-time regions. These are the Tolman solutions with/< 0; from (10a) we see that as the parameter n changes monotonically (from to In) for given R, the time t changes mono- tonically, while r goes through a maximum. We set 1 n /* 2 ,i\ 3 ' 2 -1 The we have: \(* + l)(\-cMt,\ } = ^(^ + 1 ) (rc-'Z+sin*/) K5 +I )" (where the parameter tj runs through values to 2n). In Fig 22, the curves /1C5 and A'C 'B' correspond to r = (parameter values q = In and ;/ - 0) The curves AOA' and BOB' correspond to the Schwarzschild sphere r = r g . Between A'C B and yi'Ofi' is a region of space-time in which only motion out from the center is possible, and between ACB and AOBsl region in which motion occurs only toward the center. The world line of a particle that is at rest relative to this reference system is a vertical line (R = const) It starts from r = (point a), cuts the Schwarzschild sphere at point b, and at time t = reaches its farthest distance [r = r g (R 2 lr 2 g + 1)], after which the particle again begins to fall in toward the Schwarzschild sphere, passing through it at point c and arriving once more at r = (point rf) at the time T = (7r/2)(CR 2 /r g 2 )+l) 3 ' 2 . . . This reference system is complete: both ends of the world line of any particle moving in the field lie either on the true singularity r = or go out to infinity. The metric (100.3) covers only the region t Such a system was first found by M. Kruskal, Phys. Rev. 119, 1743 (1960). The form of the solution given here (in which the reference system is synchronous) is due to I. D. Novikov (1963). % The case F = corresponds to the absence of a field ; by a suitable transformation of the variables, the metric can be brought to galilean form. 304 THE GRAVITATIONAL FIELD EQUATIONS A t B § 101 Fig. 22. to the right of AOA' (or to the left of BOB), and the "expanding" reference system covers the region to the right of BOB' (or to the left of AOA'). The system with metric (97.14) covers only the region to the right of BOA' (or to the left of A OB'). §101. The energy-momentum pseudotensor In the absence of a gravitational field, the law of conservation of energy and momentum of the material (and electromagnetic field) is expressed by the equation dT ik /dx k = 0. The generalization of this equation to the case where a gravitational field is present is equation (94.7): 1 d(T k J~g) ldg kl Tfc V- 9 dx k 2 dx* T Kl = 0. (101.1) In this form, however, this equation does not generally express any conservation law whatever.! This is related to the fact that in a gravitational field the four-momentum of the matter alone must not be conserved, but rather the four-momentum of matter plus gravita- tional field; the latter is not included in the expression for T k . To determine the conserved total four-momentum for a gravitational field plus the matter located in it, we proceed as follows. J We choose a system of coordinates of such form that t Because the integral $T k V—g dS k is conserved only if the condition ^~gT\) dx k is fulfilled, and not (101.1). This is easily verified by carrying out in curvilinear coordinates all those cal- culations which in § 29 were done in galilean coordinates. Besides it is sufficient simply to note that these calculations have a purely formal character not connected with the tensor properties of the corresponding quantities, like the proof of Gauss' theorem, which has the same form (83.17) in curvilinear as in cartesian coordinates. % One might get the notion to apply to the gravitational field the formula (94.4), substituting A = — (c 4 /167r£)G. We emphasize, however, that this formula applies only to physical systems described by quantities q different from the g ik \ therefore it cannot be applied to the gravitational field which is determined by the quantities g ik themselves. Note, by the way, that upon substituting G in place of A in (94.4) we would obtain simply zero, as is immediately clear from the relation (95.3) and the equations of the field in vacuum. § 101 THE ENERGY-MOMENTUM PSEUDOTENSOR 305 at some particular point in spacetime all the first derivatives of the g ik vanish (the g ik need not for this, necessarily have their galilean values). Then at this point the second term in equation (101.1) vanishes, and in the first term we can take V -g out from under the derivative sign, so that there remains d or, in contravariant components, - t* = dx k l ' ' -?- T ik = 0. 8x k Quantities T ik , identically satisfying this equation, can be written in the form 8 1 ~dx in ' where the r\ m are quantities antisymmetric in the indices k, /; n ikl = -ri m . Actually it is not difficult to bring T ik to this form. To do this we start from the field equation and for R ik we have, according to (92.4) R =- 2 9 9 9 \ dx m dx n + dx i dx p dx »*dx p dx l dx n j (we recall that at the point under consideration, all the It, = 0). After simple transforma- tions the tensor T ik can be put in the form The expression in the curly brackets is antisymmetric in k and /, and is the quantity which we designated above as rf kl . Since the first derivatives of g ik are zero at the point under consideration, the factor l/(-g) can be taken out from under the sign of differentia- tion d/dx l . We introduce the notation. h m -^-hn l(-9)(9 ik 9 lm -9"9 km )l (101.2) 16nk dx These quantities are antisymmetric in k and /: h ikl =-h ilk . (101.3) Then we can write = {-g)T k . ox This relation, derived under the assumption dgjdx 1 = 0, is no longer valid when we go to an arbitrary system of coordinates. In the general case, the difference dh lkl /dx l -(-g)T l 306 THE GRAVITATIONAL FIELD EQUATIONS § 101 is different from zero; we denote it by (-g)t ik . Then we have, by definition, dh ikl (-g)(T k + t ik ) = ^ r . (101.4) The quantities t ik are symmetric in i and k: t ik = t ki . (101.5) This is clear immediately from their definition, since like the tensor T ik , the derivatives dh lkl /dx l are symmetric quantities.! Expressing T ik in terms of R ik , according to the Einstein equations, and using expression (101.2) for the h ik \ one can obtain, after a rather lengthy calculation, the following expression for t ik : tik= iLc ^ 2r '- r ^- r "p r -- r "» r ^x^^ fcm -^^' m )+ +^^ m "(rf p rL+rLrf p -r„ fc p rf w -rLr^)+ ~*~9 9 K\ lp A mn + imn ^lp ~ *np ^Im ~ *lm ^np) + +9 lm g n \T\ n r k mp -T\ m r k np )l (101.6) or, in terms of derivatives of the components of the metric tensor, (~9)t ik = ~ {Q ik ,iQ lm , m -Q U ,,Q km , m + W k g lm Q ln ,pQ pm , n - - (9 il g m „ Q kn , P Q mp , 1 + g kl g m „ Q in , P Q mp , :) + g lm g n Y, „ Q km , p + __ +K2g il g km -9 ik 9 lm )(2g np g qr -g pq g„ r )Q nr , , g«, m }, (101.7) where g' fc = V -g g lk , while the index ,i denotes a simple differentiation with respect to x\ An essential property of the t ik is that they do not constitute a tensor; this is clear from the fact that in dh ikl /dx l there appears the ordinary, and not the co variant derivative. However, t lk is expressed in terms of the quantities P kl , and the latter behave like a tensor with respect to linear transformations of the coordinates (see § 85), so the same applies to the t ik . From the definition (101.4) it follows that for the sum T ik + t ik the equation -^ k (-g)(T ik + t ik ) = (101.8) is identically satisfied. This means that there is a conservation law for the quantities pi = ~ c J (-9)(T ik + t ik ) dS k . (101.9) In the absence of a gravitational field, in galilean coordinates, t ih = 0, and the integral we have written goes over into (1/c) J T ik dS k , that is, into the four-momentum of the material. Therefore the quantity (101.9) must be identified with the total four-momentum of matter plus gravitational field. The aggregate of quantities t ik is called the energy- momentum pseudo-tensor of the gravitational field. The integration in (101.9) can be taken over any infinite hypersurface, including all of the three-dimensional space. If we choose for this the hypersurface jc° = const, then P* | For just this aim we took(— g) out from under the derivative sign in the expression for T ik . If this had not been done, dh m /dx l and therefore also t lk would turn out not to be symmetric in i and k. § 101 THE ENERGY-MOMENTUM PSEUDOTENSOR 307 can be written in the form of a three-dimensional space integral: P i = \ f (_0XT ro -M , °) dV. (101.10) This fact, that the total four-momentum of matter plus field is expressible as an integral of the quantity (-g) (T ik + t ik ) which is symmetric in the indices /, k, is very important. It means that there is a conservation law for the angular momentum, defined as (see § 32)f M ik = f (x* dP k -x k dF) = 1 f {x\T kl + t kl )-x k (T il + t il )}(-g) dS t . (101.11) Thus, also in the general theory of relativity, for a closed system of gravitating bodies the total angular momentum is conserved, and, moreover, one can again define a center of inertia which carries out a uniform motion. This latter point is related to the conservation of the components M 0a (see § 14) which is expressed by the equation x o f (T «o + f «o )( _ g) dv _ j x XT 00 + t 00 )(-g) dV = const, so that the coordinates of the center of inertia are given by the formula f a (T 00 + f 00 )( _ )d y X* = ± - . (101.12) J(r oo +O(-0)^ By choosing a coordinate system which is inertial in a given volume element, we can make all the t ik vanish at any point in space-time (since then all the T l u vanish). On the other hand, we can get values of the t ik different from zero in flat space, i.e. in the absence of a gravitational field, if we simply use curvilinear coordinates instead of cartesian. Thus, in any case, it has no meaning to speak of a definite localization of the energy of the gravita- tional field in space. If the tensor T ik is zero at some world point, then this is the case for any reference system, so that we may say that at this point there is no matter or electro- magnetic field. On the other hand, from the vanishing of a pseudo-tensor at some point in one reference system it does not at all follow that this is so for another reference system, so that it is meaningless to talk of whether or not there is gravitational energy at a given place. This corresponds completely to the fact that by a suitable choice of coordinates, we can "annihilate" the gravitational field in a given volume element, in which case, from what has been said, the pseudotensor t ik also vanishes in this volume element. The quantities P l (the four-momentum of field plus matter) have a completely definite meaning and are independent of the choice of reference system to just the extent that is necessary on the basis of physical considerations. Let us draw around the masses under consideration a region of space sufficiently large so that outside of it we may say that there is no gravitational field. In the course of time, this region cuts out a "channel" in four-dimensional space-time. Outside of this channel there is no field, so that four-space is flat. Because of this we must, when calculating the energy t It is necessary to note that the expression obtained by us for the four-momentum of matter plus field is by no means the only possible one. On the contrary, one can, in an infinity of ways (see, for example, the problem in this section), form expressions which in the absence of a field reduce to T ik , and which upon integration over dS k , give conservation of some quantity. However, the choice made by us is the only one for which the energy-momentum pseudotensor of the field contains only first (and not higher) derivatives of g tk (a condition which is completely natural from the physical point of view), and is also symmetric, so that it is possible to formulate a conservation law for the angular momentum. 308 THE GRAVITATIONAL FIELD EQUATIONS § 101 and momentum of the field, choose a four-dimensional reference system such that outside the channel it goes over into a galilean system and all the t ik vanish. By this requirement the reference system is, of course, not at all uniquely determined- it can still be chosen arbitrarily in the interior of the channel. However the P', in full accord with their physical meaning, turn out to be completely independent of the choice of co- ordinate system in the interior of the channel. Consider two coordinate systems, different in the interior of the channel, but reducing outside of it to one and the same galilean system, and compare the values of the four-momentum P l and P"' in these two systems at definite moments of "time" x° and x'°. Let us introduce a third coordinate system, coinciding in the interior of the channel at the moment jc° with the first system, and at the moment x'° with the second, while outside of the channel it is galilean. But by virtue of the law of con- servation of energy and momentum the quantities P l are constant (dP^dx = 0). This is the case for the third coordinate system as well as for the first two, and from this it follows thatP f = P'\ Earlier it was mentioned that the quantities t ik behave like a tensor with respect to linear transformations of the coordinates. Therefore the quantities P l form a four- vector with respect to such transformations, in particular with respect to Lorentz transformations which, at infinity, take one galilean reference frame into another, f The four-momentum P *" can also be expressed as in integral over a distant three-dimensional surface surrounding "all space". Substituting (101.4) in (101.9), we find c) dx l dS k . This integral can be transformed into an integral over an ordinary surface by means of (6.17): pi = t§ htkldf * 1 - (10U3) If for the surface of integration in (101.9) we choose the hypersurface x° = const, then in (101.13) the surface of integration turns out to be a surface in ordinary space: J -if h i0 * df a . (101.14) To derive the analogous formula for the angular momentum, we write formula (101.2) in the form /i ifci = — A'' fc ' m ; (101.15) the expression for the quantities l iklm in terms of the components of the metric tensor is t Strictly speaking, in the definition (101.9) P l is a four-vector only with respect to linear transformations with determinant equal to unity; among these are the Lorentz transformations, which alone are of physical interest. If we also admit transformations with determinant not equal to unity, then we must introduce into the definition of P* the value of g at infinity by writing V— #«, P l in place of P l on the left side of (101.9). % The quantity df tk is the "normal" to the surface element, related to the "tangential" element df ih by (6.11): df ik = ?e iklm df lm . On the surface bounding the hypersurface which is perpendicular to the x° axis, the only nonzero components of df lm are those with /, m = 1, 2, 3, and so df* k has only those components in which one of i and k is 0. The components df * are just the components of the three-dimensional element of ordinary surface, which we denote by df a . § 101 THE ENERGY-MOMENTUM PSEUDOTENSOR 309 obvious from (101.2). Substituting (101.4) in (101.11) and integrating by parts, we obtain: M -c) \ * dx-dx" X dx>»dx») dSl if/. dx klmn k dx ilmn \ „„ i r (« dx klmn dx ilmn \ From the definition of the quantities X lklm it is easy to see that Thus the remaining integral over dS l is equal to 1 r r)7 ilnk 1 r lSh dS >=kS x "°' dff " Finally, again choosing a purely spatial surface for the integration, we obtain : M ifc = - f (x i h koa -x k h ioa + X ioak )df a . (101.16) We remind the reader that in applying formulas (101.14) and (101.16), in accordance with what was said above, the system of space coordinates should be chosen so that at inifinity the g ik tend toward their constant galilean values. Thus for the calculation according to formula (101.14) of the four-momentum of an isolated system of bodies which always remain close to the origin of coordinates, we can use for the metric at large distances from the bodies the expression (101.14), transforming it from spherical spatial coordinates to cartesian (for which we must replace dr by n a dx", where n is a unit vector along the direction of r) ; the corresponding metric tensor is 1 2km si 2kmn a n p O o = l 2-, g aP =-Kp 2 » 9o* = ®, (101.17) c r c r where m is the total mass of the system. Computing the required components of h M using formula (101.2), we obtain to the required accuracy (we keep terms ~ 1/r 2 ): h aoP = 0, 16nk dx p K9 9 } 8;r d*P \ r + r 3 / An r 2 ' Now integrating (101.14) over a sphere of radius r, we obtain finally: P" = 0, P° = mc, (101.18) a result which was to be expected. It is an expression of the equality of "gravitational" and "inertial" mass. {Gravitational mass means the mass which determines the gravitational field produced by a body; this is the mass which enters in the expression for the interval in a gravitational field, or, in particular, in Newton's law. The inertial mass determines the relation between momentum and energy of a body and, in particular, the rest energy of a body is equal to this mass multiplied by c 2 .) In the case of a constant gravitational field it turns out to be possible to derive a simple expression for the total energy of matter plus field in the form of an integral extended only over the space occupied by the matter. To obtain this expression one can, for example, R°o = -i^~^-gg io nd. 310 THE GRAVITATIONAL FIELD EQUATIONS § 101 start from the following identity, valid (as one easily verifies) when all quantities are in- dependent of x° :f d_ . / — Integrating KgV^g over (three-dimensional) space and applying Gauss' theorem, we obtain j R° ^ dV = j ^~g g io r a oi df a . We choose a sufficiently distant surface of integration and use on it the expressions for the g ik given by formula (101.17), and obtain after a simple calculation I R° ^g dV = -j- m = -^ P°. Noting also from the equations of the field that R o o=^(n-\Ty 4 -^(n-Ti-n-n), we get the required expression P° = mc= j (T{ + T 2 2 + Tl-T° )yJ~g dV. (101.19) This formula expresses the total energy of matter plus constant gravitational field (i.e. the total mass of the bodies) in terms of the energy-momentum tensor of the matter alone (R. Tolman, 1930). We recall that in the case of central symmetry of the field, we had still another expression for this same quantity, formula (97.23). PROBLEM Find the expression for the total four-momentum of matter plus gravitational field, using formula (32.5). Solution: In curvilinear coordinates one has S=jAV^gdVdt, and therefore to obtain a quantity which is conserved we must in (32.5) write AV—g in place of A, so that the four-momentum has the form H '{ -Av^^s'^a* Pl • • " ' -"' ■- dx i dq w t From (92.10), we have Ro= g *R iQ =g°i ^ +r{ r{,„-rnrS' w ), and with the aid of (86.5) and (86.8), we find that this expression can be written as r° — ±=— (V~ ^TioH^r^rSo. V-g dx l With the help of these same relations (86.8), one can easily verify that the second term on the right is identically equal to — hTiSAdg lm l%x ), and vanishes as a consequence of the fact that all quantities are independent of x°. Finally, for the same reason, replacing the summation over / in the first term by a summation over a, we obtain the formula of the text. § 102 GRAVITATIONAL WAVES 311 In applying this formula to matter, for which the quantities q w are different from the g ik , we can take V^g out from under the sign of differentiation, and the integrand turns out to be equal to V^g T k , where T k t is the energy-momentum tensor of the matter. When applying this same formula to the gravitational field, we must set A = -(c 4 /16ttA:)G, while the quantities q m are the components g ik of the metric tensor. The total four-momentum of field plus matter is thus equal to ]Jnv_^ +I |lj* oV-gS^—i——- dS k . Using the expression (93.3) for G, we can rewrite this expression in the form: The second term in the curly brackets gives the four-momentum of the gravitational field in the absence of matter. The integrand is not symmetric in the indices /, k, so that one cannot formulate a law of conservation of angular momentum. § 102. Gravitational waves Let us consider a weak gravitational field in vacuo. In a weak field the space-time metric is "almost galilean", i.e. we can choose a system of reference in which the components of the metric tensor are almost equal to their galilean values, which we denote by gS )= - 5 ^ 9^ = 0, g® = l. (102.1) We can therefore write the g ik in the form 9ik = 9$ ) + h ik , (102.2) where the h ik are small corrections, determined by the gravitational field. With small h ik , the components T kl , which are expressed in terms of the derivatives of g ik , are also small. Neglecting powers of h ik higher than the first, we may retain in the tensor R iklm (92.4) only the terms in the first bracket: _ifd 2 h im d 2 h kl d 2 h km d 2 h u \ mm 2 \dx k dx l dx l dx m dx'dx 1 dx k dx m )' K ' ) For the contracted tensor R ik , we have to this same accuracy ^ik — 9 l^limk ~ 9 ^limk or ik ~2\ 9 dx l dx m + 8x k dx l + dx i dx l dx'dxY K > where h = /i-.f We have chosen our reference system so that the g ik differ little from the g\ k ^. But this condition is also fulfilled for any infinitesimal coordinate transformation, so that we can t In accordance with the approximation, all operations of raising and lowering indices of small tensors and vectors are performed here and in the sequel using the "unperturbed" metric tensor g\%. Thus h*, = g iom hii, etc. Then we have for the contravariant components g ik : gK^gMK-h* (102.2a) (so that, to terms of first order, the condition gug lk = S k is satisfied). The determinant of the metric tensor is 9 = g™(l +g mk h ik ) = g™(l+h). (102.2b) ll* fc = 0, tf = h* t --$h. (102.5) 312 THE GRAVITATIONAL FIELD EQUATIONS § 102 still apply to the h ik four conditions (equal to the number of coordinates) which do not violate the condition that the h ik be small. We choose for these auxiliary conditions the equations It should be pointed out that even with these conditions the coordinates are not uniquely determined; let us see what transformations are still admissible. Under the transformation x n = x l + £', where the £' are small quantities, the tensor g ik goes over into 9ik-9ik ^k dy y i.e. [see formula (94.3)], in which the covariant differentiation reduces for the present case to ordinary differentiation, because of the constancy of g\ k } ). It is then easy to show that, if the h ik satisfy the condition (102.5), the h' ik will also satisfy this condition, if the ^ are solu- tions of the equation □& = 0, (102.7) where □ denotes the d'Alembertian operator , 0Mm d 2 d 2 1 d 2 n = — n^ ' = — u y dx l dx m dx 2 c 2 dt 2 ' From condition (102.5), the last three terms in the expression (102.4) for R ik cancel one another, and we find: R ik = ^ UK- Thus the equation for the gravitational field in vacuum takes on the form □ # = 0. (102.8) This is the ordinary wave equation. Thus gravitational fields, like electromagnetic fields, propagate in vacuum with the velocity of light. Let us consider a plane gravitational wave. In such a wave the field changes only along one direction in space; for this direction we choose the axis a; 1 = x. Equation (102.8) then changes to (£-?£)"- ' (io2 - 9) the solution of which is any function of t±x/c (§ 47). Consider a wave propagating in the positive direction along the x axis. Then all the quan- tities h) are functions of t—x/c. The auxiliary condition (102.5) in this case gives \[/l—ij/? = 0, where the dot denotes differentiation with respect to t. This equality can be integrated by simply dropping the sign of differentiation — the integration constants can be set equal to zero since we are here interested only (as in the case of electromagnetic waves) in the varying part of the field. Thus, among the components ij/] that are left, we have the relations xj,\ = K *i = i& ^3 = ^3, *h = K (102.10) § 102 GRAVITATIONAL WAVES 313 As we pointed out, the conditions (102.5) still do not determine the system of reference uniquely. We can still subject the coordinates to a transformation of the form x 'i = x i + ^{t-xjc). These transformations can be employed to make the four quantities ^°u ^5» *l*l* ^2 + ^3 vanish; from the equalities (102.10) it then follows that the components \j/\,il/ l 2 , \j/\, \{/o also vanish. As for the remaining quantities ^|, ^f-^i. thev cannot be made to vanish by any choice of reference system since, as we see from (102.6), these com- ponents do not change under a transformation £, = ^(t-x/c). We note that \jt = \j/\ also vanishes, and therefore xj/ 1 - = h). Thus a plane gravitational wave is determined by two quantities, h 23 and h 22 = -h 33 . In other words, gravitational waves are transverse waves whose polarization is determined by a symmetric tensor of the second rank in the yz plane, the sum of whose diagonal terms, h 22 + h 33 , is zero. We calculate the energy flux in a plane gravitational wave. The energy flux in a gravita- tional field is determined by the quantities -cgt 0a &ct 0a . In a wave propagating along the jc 1 axis, it is clear that only the component t 10 is different from zero. The pseudotensor t ik is of second order; we must calculate the t 01 only to this accuracy. A calculation making use of the formula (101.6), and the fact that in a plane wave the only components of h ik different from zero are h 23 , h 22 = — h 33 , leads to the result: 01 _ _ c 3 / dh 22 dh 22 ^33 3^33 5/J23 ^23 32nk \ dx dt dx dt dx dt If all quantities are functions only of t-xjc, then we get from this, finally, t 01 = ^-j c [hh + Xh 22 -h 33 ) 2 l (102.11) Since it has a definite energy, a gravitational wave produces around itself a certain additional gravitational field. This field is a quantity of higher (second) order compared to the field of the wave itself, since the energy producing it is a quantity of second order. As initial conditions for the arbitrary field of a gravitational wave we must assign four arbitrary functions of the coordinates : because of the transversality of the field there are just two independent components of h aP , in addition to which we must also assign their first time derivatives. Although we have made this enumeration here by starting from the properties of a weak gravitational field, it is clear that the result, the number 4, cannot be related to this assumption and applies for any free gravitational field, i.e. for any field which is not associated with gravitating masses. PROBLEMS 1. Determine the curvature tensor in a weak plane gravitational wave. Solution: Calculating R mm for (92.4) in the linear approximation in the h ik , we find the following nonzero components : — -^0202 — ^?0303 = — J?1212 = -f?0212 = -^0331 = -"3131 ~ a > -^0203 = — -^1231 — — ^0312 = ^?0231 = Mt where we use the notation O = — i/133 = i/?22> M = — i^23« 314 THE GRAVITATIONAL FIELD EQUATIONS § 103 In terms of the three-dimensional tensors A aB and B aB introduced in problem 3 of § 92, we have: /0 0\ /0 \ /4a/j = |0 — a nV B ae = lo m o J. \0 no) \0 a -nj By a suitable rotation of the x 2 , x 3 axes, we can make one of the quantities a or n vanish (at a given point of four-space); if we make a vanish in this way, we reduce the curvature tensor to the degenerate Petrov type II (type N). 2. Find the small corrections to the tensor R ik for an arbitrary "unperturbed" metric g%\ Solution: The corrections to the ChristofFel symbols are expressed in terms of 8g ik = h ik as ST i M = ¥hUi+h\ i H-h M :' t ) i which can be verified by direct calculation (all operations of raising and lowering indices, and of covariant differentiation, are done with the metric g$). For the corrections to the curvature tensor we find dRkim = 2(.h kim; i-\-h m -i c; i—hi cm ' ;l—h k - t i- m —h l U } C ;m J rhki' l ;rn)' The corrections to the Ricci tensor are: SR lk =dR l tlk = i(h 1 i : k: i+K. l:l -h lk - l :l -h :Uk ). (1) From the relation we have for the corrections to the mixed components R k : SR k =g mkl SR il -h kl R? l K (2) § 103. Exact solutions of the gravitational field equations depending on one variable In this section we shall consider the possible types of exact solutions of the gravitational field equations in vacuum, in which all the components of the metric tensor, for a suitable choice of reference system, are functions of a single variable.! This variable may have either timelike or spacelike character; to be specific, we shall assume first that it is timelike, and shall denote it by x° = t.% As we shall see, essentially different types of solutions are obtained depending on whether or not it is possible to choose a reference system for which all the components g 0a = while at the same time all other components still depend on only a single variable. The last condition obviously permits transformations of the coordinates x of the form x a -+x*+(j)Xi) where the <£* are arbitrary functions of t. For such a transformation, 9o*-+9o*+9*e¥ (where the dot denotes differentiation with respect to t). If the determinant \g ap \ ^ 0, the system of equations 9o a +9ap^ = (103.1 determines functions <jf{t) which accomplish the transformation to a reference system with g 0x = 0. By a transformation of the variable t according to V — g 00 dt->dt, we can then make g 00 equal to unity, so that we obtain a synchronous reference system, in which 9 oo = h 0o« = 0> 9ap = -y*p(t). (103.2) t Exact solutions of the field equations in vacuum, depending on a large number of variables, can be found in the paper: B. K. Harrison, Phys. Rev. 116, 1285 (1959). % In this section, to simplify the writing of formulas, we set c = 1. § 103 EXACT SOLUTIONS OF THE GRAVITATIONAL FIELD EQUATIONS 315 We can now use the equations of gravitation in the form (99. 1 1)— (99. 1 3). Since the quantities y aP , and with them the components of the three-dimensional tensor x aP = y a/? , do not depend on the coordinates x*, R% = 0. For the same reason, P ap = 0, and as a result the equations of the gravitational field in vacuum reduce to the following system : #+**£*! = 0» (103.3) 7-(Vy*&" = (103.4) vy From equation (103.4) it follows that Vy«J = 2& (103.5) where the X p a are constants. Contracting on the indices a and /?, we then obtain y 2 a y Vy from which we see that y = const • t 2 ; without loss of generality we may set the constant equal to unity (simply by a scale change of the coordinates x*); then A" = 1. Substitution of (103.5) into equation (103.3) now gives the relation %k% = 1 (103.6) which relates the constants A£. Next we lower the index /? in equations (103.5) and rewrite them as a system of ordinary differential equations: U = \%y»- ( 103 - 7 ) The set of coefficients XI may be regarded as the matrix of some linear substitution. By a suitable linear transformation of the coordinates jc 1 , x 2 , x 3 (or, what is equivalent, of g lp , g 2p , 03/?)> we can m general bring this matrix to diagonal form. We shall denote its principal values (roots of the characteristic equation) by p l ,p 2 ,p 3 , and assume that they are all real and distinct (concerning other cases, cf. below); the unit vectors along the corresponding principal axes are n (1) , n (2) and n (3) . Then the solution of equations (103.7) can be written in the form y«p = t^ni'^ + t^n^^ + t^ni 3 ^ (103.8) (where the coefficients of the powers of t have been made equal to unity by a suitable scale change of the coordinates). Finally, choosing the directions of the vectors n (1) , n (2) , n (3) as the directions of our axes (we call them x, y, z), we bring the metric to the final form (E. Kasner, 1922): ds 2 = dt 2 -t 2pi dx 2 -t 2p2 dy 2 -t 2p3 dz 2 . (103.9) Here^, p 2 and/? 3 are any three numbers satisfying the two relations Pi+P 2 +P 3 = h P 2 i + Pl+Pl = l (103.10) [the first of these follows from —g = t 2 , and the second — from (103.6)]. 316 THE GRAVITATIONAL FIELD EQUATIONS § 103 The three numbers p lf p 2 and p 3 obviously cannot all have the same value. The case where two of them are equal occurs for the triples 0,0,1 and — 1/3, 2/3, 2/3. In all other cases the numbers p u p 2 and p 3 are all different, one of them being negative and the other two positive. If we arrange them in the order p t <p 2 </>3, their values will lie in the intervals -±<Pi<0, 0<p 2 <$, l^p 3 ^l. (103.10a) Thus the metric (103.9) corresponds to a homogeneous but anisotropic space whose total volume increases (with increasing t) proportionally to t ; the linear distances along two of the axes (y and z) increase, while they decrease along the third axis (x). The moment t — is a singular point of the solution; at this point the metric has a singularity which can- not be eliminated by any transformation of the reference system. The only exception is the case where p t =p 2 = 0, p 3 = 1. For these values we simply have a flat space-time; by the transformation / sinh z = C, t cosh z = x we can bring the metric (103.9) to galilean form. A solution of the type of (103.9) also exists in the case where the parameter is^spacelike ; we need only make the appropriate changes of sign, for example, ds 2 = x 2pi dt 2 -dx 2 -x Zp2 dy 2 -x 2p3 dz 2 . However, in this case there also exist solutions of another type, which occur when the characteristic equations of the matrix A^O in equations (103.7) has complex or coincident roots (cf. problems 1 and 2). For the case of a timelike parameter t, these solutions are not possible, since the determinant g in them would not satisfy the necessary condition g < 0. A completely different type of solution corresponds to the case where the determinant of the tensor g aP which appears in equations (103.1) is equal to zero. In this case there is no reference system satisfying conditions (103.2). Instead we can now choose the reference frame so that: 9 10 = U 9oo = 920 = 930 = 0» 9*p - 9 a p(* )» where the determinant \g aP \ = 0. The variable jc° then has "lightlike" character: for dx" = 0, dx° # 0, the interval ds goes to zero; we denote this variable by x° = rj. The corresponding interval element can be represented in the form ds 2 = 2dx x dt]+g ab (dx a +g a dx 1 )(dx b + g h dx 1 ). Here in the following equations the indices a, b, c, ... run through the values 2,3; we may treat g ab as a two-dimensional tensor and g a as the components of a two-dimensional vector. Computation of the quantities R ab , which we shall omit here, gives the following field equations : Kb = -$9ac9 c 9bd9 d = (where the dot denotes differentiation with respect to rj). From this it follows that g ac g c = 0, or g c = 0, i.e., g c = const. By the transformation x a + g a x 1 -+ x a we can therefore bring the metric to the form ds 2 = 2dx l drj+g ab (ri) dx a dx\ (103.11) The determinant —g of this metric tensor coincides with the determinant \g ab \, while the only Christoffel symbols which are different from zero are the following: 1 bO ~ ? X b> x ab — 2 x ab> where we have introduced the two-dimensional tensor x ab = g ab . The only component of the tensor R ik which is not identically zero is R 00 , so that we have the equation Koo = **2+K*Z = 0. (103.12) § 103 EXACT SOLUTIONS OF THE GRAVITATIONAL FIELD EQUATIONS 317 Thus the three functions g 2 % (rj), g 33 (rj), g 23 (rj) must satisfy just one equation. There- fore two of them may be assigned arbitrarily. It is convenient to write equation (103.12) in another form, by representing the quantities g ab as g ab = -x 2 y ab , \y ab \ = 1. (103.13) Then the determinant -g = \g ab \ = % 4 , and substitution in (103.12) and a simple trans- formation gives: x+Kyacy hc )(y bd y ad )x = o (103.14) (where y ab is the two-dimensional tensor which is the inverse of y ab ). If we arbitrarily assign the functions y ab (r]) (which are related to one another by the relation \y ab \ — 1), the function X(rj) is determined by this equation. We thus arrive at a solution containing two arbitrary functions. It is easy to see that it represents a generalization of the treatment in § 102 of a weak plane gravitational wave (propagating along one direction).! The latter is obtained if we make the transformation _ t+x t t — x V2 V2 and set y ab = S ab + h ab (r]) (where the h ab are small quantities, which are subjected to the condition h 22 +h 33 = 0) and x = 1 ; a constant value of x satisfies equation (103.14) if we neglect terms of second order. Suppose that a weak gravitational wave of finite extension (a "wave packet") passes through some point x in space. Before the arrival of the packet we have h ab = and x = 1 ; after its passage we again have h ab = 0,d 2 x/dt 2 = 0, but the inclusion of second order terms in equation (103.14) leads to the appearance of a nonzero negative value of dx/dt: (where the integral is taken over the time during which the wave passes). Thus after the passage of the wave we will have x = 1 -const • /, and after some finite time interval has elapsed x will change sign. But a null value of x means that the determinant g of the metric tensor is zero, i.e. there is a singularity of the metric. However, this singularity is not physi- cally significant; it is related only to the inadequacy of the reference system furnished by the passing gravitational wave and can be eliminated by appropriate transformation; after passage of the wave the space-time will again be flat. One can show this directly. If we measure the values of the parameter r\ from its value for the singular point, then x = V, so that ds 2 = 2drj dx 1 -r] 2 [(dx 2 ) 2 + (dx i ) 2 ]. It is easily seen that for this metric R iklm = 0, so that the corresponding space-time is flat. And, in fact, after the transformation we get f The possibility of such a generalization was first pointed out by I. Robinson and H. Bondi (1957). We also cite papers in which solutions of a related character are found for a larger number of variables: A. Peres, Phys. Rev. Letters 3, 571 (1959); I. Robinson and A. Trautman, ibid. 4, 431 (1960). *ix 2 = y, 3 l e y 2 + Z 2 rjx 3 = z, x l =£ , 2rj ds 2 = 2dr]dZ-dy 2 -dz 2 , 318 THE GRAVITATIONAL FIELD EQUATIONS § 104 after which the substitution rj = (t+x)/y[2, £ = (t—x)/>/2 finally brings the metric to galilean form. This property of a gravitational wave — the appearance of a fictitious singularity — is, of course, not related to the fact that the wave is weak, but also occurs in the general solution of equation (103.12).f As in the example treated here, near the singularity % ~ r], i.e. — g ~ vf. Finally we point out that, in addition to the general solution given above, equation (103.12) also has special solutions of the form ds 2 = 2dr\ d Xi -ri 2s2 (dx 3 ) 2 -rj 2s3 (dx 2 ) 2 , (103.15) where s 2 , s 3 are numbers related to one another by the relation In these solutions the metric has a true singular point (at y\ — 0) which cannot be eliminated by any transformation of the reference system. PROBLEMS 1. Find the solution of equations (103.7) corresponding to the case where the characteristic equation of the matrix X B a has one real (p 3 ) and two complex (p lt 2 =/>'+*>") roots. Solution: In this case the parameter x°, on which all the quantities depend, must have spacelike character; we denote it by x. Correspondingly, we must now have g 00 = — 1 in (102.2). Equations (103.3-4) are not changed. The vectors n (1 \ n (2) in (103.8) become complex: n (1 - 2) = (n'±/n")/v2, where n', n" are unit vectors. Choosing the axes x 1 , x 2 , x 3 along the directions n', n", n (3) , we obtain the solution in the form -0n = 022 = * 2p/ cos Up" In ^Y ^12 = -x**' sin Up" In ^ Y 033 = -* 2p3 , -g = -0oo|0«/j| = x 2 , where a is a constant (which can no longer be eliminated by a scale change along the x axis, without changing other coefficients in the expressions given). The numbers, p u p 2 , Pa again satisfy the rela- tions (103.10), where the real numbers is either < —1/3 or > 1. 2. Do the same for the case where two of the roots coincide (p 2 —P3). Solution: We know from the general theory of linear differential equations that in this case the system (103.7) can be brought to the following canonical form: 2pi . 2p 2 . 2p 2 .A _ . 011 = 011, 02a = — 02a, 03« = — T 03«+ " 02*> O — Z, J, X X xx, where A is a constant. If A = 0, we return to (103.9). If X =£ 0, we can put X = 1 ; then g xx = -t 2 *l, 02a = d a X 2p 2, 03a = 6« t 2p * + d a X 2 ^ In X. From the condition # 23 =0 32 , we find that a 2 = 0,a 3 = b 2 . By appropriate choice of scale along the x 2 and x 3 axes, we finally bring the metric to the following form: ds 2 = -dx 2 -x 2p i (^ 1 ) 2 d=2jc 2 ^ dx 2 dx 3 ±x 2 '2 In I (dx 3 ) 2 . The numbers p u p 2 can have the values 1,0 or —1/3, 2/3. § 104. Gravitational fields at large distances from bodies Let us consider the stationary gravitational field at large distances r from the body which produces it, and determine the first terms of its expansion in powers of 1/r. t This can be shown by using equation (103.12) in exactly the same way as was done in § 100 for the analogous three-dimensional equation. § 104 GRAVITATIONAL FIELDS AT LARGE DISTANCES FROM BODIES 319 In the first approximation, to terms of order 1/r, the small corrections to the galilean values are given by the corresponding terms in the expansion of the Schwarzschild solution (97.14), i.e. by the formulas already given in (101.17): h® = ~, Ky=-~^, fcji> = o. (mi) Among the second order terms, proportional to 1/r 2 , there are terms which come from two different sources. Some of the terms arise, as a result of the nonlinearity of the equations of gravitation, from the first-order terms. Since the latter depend only on the total mass (and on no other characteristics) of the body, these second-order terms also can only depend on the total mass. It is therefore clear that these terms can be obtained by expanding the Schwarzschild solution (97.14), from which we find:f KV = 0, h $ =-*!££ n.n,. (104.2) The remaining second order terms appear as solutions of the already linearized equations of the field. To calculate them, we use the linearized equations in the form (102.8). In the stationary case, the wave equation reduces to the Laplace equation Ah* = 0. (104.3) The quantities h\ are coupled by the auxiliary conditions (102.5), which take the following form, since the h\ are independent of the time: A (fc ;_ i fc«J ) = 0> (104.4) £*-0. (104.5) The component h 00 must be given by a scalar solution of the Laplace equation. We know that such a solution, proportional to 1/r 2 , has the form a-V(l/r) where a is a constant vector. But a term of this type in h 00 can always be eliminated by a simple displacement of the coordinate origin in the first order term in 1/r. Thus the presence of such a term in h 00 would simply indicate that we had made a poor choice of the coordinate origin, and is therefore not of interest. The components h 0a are given by a vector solution of the Laplace equation, i.e. they must have the form where X aP is a constant tensor. The condition (104.5) gives: d 2 1 aP dx x dx p r~ ' t It should be noted that the specific appearance of the /&V, h ( a 2 \ h ( oo depends on the particular choice of the spatial coordinates (galilean at infinity) ; the form given in the text corresponds to just that definition of r for which the Schwarzschild solution is given by (97.14). So the transformation x' a = x a +£ a , £ a = ax a /2r results [see (102.6)] in the addition to h l „e of the term (a/r)(S aB —n a n B ), and by a suitable choice of a we can obtain: h^=-^i^l, (104.1a) c 2 r which corresponds to the Schwarzschild solution in the form given in problem 3 of § 97. 320 THE GRAVITATIONAL FIELD EQUATIONS § 104 from which it follows that A ap must have the form Kp — a afi + ^ap-> where a af} is an antisymmetric tensor. But a solution of the form A(d/dx a )/(l/r) can be eliminated by the transformation x'° = x° + £°, with £° = Xjr [see (102.6)]. Therefore the only solution which has a real meaning is of the form h 0a = a a p fap ~, a a p = ~ a Pa . (104.6) Finally, by a similar but more complicated argument, one can show that by a suitable transformation of the space coordinates one can always eliminate the quantities h a » given by a tensor (symmetric in a, /?) solution of the Laplace equation. There remains the task of examining the meaning of the tensor a ap in (104.6). For this purpose we use (101.16) to compute the total angular momentum tensor M aP in terms of the expressions we have found for the h 0a (assuming that all other components of h ik are absent). To terms of second order in the h 0a we have from formula (101.2) (we note that 9*°=-h a0 = h a0 ): _ c 4 d ~'i6nicdx'y (haod ^~ h " o5a ' i) c 4 8 16nk dx 13 ha0 A X2 d 2 1 a„ 16nk ay dx l! dx y r C 4 Zflpfly-dpy 16nk " y r 3 (where n is a unit vector along the radius vector). By means of these expressions, we find, after performing the integration over a sphere of radius r(df y = n 7 r 2 do): - c j tfh'oy-x'h' *) df y =-~j (n a n y a py -n p n y a ay ) do _ _ _^ 4jr " Znk 3 V«t a to- d to a «t> c 3 a t*p- 3k A similar calculation gives : i j X"» dfy=-^ k j (Ko dfp-h fi0 df a ) Combining these two results, we get : 67c" ap- Thus we finally have : M «" = k "• h£> = -~ M^. (104.7) § 104 GRAVITATIONAL FIELDS AT LARGE DISTANCES FROM BODIES 321 We emphasize that in the general case, when the field near the bodies may not be weak, M aP is the angular momentum of the body together with its gravitational field. Only when the field is weak at all distances can its contribution to the angular momentum be neglected. We also note that in the case of a rotating body of spherical shape, producing a weak field everywhere, formula (104.7) is valid over the whole space outside the body. Formulas (104.1), (104.2) and (104.7) solve our problem to terms of order 1/r 2 . The covariant components of the metric tensor are g ik = 9^ + KP + h^\ (104.8) To this same accuracy, the contravariant components are gik = g Wik_ h (i)ik_ h (2)ik + h (i)i ih (i)ik^ (104.9) Formula (104.7) can be written in vector form asf 2k 9 ~ c 3 r 2 nxM. (104.10) In problem 1 of § 88 it was shown that in a stationary gravitational field there acts on the particle a "Coriolis force" equal to that which would act on the particle if it were on a body rotating with angular velocity il =(c/2)V-#oo v x 3- Therefore we may say that in the field produced by a rotating body (with total angular momentum M) there acts on a particle distant from the body a force which is equivalent to the Coriolis force which would appear for a rotation with angular velocity fl^-Vx^yr [M-3n(n-M)]. 2 c r PROBLEM Determine the systematic ("secular") shift of the orbit of a particle moving in the field of a central body, associated with the rotation of the latter (J. Lense, H. Thirring, 1918). Solution: Because all the relativistic effects are small, they superpose linearly with one another, so in calculating the effects resulting from the rotation of the central body we can neglect the in- fluence of the non-Newtonian centrally symmetric force field which we considered in § 98; in other words, we can make the computations assuming that of all the h ik only the h 0a are different from zero. The orientation of the classical orbit of the particle is determined by two conserved quantities: the angular momentum of the particle, M = r xp, and the vector p _, kmtrir, A = — xM m r whose conservation is peculiar to the Newtonian field <p = —km'/r (where m' is the mass of the central body). J The vector M is perpendicular to the plane of the orbit, while the vector A is directed along the major axis of the ellipse toward the perihelion (and is equal in magnitude to kmm'e, where e is the eccentricity of the orbit). The required secular shift of the orbit can be described in terms of the change in direction of these vectors. The Lagrangian for a particle moving in the field (104.10) is ds 1km m ,. ,..,. L= -mc — =L +SL, 8L = mcg-\ =- ? - 3 M'-vxr (1) dt c 2 r 3 t To the present accuracy, the vector g a = -g Jgoo = —Qoa- For the same reason, in the definitions of vector product and curl (cf. the footnote on p. 252), we must set y = 1, so that they may be taken as usual for cartesian vectors. % See Mechanics, § 15. 322 THE GRAVITATIONAL FIELD EQUATIONS § 104 (where we denote the angular momentum of the central body by M' to distinguish it from the angular momentum M of the particle). Then the Hamiltonian is:f Ik Computing the derivative ]VI = rxp+rxp using the Hamilton equations r = 3^/8p, p = ~(8Jf?/dr), we get: 2k M = ^3 M 'x M - (2) Since we are interested in the secular variation of M, we should average this expression over the period of rotation of the particle. The averaging is conveniently done using the parametric represen- tation of the dependence of r on the time for motion in an elliptical orbit, in the form T r = a(l -e cos £), t = —^-e sin £) Ln (a and e are the semimajor axis and eccentricity of the ellipse) :% d£ 1 TJ r 3 ~2na 3 ] (T^ ecos£) 2 a 3 (l~e 2 ) 3 ' 2 ' Thus the secular change of M is given by the formula dM 2kM'xM dt c 2 a 3 (l-e 2 ) 3 ' 2 ' (3) i.e. the vector M rotates around the axis of rotation of the central cody, remaining fixed in mag- nitude. An analogous calculation for the vector A gives : 2k fik A = -=-= M' X A+ -£—. . (M • M')(r x M). c 2 r 3 c 2 mr 5 The averaging of this expression is carried out in the same way as before; from symmetry considera- tions it is clear beforehand that the averaged vector r/r 5 will be along the major axis of the ellipse, i.e. along the direction of the vector A. The computation leads to the following expression for the secular change of the vector A: ^ = " XA ' n = c 2 a 3 t M er' 2 {n, ~ Mn ' n,)} (4) (n and n' are unit vectors along the directions of M and M'), i.e. the vector A rotates with angular velocity SI, remaining fixed in magnitude; this last point shows that the eccentricity of the orbit does not undergo any secular change. Formula (3) can be written in the form — r- =nxM, dt with the same SI as in (4); in other words, SI is the angular velocity of rotation of the ellipse "as a whole". This rotation includes both the additional (compared to that considered in § 98) shift of the perihelion of the orbit, and the secular rotation of its plane about the direction of the axis of the body (where the latter effect is absent if the plane of the orbit coincides with the equatorial plane of the body). For comparison we note that to the effect considered in § 98 there corresponds _ 6nkrri ~ c 2 a(l-e 2 )T U ' f See Mechanics, § 40. t See Mechanics, § 15. § 105 RADIATION OF GRAVITATIONAL WAVES 323 § 105. Radiation of gravitational waves Let us consider next a weak gravitational field, produced by arbitrary bodies, moving with velocities small compared with the velocity of light. Because of the presence of matter, the equations of the gravitational field will differ from the simple wave equation of the form \Jh k = (102.8) by having, on the right side of the equality, terms coming from the energy-momentum tensor of the matter. We write these equations in the form |n*}-$Jtj, (los.i) where we have introduced in place of the h) the more convenient quantities \J/ k = h k -%5 k h, and where x\ denotes the auxiliary quantities which are obtained upon going over from the exact equations of gravitation to the case of a weak field in the approximation we are con- sidering. It is easy to verify that the components x% and x° a are obtained directly from the corresponding components T) by taking out from them the terms of the order of magnitude in which we are interested; as for the components x a p , they contain along with terms obtained from the Tfj, also terms of second order from R k — i<5*R. The quantities if/] satisfy the condition (102.5) d\i/\jdx k = 0. From (105.1) it follows that this same equation holds for the t* : dx k ^L = 0. (105.2) This equation here replaces the general relation T k . k = 0. Using the equations which we have obtained, let us consider the problem of the energy radiated by moving bodies in the form of gravitational waves. The solution of this problem requires the determination of the gravitational field in the "wave zone", i.e. at distances large compared with the wavelength of the radiated waves. In principle, all the calculations are completely analogous to those which we carried out for electromagnetic waves. Equation (105.1) for a weak gravitational field coincides in form with the equation of the retarded potentials (§ 62). Therefore we can immediately write its general solution in the form #t--?Je&-!x- (105 - 3) Since the velocities of all the bodies in the system are small, we can write, for the field at large distances from the system (see §§66 and 67), c R J c where R is the distance from the origin, chosen anywhere in the interior of the system. From now on we shall, for brevity, omit the index t—(R /c) in the integrand. For the evaluation of these integrals we use equation (105.2). Dropping the index on the T* and separating space and time components, we write (105.2) in the form dx ay ^a0_ n faoy dtpO _ MftS^ Multiplying the first equation by x?, we integrate over all space, 324 THE GRAVITATIONAL FIELD EQUATIONS § 105 Since the infinity x ik = 0, the first integral on the right, after transformation by Gauss' theorem, vanishes. Taking half the sum of the remaining equation and the same equation with transposed indices, we find j r aP dV = - - ^ j (r a0 x p + y x a ) dV. Next, we multiply the second equation of (105.5) by x a x p , and again integrate over all space. An analogous transformation leads to ^o j t 00 xV dv= -\ fao*' + */»<>*") dV. Comparing the two results, we find J XaP dV = 2 dx* J TooX " X ' dV ' (105-6) Thus the integrals of all the x aP appear as expressions in terms of integrals containing only the component t 00 . But this component, as was shown earlier, is simply equal to the corres- ponding component T 00 of the energy-momentum tensor and can be written to sufficient accuracy [see (96.1)] as: T 00 = VC 2 . (105.7) Substituting this in (105.6) and introducing the time t = x°/c, we find for (105.4) 2k d 2 C ^ = ~ 7r~ d? J ^ dv ' (105 ' 8) At large distances from the bodies, we can consider the waves as plane (over not too large regions of space). Therefore we can calculate the flux of energy radiated by the system, say along the direction of the x 1 axis, by using formula (102.11). In this formula there enter the components h 23 = \J/ 23 and h 22 — h 33 = ij/ 22 — ^33- From (105.8), we find for them the expressions 2k .. 2k "23 = — o 4n ^23> "22~~"33 = _ o 4n C^22 — ^33) (the dot denotes time differentiation), where we have introduced the tensor D aP = j ii(3x a x p -8 aP x 2 ) dV, (105.9) the "quadrupole moment" of the mass (see § 96). As a result, we obtain the energy flux along the x 1 axis in the form 10 ^ / "22 ^33 \ . K2 ct xv = 36nc _ / D 22 -p 33 y (105.10) Knowing the radiation in the direction of the x 1 axis, it is easy to determine the radiation in an arbitrary direction characterized by the unit vector n. To do this we must construct from the components of the tensor D afi and the vector n a a scalar, quadratic in the D afi , which for n± = 1, n 2 = n 3 = reduces to the expression in square brackets in (105.10). The result for the intensity of energy radiated into solid angle do turns out to be k Tl 1 dI = 36^? U iD ' pnatlfi)2 + 2 D2 «- D * D o n > n r do. (105.11) § 106 EQUATIONS OF MOTION, SECOND APPROXIMATION 325 The total radiation in all directions, i.e., the energy loss of the system per unit time {-dijdt), can be found by averaging the flux over all directions and multiplying the result by 4tt. The averaging is easily performed using the formulas given in the footnote on p. 189. This averaging leads to the following expression for the energy loss: _dl == k D 2 (105.12) dt 45c 5 a/I It is necessary to note that the numerical value of this energy loss, even for astronomical objects, is so small that its effects on the motion, even over cosmic time intervals, is com- pletely negligible (thus, for double stars, the energy loss in a year turns out to be ~ 10 of the total energy). PROBLEM Two bodies attracting each other according to Newton's law move in a circular orbit (around their common center of mass). Calculate the velocity of approach of the two bodies, due to the loss of energy by radiation of gravitational waves. Solution: If mi, m 2 are the masses of the bodies, and r their mutual distance (constant for motion in a circular orbit), then a calculation using (105.12) gives dS 32k i m 1 m 2 \ 2 , 6 dt 5c 5 \mr-\-mJ where co = In/T, and Tis the period of rotation. The frequency co is related to r by co 2 r 3 = k(m x + m 2 ) . Since km 1 m 2 = 2r 2 dS ~~~ 2r ' km 1 m 2 dt" 1 and we get finally , _ 64k 3 mx m 2 {m x + m 2 ) § 106. The equations of motion of a system of bodies in the second approximation The expression (105.12) found in the preceding section for the loss of energy of a system in the form of radiation of gravitational waves contains a factor l/c 5 , i.e. this loss appears only in the fifth approximation in l/c. In the first four approximations, the energy of the system remains constant. From this it follows that a system of gravitating bodies can be described by a Lagrangian correctly to terms of order l/c 4 in the absence of an electro- magnetic field, for which a Lagrangian exists in general only to terms of second order (§ 65). Here we shall give the derivation of the Lagrangian of a system of bodies to terms of second order. We thus find the equations of motion of the system in the next approximation after the Newtonian. We shall neglect the dimensions and internal structure of the bodies, regarding them as "pointlike"; in other words, we shall restrict ourselves to the zero'th approximation in the expansion in powers of the ratios of the dimensions a of the bodies to their mutual separa- tions /. To solve our problem we must start with the determination, in this same approximation, of the weak gravitational field produced by the bodies at distances large compared to their dimensions, but at the same time small compared to the wavelength A of the gravitational waves radiated by the system (a <4 r 4, X ~ lc/v). 326 THE GRAVITATIONAL FIELD EQUATIONS § 106 In the first approximation, in equations (105.1) we must neglect terms containing second time derivatives, with the factor 1/c 2 , and of all the components t& assume different from zero only the component Tq = !*c z which contains c 2 (whereas the other components contain the first or second power of the velocities of the bodies). We then obtain the equations A^ = 0, A^o = 0, A«AS = ^V c We must look for solutions of these equations which go to zero at infinity (galilean metric). It therefore follows from the first two equations that \j/ p a = 0,\J/ a = 0. Comparing the third equation with equation (96.2) for the Newtonian potential 0, we find j/^ = 40/c 2 . Then we have for the components of the tensor h\ = ^f-# 5) the following values :f 2 h i=-^<l>%, (106.1) 2 K = 0, h° = -2 cf), (106.2) and for the interval, ds 2 = (l + l<f>\ c 2 dt 2 -fl-~A (dx 2 + dy 2 + dz 2 ). (106.3) We note that first order terms containing <j> appear not only in g 00 but also in g a/} ; in § 87 it was already stated that, in the equations of motion of the particle, the correction terms in g aP give quantities of higher order than the terms coming from g 00 ; as a consequence, of this, by a comparison with the Newtonian equations of motion we can determine only 000- As will be seen from the sequel, to obtain the required equations of motion it is sufficient to know the spatial components h xp to the accuracy (~ 1/c 2 ) with which they are given in (106.1); the mixed components (which are absent in the 1/c 2 approximation) are needed to terms of order 1/c 3 , and the time component h 00 to terms in 1/c 4 . To calculate them we turn once again to the general equations of gravitation, and consider the terms of corresponding order in these equations. Disregarding the fact that the bodies are macroscopic, we must write the energy-momen- tum tensor of the matter in the form (33.4), (33.5). In curvilinear coordinates, this expression is rewritten as -, ifc ,-, m„c dx* dx k rtt =?7^^ (r -° (106 - 4) [for the appearance of the factor 1/V-flf, see the analogous transition in (90.4)]; the sum- mation extends over all the bodies in the system. The component v m a c 3 2 dt a ■sj — g dS in first approximation (for galilean g ik ) is equal to J] m a c 2 <5(r— r a ); in the next approxima- a t This result is, of course, in complete agreement with the formulas found in § 104 for h\l? [where hi)? is represented in the form (104.1a)]. § 106 EQUATIONS OF MOTION, SECOND APPROXIMATION 327 tion, we substitute for g ik from (106.3) and find, after a simple computation: Too = Ev 2 (i+f + S) <5(r-r * ) ' (106 ' 5) where v is the ordinary three-dimensional velocity (if = dxf/dt) and (f> a is the potential of the field at the point r a . (As yet we pay no attention to the fact that (p a contains an in- finite part— the potential of the self-field of the particle m a ; concerning this, see below.) As regards the components T aP , T 0a of the energy-momentum tensor, in this approxima- tion it is sufficient to keep for them only the first terms in the expansion of the expression (106.4) Tap = Z ™ a Va*VapS(r-*a), ' ^ Sit A ( 106 - 6 ) a Next we proceed to compute the components of the tensor R ik . The calculation is con- veniently done using the formula R ik = g lm R nmk with R Umk given by (92.4). Here we must remember that the quantities h aP and h 00 contain no terms of order lower than 1/c 2 , and/? « no terms lower than 1/c 3 ; differentiation with respect to x° = ct raises the order of smallness of quantities by unity. The main terms in R 00 are of order 1/c 2 ; in addition to them we must also keep terms of the next non- vanishing order 1/c 4 . A simple computation gives the result: R °° - c dt \dx* " 2c dt J + 2 °° 2 n dx*dx p 4\dx*J ldh 00 f dhj dK\ 4 dx p \ dx a dx p )' In this computation we have still not used any auxiliary condition for the quantities h ik . Making use of this freedom, we now impose the condition ff4f = 0, (106.7) dx" 2c dt as a result of which all the terms containing the components h 0a drop out of R 00 . In the remaining terms we substitute ^=-|#f, fc o= -|^ + °( c 4) R 00 = - Aftoo + -4 0A0 - -4 (V<^) 2 , (106.8) and obtain, to the required accuracy, 12 2 -A&OO+-4 0A0--4 2 c c where we have gone over to three-dimensional notation; here $ is the Newtonian potential of the system of point particles, i.e. In computing the components R 0a it is sufficient to keep only the terms of the first non- vanishing order — 1/c 3 . In similar fashion, we find: ■> i s*k i dv 1 8 *h$ i °* 2c dtdx" 2 dx'dx" 2c dtdx" 2 °* 328 THE GRAVITATIONAL FIELD EQUATIONS § 106 and then, using the condition (106.7): 1 a , 1 d 2 cf> *"-2 A *°- + 2?Sa? (m9) Using the expressions (106.5)-(106.9), we now write the Einstein equations Snk/ 1 \ R ik = -jr [T ik - - g ik T). (106.10) The time component of equation (106.10) gives: Ah 00 + - 4 0A0- ^ (V<£) 2 = -^r I m a c 2 ^1+ -^ + ^-f J <5(r-r a ); making use of the identity 4(V0) 2 = 2A(0 2 )-40A0 and the equation of the Newtonian potential A(j) = Ank £ m a d(r-r a ), (106.11) a we rewrite this equation in the form a (h 00 - 1 p) = "E».(i+f + 1?) *'-'.>• ( 106 - 12 ) After completing all the computations, we have replaced cf) a on the right side of (106.12) by m h b \r a -r b \ i.e. by the potential at the point x a of the field produced by all the particles except for the particle m a ; the exclusion of the infinite self-potential of the particles (in the method used by us, which regards the particles as pointlike) corresponds to a "renormalization" of their masses, as a result of which they take on their true values, which take into account the field produced by the particles themselves.f The solution of (106.12) can be given immediately, using the familiar relation (36.9) A - = -4n5(r). r We thus find : 2<t> 2<f> 2 2k m a <J>' a 3k m^l , 1(v;i . The mixed component of equation (105.10) gives c 2 8t dx x ' Afco« = " ^Im.ar-0- i ^-. (106.14) t Actually, if there is only one particle at rest, the right side of the equation will have simply (8nk/c 2 )m a S(r— r a ), and this equation will determine correctly (in second approximation) the field produced by the particle. § 106 EQUATIONS OF MOTION, SECOND APPROXIMATION 329 The solution of this linear equation isf _ Ak „ m a v aa _ 1_ d 2 f 0a ~ c 3 V|r-r a | c*dtdx*' where /is the solution of the auxiliary equation km„ a I 1 l a\ Using the relation Ar = 2/r, we find: /= -~E m -l r - r «l> and then, after a simple computation, we finally obtain: ho* = 2? ? [r^| [ 7 *~ + (V«0«J. (106.15) where n fl is a unit vector along the direction of the vector r-r fl . The expressions (106.1), (106.13) and (106.15) are sufficient for computing the required Langrangian to terms of second order. The Lagrangian for a single particle, in a gravitational field produced by other particles and assumed to be given, is L a = ~ m a c j= -m a c |l + noo+2no I -- c i+^- c r] • Expanding the square root and dropping the irrelevant constant -m a c 2 , we rewrite this expression, to the required accuracy, as (106.16) Here the values of all the h ik are taken at the point r fl ; again we must drop terms which become infinite, which amounts to a "renormalization" of the mass m a appearing as a coefficient inL fl . The further course of the calculations is the following. The total Lagrangian of the system is, of course, not equal to the sum of the Lagrangians L a for the individual bodies, but must be constructed so that it leads to the correct values of the forces f a acting on each of the bodies for a given motion of the others. For this purpose we compute the forces f a by differentiating the Lagrangian L a : t = •■-m... the differentiation is carried out with respect to the running coordinate r of the "field point" in the expressions for h ik ). It is then easy to form the total Lagrangian L, from which all of the forces f a are obtained by taking the partial derivatives dL/dr a . t In the stationary case, the second term on the right of equation (106.14) is absent. At large distances from the system, its solution can be written immediately by analogy with the solution (44.3) of equation (43.4) /»o« = ^ r2 (Mxn) (r (where M= J" rx/m/F= Im o r xv a is the angular momentum of the system), in complete agreement with formula (104.10). a ■£ \ 6 C r ab / a OC a b ^'ab 330 THE GRAVITATIONAL FIELD EQUATIONS § 106 Omitting the simple intermediate computations, we give immediately the final result for the Lagrangian:f rri _ i ' L = v \^,km a m b k 2 m a m b m c ■" * H ' c r o6 a 6 c **c r ab r ac where r a6 = |r fl -r 6 |, n ab is a unit vector along the direction r a -r 6 , and the prime on the summation sign means that we should omit the term with b = a or c = a. PROBLEMS 1. Find the action function for the gravitational field in the Newtonian approximation. Solution: Using the# ifc from (106.3), we find from the general formula (93.3), G = — (2/c 4 )(Vp) 2 , so that the action for the field is * — 85 /J (")"'»'*• The total action, for the field plus the masses distributed in space with density p, is : ff ~JJ[Tr-"-85 (vrt "] , ' KA (I) One easily verifies that variation of S with respect to q> gives the Poisson equation (96.2), as it should. The energy density is found from the Lagrangian density A [the integrand in (1)] by using the general formula (32.5), which reduces in the present case (because of the absence of time derivatives of <p in A) to changing the signs of the second and third terms. Integrating the energy density over all space, where we substitute n<P = (M4nk)<p/\<p in the second term and integrate by parts, we finally obtain the total energy of field plus matter in the form /[ 2. %nk KV) dV. Consequently the energy density of the gravitational field in the Newtonian theory is W = -(l/$nk)(V<p) 2 .t 2. Find the coordinates of the center of inertia of a system of gravitating bodies in the second approximation. Solution: In view of the complete formal analogy between Newton's law for gravitational interaction and Coulomb's law for electrostatic interaction, the coordinates of the center of inertia are given by the formula which is analogous to the formula found in Problem 1 of § 65. 3. Find the secular shift of the perihelion of the orbit of two gravitating bodies of comparable mass (H. Robertson, 1938). t The equations of motion corresponding to this Lagrangian were first obtained by A. Einstein, L. Infeld and B. Hoffmann (1938) and by A. Eddington and G. Clark (1938). J To avoid any misunderstanding, we state that this expression is not the same as the component (— g) t 00 of the energy-momentum pseudotensor (as calculated with the g ik from (106.3)); there is also a contribution to W from (-g) T ik . § 106 EQUATIONS OF MOTION, SECOND APPROXIMATION 331 Solution: The Lagrangian of the system of two bodies is mi v? , m 2 v\ knix m 2 , 1 , 4 , 4 v , L = -y- 1 + —^ + — y— + g^ (mi »i -f m 2 vi) + ,teim 2rv!ll 2 . _ / v „yi k 2 m 1 m 2 (m 1 +m 2 ) + ~2^r P(v?+ v2)-7Vi • v 2 -(vi • n)(v 2 -n)] ^a • Going over to the Hamiltonian function and eliminating from it the motion of the center of inertia (see problem 2 in § 65), we get: v-P 2 / 1 , 1\ kimm p* (1 1\ "~2Vmi + mJ r %c 2 \m*J m%) where p is the momentum of the relative motion. We determine the radial component of momentum p r as a function of the variable r and the parameters M (the angular momentum) and € (the energy). This function is determined from the equation & = £ (in which, in the second-order terms, we must replace p 2 by its expression from the zero'th approximation): a 1 / 1 , 1 \ / 2 , M 2 \ km im2 1 / 1 1 \ f 2m im2 \ 2 ( km^mtf The further course of the computations is analogous to that used in § 98. Having determined p r from the algebraic equation given above, we make a transformation of the variable r in the integral S r = jp,dr, so that the term containing M 2 is brought to the form M 2 jr 2 . Then expanding the expression under the square root in terms of the small relativistic corrections, we obtain: k 2c 2 r '3(^ + ^+7 \mi m 2 ) S r = j Ja + ° t -( M >-^)L* [see (98.6)], where A and B are constant coefficients whose explicit computation is not necessary. As a result we find for the shift in the perihelion of the orbit of the relative motion : _ 6 nk 2 m\ m\ _ 6 nkitrix + m 2 ) (p ~ c 2 M 2 ~ c 2 a(\-e 2 ) ' Comparing with (98.7) we see that for given dimensions and shape of the orbit, the shift in the peri- helion will be the same as it would be for the motion of one body in the field of a fixed center of mass mi+m 2 . 4. Determine the frequency of precession of a spherical top, performing an orbital motion in the gravitational field of a central body that is rotating about its axis. Solution: In the first approximation the effect is the sum of two independent parts, one of which is related to the non-Newtonian character of the centrally symmetric field, and the other to the rotation of the central body.t The first part is described by an additional term in the Lagrangian of the top, corresponding to the second term in (106.17). We write the velocities of individual elements of the top (with mass dm) in the form v = V+ co x r, where V is the velocity of the orbital motion, & is the angular velocity, and r is the radius vector of the element dm relative to the center of the top (so that the integral over the volume of the top [xdm = 0). Dropping terms independent of a and also neglecting terms quadratic in eo, we have: „„_ 3fcm' f . V-oxr , t The first effect was treated by H. Weyl (1923) and the second by L. Schiff (1960). 332 THE GRAVITATIONAL FIELD EQUATIONS § 106 where m' is the mass of the central body, i? = |R +r] is the distance from the center of the field to the element dm, R is the radius vector of the center of inertia of the top. In the expansion 1/R « l/R -(n-r/Rl) (where n = R /i? ) the integral of the first term vanishes, while integration of the second term is done using the formula x a x B dm = iIS a g where / is the moment of inertia of the top. As a result we get : 3km' 2c 2 R 2 where M = /« is the angular momentum of the top. The additional term in the Lagrangian, due to the rotation of the central body, can also be found from (106.17), but it is even simpler to calculate it using formula (1) of the problem in § 104: where M' is the angular momentum of the central body. Expanding, R n | r 3 *K + r 3 and performing the integration, we get : ^ 2) ^=^ 5 {M-M / -3(n-MXn-M')}. Thus the total correction to the Lagrangian is 6L=-M-£l, n = |~nxvo+~{3n(n-M0-M'}. To this function there corresponds the equation of motion — =fixM dt [see equation (2) of the problem in § 104]. This means that the angular momentum M of the top precesses with angular velocity Q, remaining constant in magnitude. <5 (1) £=^7T 2 M-VoXn, ,3~ ^ + ^- 3 ( r - 3n ( n ' r )) CHAPTER 12 COSMOLOGICAL PROBLEMS § 107. Isotropic space The general theory of relativity opens new avenues of approach to the solution of prob- lems related to the properties of the universe on a cosmic scale. The new possibilities which arise from the non-galilean nature of space-time are remarkable. These possibilities are the more important because Newtonian mechanics here leads to contradictions which cannot be avoided in a sufficiently general way within the framework of nonrelativistic theory. Thus, applying the Newtonian formula for the gravitational potential (96.3) to a flat (as it is in Newtonian mechanics) infinite space filled with matter having an arbitrarily distributed average density that vanishes nowhere, we find that the potential becomes infinite at every point. This would lead to infinite forces acting on the matter, which is absurd. We know that the stars are distributed in space in an extremely nonuniform fashion — they are concentrated in individual star systems (galaxies). But in studying the universe on a "large scale" we should disregard these "local" inhomogeneities which result from the agglomeration of matter into stars and star systems. Thus by the mass density we must understand the density .averaged over regions of space whose dimensions are large compared to the separations between galaxies. The solutions of the gravitational equations which are considered here, the so-called isotropic cosmological model (first found by A. Friedmann, 1922), are based on the assump- tion that the matter is distributed uniformly over all space.f Existing astronomical data do not contradict such an assumption. But by its very nature it inevitably can have only an approximate character, since the uniformity is surely violated when we go to a smaller scale. However, there is every reason to believe that the isotropic model gives, in its general features, an adequate description of the present state of the universe. We shall see that a basic feature of this model is its nonstationarity. There is no doubt that this property gives a correct explanation of such a fundamental for the entire cosmology phenomenon as the "red shift" (see § 1 10). Space uniformly filled with matter is completely homogeneous and isotropic. This means that we can choose a "world" time so that at every moment the metric of the space is the same at all points and in all directions. First we take up the study of the metric of the isotropic space as such, disregarding for the moment any possible time dependence. As we did previously, we denote the three- dimensional metric tensor by y aP , i.e. we write the element of spatial distance in the form dl 2 = y aP dx a dx p . (107.1) t We shall not consider at all equations containing the so-called cosmological constant, since there is no sufficient physical basis at present for such a change in the form of the gravitational equations. 333 334 COSMOLOGICAL PROBLEMS § 107 The curvature of the space is completely determined by its three-dimensional curvature tensor, which we shall denote by P% 3 in distinction to the four-dimensional tensor Rl lm (the properties of the tensor P Pyd are of course completely analogous to those of the tensor R l k i m ). In the case of complete isotropy, the tensor P a Pyd must clearly be expressible in terms of the metric tensor y afl alone. It is easy to see from the symmetry properties of Pp y5 that it must have the form : ^ = my S p-^y y p\ (107.2) where X is some constant. The tensor of the second rank, P ap = P 7 ayP , is accordingly equal to P a p = 2Xy aP (107.3) and the scalar curvature P = 6X. (107.4) Thus we see that the curvature properties of an isotropic space are determined by just one constant X. Correspondingly to this there are altogether three different possible cases for the spatial metric: (1) the so-called space of constant positive curvature (corresponding to a positive value of X), (2) space of constant negative curvature (corresponding to values of X < 0), and (3) space with zero curvature (X = 0). Of these, the last will be a flat, i.e. euclidean, space. To investigate the metric it is convenient to start from geometrical analogy, by considering the geometry of isotropic three-dimensional space as the geometry on a hypersurface known to be isotropic, in a fictitious four-dimensional space. f Such a space is a hypersphere; the three-dimensional space corresponding to this has a positive constant curvature. The equation of a hypersphere of radius a in the four-dimensional space x u x 2 , x 3 , x 4 , has the form X^ T"X2 -r^Cg t X4, ^ d i and the element of length on it can be expressed as dl 2 = dx\ + dx\ + dx\ + dx\. Considering x u x 2 , x 3 as the three space coordinates, and eliminating the fictitious co- ordinate Ar 4 with the aid of the first equation, we get the element of spatial distance in the form dl 2 = dx 2 + dx 2 + dx 2 + ( x i dx i+ x 2 2 dx 2 +x 3 dx 3 ) 2 ^ (l0? 5) From this expression, it is easy to calculate the constant X in (107.2). Since we know beforehand that P aP has the form (107.3) over all space, it is sufficient to calculate it only for points located near the origin, where the y aP are equal to y*p - *p+ ~^2~- Since the first derivatives of the y aP , which determine the quantities 1"^, vanish at the origin, the calculation from the general formula (92.10) turns out to be very simple and gives the result X = \. (107.6) t This four-space is understood to have nothing to do with four-dimensional space-time. R 107 ISOTROPIC SPACE 335 We may call the quantity a the "radius of curvature" of the space. We introduce in place of the coordinates x u x 2t x 3 , the corresponding "spherical" coordinates r, 9, 0. Then the line element takes the form dl 2 = -^ + r 2 (sin 2 9 d<\> 2 + d9 2 ). (107.7) ..2 1-4 a The coordinate origin can of course be chosen at any point in space. The circumference of a circle in these coordinates is equal to 2nr, and the surface of a sphere to 4nr . The "radius" of a circle (or sphere) is equal to r f-7 T -T- - = asin- ^r/a), J Vl-r 2 /a 2 o that is, is larger than r. Thus the ratio of circumference to radius in this space is less than In. Another convenient form for the dl 2 in "four-dimensional spherical coordinates" is obtained by introducing in place of the coordinate r the "angle" % according to r = a sin x (X goes between the limits to 7i).| Then dl 2 = a W + sin 2 x(sin 2 9 # 2 + d9 2 )]. (107.8) The coordinate x determines the distance from the origin, given by ax. The surface of a sphere in these coordinates equals Ana 2 sin 2 x- We see that as we move away from the origin, the surface of a sphere increases, reaching its maximum value Ana 2 at a distance of nail. After that it begins to decrease, reducing to a point at the "opposite pole" of the space, at distance na, the largest distance which can in general exist in such a space [all this is also clear from (107.7) if we note that the coordinate r cannot take on values greater than a]. The volume of a space with positive curvature is equal to 271 71 Jt V= ! I J a 3 sin 2 x sin 9 d% d9 d§. so that V = 2n 2 a\ (107.9) Thus a space of positive curvature turns out to be "closed on itself". Its volume is finite though of course it has no boundaries. It is interesting to note that in a closed space the total electric charge must be zero. Namely, every closed surface in a finite space encloses on each side of itself a finite region of space. Therefore the flux of the electric field through this surface is equal, on the one hand, to the total charge located in the interior of the surface, and on the other hand to the total charge outside of it, with opposite sign. Consequently, the sum of the charges on the two sides of the surface is zero. Similarly, from the expression (101.14) for the four-momentum in the form of a surface integral there follows the vanishing of the total four-momentum P l over all space. Thus the t The "cartesian" coordinates x lt x a , x 3 , x± are related to the four-dimensional spherical coordinates £> 0, 4>, X by tne relations: jci = asin;tsin0cos^, x 2 = a sin x sin 9 sin </>, x 3 = a sin x cos 6, x t = a cos x- 336 COSMOLOGICAL PROBLEMS § 108 definition of the total four-momentum loses its meaning, since the corresponding con- servation law degenerates into the empty identity = 0. We now go on to consider geometry of a space having a constant negative curvature. From (107.6) we see that the constant X is negative if a is imaginary. Therefore all the formulas for a space with negative curvature can be immediately obtained from the pre- ceding ones by replacing a by ia. In other words, the geometry of a space with negative curvature is obtained mathematically as the geometry on a four-dimensional pseudosphere with imaginary radius. Thus the constant X is now A =-j> (107.10) and the element of length in a space of negative curvature has, in coordinates r, 9, cf>, the form dr 2 dl 2 = ^ + r 2 (sin 2 9 dcj) 2 + d9 2 ), (107.11) 1+- a' ,2 where the coordinate r can go through all values from to oo. The ratio of the circumference of a circle to its radius is now greater than In. The expression for dl 2 corresponding to (107.8) is obtained if we introduce the coordinate % according to r = a sinh % (x here goes from to oo). Then dl 2 = a 2 {d X 2 + smh 2 x(sin 2 9 d<f) 2 + d9 2 )}. (107.12) The surface of a sphere is now equal to 4na 2 sinh 2 x and as we move away from the origin (increasing x)> it increases without limit. The volume of a space of negative curvature is, clearly, infinite. PROBLEM Transform the element of length (107.7) to a form in which it is proportional to its euclidean expression. Solution: The substitution leads to the result : i+ & dl 2 = (l + ^ J *(drl+rl d0 2 +rZ sin 2 6 ■ d<j> 2 ). § 108. Space-time metric in the closed isotropic model Going on now to the study of the space-time metric of the isotropic model, we must first of all make a choice of our reference system. The most convenient is a "co-moving" reference system, moving, at each point in space, along with the matter located at that point. In other words, the reference system is just the matter filling the space; the velocity of the matter in this system is by definition zero everywhere. It is clear that this reference system is reasonable for the isotropic model— for any other choice the direction of the § 108 SPACE-TIME METRIC IN THE CLOSED ISOTROPIC MODEL 337 velocity of the matter would lead to an apparent nonequivalence of different directions in space. The time coordinate must be chosen in the manner discussed in the preceding section, i.e. so that at each moment of time the metric is the same over all of the space. In view of the complete equivalence of all directions, the components g 0x of the metric tensor are equal to zero in the reference system we have chosen. Namely, the three com- ponents g 0a can be considered as the components of a three-dimensional vector which, if it were different from zero, would lead to a nonequivalence of different directions. Thus ds 2 must have the form ds 2 = g 00 (dx°) 2 -dl 2 . The component g 00 is here a function only of x°. Therefore we can always choose the time coordinate so that g 00 reduces to c 2 . Denoting it by t, we have ds 2 = c 2 dt 2 -dl 2 . (108.1) This time t is clearly the proper time at each point in space. Let us begin with the consideration of a space with positive curvature; from now on we shall, for brevity, refer to the corresponding solution of the equations of gravitation as the "closed model". For dl we use the expression (107.8) in which the "radius of curvature" a is, in general, a function of the time. Thus we write ds 2 in the form ds 2 = c 2 dt 2 -a 2 (t){d X 2 + sin 2 X (d9 2 + sin 2 9- d(j> 2 )}. (108.2) The function a(t) is determined by the equations of the gravitational field. For the solu- tion of these equations it is convenient to use, in place of the time, the quantity r\ defined by the relation cdt = a dt]. (108.3) Then ds 2 can be written as ds 2 = a\ n ){dn 2 - di 2 - sin 2 X (d9 2 + sin 2 9 • # 2 )}. (108.4) To set up the field equations we must begin with the calculation of the components of the tensor R ik (the coordinates x°, x l , x 2 , x 3 are rj, %, d, </>). Using the values of the com- ponents of the metric tensor, 0oo = « 2 > 0n= -a 2 , g 22 = -a 2 sin 2 x, g 33 = -a 2 sin 2 %sin 2 9, we calculate the quantities T l kl : ro a r° a n r a — — x a r° — r a — o A 00 = — » A a/3 — 3 #«/?' L Op — °P> x aO — A 00 — u > CI CI CI where the prime denotes differentiation with respect to r\. (There is no need to compute the components T a Py explicitly.) Using these values, we find from the general formula (92.10): _3 a A R°o = ~^(a' 2 -aa"). From the same symmetry arguments as we used earlier for the g 0a , it is clear from the start that R 0a = 0. For the calculation of the components R{ we note that if we separate in them the terms containing g aP (i.e. only the ¥%), these terms must constitute the components of a three-dimensional tensor -P£, whose values are already known from (107.3) and (107.6): Uj=-Pj+... = -^«J+..., where the dots represent terms containing g 00 in addition to the g aP . From the computation 338 COSMOLOGICAL PROBLEMS § 108 of these latter terms we find: so that Ri=-^(2a 2 + a' 2 + aa")dl R = R° + R* a =-^(a + a"). Since the matter is at rest in the frame of reference we are using, if = 0, u° = l/a, and we have from (94.9) Tg = e, where s is the energy density of the matter. Substituting these expressions in the equation „ n 1 „ 8nk „ we obtain: ~^ s = ^( a +a )• (108.5) Here there enter two unknown functions s and a; therefore we must obtain still another equation. For this is convenient to choose (in place of the spatial components of the field equations) the equation T l . , = 0, which is one of the four equations (94.7) contained, as we know, in the equations of gravitation. This equation can also be derived directly with the help of thermodynamic relations, in the following fashion. When in the field equations we use the expression (94.9) for the energy-momentum tensor, we are neglecting all those processes which involve energy dissipation and lead to an increase in entropy. This neglect is here completely justified, since the auxiliary terms which should be added to T) in connection with such processes of energy dissipation are negligibly small compared with the energy density s, which contains the rest energy of the material bodies. Thus in deriving the field equations we may consider the total entropy as constant. We now use the well-known thermodynamic relation di = TdS-pdV, where S, S, V, are the energy, entropy, and volume of the system, and p, T, its pressure and temperature. At constant entropy, we have simply d$ = -p dV. Introducing the energy density & = g\V we easily find ds= -(e+p)—. The volume V of the space is, according to (107.9), proportional to the cube of the radius of curvature a. Therefore dVjV = 3da/a =3d(\n a), and we can write ds 3d(ln a), s+p or, integrating, f ds 3 In a = - + const (108.6) (the lower limit in the integral is constant). If the relation between e and p (the "equation of state" of the matter) is known, then equation (108.6) determines s as a function of a. Then from (108.5) we can determine r\ § 108 SPACE-TIME METRIC IN THE CLOSED ISOTROPIC MODEL 339 in the form n=±( — da (108.7) ■J Equations (108.6), (108.7) solve, in general form, the problem of determining the metric in the closed isotropic model. If the material is distributed in space in the form of discrete macroscopic bodies, then to calculate the gravitational field produced by it, we may treat these bodies as material particles having definite masses, and take no account at all of their internal structure. Considering the velocities of the bodies as relatively small (compared with c), we can set s = nc 2 , where H is the sum of the masses of the bodies contained in unit volume. For the same reason the pressure of the "gas" made up of these bodies is extremely small compared with e, and can be neglected (from what we have said, the pressure in the interior of the bodies has nothing to do with the question under consideration). As for the radiation'present in space, its amount is relatively small, and its energy and pressure can also be neglected. Thus, to describe the present state of the Universe in terms of this model, we should use the equation of state for "dustlike" matter, e = juc 2 , p = 0. The integration in (108.8) then gives pa* = const. This equation could have been written immediately, since it merely expresses the constancy of the sum M of the masses of the bodies in all of space, which should be so for the case of dustlike matter, f Since the volume of space in the closed model is V = 2n 2 a 3 , const = M\2% 2 . Thus fia 3 = const = —j. (108.8) Substituting (108.8) in equation (108.7) and performing the integration, we get: a = a (l-cos>/), (108.9) where the constant 2kM a ° = to?' Finally, for the relation between t and r\ we find from (108.3): t = a -*(t\-s mr \\ (108.10) c The equations (108.9-10) determine the function a(t) in parametric form. The function a(t ) grows from zero at t = (q = 0) to a maximum value of a = 2a , which is reached when t = najc (rj = n), and then decreases once more to zero when t = 2na /c (rj = 2n). For >j^lwe have approximately a = a rj 2 /2, t = a r] 3 l6c, so that ^|M t 2 '\ (108.11) The matter density is 1 8xl ° 5 n0R1T> " = 6^ = -^ (108 ' 12) t To avoid misunderstandings (that might arise if one considers the remark in § 107 that the total four- momentum of a closed universe is zero), we emphasize that M is the sum of the masses of the bodies taken one by one, without taking account of their gravitational interaction. 340 COSMOLOGICAL PROBLEMS § 109 (where the numerical value is given for density in gm- cm -3 and t in sec). We call attention to the fact that in this limit the function fi(t) has a universal character in the sense that it does not depend on the parameter a . When a -* the density /i goes to infinity. But as fi -*■ oo the pressure also becomes large, so that in investigating the metric in this region we must consider the opposite case of maximum possible pressure (for a given energy density e), i.e. we must describe the matter by the equation of state e (see § 35). From formula (108.6) we then get: 3c 4 a 2 e a 4 = const = — -~ (108.13) (where a x is a new constant), after which equations (108.7) and (108.3) give the relations a, a = a 1 sin>/, t — — (1 — cos 77). c Since it makes sense to consider this solution only for very large values of e (i.e. small a), we assume rj <^ 1. Then a « a v rj, t <z a x rj 2 /2c, so that a = yJ2a 1 ct. (108.14) Then e 3 4.5 x 10 5 ,2 32nkt 2 (108.15) (which again contains no parameters). Thus, here too a -> for t -*■ 0, so that the value t = is actually a singular point of the space-time metric of the isotropic model (and the same remark applies in the closed model also to the second point at which a = 0). We also see from (108.14) that if the sign of t is changed, the quantity a(t) would become imaginary, and its square negative. All four components g ik in (108.2) would then be negative, and the determinant g would be positive. But such a metric is physically meaningless. This means that it makes no sense physically to continue the metric analytically beyond the singularity. § 109. Space-time metric for the open isotropic model The solution corresponding to an isotropic space of negative curvature ("open model") is obtained by a method completely analogous to the preceding. In place of (108.2), we now have ds 2 = c 2 dt 2 -a 2 (t){dx 2 + sinh 2 %(d0 2 + sin 2 9 d<f> 2 )}. (109.1) Again we introduce in place of / the variable //, according to c dt — a &r\ ; then we get ds 2 = a\r]){dri 2 -dx 2 -smh 2 x(rf0 2 + sin 2 0-# 2 )}. (109.2) This expression can be obtained formally from (108.4) by changing r\, 7, a respectively to irj, ix, ia. Therefore the equations of the field can also be gotten directly by this same sub- § 109 SPACE-TIME METRIC FOR THE OPEN ISOTROPIC MODEL 341 stitution in equations (108.5) and (108.6). Equation (108.6) retains its previous form: 3 In a = - f — + const, (109.3) while in place of (108.5), we have *^ & = I (a' 2 -a 2 ). (109.4) (109.5) Corresponding to this we find, instead of (108.7) f da n = ± . a / — -£a z + l V 3c 4 For material in the form of dust, we find:t a = a (cosh r\ - 1), t = - (sinh n - 1]\ (109.6) Ha* = 3 £a . (109.7) 4nk The first two determine the function a(t) in parametric form. In contrast to the closed model, here the radius of curvature changes monotonically, increasing from zero at t = (r\ = 0) to infinity for t -> oo (r\ -► oo). Correspondingly, the matter density decreases monotonically from an infinite value when t = (when r\<\, the monotonic decrease is given by the same approximate formula (108.12) as in the closed model). For large densities the solution (109.6-7) is not applicable, and we must again go to the case p = e/3. We again get the relation £ a 4 = const^^-l (109.8) and find for the function a(t): a = a, sinh r\, t = — (cosh r\ - 1) c f We note that, by the transformation r = Ae n sinh /, ex = Ae" cosh /, Ae n = Vc 2 x 2 -r 2 , tanh / = — , ex the expression (108.2) is reduced to the "conformal-galilean" form ds 2 =f(r, x)[c 2 dx 2 -dr 2 -r 2 (d0 2 +sin 2 9 d<j> 2 )]. Specifically, in the case of (109.6), setting A equal to a j2, ds 2 = (\-—J^=\{c 2 dx 2 -dr 2 -r 2 (d9 2 +sm 2 Q dj 2 )] ' \ 2Vc 2 x*-r 2 J (V. A. Fock, 1955). For large values of VcV- r 2 (which correspond to >/> 1), this metric tends toward a galilean form, as was to be expected since the radius of curvature tends toward infinity. In the coordinates r, 0, (/>, r, the matter is not at rest and its distribution is not uniform; the distribution and motion of the matter turns out to be centrally symmetric about any point of space chosen as the origin of coordinates r, 6, (j>. Sizk 342 COSMOLOGICAL PROBLEMS § 109 or, when r\ <| 1, a = yjlcii ct (109.9) [with the earlier formula (108.15) for s(t)]. Thus in the open model, also, the metric has a singularity (but only one, in contrast to the closed model). Finally, in the limiting case of the solutions under consideration, corresponding to an infinite radius of curvature of the space, we have a model with a flat (euclidean) space. The interval ds 2 in the corresponding space-time can be written in the form ds 2 = c 2 dt 2 -b 2 (t)(dx 2 + dy 2 + dz 2 ) (109.10) (for the space coordinates we have chosen the "cartesian" coordinates x, y, z). The time- dependent factor in the element of spatial distance does not change the euclidean nature of the space metric, since for a given t this factor is a constant, and can be made unity by a simple coordinate transformation. A calculation similarto those in the previous paragraph leads to the following equations: 3 (db\ 2 r ds = ¥{di)> 31n6=-J— + const. For the case of low pressures, we find Hb z = const, b = const t 2,z . (109.11) For small t we must again consider the case p = s/3, for which we find eb 4 = const, b = const V7. (109.12) Thus in this case also the metric has a singular point (/ = 0). We note that all the isotropic solutions found exist only when the matter density is different from zero; for empty space the Einstein equations have no such solutions.! We also mention that mathematically they are a special case of a more general class of solutions that contain three physically different arbitrary functions of the space coordinates (see the problem). PROBLEM Find the general form near the singular point for the metric in which the expansion of the space proceeds "quasihomogeneously", i.e. so that all components y aft ~ —g aB (in the synchronous reference system) tend to zero according to the same law. The space is filled with matter with the equation of state/* = s/3 (E. M. Lifshitz and I. M. Khalatnikov, 1960). Solution: We look for a solution near the singularity (t = 0) in the form: Yc0 = ta aB +t 2 b a0 + . . . , (1) where a a0 and b aB are functions of the (space) coordinates J ; below, we shall set c — 1 . The reciprocal tensor is y aB = - a aB —b aB , f For e = we would get from (109.5) a = a e n = ct [whereas the equations (108.7) are meaningless because the roots are imaginary]. But the metric ds 2 = c a rf/ 2 -c 2 / 2 {^ 2 +sinh a x(^ a +sin 2 9 d^ 2 )} can be transformed by the substitution r = ct sinh x, * = t cosh x, to the form ds 2 = c 2 dT*-dr 2 -r*(d9 2 +sm 2 d<f> 2 ), i.e. to a galilean space-time. % The Friedmann solution corresponds to a special choice of the functions a a0 , corresponding to a space of constant curvature. §110 THE RED SHIFT 343 where the tensor a"* is reciprocal to a aB , while b aB = a ay a B6 b y6 ; all raising and lowering of indices and covariant differentiation is done using the time-independent metric a aB . Calculating the left sides of equations (99.11) and (99.12) to the necessary order in I ft, we get -4^ + S*- 8 -?<- 4 "» +1 >' z (b : a -b B : e)= 3— eu a u (where b = Z>£). Also using the identity 1 =u t u l x u% u a u B a aB , Snke = 4f 2 -2 t > Utt = ~2 (Z>: a ~ b "'' ^ (2) The three-dimensional Christoffel symbols, and with them the tensor P aB , are independent of the time in the first approximation in \\t\ theP aB coincide with the expressions obtained when calculating simply with the metric a aB . Using this, we find that in equation (99.13) the terms of order t~ 2 cancel, while the terms ~ \jt give p «+l b « + T2 S » b = > from which 4 5 b B = — -P $ -\ S e P, (3) u a T ■* a I to a ' v J we find : (where P = a By P ey ). In view of the identity 1 2 [see (92.13)] the relation Pl:,-~P-.* = b B a -.B = gb :a is valid, so that the u a can be written in the form: t 2 u a = — b-. a . (4) Thus, all six functions a aB remain arbitrary, while the coefficients b ttS of the next term in the expansion (1) are determined in terms of them. The choice of the time in the metric (1) is completely determined by the condition f = at the singularity; the space coordinates still permit arbitrary transformations that do not involve the time (which can be used, for example, to bring a aB to diagonal form). Thus the solution contains all together three "physically different" arbitrary functions. We note that in this solution the spatial metric is inhomogeneous and anisotropic, while the density of the matter tends to become homogeneous as /->0. In the approximation (4) the three- dimensional velocity v has zero curl, while its magnitude tends to zero according to the law v 2 = v a v s y aB ~ t 3 . § 110. The red shift The main feature characteristic of the solutions we have considered is the nonstationary metric; the radius of curvature of the space is a function of the time. A change in the radius of curvature leads to a change in all distances between bodies in the space, as is already seen from the fact that the element of spatial distance dl is proportional to a. Thus as a increases 344 COSMOLOGICAL PROBLEMS § 110 the bodies in such a space "run away" from one another (in the open model, increasing a corresponds to rj > 0, and in the closed model, to < rj < n). From the point of view of an observer located on one of the bodies, it will appear as if all the other bodies were moving in radial directions away from the observer. The speed of this "running away" at a given time t is proportional to the separation of the bodies. This prediction of the theory must be compared with a fundamental astronomical fact — the red shift of lines in the spectra of galaxies. It we regard this as a Doppler shift, we arrive at the conclusion that the galaxies are receding, i.e. at the present time the Universe is expanding.f Let us consider the propagation of a light ray in an isotropic space. For this purpose it is simplest to use the fact that along the world line of the propagation of a light signal the interval ds = 0. We choose the point from which the light emerges as the origin of co- ordinates #, 0, 0. From symmetry considerations it is clear that the light ray will propagate "radially", i.e. along a line = const, 4> = const. In accordance with this, we set dd = dcj) = in (108.4) or (109.2) and obtain ds 2 = a 2 {dr\ 2 -dx 2 ). Setting this equal to zero, we find dr\ = ±dx or, integrating, X=±ri + const. (110.1) The plus sign applies to a ray going out from the coordinate origin, and the minus sign to a ray approaching the origin. In this form, equation (110.1) applies to the open as well as to the closed model. With the help of the formulas of the preceding section, we can from this express the distance traversed by the beam as a function of the time. In the open model, a ray of light, starting from some point, in the course of its propaga- tion recedes farther and farther from it. In the closed model, a ray of light, starting out from the initial point, can finally arrive at the "conjugate pole" of the space (this corresponds to a change in x from to n) ; during the subsequent propagation, the ray begins to approach the initial point. A circuit of the ray "around the space", and return to the initial point, would correspond to a change of / from to 2n. From (110.1) we see that then r\ would also have to change by 2n, which is, however, impossible (except for the one case when the light starts at a moment corresponding to r\ = 0). Thus a ray of light cannot return to the starting point after a circuit "around the space". To a ray of light approaching the point of observation (the origin of coordinates), there corresponds the negative sign on r\ in equation (1 10.1). If the moment of arrival of the ray at this point is t(rj ), then for r\ = tj we must have x = 0, so that the equation of propagation of such rays is X = 1o-r}. (110.2) From this it is clear that for an observer located at the point x = 0, only those rays of light can reach him at the time t(?/ ), which started from points located at "distances" not exceeding x = */o- This result, which applies to the open as well as to the closed model, is very essential. We see that at each given moment of time t(ri), at a given point in space, there is accessible to physical observation not all of space, but only that part of it which corresponds to f The conclusion that the bodies are running away with increasing a(t ) can only be made if the energy of interaction of the matter is small compared to the kinetic energy of its motion in the recession; this condition is always satisfied for sufficiently distant galaxies. In the opposite case the mutual separations of the bodies is determined mainly by their interactions; therefore, for example, the effect considered here should have practically no influence on the dimensions of the nebulae themselves, and even less so on the dimensions of stars. §110 THE RED SHIFT 345 X < r\. Mathematically speaking, the "visible region" of the space is the section of the four- dimensional space by the light cone. This section turns out to be finite for the open as well as the closed model (the quantity which is infinite for the open model is its section by the hypersurface t = const, corresponding to the space where all points are observed at one and the same time t). In this sense, the difference between the open and closed models turns out to be much less drastic than one might have thought at first glance. The farther the region observed by the observer at a given moment of time recedes from him, the earlier the moment of time to which it corresponds. Let us look at the spherical surface which is the geometrical locus of the points from which light started out at the time t(rj-x) and is observed at the origin at the time t(rj). The area of this surface is 4na 2 (rj-x) sin 2 x On the closed model), or Ana 2 {r\-x) sinh 2 % ( m the open model). As it recedes from the observer, the area of the "visible sphere" at first increases from zero (for x — 0) and then reaches a maximum, after which it decreases once more, dropping back to zero for x = *1 (where a{r\-x) = «(0) = 0). This means that the section through the light cone is not only finite but also closed. It is as if it closed at the point "conjugate" to the observer; it can be seen by observing along any direction in space. At this point e -> oo, so that matter in all stages of its evolution is, in principle, accessible to observation. The total amount of observed matter is equal in the open model to M„ •i = An I /za 3 sinh 2 x ' dx- o Substituting pia 3 from (109.7), we get Mobs = , ° ( smn */ cos h r I~ r f)- (1 10.3) This quantity increases without limit as rj -*■ oo. In the closed model, the increase of M obs is limited by the total mass M; in similar fashion, we find for this case: M M obs = — {r] — sin rj cos rf). (110.4) As t] increases from to n, this quantity increases from to M; the further increase of M obs according to this formula is fictitious, and corresponds simply to the fact that in a "con- tracting" universe distant bodies would be observed twice (by means of the light "circling the space" in the two directions). Let us now consider the change in the frequency of light during its propagation in an isotropic space. For this we first point out the following fact. Let there occur at a certain point in space two events, separated by a time interval dt = (l/c)a (rj) drj. If at the moments of these events light signals are sent out, which are observed at another point in space, then between the moments of their observation there elapses a time interval corresponding to the same change dr\ in the quantity r\ as for the starting point. This follows immediately from equation (1 10.1), according to which the change in the quantity r\ during the time of propaga- tion of a light ray from one point to another depends only on the difference in the co- ordinates x f° r these points. But since during the time of propagation the radius of curvature a changes, the time interval t between the moments of sending out of the two signals and the moments of their observation are different; the ratio of these intervals is equal to the ratio of the corresponding values of a. C.T.F. 12 346 COSMOLOGICAL PROBLEMS § 110 From this it follows, in particular, that the periods of light vibrations, measured in terms of the world time t, also change along the ray, proportionally to a. Thus, during the propaga- tion of a light ray, along its path, coa = const. (110.5) Let us suppose that at the time t(rj) we observe light emitted by a source located at a distance corresponding to a definite value of the coordinate x- According to (109.1), the moment of emission of this light is t{f\—y). If co is the frequency of the light at the time of emission, then from (110.5), the frequency co observed by us is (o = co — — — . (110.6) a(rj) Because of the monotonic increase of the function a(rj), we have to < co , that is, a decrease in the light frequency occurs. This means that when we observe the spectrum of light coming toward us, all of its lines must appear to be shifted toward the red compared with the spec- trum of the same matter observed under ordinary conditions. The "red shift" phenomenon is essentially the Doppler effect of the bodies' "running away" from each other. The magnitude of the red shift measured, for example, as the ratio co/co of the displaced to the undisplaced frequency, depends (for a given time of observation) on the distance at which the observed source is located [in relation (110.6) there enters the coordinate x °f the light source]. For not too large distances, we can expand a(rj—x) in a power series in x, limiting ourselves to the first two terms : — = \- ^M co a(rj) (the prime denotes differentiation with respect to n). Further, we note that the product Xa(rf) is here just the distance / from the observed source. Namely, the "radial" line element is equal to dl = a dx; in integrating this relation the question arises of how the distance is to be determined by physical observation. In determining this distance we must take the values of a at different points along the path of integration at different moments of time (integration for rj = const would correspond to simultaneous observation of all the points along the path, which is physically not feasible). But for "small" distances we can neglect the change in a along the path of integration and write simply / = ax, with the value of a taken for the moment of observation. As a result, we find for the percentage change in the frequency the following formula: c ^£o == _h_ h (m7) co c where we have introduced the notation h = e ^JjA (U0.8) a 2 (rj) a dt for the so-called "Hubble constant". For a given instant of observation, this quantity is independent of/. Thus the relative shift in spectral lines must be proportional to the distance to the observed light source. Considering the red shift as a result of a Doppler effect, one can determine the corres- ponding velocity v of recession of the body from the observer. Writing (co-o) )/a> = —vie, §110 THE RED SHIFT 347 and comparing with (1 10.7), we have v = hl (110.9) (this formula can also be obtained directly by calculating the derivative v = d(ax)/dt. Astronomical data confirm the law (110.7), but the determination of the value of the Hubble constant is hampered by the uncertainty in the establishment of a scale of cosmic distances suitable for distant galaxies. The latest determinations give the value h s 0.8 x 10" 10 yr" 1 = 0-25 x 10" 17 sec" *, (110.10) 1/ft « 4 x 10 17 sec = 1.3 x 10 9 yr. It corresponds to an increase in the "velocity of recession" by 25 km/sec for each million light years distance. Substituting in equation (109.4), e = fie 2 and h = ca'/a 2 , we get for the open model the following relation : £! = **_ 5* (iio.il) a 2 3 Combining this equation with the equality c sinh n c . f] h = — 7 — u T^ = ~„ coth V a (cosh rj — l) a 2 we obtain COShJ = *./r4-. ( 110 - 12 ) 2 V Snkfi For the closed model we would get: '^p-h 1 , (110.13) a 3 cos V - = h 7-4-. (110.14) 2 V Snkn Comparing (110.11) and (110.13), we see that the curvature of the space is negative or positive according as the difference (Snk/3)n-h 2 is negative or positive. This difference goes to zero for \i =■ \i k , where *-£ (U0 - 15) With the value (110.10), we get \i k s 1 x 10 -29 g/cm 3 . In the present state of astronomical knowledge, the value of the average density of matter in space can be estimated only with very low accuracy. For an estimate, based on the number of galaxies and their average mass, one now takes a value of about 3 x 10" 31 g/cm 3 . This value is 30 times less than p k and thus would speak in favor of the open model. But even if we forget about the doubtful reliability of this number, we should keep in mind that it does not take into account the possible existence of a metagalactic dark gas, which could greatly increase the average matter density.f t The uncertainty in the value of n does not allow any sort of exact calculation of tj, especially since even the sign of /i—p* is unknown. Setting fi = 3 x 10" 31 g/cm 3 in (110.12), and taking h from (110.10), we get t] = 5.0. If we set fi = 10 -30 g/cm 3 , then tj = 6.1. 348 COSMOLOGICAL PROBLEMS § 110 Let us note here a certain inequality which one can obtain for a given value of the quantity h. For the open model we have c sinh y\ a (cosh r\ — l) 2 ' and therefore a . sinh »7(sinh rj — rj) t = — (sinh rj-rj) = —- — — ir . c /i(cosh r\ — \y Since < r\ < oo, we must have h <%< \ (11016) Similarly, for the closed model we obtain sin r\{r\ — sin rj) h(l — cos rj) 2 To the increase of a{rj) there corresponds the interval < r\ < n ; therefore we get 0<t<^ (110.17) 3n Next we determine the intensity / of the light arriving at the observer from a source located at a distance corresponding to a definite value of the coordinate x- The flux density of light energy at the point of observation is inversely proportional to the surface of the sphere, drawn through the point under consideration with center at the location of the source ; in a space of negative curvature the area of the surface of the sphere equals Ana 2 sinh 2 x- Light emitted by the source during the interval dt = (l/c)a(rj—x)drj will reach the point of observation during a time interval . a(rj) 1 dt , " = - a(rj) drj. a{rj-x) c Since the intensity is defined as the flux of light energy per unit time, there appears in / a factor a(rj~x)/a(rj). Finally, the energy of a wave packet is proportional to its frequency [see (53.9)]; since the frequency changes during propagation of the light according to the law (110.5), this results in the factor a(rj — x)/a(t]) appearing in /once more. As a result, we finally obtain the intensity in the form g\n-i) a\rf) sinh 2 x For the closed model we would similarly obtain g 2 {n-x) a A (f}) sin 2 x These formulas determine the dependence of the apparent brightness of an observed object on fts distance (for a given absolute brightness). For small x we can set a(rj-x) ^ a(rj), and then J~ l/a 2 (rf)x 2 = l// 2 , that is, we have the usual law of decrease of intensity inversely as the square of the distance. Finally, let us consider the question of the so-called proper motions of bodies. In speaking of the density and motion of matter, we have always understood this to be the average density and average motion; in particular, in the system of reference which we have always / = const 4 y. 7^-. (110.18) / = const ™ .7. . (110.19) § HO THE RED SHIFT 349 used, the velocity of the average motion is zero. The actual velocities of the bodies will under- go a certain fluctuation around this average value. In the course of time, the velocities of proper motion of the bodies change. To determine the law of this change, let us consider a freely moving body and choose the origin of coordinates at any point along its trajectory. Then the trajectory will be a radial line, = const, <J> = const. The Hamilton- Jacobi equation (87.6), after substitution of the values of g ik , takes the form /^V_ 0?V + m 2 c V(ij) = 0. (110.20) Since x does not enter into the coefficients in this equation (i.e., / is a cyclic coordinate), the conservation law dS/dx = const is valid. The momentum p of a moving body is equal, by definition, to p = dS/dl = dS/a dx- Tnus for a moving body the product pa is constant: pa = const. (110.21) Introducing the velocity v of proper motion of the body according to mv P = we obtain /■ J "" = = const. (110.22) c z The law of change of velocity with time is determined by these relations. With increasing a, the velocity v decreases monotonically. PROBLEMS 1. Find the first two terms in the expansion of the apparent brightness of a galaxy as a function of its red shift; the absolute brightness of a galaxy varies with time according to an exponential law, I abs = const • e at (H. Robertson, 1955). Solution: The dependence on distance x of the apparent brightness of a galaxy at the "instant" tj, is given (for the closed model) by the formula / = const • e^"-"- ««>] a y~ X l . a\rj) sin' 2 x We define the red shift as the relative change in wave length: _ X— A _ cop — co _ a(ji)—a{ri~x) A (o a(tj—x) Expanding / and z in powers of x [using the functions a(tf) and t(t]) from (108.9) and (108.10)] and then eliminating x from the resulting equations, we find the result : 1 / = const z' where we have introduced the notation 1-1 ('-!+?) # = — = — >i. l+COS?7 ju k For the open model, we get the same formula with 2 =^<i. 1+ cosh 77 n k 350 COSMOLOGICAL PROBLEMS § 111 2. Find the leading terms in the expansion of the number of galaxies contained inside a "sphere" of given radius, as a function of the red shift at the boundary of the sphere (where the spatial distribution of galaxies is assumed to be uniform). Solution: The number N of galaxies at "distances" < x is (in the closed model) X N — const • sin 2 xdx^ const • x 3 - Substituting the first two terms in the expansion of the function x(z), we obtain : N = const • z 3 1 - 1 (2+g)z\ . In this form the formula also holds for the open model. § 111. Gravitational stability of an isotropic universe Let us consider the question of the behavior of small perturbations in the isotropic model, i.e. the question of its gravitational stability (E. M. Lifshitz, 1946). We shall restrict our treatment to perturbations over relatively small regions of space — regions whose linear dimensions are small compared to the radius a.f In every such region the spatial metric can be assumed to be euclidean in the first approxi- mation, i.e. the metric (107.8) or (107.12) is replaced by the metric dl 2 = a 2 (r})(dx 2 + dy 2 + dz 2 ), (111.1) where x, y, z are cartesian coordinates, measured in units of the radius a. We again use the parameter v\ as time coordinate. Without loss of generality we shall again describe the perturbed field in the synchronous reference system, i.e. we impose on the variations 5g ik of the metric tensor the conditions <5#oo = &9o* = 0- Varying the identity g ik u { \^ = 1 under these conditions (and remembering that the unperturbed values of the components of the four-velocity of the matter are u° = l/a, u a = 0),f we get g 00 u°du° = 0,so that 8u° = O.The perturbations du" are in general different from zero, so that the reference system is no longer co-moving. We denote the perturbations of the spatial metric tensor by h ttP = dy aP = — 5g aP . Then dy afi — —h ap , where the raising of indices on h aP is done by using the unperturbed metric In the linear approximation, the small perturbations of the gravitational field satisfy the equations 5R k i -idiSR = ~5Tl (111.2) In the synchronous reference system the variations of the components of the energy- momentum tensor (94.9) are : 8Ti = -3 p Jp, 5TI = a(p + e)du", 5T% = Se. (111.3) f A more detailed presentation of this question, including the investigation of perturbations over regions whose size is comparable to a, is given in Adv. in Physics 12, 208 (1963). % In this section we denote unperturbed values of quantities by letters without the auxiliary superscript (0) . §111 GRAVITATIONAL STABILITY OF AN ISOTROPIC UNIVERSE 351 Because of the smallness of 5s and dp, we can write dp = (dp/de)5s, and we obtain the relations: ST P =-S P ^ST° . (111.4) as Formulas for dR* can be gotten by varying the expression (99.10). Since the unperturbed metric tensor y aP = a 2 3 ap , the unperturbed values are _ 2d _2a! p _ 2d_ p d d c* where the dot denotes differentiation with respect to ct, and the prime with respect to r\. The perturbations of x aP and x p = x ay y yP are: Sx« P = ke = - K P , Sxl = - /i% + y Py ky = K = \ *J' a n where h p = y Py h ay . For the euclidean metric (111.1) the unperturbed values of the three- dimensional P p a are zero. The variations 5P p a are calculated from formulas (1) and (2) of problem 2 in § 102: it is obvious that SP p a is expressed in terms of the dy aP just as the four- tensor 5R ik is expressed in terms of the 5g ik , all tensor operations being done in the three- dimensional space with the metric (111.1); because this metric is euclidean, all the co variant differentiations reduce to simple differentiations with respect to the coordinates x* (for the contravariant derivatives we must still divide by a 2 ). Taking all this into account (and changing from derivatives with respect to t to derivatives with respect to */), we get, after some simple calculations: xvP L (uy^4-h p '' f -h p ' y -h' p> i — h p "- — h p ' h'S p 0K a — 2a 2 ' y y ' a ' y ' 2a 2 a 2fl SR* = ^- 2 (h'*-h p > p y, (111.5) 1 a' — h"- — 2a 2 2a' 1_ 2a~' (h = /i"). Here both the upper and lower indices following the comma denote simple differentiations with respect to the x a (we continue to write indices above and below only to retain uniformity of the notation). We obtain the final equations for the perturbations by substituting in (111.4) the com- ponents <5Tf, expressed in terms of the 5R) according to (111.2). For these equations it is convenient to choose the equations obtained from (111.4) for a ^ /?, and those obtained by contracting on a, jS. They are : (.K : l>,+hl : l-h : i-h> : r ) + hZ"+ ™ fcf = 0, a # /?, I (*S J-** (l « g) + h» + H- 1 ( 2+ 3 g) = 0. (111.6) The perturbations of the density and matter velocity can be determined from the known h p using formulas (111.2-3). Thus we have for the relative change of the density: 5e Snks ( dR °-l SR ) ' ids? (*£''-*:: + T *) (111J) 352 COSMOLOGICAL PROBLEMS § 111 Among the solutions of equations (111.6) there are some that can be eliminated by a simple transformation of the reference system (without destroying the condition of synchronism), and so do not represent a real physical change of the metric. The form of such solutions can be established by using formulas (1) and (2) in problem 3 of § 99. Substituting the unper- turbed values y ap = a 2 d aP , we get from them the following expressions for fictitious perturba- tions of the metric: «-/o:J fv + 5/o«+(/.''+/'..), (in.8 j a ci where the/ ,y^ are arbitrary (small) functions of the coordinates x, y, z. Since the metric in the small regions of space we are considering is assumed to be euclidean, an arbitrary perturbation in such a region can be expanded in plane waves. Using x, y, z for cartesian coordinates measured in units of a, we can write the periodic space factor for the plane waves in the form e inr , where n is a dimensionaless vector, which represents the wave vector measured in units of \\a (the wave vector is k = n/a). If we have a perturbation over a portion of space of dimensions ~ /, the expansion will involve waves of length k = 2na/n ~ /. If we restrict the perturbations to regions of size / <4 a, we automatically assume the number n to be quite large (n > 2tt). Gravitational perturbations can be divided into three types. This classification reduces to a determination of the possible types of plane waves in terms of which the symmetric tensor h afi can be represented. We thus obtain the following classification: 1. Using the scalar function Q = e inr , (111.9) we can form the vector P = nQ and the tensorsf 0& = \%Q, ^ = (3^-^)2- (U1.10) These plane waves correspond to perturbations in which, in addition to the gravitational field, there are changes in the velocity and density of the matter, i.e. we are dealing with per- turbations accompanied by condensations or rarefactions of the matter. The perturbation of /i£ is expressed in terms of the tensors Q p a and P£ , the perturbation of the velocity is expressed in terms of the vector P, and the perturbation of the density, in terms of the scalar Q. 2. Using the transverse vector wave S = se' nr , sn = 0, (111.11) we can form the tensor (n fi S a +n a S p ); the scalar corresponding to this does not exist, since n • S = 0. These waves correspond to perturbations in which, in addition to the gravita- tional field, we have a change in velocity but no change of the density of the matter; they may be called rotational perturbations. 3. The transverse tensor wave Gj = rfe fa " r , 0fn, = O. (111.12) We can construct neither a vector nor a scalar by using it. These waves correspond to perturbations of the gravitational field in which the matter remains at rest and uniformly distributed throughout space. In other words, these are gravitational waves in an isotropic universe. t We write upper and lower indices on the components of ordinary cartesian tensors only to preserve uniformity of notation. §111 GRAVITATIONAL STABILITY OF AN ISOTROPIC UNIVERSE 353 The perturbations of the first type are of principal interest. We set hl = Krj)Pt+iJto)Qt, h = nQ. (111.13) From (111.7) we find for the relative change of the density de i ~i n 2 (A+n)+ — n' a (111.14) e 24nksa 2 The equations for determining X and \i are gotten by substituting (111.13) in (111.6): 2a! n 2 k"+ — A'--(A + /z) = 0, a 3 "" + "'l( 2+3 l) + T (A+ ' ,) ( 1+3 *) = a < 11MS > These equations have the following two partial integrals, corresponding to those fictitious changes of metric that can be eliminated by transforming the reference system : X= -fi = const, (111.16) dr\ 3a' J a J a a (111.17) [the first of these is gotten from (11 1.8) by choosing f = 0,f a = P a ; the second by choosing / O =0, /« = <>]. In the early stages of expansion of the universe, when the matter is described by the equation of state p = s/3, we have a x a x rj, r\ <^ 1 (in both the open and closed models). Equations (111.15) take the form: In 2 3 In 2 k" + -k'- — (k+pi) = 0, n"+ - n'+ — (k+fi) = 0. (111.18) r\ 3 rj 3 These equations are conveniently investigated separately for the two limiting cases depend- ing on the ratio of the large quantities n and 1/rj. Let us assume first that n is not too large (or that rj is sufficiently small), so that nrj <^ 1 . To the order of accuracy for which the equations (111.18) are valid, we find from them for this case : k = -^ + c 2 [\+- n 2 y n=-^c, n +c 2 (i-- n 2 y where C l5 C 2 are constants; solutions of the form (111.16) and (111.17) are excluded (in the present case these are the solution with k— n = const and the one with k+n~ l/// 2 ). Calculating 8s/e from (111.14) and (108.15), we get the following expressions for the per- turbations of the metric and the density: n ~ = j(Cxn + C 2 r\ 2 )Q, for p = -, *]<-. (111.19) The constants C x and C 2 must satisfy conditions expressing the smallness of the perturba- tion at the time rj of its start: we must have /if <^ 1 (so that k <^ 1 and \i 4. 1) and Ss/s <| 1. As applied to (111.19) these conditions give the inequalities C x -4, r\ , C 2 <^ 1. 354 COSMOLOGICAL PROBLEMS § 111 In (111.19) there are various terms that increase in the expanding universe like different powers of the radius a = a t t]. But this growth does not cause the perturbation to become large: if we apply formula (111.19) for an order of magnitude to rj ~ l/«, we see that (because of the inequalities found above for C 1 and C 2 ) the perturbations remain small even at the upper limit of application of these formulas. Now suppose that n is so large that nrj > 1. Solving (111.18) for this condition, we find that the leading terms in A and p, are:f A = - ? = const • -= e inM ^. 2 r\ l We then find for the perturbations of the metric and the density: for hi = -^ (Pj-2G2y"^ - = - £ Qe in ^ n n s 9 ' « (111.20) e 1 * 3 n where C is a complex constant satisfying the condition \C\ <^ 1. The presence of a periodic factor in these expressions is entirely natural. For large n we are dealing with a perturbation whose spatial periodicity is determined by the large wave vector k = n/a. Such perturbations must propagate like sound waves with velocity — J; dp d(8/c 2 ) V3' Correspondingly the time part of the phase is determined, as in geometrical acoustics, by the large integral \kudt = nr\j\[?>. As we see, the amplitude of the relative change of density remains constant, while the amplitude of the perturbations of the metric itself decreases like a -2 in the expanding universe.} Now we consider later stages of the expansion, when the matter is already so rarefied that we can neglect its pressure (p = 0). We shall limit ourselves to the case of small rj, corresponding to that stage of the expansion when the radius a was still small compared to its present value, but the matter was already quite rarefied. For/? = and rj < 1, we have a « a rj 2 /2, and (111.15) takes the form: 4 n 2 A"+-A'--(A + ju) = 0, v\ 3 4 n 2 rj 3 The solution of these equations is 6C 2 , 2 / c i^ 2 4C a\ 1+fi = 2d- -^, k-H = n 2 [-^- + -^y f The factor \\rf in front of the exponential is the first term in the expansion in powers of 1/ntj. To find it we must consider the first two terms in the expansion simultaneously [which is justified within the limits of accuracy of (1 1 1 . 1 8)]. . % It is easy to verify that (for p = e/3) nt] ~Ljk, where L ~ u/Vke/c 2 . It is natural that the characteristic length L, which determines the behavior of perturbations with wave length X<^a, contains only hydro- dynamic' quantities— the matter density e/c 2 and the sound velocity u (and the gravitational constant k). We note that there is a growth of the perturbations when ^>L [in (111.19)]. § 112 HOMOGENEOUS SPACES 355 Also calculating dsjs by using (111.14) and (108.12), we find: fci-c^pj+Qft+^cpj-ea for n <{, 2 « = c 1 n¥(«-QJ)+ ?!L ^ ? «-G!!) for \<n<U (m.2i) 5s _ Cj n V C 2 n 2 7~ 30 + r] 2 ' We see that 5e/e contains terms that increase proportionally with a.| But if nv\ <^ l,then de/a does not become large even for rj ~ 1/w because of the condition C t <l. If, however, r\n>\, then for f7 ~ 1 the relative change of density becomes of order C t « 2 , while the smallness of the initial perturbation requires only that C x n 2 r\\ 4, 1. Thus, although the growth of the perturbation occurs slowly, nevertheless its total growth may be considerable, so that it becomes quite large. One can similarly treat perturbations of the second and third types listed above. But the laws for the damping of these perturbations can also be found without detailed calculations by starting from the following simple arguments. If over a small region of the matter (with linear dimensions /) there is a rotational per- turbation with velocity 5v, the angular momentum of this region is ~ (e/c 2 )/ 3 • / • v. During the expansion of the universe / increases proportionally with a, while e decreases like a~ 3 (in the case of;? = 0) or like a~ 4 (for/? = e/3). From the conservation of angular momentum, we have 5v = const for p = s/3,dv ~ - for p = 0. (Ill .22) a Finally, the energy density of gravitational waves must decrease during the expansion of the universe like a~ 4 . On the other hand, this density is expressed in terms of the pertur- bation of the metric by ~ k 2 (h%) 2 , where k = n\a is the wave vector of the perturbation. It then follows that the amplitude of perturbations of the type of gravitational waves decreases with time like I J a. §112. Homogeneous spaces The assumption of homogeneity and isotropy of space determines the metric completely (leaving free only the sign of the curvature). Considerably more freedom is left if one assumes only homogeneity of space, with no additional symmetry. Let us see what metric properties a homogeneous space can have. We shall be discussing the metric of a space at a given instant of time t. We assume that the space-time reference system is chosen to be synchronous, so that t is the same synchro- nized time for the whole space. t A more detailed analysis taking into account the small pressure ^(e) shows that the po ssibility of neglect- ing the pressure requires that one satisfy the condition ufjn/c <^ 1 (where u = cVdplde is the small sound velocity); it is easy to show that in this case also it coincides with the condition 2./L^> 1. Thus, growth of the perturbation always occurs if A/L^> 1. 356 COSMOLOGICAL PROBLEMS § 112 Homogeneity implies identical metric properties at all points of the space. An exact definition of this concept involves considering sets of coordinate transformations that trans- form the space into itself, i.e. leave its metric unchanged: if the line element before trans- formation is dl 2 = y aP (x l , x 2 , x 3 ) dx a dx p , then after transformation the same line element is dl 2 = y afi (x'\ x' 2 , x' 3 ) dx' a dx' p , with the same functional dependence of the y aP on the new coordinates. A space is homo- geneous if it admits a set of transformations (a group of motions) that enables us to bring any given point to the position of any other point. Since space is three-dimensional the different transformations of the group are labelled by three independent parameters. Thus, in euclidean space the homogeneity of space is expressed by the invariance of the metric under parallel displacements (translations) of the cartesian coordinate system. Each translation is determined by three parameters — the components of the displacement vector of the coordinate origin. All these transformations leave invariant the three independent differentials (dx, dy, dz) from which the line element is constructed. In the general case of a noneuclidean homogeneous space, the transformations of its group of motions again leave invariant three independent linear differential forms, which do not, however, reduce to total differentials of any coordinate functions. We write these forms as e a a dx«, (112.1) where the Latin index a labels three independent vectors (coordinate functions); we call these vectors a frame. Using the forms (1 12.1) we construct a spatial metric invariant under the given group of motions : dl 2 = y ab (e a a dx*)(e b dx l! ), i.e. the metric tensor is y a p = y ab e a a e b p . (112.2) where the coefficients y ab , which are symmetric in the indices a and b, are functions of the time.f The contravariant components of the metric tensor are written as y aP = y ab e a a e p b , (112.3) where the coefficients y ab form a matrix reciprocal to the matrix y ab (y ac y cb = dl), while the quantities e* a form three vectors, "reciprocal" to the vectors e a a \ e a a e b a = S b a , 44 = % (112.4) (each of these equations following automatically from the other). We note that the relation between e a a and e a a can be written explicitly as 111 e 1 =-e 2 xe 3 , e, = -e 3 xe l , e 3 = -e 1 xe 2 , (112.5) v v v where v = e 1 • e 2 x e 3 , while e a and e a are to be regarded as cartesian vectors with com- t Throughout this section we sum over repeated indices, both Greek indices and the Latin indices (a, b, c, . . .) that label the frame vectors. §112 HOMOGENEOUS SPACES 357 ponents e* and e a a respectively.f The determinant of the metric tensor (112.2) is y = MK\ 2 = \y*eK < 112 - 6) where \y aP \ is the determinant of the matrix y ap . The invariance of the differential forms (112.1) means that e a a {x) dx a = e%x') dx'\ (112.7) where the e a a on the two sides of the equation are the same functions of the old and new coordinates, respectively. Multiplying this equation by e p (x'), setting dx' p dx' p = — dx\ dx* and comparing coefficients of the same differentials dx a , we find ~ = ef(x'K(x). (U2.8) These equations are a system of differential equations that determine the functions x' p (x) for a given frame. J In order to be integrable, the equations (112.8) must satisfy identically the conditions d 2 x' p _ d 2 x' p dx a dx y ~ dx y dx a ' Calculating the derivatives, we find Multiplying both sides of the equations by e%x)e y c {x)e f p {x') and shifting the differentiation from one factor to the other by using (112.4), we get for the left side: ei(x) \rt^ e%x) ~ ~t^ e%x) \ = 4(x )d(x } b^~ " ~^rr and for the right, the same expression in the variable x. Since x and x' are arbitrary, these expressions must reduce to constants : ©-§)«"-<*• (ili9) The constants C c ab are called the structure constants of the group. Multiplying by e 7 c , we can rewrite (112.9) in the form - del « del "aMa?-' 5 "'- (lmo) t Do not confuse the el with the contravariant components of the vectors e«! The latter are equal to: e aa = y aP ee = y ab e a b . % For a transformation of the form x'" = x p +Z e , where the Z? are small quantities, we obtain from (112.8) the equations ^! = £r<^ e « (112.8a) The three linearly independent solutions of these equations, £\(b = 1, 2, 3), determine the infinitesimal transformations of the group of motions of the space. The vectors £% are called the Killing vectors. 358 COSMOLOGICAL PROBLEMS § 112 As we see from their definition, the structure constants are antisymmetric in their lower indices: C c ab =-C c ba . (112.11) We can obtain still another condition on them by noting that (1 12.10) can be written in the form of commutation relations [X a , X h ~\ = X a X b -X b X a = C c ab X c (112.12) or the linear differential operatorsf X °= e «i^- ( ll2 - 13 ) Then the condition mentioned above follows from the identity [[x a , x„i x c -\ + [[z 6 , x c i x fl ] + [[x c , x a i x b -\ = (the Jacobi identity), and has the form: C e ab C d ec +C e bc C d ea + C e ca C d eb = 0. (112.14) We note that equation (112.9) can be written in vector form as (e a xe„)-curle c = -C c ab , where again the vectorial operations are carried out as if the coordinates jc" were cartesian. Using (112.5) we then have - e 1 • curl e 1 = C\ 2 , - e 2 -curl e 1 = C{ 3 , -e 3 - curie 1 = C\ u (112.15) and six other equations, obtained by cyclic permutation of the indices 1, 2, 3. The Einstein equations for a universe with a homogeneous space can be written as a system of ordinary differential equations containing only functions of the time. To do this all three-dimensional vectors and tensors must be expanded in the triple of frame vectors of the given space. Denoting the components of this expansion by indices a,b, . , we have, by definition: R ab = K p e a a 4, R 0a = R 0a e* a , u a = u"e a a , where all these quantities are functions only of t (as are the scalar quantities e and p). Any further raising or lowering of indices is done using the y ab : R b a = y bc R ac , u a = y ab u b etc. According to (99.11-13) the Einstein equations in the synchronous reference system are given in terms of the three-dimensional tensors yc ap and P aP . For the first of these we have simply x ab = y a b (112.16) (the dot denoting differentiation with respect to 0- The components P ab can be expressed in terms of the quantities y ab and the structure constants of the group: Rab = ~ a ad a bc~Cdc a ab> alb = $(C c ab + C e bd y ea y dc - C e da y eb y dc ). (112.17) t The results presented belong to the mathematical theory of continuous groups (Lie groups). In this theory the operators X a satisfying conditions of the form (1 12.12) are called the generators of the group. We mention, however (to avoid confusion when comparing with other presentations), that the systematic theory usually starts from operators defined using the Killing vectors: § 112 HOMOGENEOUS SPACES 359 The covariant derivatives %£. y [which appear in (99.12)] are also expressed in terms of these quantities, and we find for R° : R° a = -^y bc y b XC c da -5 c a C e ed ). (112.18) We emphasize that, in forming the Einstein equations, there is thus no need to use explicit expressions for the frame vectors as functions of the coordinates.f The choice of the three frame vectors in the differential forms (112.1) and, with them, of the operators (112.13), is clearly not unique. They can be subjected to any linear trans- formation with constant (real) coefficients: e' a * = A b a et. (112.19) Relative to such transformations the quantities y ab behave like a covariant tensor, and the constants C c ab like a tensor covariant in the indices a, b and contravariant in the index c. The conditions (112.11) and (112.14) are the only ones that the structure constants must satisfy. But among the sets of constants admissible under these conditions there are equivalent ones, in the sense that their difference is caused only by a transformation (1 12. 19). The problem of the classification of homogeneous spaces reduces to the determination of all nonequivalent sets of structure constants. A simple procedure for doing this is to make use of the "tensor" properties of the con- stants C c ab , expressing these nine quantities in terms of the six components of a symmetric "tensor" n ab and the three components of a "vector" a c as C c ah = e abd n dc +5 c b a a -5 c a a b . (112.20) where e abd is the unit antisymmetric "tensor" (C. G. Behr, 1962). The condition for anti- symmetry of (112.11) has already been met, while the Jacobi identity (112.14) gives the condition n ab a b = 0. (112.21) By means of the transformations (112.19) the symmetric "tensor" n ah can be brought to diagonal form: let n (1) , ra (2) , « (3) be its eigenvalues. Equation (1 12.21) shows that the "vector" a b (if it exists) lies along one of the principal directions of the "tensor" n ab , the one corres- ponding to the eigenvalue zero. Without loss of generality we can therefore set a b = (a, 0, 0). Then (112.21) reduces to an w = 0, i.e. one of the quantities a or n (1) must be zero. The com- mutation relations (112.12) take the form: lX l ,X 2 ] = aX 2 + n^X 3 , iX 2 ,X 3 -] = n^X l , (112.22) [_X 3 ,X 1 -] = n^X 2 -aX 3 . The only remaining freedom is a change of sign of the operators X a and arbitrary scale transformations of them (multiplication by constants). This permits us simultaneously to change the sign of all the n {a) and also to make the quantity a positive (if it is different from zero). We can also make all the structure constants equal to ±1, if at least one of the quantities a, n (2) , n (3) vanishes. But if all three of these quantities differ from zero, the scale transformations leave invariant the ratio a 2 /« (2) w (3) . t The derivation of formulas (1 12.17-18) can be found in the paper of E. Schiicking in the book Gravita- tion: an Introduction to Current Research, ed. L. Witten, J. Wiley, New York, 1962, p. 454. 360 COSMOLOGICAL PROBLEMS § 113 Thus we arrive at the following list of possible types of homogeneous spaces; in the first column of the table we give the roman numeral by which the type is labelled according to the Bianchi classification (L. Bianchi, 191 8) :f Type a n (1 > n w H< 2 > I II VII 1 VI -1 IX 1 1 VIII 1 -1 V 1 IV 1 1 VII a 1 1 III (a VI (a = D1 a 1 -1 Type I is euclidean space (all components of the spatial curvature tensor vanish). In addition to the trivial case of a galilean metric, the metric (103.9) belongs to this type. If for the space of type IX one puts y ab = (a 2 /4)S ab , one finds for the Ricci tensor P ab = \b ab and hence : "afi = °ab e a e p = ~~2 7ap> which corresponds to a space of constant positive curvature [cf. (107.3), (107.6)]; this space is thus contained in type IX as a special case. Similarly the space of constant negative curvature is contained as a special case in type V. This is easily seen by transforming the structure constants of this group by the sub- stitution X 2 + X 3 = X 2 , X 2 -X 3 = X 3 , X 1 =X' 1 . Then [X' Xi X' 2 ~\ = X' 2i [X' 2 ,X' 3 ] = 0, [Z' 3 ,Zi] = -X 3 , and if one puts y ab = a 2 d ab , the Ricci tensor becomes P ah =-25 ab , P a p = — (2/a 2 )y a p which corresponds to a space of constant negative curvature. § 113. Oscillating regime of approach to a singular point On the model of a universe with a homogeneous space of type IX we shall study the time singularity of the metric, whose character is basically different from that of the singularity in the homogeneous and isotropic model (V. A. Belinskii, I. M. Khalatnikov, E. M. Lifshitz, 1969; C. W. Misner, 1969). We shall see in the next section that such a situation has a very general significance. We shall be interested in the behavior of the model near the singularity (which we choose as the time origin t = 0). We shall see later that the presence of matter does not affect the qualitative properties of his behavior. For simplicity we shall therefore assume at first t The parameter a runs through all positive values. The corresponding types are actually a one-parameter family of different groups. From given structure constants one can find the basis vectors by solving the differential equations (1 12.10). They have been given for all types (together with the corresponding Killing vectors) in the paper of A. H. Taub, Ann. Math. 53, 472, 1951. § 113 OSCILLATING REGIME OF APPROACH TO A SINGULAR POINT 361 that the space is empty. A physical singularity for such a space means that the invariants of the four-dimensional curvature tensor go to infinity at t = 0. We take the quantities yjt) in (112.2) to be diagonal, denoting the diagonal elements by a 2 , b 2 , c 2 ; we here denote the three frame vectors e 1 , e 2 , e 3 by 1, m, n. Then the spatial metric is written as: n y a p = a 2 U p + b 2 m a m p + c 2 n a n p . U^-i) For a space of type IX the structure constants are:| C\, = C 2 , = C\ 2 = 1. (113-2) From (112, 16-18) it can be seen that for these constants and a diagonal matrix y ab , the components Hj, Uj, R°„, R?, *?, K of the Ricci tensor vanish identically in the synchronous reference system. The remaining components of the Einstein equations give the following system of equations for the functions a, b, c: (abc)' 1 abc ,2l2„2 l(Xa 2 -vc 2 ) 2 -fi 2 b% (H3.3) [(Afl 2 -|ib 2 ) 2 -vV], a h c + -+- = 0. (H3.4) abc [Equations (113.3) are the equation set R\ = K£ = R n n = 0; equation (113.4) is the equation R o _ 0> ) The letters X, \i, v here denote the structure constants C\ 3 , C\ u C 12 ; although they are set equal to 1 everywhere from now on, they here illustrate the origin of the different terms in the equations. The time derivatives in the system (113.3-4) take on a simpler form if we introduce m place of the functions a, b, c, their logarithms a, /?, y : a = e\ b = e p , c = e\ (113.5) and in place of /, the variable t: dt = abcdx. (113.6) Then: 2a„ = (b 2 -c 2 ) 2 -a 4 , 2p xx = (a 2 -c 2 ) 2 -b\ (113.7) 2y„ = (a 2 -b 2 ) 2 -c 4 ; K«+0+y)« = «.&+««7t+&7t> ( 113 - 8 > t The frame vectors corresponding to these constants are: 1 = (sin x 3 , -cos x 3 sin x\ 0), m = (cos x 3 , sin x 3 sin x\ 0), n = (0, cosx 1 , 1). The element of volume is : dV= Vy dx 1 dx 2 dx 3 = abc sin x 1 dx 1 dx 2 dx 3 . The coordinates run through values in the ranges ^ x 1 ^ n, < x 2 ^ lit, ^ x 3 < 4tt. The space is closed, and its volume V= I6n 2 abc (when a = b = c it goes over into a space of constant positive curvature with radius of curvature 2d). 362 COSMOLOGICAL PROBLEMS § 113 where the subscript t denotes differentiation with respect to t. Adding equations (113.7) and replacing the sum of second derivatives on the left by (113.8), we obtain: aJ x + a T y t +P x y T = Ka 4 +b 4 +c*-2a 2 b 2 -2a 2 c 2 -2b 2 c 2 ). (113.9) This relation contains only first derivatives, and is a first integral of the equations (113.7). Equations (113.3-4) cannot be solved exactly in analytic form, but permit a detailed qualitative study. We note first that if the right sides of equations (1 13.3) were absent, the system would have an exact solution, in which a~t p >, b~t Pm , c~t Pn , (113.10) where p h p m , and p„ are numbers connected by the relations Pl + Pm + Pn = Pf+Pl + Pn = 1 (113.11) [the analog of the Kasner solution (103.9)]. We have denoted the exponents by p h p m , p„, without assuming any order of their size; we shall retain the notation Pi,p 2 ,p 3 of § 103 for the triple of numbers arranged in the order p^<p 2 < P3 and taking on values in the intervals (103.10a) respectively. These numbers can be written in parametric form as *«-ITT^' P ^ = TTI+?' ft(s) = ITiT?- (113 - 12) All the different values of the p u p 2 , p 3 (preserving the assumed order) are obtained if the parameter s runs through values in the range s ^ 1. The values s < 1 are reduced to this same region as follows: Pi (-) = Pi(«), Pi (-) = Pi(s), p 3 (-) = p 2 (s). (113.13) Let us assume that within some time interval the right sides of equations (113.3) are small, so that they can be neglected and we have the "Kasner-like" regime (113.10). To be specific, let us suppose that the exponent in the function a is negative: p t =p t <0. We shall follow the evolution of the metric in the direction of decreasing t. The left sides of (113.3) have a "potential" order of magnitude ~ t~ 2 . Noting that in the regime (113.10), abc ~ t, we see that on the right sides, all the terms increase (for/-»0) more slowly than t~ 2 , except for the term sa*/a 2 b 2 c 2 ~ t~ 2 t~ 4lpi1 . These are the terms that will play the role of a perturbation that destroys the Kasner regime. The terms a 4 on the right sides of (113.7) correspond to them. Keeping only these terms, we find **,= -&*, P X r = y,t = ^ (113.14) To the "initial" statef (113.10) there correspond the conditions Vt = Pi> Px = Pm> yx = Pn- The first of equations (113.14) has the form of the equation of one-dimensional motion of a particle in the field of an exponential potential wall, where a plays the role of the co- ordinate. In this analogy, to the initial Kasner regime there corresponds a free motion with constant velocity a T =p l . After reflection from the wall, the particle will again move freely with the opposite sign of the velocity: a t = — p t . We also note that from equations (113.14), a z +P x = const, and a T +y t = const, hence we find that fl x and y x take the values Pr = Pm + 2p h V t = P» + 2p l . f We emphasize once again that we are considering the evolution of the metric as / -> 0; thus the "initial" conditions refer to a later, and not an earlier time. § 113 OSCILLATING REGIME OF APPROACH TO A SINGULAR POINT 363 Now determining a, p, y, and then t, using (113.6), we find e a ~ e~ PlX e p ~ e {Pm+2pi)x , e y ~ e (Pn+2pi)x i.e. a ~ t p '\ b ~ t p ' m , c ~ t p '", where Pl = i^ip-; Pm ~ Tfw Pn ~ i+2 ft - (113>15) If we had p l <p m < p„, Pi < 0, then now p' m <p' l < p'„, p' m <0; the function b, which was decreasing (for t -> 0) begins to increase, the rising function a now drops, while the function c continues to fall. The perturbation itself [~ a 4 in (1 13.7)], which previously was increasing, now damps out. The law of change of the exponents (113.15) is conveniently represented using the para- metrization (113.12): if Pi = Pl(s)> Pm = Pl(s), Pn = PaOO, then p' l = p 2 (s-l), p' m = p 1 (s-l), p„ = p 3 (s-l). (113.16) The larger of the two positive exponents remains positive. Thus the action of the perturbation results in the replacement of one Kasner regime by another, with the negative power shifting from the direction 1 to the direction m. Further evolution of the metric leads in an analogous way to an increase in the perturbation given by the terms ~ b 4 in (113.7), another shift of the Kasner regime, etc. The successive shifts (113.16) with bouncing of the negative exponent/?! between the direc- tions 1 and m continue so long as s remains greater than 1. Values s < 1 are transformed into s > 1 according to (113.13); at this moment either p t or p m is negative, while />„ is the smaller of the two positive numbers (p„ = p 2 )- The following series of shifts will now bounce the negative exponent between the directions n and I or between n and m. For an arbitrary (irrational) initial value of s, the process of shifting continues without end. In an exact solution of the equations, the powers/?,, p m ,p n will, of course, lose their literal meaning. But the regularities in the shifting of exponents allow one to conclude that the course of change of the metric as we approach the singularity will have the following qualitative properties. The process of evolution of the metric is made up of successive periods (we shall call them eras), during which distances along two of the axes oscillate, while distances along the third axis decrease. On going from one era to the next, the direc- tion along which distances decrease monotonically bounces from one axis to another. The order of this bouncing acquires asymptotically the character of a random process. The successive eras crowd together as we approach t = 0. But the natural variable for describing the behavior of this time evolution appears to be not the time t, but its logarithm, In t, in terms of which the whole process of approach to the singularity is stretched out to — oo. The qualitative analysis presented above must, however, be supplemented with respect to the following point. In this analysis there correspond to the «'th era values of the numbers s {n) starting from some largest value sj£> x down to some smallest, s^} n < 1. The length of the era (as measured by the number of oscillations) is the integer s<£> x - s£in- For the next era, s£i 1) = l/ s min- In the infinite sequence of numbers formed in this way one will find arbitrarily small (but 364 COSMOLOGICAL PROBLEMS § 113 never zero) values of sg? n and correspondingly arbitrarily large values of s££ 1} ; such values correspond to "long" eras. But to large values of the parameter s there correspond exponents (Pi> Pi, Pz) close to the values (0, 0, 1). Two of the exponents which are close to zero are thus close to one another, hence also the laws of the change of two of the functions a, b, c, are close to one another. If in the beginning of such a long era these two functions happen to be close also in their absolute magnitude, they shall remain to be such during the larger part of the entire era. In such a case it becomes necessary to keep not one term (a 4 ) on the right sides of (113.7), but two terms. Let c be that one of the functions a, b, c that decreases monotonically in the course of a long era. It then rapidly becomes smaller than the other two; let us consider the solution of equations (113.7-8) in just that region of the variable x where we can neglect c compared to a and b. Let the upper limit of this region be x = t . In this case the first two equations of (113.7) give arr+ft t = 0, (113.17) a tt -P tx =-e 4a +e 4li , (113.18) while for the third equation we use (113.9), which gives: y T (a r +&)= - a J x + \{e 2 *-e 2l >) 2 . (113.19) We write the solution of (113.17) in the form 2a 2 a+0 = — (T-T ) + 21na , where a and £ are positive constants. It will be convenient to introduce in place of t a new variable f = £ exp Then 2a 2 " ^-(t-t ) Co (113.20) cc + P = In— +2lna . (113.21) Co We also transform equations (113.18-19), introducing the notation x = a-/? 1 2 1 1 fe+ * Xt+ ~ sinh 2 X = 0, (113.22) 1 £ , 7t = - ^ + g (2/| + cosh 2x-l). (113.23) To the decrease of t from oo to there corresponds the drop of x from oo to — oo ; corres- pondingly f drops from oo to 0. As we shall see later, a long era is obtained if £ (the value of f corresponding to the instant t = t ) is a very large quantity. We shall consider the solution of equations (113.22-23) in the two regions £ > 1 and ^ -4 1. For large £ the solution of (113.22) in first approximation (in 1/f) is: X = a-p = - r ~sm(£-£ ) (113.24) (where A is a constant); the factor 1/Vc makes x a small quantity, so that we can make the § 113 OSCILLATING REGIME OF APPROACH TO A SINGULAR POINT 365 substitution sinh 2 X « 2/ in (113.22). From (113.23) we now find: 7i ~ \(xt + X 2 ) = A 2 , 1 = ^ 2 (^-^o) + const. Having determined a and from (1 13.21) and (1 13.24) and expanded e* and e> in accordance with our approximation, we finally obtain: ""-Jlk-Tt* 1 ™]- (113 ' 25) c = c oe -^°-^ The relation of £ to the time t is gotten by integrating the defining equation (113.6), and is given by the formula 1 = e -;i 2 «o-« (113.26) The constant c (the value of c when £ = £ ) must satisfy c < a . We now turn to the region f < 1. Here the leading terms in the solution of (113.22) are: z==a -j3 = Kln£ + const, (113.27) where k is a constant lying in the range - 1< k < + 1 ; this condition assures the smallness of the last term in (113.22) (since sinh 2/ contains <f K and £ 2k ) compared to the first two (~ r 2 )- Having determined a and p from (113.27) and (113.21), y from (113.23) and t from (113.6), we get 2 t~r^. (113-28) This is again a Kasner regime, where the negative power of t appears in the function c(t). Thus we arrive again at a picture of the same qualitative character. Over a long period of time (corresponding to large decreasing values of two of the functions (a and b) oscillate, while equation (113.25) shows, in addition, that these oscillations proceed on the background of a slow (~ VI) fall off of their mean values. Throughout all this time the functions a and b remain close in value. The third function c falls monotonically, the decrease following the law c = c t/t . This evolution lasts until £ ~ 1, when formulas (113.25-26) become inapplicable. After this, as we see from (113.28), the decreasing function c begins to rise, and the functions a and b drop. This will continue until the terms ~ c 2 /a 2 b 2 on the right sides of (113.3) become ~ t~ 2 , when the next series of oscillations begin. These qualitative features of the behavior of the metric near a singular point are not changed by the presence of matter; near the singularity the matter can be "written into" the metric of empty space, neglecting its back reaction on the gravitational field. In other words, the evolution of the matter is determined simply by the equations of its motion in 366 COSMOLOGICAL PROBLEMS § 113 the given field. These equations are the hydrodynamic equations 1 d , , — . -7=~i(v-g(Tu l ) = 0, V —g ox 1 (cf. Fluid Mechanics, § 126). Here a is the entropy density; near the singularity we must use the ultrarelativistic equation of state p = e/3, when a ~ e 3/4 . Applying these equations to the motion of matter in the Kasner metric and the metric (113.25), we find that in both cases the energy density increases monotonically (cf. the Problems). This proves that the energy density tends to infinity (when t -»• 0) in this model. PROBLEMS 1. Find the law of variation with time of the density of matter, uniformly distributed in a space with the metric (103.9), for small t. Solution: Denote the time factors in (103.9) by a = t PX , b = t P2 , c = t P3 . Since all quantities depend only on the time, and — g = abc, equations (113.29) give ~(abcu Q e 3 ") = t Ae^+uJ-^O. Then abcuo e 3 /4 = const, (U lias 1 ' 4 — const. (2) According to (2) all the covariant components u a have the same order of magnitude. Among the contravariant components, the largest (when /->0) is u 3 =u 3 /c 2 . Keeping only the largest terms in the identity 11,11' = 1, we get u% x u 3 u 3 = uj 2 c 2 , and then, from (1) and (2), e ~ l/a 2 b 2 , u a ~ Vd>, (3) or As it should, e goes to infinity when t^>0 for all values ofp 3 except p 3 = 1, in accordance with the fact that the singularity in the metric with exponents (0, 0, 1) is unphysical. The validity of this approximation is verified by estimating the components T l k omitted on the right sides of (103.3-4). The leading terms are: r° ~ eul ~ t -a+P3\ T\~ s~t -aa-»*>, Tl ~ su 2 u 2 - / -a+2 P2 - P3 ) 5 ti „ SU3ll 3 „ t -a+p 3 )_ They all actually increase more slowly, when /->0, than the left sides of the equations, which increase like t~ 2 . 2. The same problem for the metric (113.25). Solution: With the functions a and b from (113.25), we find from (3): e ~ £~ 2 . Through the whole time when £ varies from £ to £ ~ 1, the density increases by a factor k%- Considering the connection (113.26) between £ and /, this means an increase by the factor In 2 (/ /'i), where t and t 1 are the upper and lower limts of the era in terms of the time /. § 114. The character of the singularity in the general cosmological solution of the gravitational equations The adequacy of the isotropic model for describing the present state of the Universe is no basis for expecting that it is equally suitable for describing its early stages of evolution. § 114 THE GENERAL COSMOLOGICAL SOLUTION 367 One may even ask to what extent the existence of a time singularity (i.e. finiteness of the time) is a necessary general property of cosmological models, or whether it is really caused by the specific simplifying assumptions on which the models are based. If the presence of the singularity were independent of these assumptions, it would mean that it is inherent not only to special solutions, but also to the general solution of the gravita- tional equations.! Finding such a solution in exact form, for all space and over all time, is clearly impossible. But to solve our problem it is sufficient to study the form of the solution only near the singularity. The criterion of generality of the solution is the number of "physically arbitrary" functions of the space coordinates contained in it. In the general solution the number of such functions must be sufficient for arbitrary assignment of initial conditions at any chosen time [4 for empty space, 8 for space filled with matter (cf. § 95)].J The singularity of the Friedmann solution for t = is characterized by the fact that the vanishing of spatial distances occurs according to the same law in all directions. This type of singularity is not, however, sufficiently general: it is typical of a class of solutions that contain only three physically arbitrary coordinate functions (cf. the problem in § 109). We also note that these solutions exist only for a space filled with matter. The singularity that is characteristic of the Kasner solution (103.9) has a much more general character.§ It belongs to the class of solutions in which the leading terms in the expansion of the spatial metric tensor ( in the synchronous reference system) near the singular point t = have the form ?«„ = t 2p %l p + t 2pm m a m p + t 2p "n a n p , (114.1) where 1, m and n are three vector functions of the coordinates, while p h p m and p„ are co- ordinate functions related by the two equations (113.11). For the metric (114.1), the equation R% = for the field in vacuum is satisfied automatically in its leading terms. Satisfying the equations Rf = requires fulfilment of the additional condition l-curll = (H4.2) for that one of the vectors 1, m, n that has a negative power in (1 14.1) (which we have chosen to be Pi = p x < 0). The origin of this condition can be followed using the equations (113.3) of the preceding section which correspond to a definite choice of the vectors 1, m, n. These equations could have the solution (113.10) valid down to t = only under the condition X = 0, when on the right sides of the equations the terms a 2 /2b 2 c 2 , that grow faster than t ~ 2 for t -> 0, would vanish. But according to (1 12.15), the requirement that the structure constant X = C\ 3 vanish precisely implies the condition (114.2). As for the equations i?° = 0, which contain only first time derivatives of the components of the tensor y af , they lead to three more relations (not containing the time) that must be imposed on the coordinate functions in (114.1). Together with (114.2) there are thus all t When we speak of a singularity in the cosmological solution we have in mind a singularity that is attainable in all of the space (and not over some restricted part, as in the gravitational collapse of a finite body). . t We emphasize that for a system of nonlinear equations, such as the Einstein equations, the notion oi a general solution is not unambiguous. In principle more than one general integral may exist, each of the integrals covering not the entire manifold of conceivable initial conditions, but only some finite part of it. Each such integral will contain the whole required set of arbitrary functions, but they may be subject to specific conditions in the form of inequalities. The existence of a general solution possessing a singularity does not therefore preclude the existence of other solutions that do not have a singularity. § This section gives only a general outline of the situation. For a more detailed presentation, cf. I. M. Khalatnikov and E. M. Lifshitz, Adv. in Phys. 12, 185, 1963; V. A. Belinskii, I. M. Khalatmkov, and E. M. Lifshitz, Adv. in Phys. 1970. 368 COSMOLOGICAL PROBLEMS § \\4 together four conditions. These conditions connect ten different coordinate functions: three components of each of the three vectors 1, m and n, and one of the functions that appears as a power of / [any one of the three functions p u p m and p n , which are related by the two equa- tions (113.11)]. In determining the number of physically arbitrary functions we must also remember that the synchronous reference system used still allows arbitrary transformations of the three spatial coordinates, not affecting the time. Thus the solution (114.1) contains altogether 10-4-3 = 3 physically arbitrary functions, which is one fewer than required for the general solution in empty space. The degree of generality achieved is not reduced when matter is introduced: the matter is "written into" the metric (1 14. 1) with its four new coordinate functions needed for assigning the initial distribution of the matter density and the three velocity components (cf the problem in § 113). Of the four conditions that must be imposed on the coordinate functions in (114.1), the three conditions that arise from the equations R° a = are "natural"; they follow from the very structure of the equations of gravitation. The imposition of the additional condition (1 14.2) results in the "loss" of one arbitrary function. By definition the general solution is completely stable. Application of any perturbation is equivalent to changing the initial conditions at some moment of time, but since the general solution admits arbitrary initial conditions, the perturbation cannot change its character. But for the solution (114.1) the presence of the restrictive condition (114.2) means, in other words, instability with respect to perturbations that violate this condition. The application of such a perturbation should carry the model into a different regime, which ipso facto will be completely general. This is precisely the study made in the previous section for the special case of the homo- geneous model. The structure constants (113.2) mean precisely that for the homogeneous space of type IX all three products 1 -curl I, m -curl m, n -curl n are different from zero [cf. (112.15)]. Thus the condition (114.2) cannot be fulfilled, no matter which direction we assign the negative power of the time. The discussion given in § 113 of the equations (113.3-4) consisted in an explanation of the effects produced on the Kasner regime by the perturbation associated with a nonvanishing X = (1 -curl \)/v. Although the investigation of a special case cannot exhibit all the details of the general case, it does give a basis for concluding that the singularity in the general cosmological solution has the oscillating character described in § 113. We emphasize once more that this character is not related to the presence of matter, and is already a feature of empty space- time itself. The oscillating regime of approach to the singularity gives a whole new aspect to the con- cept of finiteness of time. An infinite set of oscillations are included between any finite moment of world time t and the moment t = 0. In this sense the process has infinite charac- ter. Instead of the time t, a more natural variable (as already noted in § 113) appears to be the logarithm In t, in terms of which the process is stretched out to - oo. We have spoken throughout of the direction of approach to the singularity as the direc- tion of decreasing time; but in view of the symmetry of the equations of gravitation under time reversal, we could equally well have talked of an approach to the singularity in the direction of increasing time. Actually, however, because of the physical inequivalence of future and past, there is an essential difference between these two cases with respect to the formulation of the problem. A singularity in the future can have physical meaning only if it is attainable from arbitrary initial conditions, assigned at any previous instant of time. It is § 114 THE GENERAL COSMOLOGICAL SOLUTION 369 clear that there is no reason why the distribution of matter and field that is attainable at some instant in the process of evolution of the Universe should correspond to the specific conditions required for the existence of some particular solution of the gravitational equations. As for the question of the type of singularity in the past, an investigation based solely on the equations of gravitation can hardly give an unambiguous answer. It is natural to think that the choice of the solution corresponding to the real universe is connected with some profound physical requirements, whose establishment solely on the basis of the present theory is impossible and whose clarification will come only from a further synthesis of physical theories. In this sense it could, in principle, turn out that this choice corresponds to some special (for example, isotropic) type of singularity. Nevertheless, it appears more natural a priori to suppose that, in view of the general character of the oscillating regime, just this regime should describe the early stages of evolution of the universe. INDEX Aberration of light 13 Absolute future 6 Absolute past 6 Action function 24, 67, 266 Adiabatic invariant 54 Airy function 149, 183, 200 Angular momentum 40, 79 Antisymmetric tensor 16, 21 Astigmatism 136 Asymptotic series 152 Axial vector 18, 47 Babinet's principle 155 Bessel functions 182 Bianchi classification 360 Bianchi identity 261 Binding energy 30 Biot-Savart.law 103 Bremsstrahlung 179 magnetic 197 Caustic 133, 148 Center of inertia 41 Center-of-mass system 31 Centrally symmetric gravitational field 282, 287 Characteristic vibrations 141 Charge 44 density 69 Christoffel symbols 238 Circular polarization 114 Circulation 67 Classical mechanics 2 Closed model 336 Coherent scattering 220 Collapse 296 Combinational scattering 220 Comoving reference system 293, 294, 301, 336 Conformal-galilean system 341 Contraction of a field 92 of a tensor 16 Contravariant derivative 239 tensor 16, 229 vector 14 Coriolis force 252, 321 Coulomb field 88 Coulomb law 88 Covariant derivative 236 tensor 16, 229, 313 vector 14 Cross section 34, 215 C-system 31 Current four-vector 69 Curvature tensor 258, 295 canonical forms of 264 Curved space-time 227 Curvilinear coordinates 229 D'Alembert equation 109 D'Alembertian 109 Decay of particles 30 Degree of polarization 121 Delta function 29, 70 Depolarization, coefficient 122 Diffraction 145 Dipole moment 96 radiation 173 Displacement current 75 Distribution function 29 Doppler effect 1 1 6, 254 Drift velocity 55, 59 Dual tensor 17 Dustlike matter 301 Effective radiation 177 Eikonal 129, 145 angular 134 equation 130 Elastic collision 36 Electric dipole moment 96 Electric field intensity 47 371 372 INDEX Electromagnetic field tensor 60 Electromagnetic waves 108 Electromotive force 67 Electrostatic energy 89 Electrostatic field 88 Element of spatial distance 233 Elementary particles 26, 43 Elliptical polarization 115 Energy 26 density 75 flux 75 Energy-momentum pseudotensor 304 Energy-momentum tensor 77, 80, 268 for macroscopic bodies 97 Equation of continuity 71 Era 363 Euler constant 185 Events 3 Exact solutions of gravitational equations 314 Fermat's principle 132, 252 Field constant 50 Lorentz transformation of 62 quasiuniform 54 uniform electric 52 uniform magnetic 53 Flat space-time 227 Flux 66 Focus 133 Four-acceleration 22 Four-dimensional geometry 4 Four-force 28 Four-gradient 19 Four-momentum 27 Four-potential 45 Four-scalar 15 Four- vector 14 Four-velocity 21 Fourier resolution 119,124,125 Frame vector 356 Fraunhofer diffraction 153 Frequency 114 Fresnel diffraction 150, 154 integrals 152 Friedmann solution 333 Galilean system 228 Galileo transformation 9 Gamma function 185 Gauge invariance 49 Gauss' theorem 20 Gaussian curvature 262 Gaussian system of units 69 General cosmological solution 367 Generalized momentum 45 Geodesic line 244 Geometrical optics 129 Gravitational collapse 296 Gravitational constant 266 Gravitational field 225, 274 centrally symmetric 282, 287 Gravitational mass 309 Gravitational potential 226 Gravitational radius 284 Gravitational stability 350 Gravitational waves 311, 352 Group of motions 356 Guiding center 55, 59 28,46,94, 112, 130, Hamiltonian 26 Hamilton-Jacobi equation 349 Hankel function 149 Heaviside system 69 Homocentric bundle 136 Homogeneous space 355 Hubble constant 346 Huygens' principle 146 Hypersurface 19, 73 Impact parameter 95 Incoherent scattering 220 Incoherent waves 122 Inertial mass 309 Inertial system 1 Interval 3 Invariants of a field 63 Isotropic coordinates 287 Isotropic space 333 Jacobi identity 358 Kasner solution 315, 362 Killing equations 269 Killing vector 357 Laboratory system 31 Lagrangian 24, 69 density 77 to fourth order 236, 367 to second order 165 Laplace equation 88 Larmor precession 107 Larmor theorem 105 Legendre polynomials 98 Lens 137 Lie group 358 Lienard-Wiechert potentials 160, 171, 179 Light aberration of 13 cone 7 pressure 112 Linearly polarized wave 116 Locally-geodesic system 240, 260 Locally-inertial system 244 INDEX 373 Longitudinal waves 125 Lorentz condition 109 Lorentz contraction 1 1 Lorentz force 48 Lorentz frictional force 204 Lorentz gauge 109 Lorentz transformation 9, 62 L-system 31 MacDonald function 183 Macroscopic bodies 85 Magnetic bremsstrahlung 197 Magnetic dipole radiation 188 Magnetic field intensity 47 Magnetic lens 140 Magnetic moment 103 Magnification 142 Mass current vector 83 Mass density 82 Mass quadrupole moment tensor 280 Maupertuis' principle 51, 132 Maxwell equations 66, 73, 254 Maxwell stress tensor 82, 112 Metric space-time 227 tensor 16, 230 Mirror 137 Mixed tensor 230 Moment-of-inertia tensor 280 Momentum 25 density 79 four-vector of 27 space 29 Monochromatic wave 114 Multipole moment 97 Natural light 121 Near zone 190 Newtonian mechanics Newton's law 278 Nicol prism 120 Null vector 1 5 Observed matter 345 Open model 340 Optic axis 136 Optical path length 134 Optical system 134 Oscillator 55 Parallel translation 237 Partially polarized light 1 19 Pascal's law 85 Petrov classification 263 Phase 114 Plane wave 110,129 Poisson equation 88 Polarization circular 114 elliptic 114 tensor 120 Polar vector 18, 47 Poynting vector 76 Principal focus 138 Principal points 138 Principle of equivalence 225 of least action 24, 27 of relativity 1 superposition 68 Proper acceleration 52 length 11 time 7 volume 1 1 Pseudo-euclidean geometry 4, 10 Pseudoscalar 17, 68 Pseudotensor 17 Quadrupole moment 98 Quadrupole potential 97 Quadrupole radiation 188 Quantum mechanics 90 Quasicontinuous spectrum 200 Radiation damping 203,208 of gravitational waves 323 Radius of electron 90 Rays 129 Real image 136 Recession of nebulae 343 Red shift 249,343 Reference system 1 Renormalization 90, 208 Resolving power 144 Rest energy 26 Rest frame 34 Retarded potentials 158 Ricci tensor 261, 360 Riemann tensor 259 Rotation 253 Rutherford formula 95 Scalar curvature 262 Scalar density 232 Scalar potential 45 Scalar product 15 Scattering 215 Schwarzschild sphere 296 Secular shift of orbit 321 of perihelion 330 Self-energy 90 Signal velocity 2 Signature 228 374 INDEX Space component 15 Spacelike interval 6 Spacelike vector 15 Spatial curvature 286 distance 233 metric tensor 250 Spectral resolution 118, 1 63, 21 1 Spherical harmonics 98 Static gravitational field 247 Stationary gravitational field 247 Stokes' parameters 122 Stokes' theorem 20 Stress tensor 80 Structure constants 357 Symmetric tensor 16, 21 Synchronization 236 Synchronous reference system 290 Synchrotron radiation 197, 202 Telescopic imaging 139 Tensor 15 angular momentum 41 antisymmetric 16, 21 completely antisymmetric unit 17 contraction of 16 contravariant 16, 229 covariant 16, 229, 313 density 232 dual 17 electromagnetic field 60 hermitian 120 irreducible 98 mixed 230 moment-of-inertia 280 symmetric 16, 21 Thomson formula 216 Time component 15 Timelike interval 5 Transverse vector 15 wave 111 Trochoidal motion 57 Ultrarelativistic region 27, 195, 201 Unit four-tensor 16 Unpolarized light 121 Vector axial 18, 47 density 232 polar 18, 47 potential 45 Poynting 76 Velocity space 35 Virial theorem 84 Virtual image 136 Wave equation 108 length 114 packet 131 surface 129 vector 116 zone 170 World line 4 point 4 time 247 THE JOURNAL OF THE FRANKLIN INSTITUTE The Journal of the Franklin Institute covers the traditional branches of mathematics and the physical sciences, both pure and applied, as well as the new composite sciences, combining two or more disciplines. Dedicated to honour Ben Franklin, America's great inventor, writer and scientist, the journal provides a platform for the dissemination of scientific ideas and research, and draws its authors and readership from more than sixty countries throughout the world. Major papers describing theoretical and experimental researches are accepted for publication on the basis of their lasting value. The journal also publishes brief communications of exceptional interest and reviews a number of current books in each issue. Demonstrating the extreme flexibility of the editorial policy, special issues have been published on topics that are timely and fall within the broad range of interest of the journal. Since its initial publication in 1826, the journal has proved its ability to relate to the times by bridging the gap from one era to another, becoming one of the most highly respected publications in the world of science and engineering. Today, the Journal of the Franklin Institute remains as relevant, for present and future generations of scientific workers, as it was for its founders, almost fifteen decades ago. Write for a specimen copy of the latest issue and details of related Pergamon journals. Pergamon Press Headington Hill Hall, Oxford 0X3 OBW Maxwell House, Fairview Park, Elmsford, New York 10523 207 Queen's Quay West, Toronto 1 19a Boundary Street, Rushcutters Bay. NSW 2011, Australia Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Printed in Great Britain/Bradley COURSE OF THEORETICAL PHYSICS by L D. LANDAU {Deceased) and E. M. LIFSHITZ Institute of Physical Problems, USSR Academy of Sciences The complete Course of Theoretical Physics by Landau and Lifshitz, recognized as two of the world's outstanding physicists, is being published in full by Pergamon Press. It comprises nine volumes, covering all branches of the subject; translations from the Russian are by leading scientists. Typical of the many statements made by experts, reviewing the series, are the following : "The titles of the volumes in this series cover a vast range of topics, and there seems to be little in physics on which the authors are not very well informed. " Nature "The remarkable nine-volume Course of Theoretical Physics . . . the clearness and accuracy of the authors' treatment of theoretical physics is well maintained. " Proceedings of the Physical Society Landau Lifshitz Of individual volumes, reviewers have written : MECHANICS "The entire book is a masterpiece of scientific writing. There is not a superfluous sentence and the authors know exactly where they are going. ... It is certain that this volume will be able to hold its own amongst more conventional texts in classical mechanics, as a scholarly and economic exposition of the subject." 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Journal of the Royal Aeronautical Society ELECTRODYNAMICS OF CONTINUOUS MEDIA "Within the volume one finds everything expected of a textbook on classical electricity and magnetism, and a great deal more. It is quite certain that this book will remain unique and indispensable for many years to come." Science Progress "The volume on electrodynamics conveys a sense of mastery of the subject on the part of the authors which is truly astonishing." Nature CD CD So O CO ■*2. _! 0) CD — E 08 01 601 9 Pergamon