GIFT OF
MATVU STAT.
SEMICENTENNIAL PUBLICATIONS
OF THE
UNIVERSITY OF CALIFORNIA
THE THEORY OF THE RELATIVITY OF MOTION
BY
RICHARD CHACE TOLMAN
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY
1918
SEMICENTENNIAL PUBLICATIONS
OF THE
UNIVERSITY OF CALIFORNIA
1868-1918
THE THEORY OF THE RELATIVITY OF MOTION
BY
RICHARD C. TOLMAN
UNIVERSITY OF CALIFORNIA PRESS
BERKELEY
1917
PRESS OF
THE NEW ERA PRINTING COMPANY LANCASTER. PA
TO
H. E.
383389
THE THEORY OF THE RELATIVITY OF MOTION.
BY
RICHARD C. TOLMAN, PH.D.
TABLE OF CONTENTS.
PREFACE 1
CHAPTER I. Historical Development of Ideas as to the Nature of Space and
Time 5
Part I. The Space and Time of Galileo and Newton 5
Newtonian Time 6
Newtonian Space 7
The Galileo Transformation Equations 9
Part II. The Space and Time of the Ether Theory 10
Rise of the Ether Theory 10
Idea of a Stationary Ether 12
Ether in the Neighborhood of Moving Bodies 12
Ether Entrained in Dielectrics 13
The Lorentz Theory of a Stationary Ether 13
Part III. Rise of the Einstein Theory of Relativity 17
The Michelson-Morley Experiment 17
The Postulates of Einstein 18
CHAPTER II. The Two Postulates 01 the Einstein Theory of Relativity 20
The First Postulate of Relativity 20
The Second Postulate of the Einstein Theory of Relativity 21
Suggested Alternative to the Postulate of the Independence of the
Velocity of Light and the Velocity of the Source 23
Evidence against Emission Theories of Light 24
Different Forms of Emission Theory 25
Further Postulates of the Theory of Relativity 27
CHAPTER III. Some Elementary Deductions 28
Measurements of Time in a Moving System 28
Measurements of Length in a Moving System 30
The Setting of Clocks in a Moving System 33
The Composition of Velocities
The Mass of a Moving Body 37
The Relation between Mass and Energy 39
CHAPTER IV. The Einstein Transformation Equations for Space and Time . . 42
The Lorentz Transformation 42
Deduction of the Fundamental Transformation Equations 43
The Three Conditions to be Fulfilled 44
The Transformation Equations
Further Transformation Equations 47
Transformation Equations for Velocity
Transformation Equations for the Function — = 47
vi Table of Contents.
Transformation Equations for Acceleration 48
CHAPTER V. Kinematical Applications 49
The Kinematical Shape of a Rigid Body 49
The Kinematical Rate of a Clock 50
The Idea of Simultaneity 51
The Composition of Velocities 52
The Case of Parallel Velocities 52
Composition of Velocities in General 53
Velocities Greater than that of Light 54
Applications to Optical Problems 56
The Doppler Effect 57
The Aberration of Light 59
Velocity of Light in Moving Media 60
Group Velocity 61
CHAPTER VI. The Dynamics of a Particle 62
The Laws of Motion 62
Difference between Newtonian and Relativity Mechanics 62
The Mass of a Moving Particle 63
Transverse Collision 63
Mass the Same in all Directions 66
Longitudinal Collision : 67
Collision of any Type 68
Transformation Equations for Mass 72
The Force Acting on a Moving Particle 73
Transformation Equations for Force 73
The Relation between Force and Acceleration 74
Transverse and Longitudinal Acceleration 76
The Force Exerted by a Moving Charge 77
The Field around a Moving Charge 79
Application to a Specific Problem 80
Work 81
Kinetic Energy 81
Potential Energy 82
The Relation between Mass and Energy 83
Application to a Specific Problem * 85
CHAPTER VII. The Dynamics of a System of Particles 88
On the Nature of a System of Particles 88
The Conservation of Momentum 89
The Equation of Angular Momentum 90
The Function T 92
The Modified Lagrangian Function 93
The Principle of Least Action 93
Lagrange's Equations 95
Equations of Motion in the Hamiltonian Form 96
Value of the Function T' 97
The Principle of the Conservation of Energy 99
On the Location of Energy in Space 100
Table of Contents. vii
CHAPTER VIII. The Chaotic Motion of a System of Particles 102
The Equations of Motion 102
Representation in Generalized Space 103
Liouville's Theorem .103
A System of Particles • 104
Probability of a Given Statistical State 105
Equilibrium Relations 106
The Energy as a Function of the Momentum 108
The Distribution Law 109
Polar Coordinates
The Law of Equipartition
Criterion for Equality of Temperature 112
Pressure Exerted by a System of Particles . .
The Relativity Expression for Temperature
The Partition of Energy
Partition of Energy for Zero Mass
Approximate Partition for Particles of any Mass 118
CHAPTER IX. The Principle of Relativity and the Principle of Least Action . 121
The Principle of Least Action 121
The Equations of Motion in the Lagrangian Form 122
Introduction of the Principle of Relativity 124
Relation between f W'dt' and f Wdt .124
Relation between H' and H -127
CHAPTER X. The Dynamics of Elastic Bodies
On the Impossibility of Absolutely Rigid Bodies
Part I. Stress and Strain
Definition of Strain
Definition of Stress
Transformation Equations for Strain
Variation in the Strain
Part II. Introduction of the Principle of Least Action . .
The Kinetic Potential for an Elastic Body . .
Lagrange's Equations
Transformation Equations for Stress . .
Value of E°
The Equations of Motion in the Lagrangian Form
Density of Momentum
Density of Energy
Summary of Results from the Principle of Least Action
Part III. Some Mathematical Relations . .
The Unsymmetrical Stress Tensor t
The Symmetrical Tensor p
Relation between div t and t» . . .
The Equations of Motion in the Eulerian Form . . . Part IV. Applications of the Results
Relation between Energy and Momentum
The Conservation of Momentum
viii Table of Contents.
The Conservation of Angular Momentum 150
Relation between Angular Momentum and the Unsymmetrical
Stress Tensor 151
The Right- Angled Lever 152
Isolated Systems in a Steady State 154
The Dynamics of a Particle 154
Conclusion 154
CHAPTER XI. The Dynamics of a Thermodynamic System '. 156
The Generalized Coordinates and Forces 156
Transformation Equation for Volume 156
Transformation Equation for Entropy 157
Introduction of the Principle of Least Action. The Kinetic
Potential 157
The Lagrangian Equations 158
Transformation Equation for Pressure 159
Transformation Equation for Temperature 159
The Equations of Motion for Quasistationary Adiabatic Accelera tion 160
The Energy of a Moving Thermodynamic System 161
The Momentum of a Moving Thermodynamic System 161
The Dynamics of a Hohlraum 162
CHAPTER XII. Electromagnetic Theory 164
The Form of the Kinetic Potential 164
The Principle of Least Action : 165
The Partial Integrations 165
Derivation of the Fundamental Equations of Electromagnetic
Theory 166
The Transformation Equations for e, h and p 168
The Invariance of Electric Charge 170
The Relativity of Magnetic and Electric Fields 171
Nature of Electromotive Force 172
Derivation of the Fifth Fundamental Equation 172
Difference between the Ether and the Relativity Theories of Electro magnetics 173
Applications to Electromagnetic Theory 176
The Electric and Magnetic Fields around a Moving Charge 176
The Energy of a Moving Electromagnetic System 178
Relation between Mass and Energy 180
The Theory of Moving Dielectrics 181
Relation between Field Equations for Material Media and
Electron Theory 182
Transformation Equations for Moving Media 183
Theory of the Wilson Experiment 186
CHAPTER XIII. Four-Dimensional Analysis 188
Idea of a Time Axis 188
Non-Euclidean Character of the Space - .... 189
Table of Contents. ix
Part I. Vector Analysis of the Non-Euclidean Four-Dimensional Mani fold 191
Space, Time and Singular Vectors 192
Invariance of z2 + y2 + z2 — c-t2 192
Inner Product of One- Vectors 193
Non-Euclidean Angle 194
Kinematical Interpretation of Angle in Terms of Velocity 194
Vectors of Higher Dimensions 195
Outer Products 195
Inner Product of Vectors in General 198
The Complement of a Vector 198
The Vector Operator, O or Quad 199
Tensors 200
The Rotation of Axes 201
Interpretation of the Lorentz Transformation as a Rotation of
Axes 206
Graphical Representation 208
Part II. Applications of the Four-Dimensional Analysis 211
Kinematics 211
Extended Position 211
Extended Velocity 212
Extended Acceleration 213
The Velocity of Light .214
The Dynamics of a Particle 214
Extended Momentum 214
The Conservation Laws 215
The Dynamics of an Elastic Body 216
The Tensor of Extended Stress 216
The Equation of Motion 216
Electromagnetics 217
Extended Current 218
The Electromagnetic Vector M 217
The Field Equations 217
The Conservation of Electricity 218
The Product M-q
The Extended Tensor of Electromagnetic Stress 219
Combined Electrical and Mechanical Systems 221
Appendix I. Symbols for Quantities
Scalar Quantities
Vector Quantities 223
Appendix II. Vector Notation 224
Three Dimensional Space 224
Non-Euclidean Four Dimensional Space 225
PREFACE.
Thirty or forty years ago, in the field of physical science, there was a widespread feeling that the days of adventurous discovery had passed forever, and the conservative physicist was only too happy to devote his life to the measurement to the sixth decimal place of quantities whose significance for physical theory was already an -old story. The passage of time, however, has completely upset such bourgeois ideas as to the state of physical science, through the dis covery of some most extraordinary experimental facts and the develop ment of very fundamental theories for their explanation.
On the experimental side, the intervening years have seen the discovery of radioactivity, the exhaustive study of the conduction of electricity through gases, the accompanying discoveries of cathode, canal and X-rays, the isolation of the electron, the study of the distribution of energy in the hohlraum. and the final failure of all attempts to detect the earth's motion through the supposititious ether. During this same time, the theoretical physicist has been working hand in hand with the experimenter endeavoring to correlate the facts already discovered and to point the way to further research. The theoretical achievements, which have been found particularly helpful in performing these functions of explanation and prediction, have been the development of the modern theory of electrons, the application of thermodynamic and statistical reasoning to the phe nomena of radiation, and the development of Einstein's brilliant theory of the relativity of motion.
It has been the endeavor of the following book to present an introduction to this theory of relativity, which in the decade since the publication of Einstein's first paper in 1905 (Annalen der Physik) has become a necessary part of the theoretical equipment of every physicist. Even if we regard the Einstein theory of relativity merely as a convenient tool for the prediction of electromagnetic and optical phenomena, its importance to the physicist is very great, not only because its introduction greatly simplifies the deduction of many
2 1
2 Preface.
theorems which were already familiar in the older theories based on a stationary ether, but also because it leads simply and directly to cor rect conclusions in the case of such experiments as those of Michelson and Morley, Trouton and Noble, and Kaufman and Bucherer, which can be made to agree with the idea of a stationary ether only by the introduction of complicated and ad hoc assumptions. Regarded from a more philosophical point of view, an acceptance of the Einstein theory of relativity shows us the advisability of completely remodelling some of our most fundamental ideas. In particular we shall now do well to change our concepts of space and time in such a way as to give up the old idea of their complete independence, a notion which we have received as the inheritance of a long ancestral experience with bodies moving with slow velocities, but which no longer proves pragmatic when we deal with velocities approaching that of light.
The method of treatment adopted in the following chapters is to a considerable extent original, partly appearing here for the first time and partly already published elsewhere.* Chapter III follows a method which was first developed by Lewis and Tolman,|and the last chapter a method developed by Wilson and Lewis.J The writer must also express his special obligations to the works of Einstein, Planck, Poincare, Laue, Ishiwara and Laub.
It is hoped that the mode of presentation is one that will be found well adapted not only to introduce the study of relativity theory to those previously unfamiliar with the subject but also to provide the necessary methodological equipment for those who wish to pursue the theory into its more complicated applications.
After presenting, in the first chapter, a brief outline of the historical development of ideas as to the nature of the space and time of science, we consider, in Chapter II, the two main postulates upon which the theory of relativity rests and discuss the direct experimental evidence for their truth. The third chapter then presents an elementary and
* Philosophical Magazine, vol. 18, p. 510 (1909); Physical Review, vol. 31, p. 26 (1910); Phil. Mag., vol. 21, p. 296 (1911); ibid., vol. 22, p. 458 (1911); ibid., vol. 23, p. 375 (1912); Phys. Rev., vol. 35, p. 136 (1912); Phil. Mag., vol. 25, p. 150 (1913); ibid., vol. 28, p. 572 (1914); ibid., vol. 28, p. 583 (1914).
t Phil. Mag., vol. 18, p. 510 (1909).
J Proceedings of the American Academy of Arts and Sciences, vol. 48, p. 389 (1912).
Preface. 3
non-mathematical deduction of a number of the most important consequences of the postulates of relativity, and it is hoped that this chapter will prove especially valuable to readers without unusual mathematical equipment, since they will there be able to obtain a real grasp of such important new ideas as the change of mass with velocity, the non-additivity of velocities, and the relation of mass and energy, without encountering any mathematics beyond the elements of analysis and geometry.
In Chapter IV we commence the more analytical treatment of the theory of relativity by obtaining from the two postulates of relativity Einstein's transformation equations for space and time as well as transformation equations for velocities, accelerations, and for an important function of the velocity. Chapter V presents various kinematical applications of the theory of relativity following quite closely Einstein's original method of development. In par ticular we may call attention to the ease with which we may handle the optics of moving media by the methods of the theory of relativity as compared with the difficulty of treatment on the basis of the ether theory.
In Chapters VI, VII and VIII we develop and apply a theory of the dynamics of a particle which is based on the Einstein trans formation equations for space and time, Newton's three laws of motion, and the principle of the conservation of mass.
We then examine, in Chapter IX, the relation between the theory of relativity and the principle of least action, and find it possible to introduce the requirements of relativity theory at the very start into this basic principle for physical science. We point out that we might indeed have used this adapted form of the principle of least action, for developing the dynamics of a particle, and then proceed in Chapters X, XI and XII to develop the dynamics of an elastic body, the dynamics of a thermodynamic system, and the dynamics of an electromagnetic system, all on the basis of our adapted form of the principle of least action.
Finally, in Chapter XIII, we consider a four-dimensional method of expressing and treating the results of relativity theory. This chapter contains, in Part I, an epitome of some of the more important methods in four-dimensional vector analysis and it is hoped that it
4 Preface.
can also be used in connection with the earlier parts of the book as a convenient reference for those who are not familiar with ordinary three-dimensional vector analysis.
In the present book, the writer has confined his considerations to cases in which there is a uniform relative velocity between systems of coordinates. In the future it may be possible greatly to extend the applications of the theory of relativity by considering accelerated systems of coordinates, and in this connection Einstein's latest work on the relation between gravity and acceleration is of great interest. It does not seem wise, however, at the present time to include such considerations in a book which intends to present a survey of accepted theory.
The author will feel amply repaid for the work involved in the preparation of the book if, through his efforts, some of the younger American physicists can be helped to obtain a real knowledge of the important work of Einstein. He is also glad to have this opportunity to add his testimony to the growing conviction that the conceptual space and time of science are not God-given and unalterable, but are rather in the nature of human constructs devised for use in the de scription and correlation of scientific phenomena, and that these spatial and temporal concepts should be altered whenever the discovery of new facts makes such a change pragmatic.
The writer wishes to express his indebtedness to Mr. William H. Williams for assisting in the preparation of Chapter I.
CHAPTER I.
HISTORICAL DEVELOPMENT OF IDEAS AS TO THE NATURE OF
SPACE AND TIME.
1. Since the year 1905, which marked the publication of Einstein's momentous article on the theory of relativity, the development of scientific thought has led to a complete revolution in accepted ideas as to the nature of space and time, and this revolution has in turn profoundly modified those dependent sciences, in particular mechanics and electromagnetics, which make use of these two fundamental concepts in their considerations.
In the following pages it will be our endeavor to present a de scription of these new notions as to the nature of space and time,* and to give a partial account of the consequent modifications which have been introduced into various fields of science. Before pro ceeding to this task, however, we may well review those older ideas as to space and time which until now appeared quite sufficient for the correlation of scientific phenomena. We shall first consider the space and time of Galileo and Newton which were employed in the development of the classical mechanics, and then the space and time of the ether theory of light.
PART I. THE SPACE AND TIME OF GALILEO AND NEWTON.
2. The publication in 1687 of Newton's Principia laid down so satisfactory a foundation for further dynamical considerations, that it seemed as though the ideas of Galileo and Newton as to the nature of space and time, which were there employed, would certainly remain forever suitable for the interpretation of natural phenomena. And indeed upon this basis has been built the whole structure of classical mechanics which, until our recent familiarity with very high velocities, has been found completely satisfactory for an extremely large number of very diverse dynamical considerations.
* Throughout this work by "space" and "time" we shall mean the conceptual space and time of science.
5
6 Chapter One.
An examination of the fundamental laws of mechanics will show how the concepts of space and time entered into the Newtonian system of mechanics. Newton's laws of motion, from which the whole of the classical mechanics could be derived, can best be stated with the help of the equation
P-|(«u). (1)
This equation defines the force F acting on a particle as equal to the rate of change in its momentum (i. e., the product of its mass m and its velocity u), and the whole of Newton's laws of motion may be summed up in the statement that in the case of two interacting par ticles the forces which they mutually exert on each other are equal in magnitude and opposite in direction.
Since in Newtonian mechanics the mass of a particle is assumed constant, equation (1) may be more conveniently written
du d (dr\
F = - = - -
or
m ,
dt dt\dt J
d dx
'-4(1).
and this definition of force, together with the above-stated principle of the equality of action and reaction, forms the starting-point for the whole of classical mechanics.
The necessary dependence of this mechanics upon the concepts of space and time becomes quite evident on an examination of this fundamental equation (2), in which the expression for the force acting on a particle is seen to contain both the variables x, y, and z, which specify the position of the particle in space, and the variable tt which specifies the time.
3. Newtonian Time. To attempt a definite statement as to the
Historical Development. 7
meaning of so fundamental and underlying a notion as that of time is a task from which even philosophy may shrink. In a general way, conceptual time may be thought of as a one-dimensional, uni directional, one-valued continuum. This continuum is a sort of frame work in which the instants at which actual occurrences take place find an ordered position. Distances from point to point in the continuum, that is intervals of time, are measured by the periods of certain continually recurring cyclic processes such as the daily rota tion of the earth. A unidirectional nature is imposed upon the time continuum among other things by an acceptance of the second law of thermodynamics, which requires that actual progression in time shall be accompanied by an increase in the entropy of the material world, and this same law requires that the continuum shall be one- valued since it excludes the possibility that time ever returns upon itself, either to commence a new cycle or to intersect its former path even at a single point.
In addition to these characteristics of the time continuum, which have been in no way modified by the theory of relativity, the New tonian mechanics always assumed a complete independence of time and the three-dimensional space continuum which exists along with it. In dynamical equations time entered as an entirely independent vari able in no way connected with the variables whose specification determines position in space. In the following pages, however, we shall find that the theory of relativity requires a very definite inter relation between time and space, and in the Einstein transformation equations we shall see the exact way in which measurements of time depend upon the choice of a set of variables for measuring position in space.
4. Newtonian Space. An exact description of the concept of space is perhaps just as difficult as a description of the concept of time. In a general way we think of space as a three-dimensional, homo geneous, isotropic continuum, and these ideas are common to the conceptual spaces of Newton, Einstein, and the ether theory of light. The space of Newton, however, differs on the one hand from that of Einstein because of a tacit assumption of the complete independence of space and time measurements; and differs on the other hand from that of the ether theory of light by the fact that " free " space was
8 Chapter One.
assumed completely empty instead of filled with an all-pervading quasi-material medium — the ether. A more definite idea of the par ticularly important characteristics of the Newtonian concept of space may be obtained by considering somewhat in detail the actual methods of space measurement.
Positions in space are in general measured with respect to some arbitrarily fixed system of reference which must be threefold in character corresponding to the three dimensions of space. In par ticular we may make use of a set of Cartesian axes and determine, for example, the position of a particle by specifying its three Cartesian coordinates x, y and z.
In Newtonian mechanics the particular set of axes chosen for specifying position in space has in general been determined in the first instance by considerations of convenience. For example, it is found by experience that, if we take as a reference system lines drawn upon the surface of the earth, the equations of motion based on New ton's laws give us a simple description of nearly all dynamical phe nomena which are merely terrestrial. When, however, we try to interpret with these same axes the motion of the heavenly bodies, we meet difficulties, and the problem is simplified, so far as planetary motions are concerned, by taking a new reference system determined by the sun and the fixed stars. But this system, in its turn, becomes somewhat unsatisfactory when we take account of the observed motions of the stars themselves, and it is finally convenient to take a reference system relative to which the sun is moving with a velocity of twelve miles per second in the direction of the constellation Hercules. This system of axes is so chosen that the great majority of stars have on the average no motion with respect to it, and the actual motion of any particular star with respect to these coordinates is called the peculiar motion of the star.
Suppose, now, we have a number of such systems of axes in uni form relative motion; we are confronted by the problem of finding some method of transposing the description of a given kinematical occurrence from the variables of one of these sets of axes to those of another. For example, if we have chosen a system of axes S and have found an equation in x, y, z, and t which accurately describes the motion of a given point, what substitutions for the quantities involved
Historical Development. 9
can be made so that the new equation thereby obtained will again correctly describe the same phenomena when we measure the dis placements of the point relative to a new system of reference S' which is in uniform motion with respect to £? The assumption of Galileo and Newton that " free " space is entirely empty, and the further tacit assumption of the complete independence of space and time, led them to propose a very simple solution of the problem, and the transformation equations which they used are generally called the Galileo Transformation Equations to distinguish them from the Einstein Transformation Equations which we shall later consider.
5. The Galileo Transformation Equations. Consider two systems of right-angled coordinates, S and S', which are in relative motion in the X direction with the velocity V; for convenience let the X axes, OX and O'X', of the two systems coincide in direction, and for further simplification let us take as our zero point for time measurements the instant when the two origins 0 and 0' coincide. Consider now a point which at the time t has the coordinates x, y and z measured in system S. Then, according to the space and time considerations of Galileo and Newton, the coordinates of the point with reference to system S' are given by the following transformation equations :
x' = x - Vt, (3)
y' = y, (4)
(5) (6)
These equations are fundamental for Newtonian mechanics, and may appear to the casual observer to be self-evident and bound up with necessary ideas as to the nature of space and time. Nevertheless, the truth of the first and the last of these equations is absolutely dependent on the unsupported assumption of the complete inde pendence of space and time measurements, and since in the Einstein theory we shall find a very definite relation between space and time measurements we shall be led to quite a different set of transformation equations. Relations (3), (4), (5) and (6) will be found, however, to be the limiting form which the correct transformation equations as sume when the velocity between the systems V becomes small com-
10 Chapter One.
pared with that of light. Since until very recent times the human race in its entire past history has been familiar only with velocities that are small compared with that of light, it need not cause surprise that the above equations, which are true merely at the limit, should appear so self-evident.
6. Before leaving the discussion of the space and time system of Newton and Galileo we must call attention to an important charac teristic which it has in common with the system of Einstein but which is not a feature of that assumed by the ether theory. If we have two systems of axes such as those we have just been considering, we may with equal right consider either one of them at rest and the other moving past it. All we can say is that the two systems are in relative 'motion; it is meaningless to speak of either one as in any sense " absolutely " at rest. The equation x' = x — Vt which we use in transforming the description of a kinematical event from the variables of system S to those of system S' is perfectly symmetrical with the equation x = x' + Vt' which we should use for a trans formation in the reverse direction. Of all possible systems no par ticular set of axes holds a unique position among the others. We shall later find that this important principle of the relativity of motion is permanently incorporated into our system of physical science as the first postulate of relativity. This principle, common both to the space of Newton and to that of Einstein, is not characteristic of the space assumed by the classical theory of light. The space of this theory was supposed to be filled with a stationary medium, the luminiferous ether, and a system of axes stationary with respect to this ether would hold a unique position among the other systems and be the one peculiarly adapted for use as the ultimate system of reference for the measurement of motions.
We may now briefly sketch the rise of the ether theory of light and point out the permanent contribution which it has made to physical science, a contribution which is now codified as the second postulate of relativity.
PART II. THE SPACE AND TIME OF THE ETHER THEORY.
7. Rise of the Ether Theory. Twelve years before the appearance of the Principia, Homer, a Danish astronomer, observed that an
Historical Development. 11
eclipse of one of the satellites of Jupiter occurred some ten minutes later than the time predicted for the event from the known period of the satellite and the time of the preceding eclipse. He explained this delay by the hypothesis that it took light twenty-two minutes to travel across the earth's orbit. Previous to Romer's discovery, light was generally supposed to travel with infinite velocity. Indeed Galileo had endeavored to find the speed of light by direct experiments over distances of a few miles and had failed to detect any lapse of time between the emission of a light flash from a source and its ob servation by a distant observer. Romer's hypothesis has been re peatedly verified and the speed of light measured by different methods with considerable exactness. The mean of the later determinations is 2.9986 X 1010 cm. per second.
8. At the time of Romer's discovery there was much discussion as to the nature of light. Newton's theory that it consisted of par ticles or corpuscles thrown out by a luminous body was attacked by Hooke and later by Huygens, who advanced the view that it was something in the nature of wave motions in a supposed space-filling medium or ether. By this theory Huygens was able to explain reflection and refraction and the phenomena of color, but assuming longitudinal vibrations he was unable to account for polarization. Diffraction had not yet been observed and Newton contested the Hooke-Huygens theory chiefly on the grounds that it was contra dicted by the fact of rectilinear propagation and the formation of shadows. The scientific prestige of Newton was so great that the emission or corpuscular theory continued to hold its ground for a hundred and fifty years. Even the masterly researches of Thomas Young at the beginning of the nineteenth century were unable to dislodge the old theory, and it was not until the French physicist, Fresnel, about 1815, was independently led to an undulatory theory and added to Young's arguments the weight of his more searching mathematical analysis, that the balance began to turn. From this time on the wave theory grew in power and for a period of eighty years was not seriously questioned. This theory has for its essential postulate the existence of an all-pervading medium, the ether, in which wave disturbances can be set up and propagated. And the physical properties of this medium became an enticing field of inquiry and speculation.
12 Chapter One.
9. Idea of a Stationary Ether. Of all the various properties with which the physicist found it necessary to endow the ether, for us the most important is the fact that it must apparently remain stationary, unaffected by the motion of matter through it. This conclusion was finally reached through several lines of investigation. We may first consider whether the ether would be dragged along by the motion of nearby masses of matter, and, second, whether the ether enclosed in a moving medium such as water or glass would partake in the latter's motion.
10. Ether in the Neighborhood of Moving Bodies. About the year 1725 the astronomer Bradley, in his efforts to measure the parallax of certain fixed stars, discovered that the apparent position of a star continually changes in such a way as to trace annually a small ellipse in the sky, the apparent position always lying in the plane determined by the line from the earth to the center of the ellipse and by the direction of the earth's motion. On the corpuscular theory of light this admits of ready explanation as Bradley himself discovered, since we should expect the earth's motion to produce an apparent change in the direction of the oncoming light, in just the same way that the motion of a railway train makes the falling drops of rain take a slanting path across the window pane. If c be the velocity of a light particle and v the earth's velocity, the apparent or relative velocity would be c — v and the tangent of the angle of
aberration would be - . c
Upon the wave theory, it is obvious that we should also expect a similar aberration of light, provided only that the ether shall be quite stationary and unaffected by the motion of the earth through it, and this is one of the important reasons that most ether theories have assumed a stationary ether unaffected by the motion of neighboring matter. *
In more recent years further experimental evidence for assuming that the ether is not dragged along by the neighboring motion of large masses of matter was found by Sir Oliver Lodge. His final experiments were performed with a large rotating spheroid of iron
* The most notable exception is the theory of Stokes, which did assume that the ether moved along with the earth and then tried to account for aberration with the help of a velocity potential, but this led to difficulties, as was shown by Lorentz.
Historical Development. 13
with a narrow groove around its equator, which was made the path for two rays of light, one travelling in the direction of rotation and the other in the opposite direction. Since by interference methods no difference could be detected in the velocities of the two rays, here also the conclusion was reached that the ether was not appreciably dragged along by the rotating metal.
11. Ether Entrained in Dielectrics. With regard to the action of a moving medium on the ether which might be entrained within it, experimental evidence and theoretical consideration here too finally led to the supposition that the ether itself must remain perfectly stationary. The earlier view first expressed by Fresnel, in a letter written to Arago in 1818, was that the entrained ether did receive a fraction of the total velocity of the moving medium. Fresnel gave
2 "1
to this fraction the value — , where AC is the index of refraction of
M2
the substance forming the medium. On this supposition, Fresnel was able to account for the fact that Arago's experiments upon the reflection and refraction of stellar rays show no influence whatever of the earth's motion, and for the fact that Airy found the same angle of aberration with a telescope filled with water as with air. More over, the later work of Fizeau and the accurate determinations of Michelson and Morley on the velocity of light in a moving stream of water did show that the speed was changed by an amount corre sponding to Fresnel's fraction. The fuller theoretical investigations of Lorentz, however, did not lead scientists to look upon this increased velocity of light in a moving medium as an evidence that the ether is pulled along by the stream of water, and we may now briefly sketch the developments which culminated in the Lorentz theory of a com pletely stationary ether.
12. The Lorentz Theory of a Stationary Ether. The considera tions of Lorentz as to the velocity of light in moving media became possible only after it was evident that optics itself is a branch of the wider science of electromagnetics, and it became possible to treat transparent media as a special case of dielectrics in general. In 1873, in his Treatise on Electricity and Magnetism, Maxwell first advanced the theory that electromagnetic phenomena also have their seat in the luminiferous ether and further that light itself is merely an electro-
14 Chapter One.
magnetic disturbance in that medium, and Maxwell's theory was confirmed by the actual discovery of electromagnetic waves in 1888 by Hertz.
The attack upon the problem of the relative motion of matter and ether was now renewed with great vigor both theoretically and experi mentally from the electromagnetic side. Maxwell in his treatise had confined himself to phenomena in stationar}^ media. Hertz, however, extended Maxwell's considerations to moving matter on the assump tion that the entrained ether is carried bodily along by it. It is evi dent, however, that in the field of optical theory such an assumption could not be expected to account for the Fizeau experiment, which had already been explained on the assumption that the ether receives only a fraction of the velocity of the moving medium; while in the field of electromagnetic theory it was found that Hertz's assumptions would lead us to expect no production of a magnetic field in the neighborhood of a rotating electric condenser providing the plates of the condenser and the dielectric move together with the same speed and this was decisively disproved by the experiment of Eichenwald. The conclusions of the Hertz theory were also out of agreement with the important experiments of H. A. Wilson on moving dielectrics. It remained for Lorentz to develop a general theory for moving dielectrics which was consistent with the facts.
The theory of Lorentz developed from that of Maxwell by the addition of the idea of the electron, as the atom of electricity, and his treatment is often called the " electron theory." This atomistic conception of electricity was foreshadowed by Faraday's discovery of the quantitative relations between the amount of electricity asso ciated with chemical reactions in electrolytes and the weight of substance involved, a relation which indicates that the atoms act as carriers of electricity and that the quantity of electricity carried by a single particle, whatever its nature, is always some small multiple of a definite quantum of electricity, the electron. Since Faraday's time, the study of the phenomena accompanying the conduction of elec tricity through gases, the study of radioactivity, and finally indeed the isolation and exact measurement of these atoms of electrical charge, have led us to a very definite knowledge of many of the properties of the electron.
Historical Development. 15
While the experimental physicists were at work obtaining this more or less first-hand acquaintance with the electron, the theoretical physicists and in particular Lorentz were increasingly successful in explaining the electrical and optical properties of matter in general on the basis of the behavior of the electrons which it contains, the properties of conductors being accounted for by the presence of mov able electrons, either free as in the case of metals or combined with atoms to form ions as in electrolytes, while the electrical and optical properties of dielectrics were ascribed to the presence of electrons more or less bound by quasi-elastic forces to positions of equilibrium. This Lorentz electron theory of matter has been developed in great mathematical detail by Lorentz and has been substantiated by nu merous quantitative experiments. Perhaps the greatest significance of the Lorentz theory is that such properties of matter as electrical conductivity, magnetic permeability and dielectric inductivity, which occupied the position of rather accidental experimental constants in Maxwell's original theory, are now explainable as the statistical result of the behavior of the individual electrons.
With regard now to our original question as to the behavior of moving optical and dielectric media, the Lorentz theory was found capable of accounting quantitatively for all known phenomena, in cluding Airy's experiment on aberration, Arago's experiments on the reflection and refraction of stellar rays, FresneFs coefficient for the velocity of light in moving media, and the electromagnetic experi ments upon moving dielectrics made by Rontgen, Eichenwald, H. A. Wilson, and others. For us the particular significance of the Lorentz method of explaining these phenomena is that he does not assume, as did Fresnel, that the ether is partially dragged along by moving matter. His investigations show rather that the ether must remain perfectly stationary, and that such phenomena as the changed velocity of light in moving media are to be accounted for by the modifying influence which the electrons in the moving matter have upon the propagation of electromagnetic disturbances, rather than by a dragging along of the ether itself.
Although it would not be proper in this place to present the mathematical details of Lorentz 's treatment of moving media, we may obtain a clearer idea of what is meant in the Lorentz theory by a
16 Chapter One.
stationary ether if we look for a moment at the five fundamental equations upon which the theory rests. These familiar equations, of which the first four are merely Maxwell's four field equations, modified by the introduction of the idea of the electron, may be written
1 de u
i ah curle= -caT'
div e = p, div h = 0,
in which the letters have their usual significance. (See Chapter XII.) Now the whole of the Lorentz theory, including of course his treat ment of moving media, is derivable from these five equations, and the fact that the idea of a stationary ether does lie at the basis of his theory is most clearly shown by the first and last of these equa tions, which contain the velocity u with which the charge in question is moving, and for Lorentz this velocity is to be measured with respect to the assumed stationary ether.
We have devoted this space to the Lorentz theory, since his work marks the culmination of the ether theory of light and electromag- netism, and for us the particularly significant fact is that by this line of attack science was inevitably led to the idea of an absolutely immovable and stationary ether.
13. We have thus briefly traced the development of the ether theory of light and electromagnetism. We have seen that the space continuum assumed by this theory is not empty as was the space of Newton and Galileo but is assumed filled with a stationary medium, the ether, and in conclusion should further point out that the time continuum assumed by the ether theory was apparently the same as that of Newton and Galileo, and in particular that the old ideas as to the absolute independence of space and time were all retained.
Historical Development. 17
PART III. RISE OF THE EINSTEIN THEORY OF RELATIVITY.
14. The Michelson-Morley Experiment. In spite of all the brilli ant achievements of the theory of a stationary ether, we must now call attention to an experiment, performed at the very time when the success of the ether theory seemed most complete, whose result was in direct contradiction to its predictions. This is the celebrated Michelson-Morley experiment, and to the masterful interpretation of its consequences at the hands of Einstein we owe the whole theory of relativity, a theory -which will nevermore permit us to assume that space and time are independent.
If the theory of a stationary ether were true we should find, con trary to the expectations of Newton, that systems of coordinates in relative motion are not symmetrical, a system of axes fixed relatively to the ether would hold a unique position among all other systems moving relative to it and would be peculiarly adapted for the measure ment of displacements and velocities. Bodies at rest with respect to this system of axes fixed in the ether would be spoken of as " ab solutely " at rest and bodies in motion through the ether would be said to have " absolute " motion. From the point of view of the ether theory one of the most important physical problems would be to determine the velocity of various bodies, for example that of the earth, through the ether.
Now the Michelson-Morley experiment was devised for the very purpose of determining the relative motion of the earth and the ether. The experiment consists essentially in a comparison of the velocities of light parallel and perpendicular to the earth's motion in its orbit. A ray of light from the source S falls on the half silvered mirror A, where it is divided into two rays, one of which travels to the mirror B and the other to the mirror C, where they are totally reflected. The rays are recombined and produce a set of interference fringes at 0. (See figure 1.)
We may now think of the apparatus as set so that one of the divided paths is parallel to the earth's motion and the other per pendicular to it. On the basis of the stationary ether theory, the velocity of the light with reference to the apparatus would evidently be different over the two paths, and hence on rotating the apparatus
3
18
Chapter One.
through an angle of ninety degrees we should expect a shift in the position of the fringes. Knowing the magnitude of the earth's velocity in its orbit and the dimensions of the apparatus, it is quite possible to calculate the magnitude of the expected shift, a quantity
o
FIG. 1.
entirely susceptible of experimental determination. Nevertheless the most careful experiments made at different times of day and at different seasons of the year entirely failed to show any such shift at all.
This result is in direct contradiction to the theory of a stationary ether and could be reconciled with that theory only by very arbitrary assumptions. Instead of making such assumptions, the Einstein theory of relativity finds it preferable to return in part to the older ideas of Newton and Galileo.
15. The Postulates of Einstein. In fact, in accordance with the results of this work of Michelson-Morley and other confirmatory experiments, the Einstein theory takes as its first postulate the idea familiar to Newton of the relativity of all motion. It states that there is nothing out in space in the nature of an ether or of a fixed set of coordinates with regard to which motion can be measured, that there is no such thing as absolute motion, and that all we can speak of is the relative motion of one body with respect to another.
Historical Development. 19
Although we thus see that the Einstein theory of relativity has returned in part to the ideas of Newton and Galileo as to the nature of space, it is not to be supposed that the ether theory of light and electromagnetism has made no lasting contribution to physical science. Quite on the contrary, not only must the ideas as to the periodic and polarizable nature of the light disturbance, which were first appre ciated and understood with the help of the ether theory, always remain incorporated in every optical theory, but in particular the Einstein theory of relativity takes as the basis for its second postulate a principle that has long been familiar to the ether theory, namely that the velocity of light is independent of the velocity of the source. We shall see in following chapters that it is the combination of this principle with the first postulate of relativity that leads to the whole theory of relativity and to our new ideas as to the nature and inter relation of space and titne.
CHAPTER II.
THE TWO POSTULATES OF THE EINSTEIN THEORY OF RELATIVITY.
16. There are two general methods of evaluating the theoretical development of any branch of science. One of these methods is to test by direct experiment the fundamental postulates upon which the theory rests. If these postulates are found to agree with the facts, we may feel justified in assuming that the whole theoretical structure is a valid one, providing false logic or unsuspected and incorrect assumptions have not later crept in to vitiate the conclusions. The other method of testing a theory is to develop its interlacing chain of propositions and theorems and examine the results both for their internal coherence and for their objective validity. If we find that the conclusions drawn from the theory are neither self-contradictory nor contradictory of each other, and furthermore that they agree with the facts of the external world, we may again feel that our theory has achieved a measure of success. In the present chapter we shall present the two main postulates of the theory of relativity, and indicate the direct experimental evidence in favor of their truth. In following chapters we shall develop the consequences of these postulates, show that the system of consequences stands the test of internal coherence, and wherever possible compare the predictions of the theory with experimental facts.
The First Postulate of Relativity.
17. The first postulate of relativity as originally stated by Newton was that it is impossible to measure or detect absolute translatory motion through space. No objections have ever been made to this statement of the postulate in its original form. In the development of the theory of relativity, the postulate has been modified to include the impossibility of detecting translatory motion through any medium or ether which might be assumed to pervade space.
In support of the principle is the general fact that no effects due to the motion of the earth or other body through the supposed ether
20
The Two Postulates. 21
have ever been observed. Of the many unsuccessful attempts to detect the earth's motion through the ether we may call attention to the experiments on the refraction of light made by Arago, Respighi, Hoek, Ketteler and Mascart, the interference experiments of Ketteler and Mascart, the work of Klinkerfuess and Haga on the position of the absorption bands of sodium, the experiment of Nordmeyer on the intensity of radiation, the experiments of Fizeau, Brace and Strasser on the rotation of the plane of polarized light by transmission through glass plates, the experiments of Mascart and of Rayleigh on the rotation of the plane of polarized light in naturally active substances, the electromagnetic experiments of Rontgen, Des Coudres, J. Koenigs- berger, Trouton, Trouton and Noble, and Trouton and Rankine, and finally the Michelson and Morley experiment, with the further work of Morley and Miller. For details as to the nature of these experi ments the reader may refer to the original articles or to an excellent discussion by Laub of the experimental basis of the theory of rela tivity. *
In none of the above investigations was it possible to detect any effect attributable to the earth's motion through the ether. Never theless a number of these experiments are in accord with the final form given to the ether theory by Lorentz, especially since his work satisfactorily accounts for the Fresnel coefficient for the changed velocity of light in moving media. Others of the experiments men tioned, however, could be made to accord with the Lorentz theory only by very arbitrary assumptions, in particular those of Michelson and Morley, Mascart and Rayleigh, and Trouton and Noble. For the purposes of our discussion we shall accept the principle of the relativity of motion as an experimental fact.
The Second Postulate of the Einstein Theory of Relativity.
18. The second postulate of relativity states that the velocity of light in free space appears the same to all observers regardless of the relative motion of the source of light and the observer. This postulate may be obtained by combining the first postulate of relativity with a principle which has long been familiar to the ether theory of light. This principle states that the velocity of light is unaffected by a motion of the emitting source, in other words, that the velocity with
* Jahrbuch der Radioaktivitdt, vol. 7, p. 405 (1910).
22 Chapter Two.
which light travels past any observer is not increased by a motion of the source of light towards the observer. The first postulate of relativity adds the idea that a motion of the source of light towards the observer is identical with a motion of the observer towards the source. The second postulate of relativity is seen to be merely a combination of these two principles, since it states that the velocity of light in free space appears the same to all observers regardless both of the motion of the source of light and of the observer.
19. It should be pointed out that the two principles whose com bination thus leads to the second postulate of Einstein have come from very different sources. The first postulate of relativity prac tically denies the existence of any stationary ether through which the earth, for instance, might be moving. On the other hand, the principle that the velocity of light is unaffected by a motion of the source was originally derived from the idea that light is transmitted by a stationary medium which does not partake in the motion of the source. This combination of two principles, which from a historical point of view seem somewhat contradictory in nature, has given to the second postulate of relativity a very extraordinary content. Indeed it should be particularly emphasized that the remarkable conclusions as to the nature of space and time forced upon science by the theory of relativity are the special product of the second postulate of relativity.
A simple example of the conclusions which can be drawn from this postulate will make its extraordinary nature evident.
a'
b B V
FIG. 2.
S is a source of light and A and B two moving systems. A is moving towards the source S, and B away from it. Observers on the systems mark off equal distances aaf and W along the path of the light and determine the time taken for light to pass from a to a' and b to b' respectively. Contrary to what seem the simple conclusions of common sense, the second postulate requires that the time taken
The Two Postulates. 23
for the light to pass from a to a' shall measure the same as the time for the light to go from 6 to &'. Hence if the second postulate of relativity is correct it is not surprising that science is forced in general to new ideas as to the nature of space and time, ideas which are in direct opposition to the requirements of so-called common sense.
Suggested Alternative to the Postulate of the Independence of the
Velocity of Light and the Velocity of the Source.
20. Because of the extraordinary conclusions derived by com bining the principle of the relativity of motion with the postulate that the velocity of light is independent of the velocity of its source, a number of attempts have been made to develop so-called emission theories of relativity based on the principle of the relativity of motion and the further postulate that the velocity of light and the velocity of its source are additive.
Before examining the available evidence for deciding between the rival principles as to the velocity of light, we may point out that this proposed postulate, of the additivity of the velocity of source and light, would as a matter of fact lead to a very simple kind of relativity theory without requiring any changes in our notions of space and time. For if light or other electromagnetic disturbance which is being emitted from a source did partake in the motion of that source in such a way that the velocity of the source is added to the velocity of emission, it is evident that a system consisting of the source and its surrounding disturbances would act as a whole and suffer no permanent change in configuration if the velocity of the source were changed. This result would of course be in direct agree ment with the idea of the relativity of motion which merely requires that the physical properties of a system shall be independent of its velocity through space.
As a particular example of the simplicity of emission theories we may show, for instance, how easily they would account for the nega tive result of the Michelson-Morley experiment. If 0, figure 3, is a source of light and A and B are mirrors placed a meter away from 0, the Michelson-Morley experiment shows that the time taken for light to travel to A and back is the same as for the light to travel to B and back, in spite of the fact that the whole apparatus is moving through space in the direction 0 — B, due to the earth's motion around the sun.
24 Chapter Two.
The basic assumption of emission theories, however, would require exactly this result, since it says that light travels out from 0 with a
constant velocity in all directions with respect to 0. and not with respect to some ether through which 0 is supposed to be moving.
Direction of Earth's Mot ion The problem now before us is to
decide between the two rival principles as to the velocity of light, and we shall \B find that the bulk of the evidence is all FlG 3 in favor of the principle which has led
to the Einstein theory of relativity with
its complete revolution in our ideas as to space and time, and against the principle which has led to the superficially simple emission theo ries of relativity.
21. Evidence Against Emission Theories of Light. All emission theories agree in assuming that light from a moving source has a velocity equal to the vector sum of the velocity of light from a sta tionary source and the velocity of the source itself at the instant of emission. And without first considering the special assumptions which distinguish one emission theory from another we may first present certain astronomical evidence which apparently stands in contradiction to this basic assumption of all forms of emission theory. This evidence was pointed out by Comstock* and later by de Sitter, f
Consider the rotation of a binary star as it would appear to an observer situated at a considerable distance from the star and in its plane of rotation. (See figure 4.) If an emission theory of light be true, the velocity of light from the star in position A will be c + u, where u is the velocity of the star in its orbit, while in the position B the velocity will be c — u. Hence the star will be observed to arrive
in position A, - - seconds after the event has actually occurred, and c "T~ u
in position J5, _ seconds after the event has occurred. This will
* Phys. Rev., vol. 30, p. 291 (1910).
t Phys. Zeitschr., vol. 14, pp. 429, 1267 (1913).
The Two Postulates. 25
make the period of half rotation from A to B appear to be
where A2 is the actual time of a half rotation in the orbit, which for
. / Observer
Observer
FIG. 4.
simplicity may be taken as circular. On the other hand, the period of the next half rotation from B back to A would appear to be
2ul Now in the case of most spectroscopic binaries the quantity —^
C"
is not only of the same order of magnitude as At but oftentimes prob ably even larger. Hence, if an emission theory of light were true, we could hardly expect without correcting for the variable velocity of light to find that these orbits obey Kepler's laws, as is actually the case. This is certainly very strong evidence against any form of emission theory. It may not be out of place, however, to state briefly the different forms of emission theory which have been tried. 22. Different Forms of Emission Theory. As we have seen, emis sion theories all agree in assuming that light from a moving source
26 Chapter Two.
has a velocity equal to the vector sum of the velocity of light from a stationary source and the velocity of the source itself at the instant of emission. Emission theories differ, however, in their assumptions as to the velocity of light after its reflection from a mirror. The three assumptions which up to this time have been particularly considered are (1) that the excited portion of the reflecting mirror acts as a new source of light and that the reflected light has the same velocity c with respect to the mirror as has original light with respect to its source ; (2) that light reflected from a mirror acquires a component of velocity equal to the velocity of the mirror image of the original source, and hence has the velocity c with respect to this mirror image; and (3) that light retains throughout its whole path the component of velocity which it obtained from its original moving source, and hence after reflection spreads out with velocity c in a spherical form around a center which moves with the same speed as the original source.
Of these possible assumptions as to the velocity of reflected light, the first seems to be the most natural and was early considered by the author but shown to be incompatible, not only with an experiment which he performed on the velocity of light from the two limbs of the sun,* but also with measurements of the Stark effect in canal rays.f The second assumption as to the velocity of light was made by Stewart,t but has also been shown f to be incompatible with measurements of the Stark effect in canal rays. Making use of the third assumption as to the velocity of reflected light, a somewhat complete emission theory has been developed by Ritz,§ and un fortunately optical experiments for deciding between the Einstein and Ritz relativity theories have never been performed, although such experiments are entirely possible of performance.! Against the Ritz theory, however, we have of course the general astronomical evidence of Comstock and de Sitter which we have already described above.
For the present, the observations described above, comprise the whole of the direct experimental evidence against emission theories
* Phys. Rev., vol. 31, p. 26 (1910). f Phys. Rev., vol. 35, p. 136 (1912). J Phys. Rev., vol. 32, p. 418 (1911).
§ Ann. de chim. et phys., vol. 13, p. 145 (1908); Arch, de Geneve, vol. 26, p. 232 (1908); Sdentia, vol. 5 (1909).
The Two Postulates. 27
of light and in favor of the principle which has led to the second postulate of the Einstein theory. One of the consequences of the Einstein theory, however, has been the deduction of an expression for the mass of a moving body which has been closely verified by the Kaufmann-Bucherer experiment. Now it is very interesting to note, that starting with what has thus become an experimental expression for the mass of a moving body, it is possible to work backwards to a derivation of the second postulate of relativity. For the details of the proof we must refer the reader to the original article.*
Further Postulates of the Theory of Relativity.
23. In the development of the theory of relativity to which we shall now proceed we shall of course make use of many postulates. The two which we have just considered, however, are the only ones, so far as we are aware, which are essentially different from those common to the usual theoretical developments of physical science. In particular in our further work we shall assume without examination all such general principles as the homogeneity and isotropism of the space continuum, and the unidirectional, one-valued, one-dimensional character of the time continuum. In our treatment of the dynamics of a particle we shall also assume Newton's laws of motion, and the principle of the conservation of mass, although we shall find, of course, that the Einstein ideas as to the connection between space and time will lead us to a non-Newtonian mechanics. We shall also make extensive use of the principle of least action, which we shall find a powerful principle in all the fields of dynamics.
* Phys. Rev., vol. 31, p. 26 (1910).
CHAPTER III. SOME ELEMENTARY DEDUCTIONS.
24. In order gradually to familiarize the reader with the conse quences of the theory of relativity we shall now develop by very elementary methods a few of the more important relations. In this preliminary consideration we shall lay no stress on mathematical elegance or logical exactness. It is believed, however, that the chapter will present a substantially correct account of some of the more important conclusions of the theory of relativity, in a form which can be understood even by readers without mathematical equipment.
Measurements of Time in a Moving System.
25. We may first derive from the postulates of relativity a relation connecting measurements of time intervals as made by observers in systems moving with different velocities. Consider a system S (Fig. 5) provided with a plane mirror a a, and an observer A, who
FIG. 5.
has a clock so that he can determine the time taken for a beam of light to travel up to the mirror and back along the path Am A. Consider also another similar system S', provided with a mirror b 6, and an observer B, who also has a clock for measuring the time it takes for light to go up to his mirror and back. System S' is moving past S with the velocity V, the direction of motion being parallel to the mirrors a a and b b, the two systems being arranged, more-
28
Some Elementary Deductions.
29
over, so that when they pass one another the two mirrors a a and b b will coincide, and the two observers A and B will also come into coincidence.
A, considering his system at rest and the other in motion, measures the time taken for a beam of light to pass to his mirror and return, over the path A m A, and compares the time interval thus obtained with that necessary for the performance of a similar experiment by B, in which the light has to pass over a longer path such as B n B', shown in figure 6, where B B' is the distance through which the
— a
|
v — — i 1 |
[ \ |
|
1 |
\ |
|
1 |
\ |
|
1 |
\ |
|
1 |
\ \ \ |
|
1 |
\ |
|
\ |
|
|
\ |
|
|
\ |
|
|
\ |
|
|
T> _/ |
\ |
|
B - - - |
0 |
|
FIG. 6. |
A
observer B has moved during the time taken for the passage of the light up to the mirror and back.
Since, in accordance with the second postulate of relativity, the velocity of light is independent of the velocity of its source, it is evident that the ratio of these two time intervals will be proportional to the ratio of the two paths A m A and B n B', and this can easily be calculated in terms of the velocity of light c and the velocity V of the system S'.
From figure 6 we have
(A m)2 = (p n)2 = (B n)2 - (B p)2. Dividing by (B n)2,
(Am)* _ (Pjtf (BnY '' " IB n)2 '
But the distance B p is to B n as 7 is to c, giving us
A m
30 Chapter Three.
and hence A will find, either by calculation or by direct measurement if he has arranged clocks at B and B', that it takes a longer time for the performance of B's experiment than for the performance of his
/
: -\/ 1 —
own in the ratio 1
It is evident from the first postulate of relativity, however, that B himself must find exactly the same length of time for the light to pass up to his mirror and come back as did A in his experiment, because the two systems are, as a matter of fact, entirely symmetrical and we could with equal right consider B's system to be the one at rest and A's the one in motion.
We thus find that two observers, A and B, who are in relative motion will not in general agree in their measurements of the time interval neces sary for a given event to take place, the event in this particular case, for example, having been the performance of B's experiment; indeed, making use of the ratio obtained in a preceding paragraph, we may go further and make the quantitative statement that measurements of time intervals made with a moving clock must be multiplied by the quantity
in order to agree with measurements made with a stationary
system of clocks.
It is sometimes more convenient to state this principle in the form: A stationary observer using a set of stationary clocks will
/ y2 obtain a greater measurement in the ratio 1 : -v/1 — — for a given
time interval than an observer who uses a clock moving with the velocity V.
Measurements of Length in a Moving System.
26. We may now extend our considerations, to obtain a relation between measurements of length made in stationary and moving systems.
As to measurements of length perpendicular to the line of motion of the two systems S and Sf, a little consideration will make it at once evident that both A and B must obtain identical results. This is true because the possibility is always present of making a direct com-
Some Elementary Deductions. 31
parison of the meter sticks which A and B use for such measurements by holding them perpendicular to the line of motion. When the relative motion of the two systems brings such meter sticks into juxtaposition, it is evident from the first postulate of relativity that A's meter and B's meter must coincide in length. Any difference in length could be due only to the different velocity of the two systems through space, and such an occurrence is ruled out by our first postulate. Hence measurements made with a moving meter stick held perpendicular to its line of motion will agree with those made with stationary meter sticks.
27. With regard to measurements of length parallel to the line of motion of the systems, the affair is much more complicated. Any direct comparison of the lengths of meter sticks in the two systems would be very difficult to carry out; the consideration, however, of a simple experiment on the velocity of light parallel to the motion of the systems will lead to the desired relation.
Let us again consider two systems S and S' (fig. 7), S' moving past S with the velocity V.
m
B
FIG. 7.
A and B are observers on these systems provided with clocks and meter sticks. The two observers lay off, each on his own system, paths for measuring the velocity of light. A lays off a distance of one meter, A m, so that he can measure the time for light to travel to the mirror m and return, and B, using a meter stick which has the same length as A's when they are both at rest, lays off the dis tance B n.
Each observer measures the length of time it takes for light to travel to his mirror and return, and will evidently have to find the same length of time, since the postulates of relativity require that the velocity of light shall be the same for all observers.
32 Chapter Three.
Now the observer A, taking himself as at rest, finds that B's light travels over a path B nf B' (fig. 8), where n n' is the distance
B B' n n'
FIG. 8.
through which the mirror n moves while the light is travelling up to it, and B B' is the distance through which the source travels before the light gets back. It is easy to calculate the length of this path. We have
nn' _ V B n' " c and
BB' V Bne B'=~~ c' Also, from the figure,
B n' = B n + n n', Bri B' = BnB + 2nri - BB'.
Combining, we obtain
Bri B' _ _l_ BnB '' V^'
c2
Thus A finds that the path traversed by B's light, instead of being exactly two meters as was his own, will be longer in the ratio of
/ F2\
1 : ( 1 — — ) . For this reason A is rather surprised that B does
not report a longer time interval for the passage of the light than he himself found. He remembers, however, that he has already found that measurements of time made with a moving clock must be multi plied by the quantity —. — in order to agree with his own, and
C'
sees that this will account for part of the discrepancy between the expected and observed results. To account for the remaining dis crepancy the further conclusion is now obtained that measurements of
Some Elementary Deductions. 33
length made with a moving meter sticky parallel to its motion, must be
I V2 multiplied by the quantity -y 1 in order to agree with those made
in a stationary system.
In accordance with this principle, a stationary observer will obtain a smaller measurement for the length of a moving body than will an observer moving along with the object. This has been called the Lorentz shortening, the shortening occurring in the ratio
in the line of motion.
The Setting of Clocks in a Moving System.
28. It will be noticed that in our considerations up to this point we have considered cases where only a single moving clock was needed in performing the desired experiment, and this was done purposely, since we shall find, not only that a given time interval measures shorter on a moving clock than on a system of stationary clocks, but that a system of moving clocks which have been set in synchronism by an observer moving along with them will not be set in synchronism for a stationary observer.
Consider again two systems S and S' in relative motion with the velocity V. An observer A on system S places two carefully com pared clocks, unit distance apart, in the line of motion, and has the time on each clock read when a given point on the other system passes it. An observer B on system S' performs a similar experiment. The time interval obtained in the two sets of readings must be the same, since the first postulate of relativity obviously requires that the relative velocity of the two systems V shall have the same value for both observers.
The observer A, however, taking himself as at rest, and familiar with the change in the measurements of length and time in the moving system which have already been deduced, expects that the velocity as measured by B will be greater than the value that he himself
obtains in the ratio ^ , since any particular one of B's clocks
34 Chapter Three.
gives a shorter value for a given time interval than his own, while B's measurements of the length of a moving object are greater than
his own, each by the factor *\jl — — •- . In order to explain the actual
result of B's experiment he now has to conclude that the clocks which for B are set synchronously are not set in synchronism for himself.
From what has preceded it is easy to see that in the moving system, from the point of view of the stationary observer, clocks must be set further and further ahead as we proceed towards the rear of the system, since otherwise B would not obtain a great enough difference in the readings of the clocks as they come opposite the given point on the other system. Indeed, if two clocks are situated in the moving system, S', one in front of the other by the distance l't as measured by B, then for A it will appear as though B had set his rear clock ahead
I'V
by the amount — . c
29. We have now obtained all the information which we shall need in this chapter as to measurements of time and length in systems moving with different velocities. We may point out, however, before proceeding to the application of these considerations, that our choice of A' s system as the one which we should call stationary was of course entirely arbitrary and immaterial. We can at any time equally well take B's system as the one to which we shall ultimately refer all our measurements, and indeed all that we shall mean when we call one of our systems stationary is that for reasons of convenience we have picked out that particular system as the one with reference to which we particularly wish to make our measurements. We may also point out that of course B has to subject A' 8 measurements of time
and length to just the same multiplications by the factor — p=
as did A in order to make them agree with his own.
These conclusions as to measurements of space and time are of course very startling when first encountered. The mere fact, however, that they appear strange to so-called " common sense " need cause us no difficulty, since the older ideas of space and time were obtained from an ancestral experience which never included experiments with
Some Elementary Deductions. 35
72
high relative velocities, and it is only when the ratio — becomes
c
•
appreciable that we obtain unexpected results. To those scientists who do not wish to give up their " common sense " ideas of space and time we can merely say that if they accept the two postulates of relativity then they will also have to accept the consequences which can be deduced therefrom. The remarkable nature of these consequences merely indicates the very imperfect nature of our older conceptions of space and time.
The Composition of Velocities.
30. Our conclusions as to the setting of clocks make it possible to obtain an important expression for the composition of velocities. Suppose we have a system S, which we shall take as stationary, and on the system an observer A. Moving past S with the velocity V is another system S' with an observer B, and finally moving past S' in the same direction is a body whose velocity is u' as measured by observer B. What will be the velocity u of this body as measured by A?
Our older ideas led us to believe in the simple additivity of veloci ties and we should have calculated u in accordance with the simple expression
u = V + u'.
We must now allow, however, for the fact that u' is measured with clocks which to A appear to be set in a peculiar fashion and running at a different rate from his own, and with meter sticks which give longer measurements than those used in the stationary system.
The determination of u' by observer B would be obtained by measuring the time interval necessary for the body in question to move a given distance I' along the system S'. If t' is the difference in the respective clock readings when the body reaches the ends of the line I', we have
I'V
We have already seen, however, that the two clocks are for A set —
units apart and hence for clocks set together the time interval would
36 Chapter Three.
I'V have measured t' H 5- . Furthermore these moving clocks give
V2
time measurements which are shorter in the ratio \/ 1 — — : 1 than
\ c2
those obtained by A, so that for A the time interval for the body to move from one end of V to the other would measure
V*'
'
furthermore, owing to the difference in measurements of length, this
I yj line I' has for A the length I' \| 1 .
* * C
body is moving past S' with the velocity,
V2
line I' has for A the length I' \j 1 — — . Hence A finds that the
c
I'V I'V u'V
This makes the total velocity of the body past S equal to the sum
u = V +
or
u'F
u'V
This new expression for the composition of velocities is extremely important. When the velocities u' and V are small compared with the velocity of light c, we observe that the formula reduces to the simple additivity principle which we know by common experience to be true
Some Elementary Deductions. 37
for all ordinary velocities. Until very recently the human race has had practically no experience with high velocities and we now see that for velocities in the neighborhood of that of light, the simple additivity principle is nowhere near true.
In particular it should be noticed that by the composition of velocities which are themselves less than that of light we can never obtain any velocity greater than that of light. As an extreme case, suppose for example that the system S' were moving past S itself with the velocity of light (i. e., V = c) and that in the system S' a particle should itself be given the velocity of light in the same direc tion (i. e., u' = c); we find on substitution that the particle still has only the velocity of light with respect to S. We have
c + c 2c
By the consideration of such conclusions as these the reader will appreciate more and more the necessity of abandoning his older naive ideas of space and time which are the inheritance of a long human experience with physical systems in which only slow velocities were encountered.
The Mass of a Moving Body.
31. We may now obtain an important relation for the mass of a moving body. Consider again two similar systems, S at rest and S' moving past with the velocity V. The observer A on system S has a sphere made from some rigid elastic material, having a mass of m grams, and the observer B on system Sf is also provided with a similar sphere. The two spheres are made so that they are exactly alike when both are at rest; thus B's sphere, since it is at rest with respect to him, looks to him just the same as the other sphere does to A- As the two systems pass each other (fig. 9) each of these clever experi menters rolls his sphere towards the other system with a velocity of u cm. per second, so that they will just collide and rebound in a line perpendicular to the direction of motion. Now, from the first postu late of relativity, system S' appears to B just the same as system S appears to A, and jETs ball appears to him to go through the same evolutions that A finds for his ball. A finds that his ball on collision
38
Chapter Three.
undergoes the algebraic change of velocity 2u, B finds the same change in velocity 2u for his ball. B reports this fact to "A, and A knowing that B's measurements of length agree with his own in this transverse
FIG. 9.
direction, but that his clock gives time intervals that are shorter than
/ 72 his own in the ratio \fl — — : 1, calculates that the change in veloc
ity of B's ball must be 2u
From the principle of the conservation of momentum, however, A knows that the change in momentum of B's ball must be the same as that of his own and hence can write the equation
mau =
where ma is the mass of A's ball and ra& is the mass of B's ball. . Solv ing we have
ma
nib =
In other words, B's ball, which had the same mass ma as A's when
Some Elementary Deductions. 39
both were at rest, is found to have the larger mass — - — - -- when
placed in a system that is moving with the velocity V.*
The theory of relativity thus leads to the general expression
m
for the mass of a body moving with the velocity u and having the mass ?tto when at rest.
Since we have very few velocities comparable with that of light
I ^2 it is obvious that the quantity \/l seldom differs much from
unity, which makes the experimental verification of this expression difficult. In the case of electrons, however, which are shot off from radioactive substances, or indeed in the case of cathode rays produced with high potentials, we do have particles moving with velocities comparable to that of light, and the experimental work of Kaufmann, Bucherer, Hupka and others in this field provides one of the most striking triumphs of the theory of relativity.
The Relation Between Mass and Energy.
32. The theory of relativity has led to very important conclusions as to the nature of mass and energy. In fact, we shall see that matter and energy are apparently different names for the same fundamental entity.
When we set a body in motion it is evident from the previous section that we increase both its mass as well as its energy. Now we can show that there is a definite ratio between the amount of energy that we give to the body and the amount of mass that we give to it.
If the force / acts on a particle which is free to move, its increase in kinetic energy is evidently
A# = ffdl.
But the force acting, is by definition, equal to the rate of increase in
* In carrying out this experiment the transverse velocities of the balls should be made negligibly small in comparison with the relative velocity of the systems V.
40 Chapter Three.
the momentum of the particle
/=|(m«). Substituting we have
We have, however, from the previous section,
m =
which, solved for u, gives us
Substituting this value of u in our equation for AE we obtain, after simplification,
AE = fc*dm = c2Aw.
This says that the increase of the kinetic energy of the particle, in ergs, is equal to the increase in mass, in grams, multiplied by the square of the velocity of light. If now we bring the particle to rest it will give up both its kinetic energy and its excess mass. Accepting the principles of the conservation of mass and energy, we know, how ever, that neither this energy nor the mass has been destroyed; they have merely been passed on to other bodies. There is, moreover, every reason to believe that this mass and energy, which were asso ciated together when the body was in motion and left the body when it was brought to rest, still remain always associated together. For example, if the body should be brought to rest by setting another body into motion, it is of course a necessary consequence of our con siderations that the kinetic energy and the excess mass both pass on together to the new body which is set in motion. A similar con clusion would be true if the body is brought to rest by frictional forces, since the heat produced by the friction means an increase in the kinetic energies of ultimate particles.
Some Elementary Deductions. 41
In general we shall find it pragmatic to consider that matter and energy are merely different names for the same fundamental entity. One gram of matter is equal to 1021 ergs of energy.
c2 = (2.9986 X 1010)2 = approx. 1021.
This apparently extraordinary conclusion is in reality one which produces the greatest simplification in science. Not to mention numerous special applications where this principle is useful, we may call attention to the fact that the great laws of the conservation of mass and of energy have now become identical. We may also point out that those opposing camps of philosophic materialists who defend matter on the one hand or energy on the other as the fundamental entity of the universe may now forever cease their unimportant bicker ings.
CHAPTER IV.
THE EINSTEIN TRANSFORMATION EQUATIONS FOR SPACE
AND TIME.
The Lorentz Transformation.
33. We may now proceed to a systematic study of the consequences of the theory of relativity.
The fundamental problem that first arises in considering spatial and temporal measurements is that of transforming the description of a given kinematical occurrence from the variables of one system of coordinates to those of another system which is in motion relative to the first.
Consider two systems of right-angled Cartesian coordinates S and Sf (fig. 10) in relative motion in the X direction with the velocity V.
-x v) x'
'Z
FIG. 10.
The position of any given point in space can be determined by speci fying its coordinates x, y, and z with respect to system S or its coordi nates x', y' and z' with respect to system S'. Furthermore, for the purpose of determining the time at which any event takes place, we may think of each system of coordinates as provided with a whole series of clocks placed at convenient intervals throughout the system, the clocks of each series being set and regulated* by observers in the
* We may think of the clocks as being set in any of the ways that are usual in practice. Perhaps the simplest is to consider the clocks as mechanisms which have been found to "keep time" when they are all together where they can be examined by one individual observer. The assumption can then be made, in ac-
42
Transformation Equations for Space and Time. 43
corresponding system. The time at which the event in question takes place may be denoted by t if determined by the clocks belonging to system S and by tr if determined by the clocks of system Sr.
For convenience the two systems S and S' are chosen so that the axes OX and O'X' lie in the same line, and for further simplification we choose, as our starting-point for time measurements, t and t' both equal to zero when the two origins come into coincidence.
The specific problem now before us is as follows: If a given kine- matical occurrence has been observed and described in terms of the variables x', y', z' and t', what substitutions must we make for the values of these variables in order to obtain a correct description of the same kinematical event in terms of the variables x, y, z and J? In other words, we want to obtain a set of transformation equations from the variables of system Sf to those of system S. The equations which we shall present were first obtained by Lorentz, and the process of changing from one set of variables to the other has generally been called the Lorentz transformation. The significance of these equa tions from the point of view of the theory of relativity was first appre ciated by Einstein.
Deduction of the Fundamental Transformation Equations.
34. It is evident that these transformation equations are going to depend on the relative velocity V of the two systems, so that we may write for them the expressions
x' = F^V, x, y, z, 0,
y' = F*(V, x, y, z, t),
z' = Fi(V, x, y, z, t),
t' = Ft(V, x, y, z} t),
where Fi, F2, etc., are the unknown functions whose form we wish to determine.
It is possible at the outset, however, greatly to simplify these relations. If we accept the idea of the homogeneity of space it is evident that any other line parallel to OXX' might just as well have been chosen as our line of X-axes, and hence our two equations for x' and t' must be independent of y and z. Moreover, as to the equa-
cordance with our ideas of the homogeneity of space, that they will continue to "keep time" after they have been distributed throughout the system.
44 Chapter Four.
tions for y' and z' it is at once evident that the only possible solutions are y' = y and z' = z. This is obvious because a meter stick held in the system S' perpendicular to the line of relative motion, OX', of the system can be directly compared with meter sticks held similarly in system S, and in accordance with the first postulate of relativity they must agree in length in order that the systems may be entirely symmetrical. We may now rewrite our transformation equations in the simplified form
t' = F2(V, t, a),
and have only two functions, Fi and F2, whose form has to be de termined.
To complete the solution of the problem we may make use of three further conditions which must govern the transformation equations.
35. Three Conditions to be Fulfilled. In the first place, when the velocity V between the systems is small, it is evident that the trans formation equations must reduce to the form that they had in New tonian mechanics, since we know both from measurements and from everyday experience that the Newtonian concepts of space and time are correct as long as we deal with slow velocities. Hence the limiting form of the equations as V approaches zero will be (cf. Chapter I, equations 3-4-5-6)
x' = x - Vt,
y' = y,
Zf = 2, t' = t.
36. A second condition is imposed upon the form of the functions Fi and Fz by the first postulate of relativity, which requires that the two systems S and Sf shall be entirely symmetrical. Hence the transformation equations for changing from the variables of system S to those of system S' must be of exactly the same form as those used in the reverse transformation, containing, however, — V wherever H- V occurs in the latter equations. Expressing this requirement in
Transformation Equations for Space and Time. 45
mathematical form, we may write as true equations
where FI and Fz must have the same form as above.
37. A final condition is imposed upon the form of FI and F* by the second postulate of relativity, which states that the velocity of a beam of light appears the same to all observers regardless of the motion of the source of light or of the observer. Hence our trans formation equations must be of such a form that a given beam of light has the same velocity, c, when measured in the variables of either system. Let us suppose, for example, that at the instant t = t' = 0, when the two origins come into coincidence, a light impulse is started from the common point occupied by 0 and 0' '. Then, measured in the coordinates of either system, the optical disturbance which is generated must spread out from the origin in a spherical form with the velocity c. Hence, using the variables of system S, the coordinates of any point on the surface of the disturbance will be given by the expression
x2 + 2/2 + z2 = c2/2, (7)
while using the variables of system Sf we should have the similar expression
x'2 + 2/'2 + z'2 = cH>\ (8)
Thus we have a particular kinematical occurrence, the spreading out of a light disturbance, whose description is known in the variables of either system, and our transformation equations must be of such a form that their substitution will change equation (8) to equation (7). In other words, the expression x2 + y2 + z2 - c2*2 is to be an invariant for the Lorentz transformation.
38. The Transformation Equations. The three sets of conditions which, as we have seen in the last three paragraphs, are imposed upon the form of FI and F2 are sufficient to determine the solution of the problem. The natural method of solution is obviously that of trial,
46 Chapter Four.
and we may suggest the solution :
x' = -j=^= (x - Vt) = K(X - Vt), (9)
y' = y, (io)
z' = z, (11)
(12)
1 / V \ ( V \
''"lr^V~H-«('~H'
where we have placed K to represent the important and continually
1
recurring quantity
It will be found as a matter of fact by examination that these solutions do fit all three requirements which we have stated. Thus, when V becomes small compared with the velocity of light, c, the equations do reduce to those of Galileo and Newton. Secondly, if the equations are solved for the unprimed quantities in terms of the primed, the resulting expressions have an unchanged form except for the introduction of — V in place of + V, thus fulfilling the require ments of symmetry imposed by the first postulate of relativity. And finally, if we substitute the expressions for x', y', z' and t' in the poly nomial a/2 + y'2 + z'2 = c2t'2} we shall obtain the expression x2 + 2/2 + z2 — c2t2 and have thus secured the in variance of x2 + y2 + z2 — cH2 which is required by the second postulate of relativity.
We may further point out that the whole series of possible Lorentz transformations form a group such that the result of two successive transformations could itself be represented by a single transformation provided we picked out suitable magnitudes and directions for the velocities between the various systems.
It is also to be noted that the transformation becomes imaginary for cases where V > c, and we shall find that this agrees with ideas obtained in other ways as to the speed of light being an upper limit for the magnitude of all velocities.
Transformation Equations for Space and Time. 47
Further Transformation Equations.
39. Before making any applications of our equations we shall find it desirable to obtain by simple substitutions and differentiations a series of further transformation equations which will be of great value in our future work.
By the simple differentiation of equation (12) we can obtain
dx where we have put x for -r." •
40. Transformation Equations for Velocity. By differentiation of the equations for x' ', yr and z', nos. (9), (10) and (11), and substitution
of the value just found for -j- we may obtain the following transfor
mation equations for velocity:
x-V , ux-V
X ~ °r Ux~~ ~'
' c* c2
v-^
- xV
--?
where the placing of a dot has the familiar significance of differentiation with respect to time, -r- being represented by x and ^7 by x'.
The significance of these equations for the transformation of velocities is as follows: If for an observer in system S a point appears to be moving with the uniform velocity (x, y, z) its velocity (x', ?/', z')t as measured by an observer in system S'} is given by these expressions
(14), (15) and (16).
41. Transformation Equations for the Function — j= =. These
48 Chapter Four.
three transformation equations for the velocity components of a point, permit us to obtain a further transformation equation for an important function of the velocity which we shall find continually recurring in
our later work. This is the function — j -- , where we have indi-
cated the total velocity of the point by u, according to the expression ui = ±2 + y2 + 22- By the substitution of equations (14), (15) and (16) we obtain the transformation equation
(17)
42. Transformation Equations for Acceleration. By further dif ferentiating equations (14), (15) and (16) and simplifying, we easily obtain three new equations for transforming measurements of accel eration from system S' to S, viz. :
x'=(i-~y*-*x, as)
$> = Ci _^y~2 ^ + 2/^(1 -^)~3/c~2*> (19)
z' = (l -^} * K-*z + z^2(l -~^J 3 K~*X, (20)
or
/ i/ T7\-3
K-*iix, (18)
V f 11 V
V ' U'V • (20)
CHAPTER V. KINEMATICAL APPLICATIONS.
43. The various transformation equations for spatial and temporal measurements which were derived in the previous chapter may now be used for the treatment of a number of kinematical problems. In particular it will be shown in the latter part of the chapter that a number of optical problems can be handled with extraordinary facility by the methods now at our disposal.
The Kinematical Shape of a Rigid Body.
44. We may first point out that the conclusions of relativity theory lead us to quite new ideas as to what is meant by the shape of a rigid body. We shall find that the shape of a rigid body will depend entirely upon the relative motion of the body and the observer who is making measurements on it.
Consider a rigid body which is at rest with respect to system Sf. Let Xit 2//> z/ and x2', 2/2', £2' be the coordinates in system S' of two points in the body. The coordinates of the same points as measured in system S can be found from transformation equations (9), (10) and (11), and by subtraction we can obtain the following expressions
V2
(2/2 - 2/0 = (2/2' - 2/i'), (22)
(*2 - 2/2) = (zj - */), (23)
connecting the distances between the pair of points as viewed in the two systems. In making this subtraction terms containing t have been cancelled out since we are interested in the simultaneous positions of the points. In accordance with these equations we may distinguish then between the geometrical shape of a body, which is the shape that it has when measured on a system of axes which are at rest relative to it, and its kinematical shape, which is given by the coordinates which 5 49
50 Chapter Five.
express the simultaneous positions of its various points when it is in motion with respect to the axes of reference. We see that the kine- matical shape of a rigid body differs from its geometrical shape by a shortening of all its dimensions in the line of motion in the ratio
/ V2 A/1 — — : 1; thus a sphere, for example, becomes a Heaviside ellipsoid.
In order to avoid incorrectness of speech we must be very care ful not to give the idea that the kinematical shape of a body is in any sense either more or less real than its geometrical shape. We must merely learn to realize that the shape of a body is entirely de pendent on the particular set of coordinates chosen for the making of space measurements.
The Kinematical Rate of a Clock.
45. Just as we have seen that the shape of a body depends upon our choice of a system of coordinates, so we shall find that the rate of a given clock depends upon the relative motion of the clock and its observer. Consider a clock or any mechanism which is performing a periodic action. Let the clock be at rest with respect to system S' and let a given period commence at t\ and end at £2', the length of the interval thus being At' = t-2' — t\.
From transformation equation (12) we may obtain
and by subtraction, since x% — Xi is obviously equal to Vt, we have
r
Vi-
At
Yl
c2
i
Kinematical Applications. 51
Thus an observer who is moving past a clock finds a longer period for
/ T2 the clock in the ratio 1 : \J 1 — — than an observer who is stationary
with respect to it. Suppose, for example, we have a particle which is turning alternately red and blue. For an observer who is moving past the particle the periods for which it remains a given color measure
/ 7* longer in the ratio 1 : A/ 1 — — than they do to an observer who is
stationary with respect to the particle.
46. A possible opportunity for testing this interesting conclusion of the theory of relativity is presented by the phenomena of canal rays. We may regard the atoms which are moving in these rays as little clocks, the frequency of the light which they emit corresponding to the period of the clock. If now we should make spectroscopic observations on canal rays of high velocity, the frequency of the emitted light ought to be less than that of light from stationary atoms of the same kind if our considerations are correct. It would of course be necessary to view the canal rays at right angles to their direction of motion, to prevent a confusion of the expected shift in the spectrum with that produced by the ordinary Doppler effect (see Section 54).
The Idea of Simultaneity.
47. We may now also point out that the idea of the absolute simul taneity of two events must henceforth be given up. Suppose, for example, an observer in the system S is interested in two events which take place simultaneously at the time t. Suppose one of these events occurs at a point having the X coordinate x\ and the other at a point having the coordinate z2; then by transformation equation (12) it is evident that to an observer in system S', which is moving relative to S with the velocity 7, the two events would take place respectively at the times
and
1 / V \
T— V>?*J
v1-?
52 Chapter Five.
or the difference in time between the occurrence of the events would appear to this other observer to be
-»,)• (25)
The Composition of Velocities.
48. The Case of Parallel Velocities. We may now present one of the most important characteristics of Einstein's space and time, which can be best appreciated by considering transformation equation (14). or more simply its analogue for the transformation in the reverse direction
c*
Consider now the significance of the above equation. If we have a particle which is moving in the X direction with the velocity Ux as measured in system S', its velocity ux with respect to system S is to be obtained by adding the relative velocity of the two systems V
ux'V
and dividing the sum of the two velocities by 1 H -- ^— . Thus we see
c
that we must completely throw overboard our old naive ideas of the direct additivity of velocities. Of course, in the case of very slow velocities, when ux' and V are both small compared with the velocity
ux'V of light, the quantity — ^-- is very nearly zero and the direct addition
of velocities is a close approximation to the truth. In the case of velocities, however, which are in the neighborhood of the speed of light, the direct addition of velocities would lead to extremely er roneous results.
49. In particular it should be noticed that by the composition of velocities which are themselves less than that of light we can never obtain any velocity greater than that of light. Suppose, for example, that the system S' were moving past S with the velocity of light (i. e., V = c), and that in the system S' a particle should itself be given the velocity of light in the X direction (i. e., ux' — c); we find on substitution that the particle still has only the velocity of light
Kinematical Applications. 53
with respect to S. We have
c_+_c 2c
1 +
ux= — = - = c.
c If the relative velocity between the systems should be one half
r
the velocity of light, - , and an experimenter on Sf should shoot off a
particle in the X direction with half the velocity of light, the total velocity with respect to S would be
ic + \c 4 u* = --- TT2= *c.
•
50. Composition of Velocities in General. In the case of particles which have components of velocity in other than the X direction it is obvious that our transformation equations will here also provide methods of calculation to supersede the simple addition of velocities. If we place
U? = UX2 + Uy2 + USZ,
U" = «," + «," + tt.",
we may obtain by the substitution of equations (14), (15) and (16)
where a is the angle in the system S' between the X' axis and the velocity of the particle uf. For the particular case that V and u' are in the same direction, the equation obviously reduces to the simpler form
u' + V
u=i + ^'
c2
which we have already considered.
51. We may also call attention at this point to an interesting char acteristic of the equations for the transformation of velocities. It will
54 Chapter Five.
be noted from an examination of these equations that if to any ob server a particle appears to have a constant velocity, i. e., to be unacted on by any force, it will also appear to have a uniform although of course different velocity to any observer who is himself in uniform motion with respect to the first. An examination, however, of the transformation equations for acceleration (18), (19), (20) will show that here a different state of affairs is true, since it will be seen that a point which has uniform acceleration (x, y, z) with respect to an ob server in system S will not in general have a uniform acceleration in another system Sf, since the acceleration in system S' depends not only on the constant acceleration but also on the velocit3r in system S, which is necessarily varying.
Velocities Greater than that of Light.
52. In the preceding section we have called attention to the fact that the mere composition of velocities which are not themselves greater than that of light will never lead to a speed that is greater than that of light. The question naturally arises whether velocities which are greater than that of light could ever possibly be obtained in any way.
This problem can be attacked in an extremely interesting manner. Consider two points A and B on the X axis of the system S, and suppose that some impulse originates at A, travels to B with the velocity u and at B produces some observable phenomenon, the start ing of the impulse at A and the resulting phenomenon at B thus being connected by the relation of cause and effect.
The time elapsing between the cause and its effect as measured in the units of system S will evidently be
At = ts-tA = Xj±, (28)
where XA and XB are the coordinates of the two points A and B.
Now in another system £', which has the velocity V with respect to S, the time elapsing between cause and effect would evidently be
^pv* ">**)"
Kinematical Applications. 55
where we have substituted for t'B and t'A in accordance with equation (12). Simplifying and introducing equation (28) we obtain
uV
1 ~~ c2
(29)
Let us suppose now that there are no limits to the possible magni tude of the velocities u and V, and in particular that the causal im pulse can travel from A to B with a velocity u greater than that of light. It is evident that we could then take a velocity u great enough
uV
so that - - would be greater than unity and At' would become nega- c-
tive. In other words, for an observer in system S' the effect which occurs at B would precede in time its cause which originates at A. Such a condition of affairs might not be a logical impossibility; never theless its extraordinary nature might incline us to believe that no causal impulse can travel with a velocity greater than that of light. We may point out in passing, however, that in the case of kine matic occurrences in which there is no causal connection there is no reason for supposing that the velocity must be less than that of light. Consider, for example, a set of blocks arranged side by side in a long row. For each block there could be an independent time mechanism like an alarm clock which would go off at just the right instant so that the blocks would fall down one after another along the line. The velocity with which the phenomenon would travel along the line of blocks could be arranged to have any value. In fact, the blocks could evidently all be fixed to fall just at the same instant, which would correspond to an infinite velocity. It is to be noticed here, however, that there is no causal connection between the falling of one block and that of the next, and no transfer of energy. Application of the Principles of Kinematics to Certain Optical Prob lems.
53. Let us now apply our kinematical considerations to some problems in the field of optics. We may consider a beam of light as a periodic electromagnetic disturbance which is propagated through a vacuum with the velocity c. At any point in the path of a beam of
56 Chapter Five.
light the intensity of the electric and magnetic fields will be undergoing periodic changes in magnitude. Since the intensities of both the electric and the magnetic fields vary together, the statement of a single vector is sufficient to determine the instantaneous condition at any point in the path of a beam of light. It is customary to call this vector (which might be either the strength of the electric or of the magnetic field) the light vector.
For the case of a simple plane wave (i. e., a beam of monochromatic light from a distant source) the light vector at any point in the path of the light may be put proportional to
Ix + my + nz,
•
where x, y and z are the coordinates of the point under observation, t is the time, Z, m and n are the cosines of the angles a, /5 and 7 which determine the direction of the beam of light with reference to our system, and w is a constant which determines the period of the light. If now this same beam of light were examined by an observer in system S' which is moving past the original system in the X direction with the velocity V, we could write the light vector proportional to
,/, ZV + mV + nV\
CO I t — - I .
V c J
sin a/ 1*'-- ^f- -1. (31)
It is not difficult to show that the transformation equations which we have already developed must lead to the following relations between the measurements in the two systems*
* Methods for deriving the relation between the accented and unaccented quantities will be obvious to the reader. For example, consider the relation between co and to'. At the origin of coordinates x = y = z = Qin system S, we shall have in accordance with expression (30) the light vector proportional to sin cot, and hence similarly at the point 0', which is the origin of coordinates in system Sf, we shall have the light vector proportional to sin coY. But the point 0' as observed from system S moves with the velocity V along the X-axis and at any instant has the position x = Vt; hence substituting in expression (30) we have the light vector at the point 0' as measured in system S proportional to
sin wt ( 1 — Z - J , (36)
while as measured in system S' the intensity is proportional to
sin coT. (37)
Kinematical Applications. 57
(32)
(33) (34)
F
1 - z-
c
(35)
(-1)'
With the help of these equations we may now treat some important optical problems.
54. The Doppler Effect. At the origin of coordinates, x = y = z = 0, in system S we shall evidently have from expression (30) the light vector proportional to sin wt. That means that the vector becomes zero whenever ut = 2N TT, where N is any integer; in other
2?r words, the period of the light is p = — or the frequency
Similarly the frequency of the light as measured by an observer in system S' would be
We have already obtained, however, a transformation equation for t', namely,
and further may place x = Vt. Making these substitutions and comparing ex pressions (36) and (37) we see that we must have the relation
to' = «ic(l - lj)«
Methods of obtaining the relation between the cosines I, m and n and the corre sponding cosines V, m', and n' as measured in system S' may be 1<
58 Chapter Five.
Combining these two equations and substituting the equation con necting co and co' we have
v'
This is the relation between the frequencies of a given beam of light as it appears to observers who are in relative motion.
If we consider a source of light at rest with respect to system S' and at a considerable distance from the observer in system S, we may substitute for v' the frequency of the source itself, VQ, and for I we may write cos 0, where 0 is the angle between the line connecting source and observer and the direction of motion of the source, leading to the expression
Vn
(38)
(l-cos*^)
This is the most general equation for the Doppler effect. When the source of light is moving directly in the line connecting source and observer, we have cos 0 = 1, and the equation reduces to
(39)
K I 1 — —
which except for second order terms is identical with the older ex pressions for the Doppler effect, and hence agrees with experimental determinations.
We must also observe, however, that even when the source of light moves at right angles to the line connecting source and observer there still remains a second-order effect on the observed frequency, in contradiction to the predictions of older theories. We have in this case cos 0 = 0,
I yz
~ ~j - (40)
This is the change in frequency which we have already considered when we discussed the rate of a moving clock. The possibilities of
Kinematical Applications. 59
direct experimental verification should not be overlooked (see sec tion 46).
55. The Aberration of Light. Returning now to our transforma tion equations, we see that equation (33) provides an expression for calculating the aberration of light. Let us consider that the source of light is stationary with respect to system S, and let there be an observer situated at the origin of coordinates of system S' and thus moving past the source with the velocity V in the X direction. Let <£ be the angle between the X-axis and the line connecting source of light and observer and let <£' be the same angle as it appears to the moving observer; then we can obviously substitute in equation (33), cos <£ = /, cos 0' = lf, giving us
V
cos 0 - -
cos <f>f = - — y . (41)
1 — COS cj> —
c
This is a general equation for the aberration of light.
For the particular case that the direction of the beam of light is perpendicular to the motion of the observer we have cos <£ = 0
cos *'---, (42)
C
which, except for second-order differences, is identical with the familiar expression which makes the tangent of the angle of aberration nu merically equal to V/c. The experimental verification of the formula by astronomical measurements is familiar.
56. Velocity of Light in Moving Media. It is also possible to treat very simply by kinematic methods the problem of the velocity of light in moving media. We shall confine ourselves to the particular case of a beam of light in a medium which is itself moving parallel to the light.
Let the medium be moving with the velocity V in the X direction, and let us consider the system of coordinates S' as stationary with respect to the medium. Now since the medium appears to be sta tionary with respect to observers in S' it is evident that the velocity of the light with respect to S' will be c/V, where M is index of refraction
60 Chapter Five.
for the medium. If now we use our equation (26) for the addition of velocities we shall obtain for the velocity of light, as measured by observers in S,
c-+v
Carrying out the division and neglecting terms of higher order we obtain
. (44)
The equation thus obtained is identical with that of Fresnel, the
/V - 1\ quantity ( — 1 being the well-known Fresnel coefficient. The
empirical verification of this equation by the experiments of Fizeau and of Michelson and Morley is too well known to need further mention.
For the case of a dispersive medium we should obviously have to substitute in equation (44) the value of /z corresponding to the par ticular frequency, *>', which the light has in system S'. It should be noticed in this connection that the frequencies / and v which the light has respectively in system S and system S', although nearly enough the same for the practical use of equation (44), are in reality connected by an expression which can easily be shown (see section 54) to have the form
•K\1-7)'-
57. Group Velocity. In an entirely similar way we may treat the problem of group velocity and obtain the equation
(46)
where Gr is the group velocity as it appears to an observer who is
Kinematical Applications. 61
stationary with respect to the medium. G' is, of course, an experi mental quantity, connected with frequency and the properties of the medium, in a way to be determined by experiments on the stationary medium.
In conclusion we wish to call particular attention to the extra ordinary simplicity of this method of handling the optics of moving media as compared with those that had to be employed before the introduction of the principle of relativity.
CHAPTER VI. THE DYNAMICS OF A PARTICLE.
58. In this chapter and the two following, we shall present a system of " relativity mechanics " based on Newton's three laws of motion, the* Einstein transformation equations for space and time, and the principle of the conservation of mass.
The Laws of Motion.
Newton's laws of motion may be stated in the following form:
I. Every particle continues in its state of rest or of uniform motion in a straight line, unless it is acted upon by an external force.
II. The rate of change of the momentum of the particle is equal to the force acting and is in the same direction.
III. For the action of every force there is an equal force acting in the opposite direction.
Of these laws the first two merely serve to define the concept of force, and their content may be expressed in mathematical form by the following equation of definition
d , • du dm
F = dt(mu) = mdt+-dttt' (47)
where F is the force acting on a particle of mass m which has the velocity u, and hence the momentum mu.
Quite different in its nature from the first two laws, which merely give us a definition of force, the third law states a very definite physical postulate, since it requires for every change in the momentum of a body an equal and opposite change in the momentum of some other body. The truth of this postulate will of course be tested by com paring with experiment the results of the theory of mechanics which we base upon its assumption.
Difference between Newtonian and Relativity Mechanics.
59. Before proceeding we may point out the particular difference between the older Newtonian mechanics, which were based on the laws of motion and the Galilean transformation equations for space
62
Dynamics of a Particle. 63
and time, and our new system of relativity mechanics based on those same laws of motion and the Einstein transformation equations. In the older mechanics there was no reason for supposing that the mass of a body varied in any way with its velocity, and hence force could be defined interchangeably as the rate of change of momentum or as mass times acceleration, since the two were identical. In rela tivity mechanics, however, we shall be forced to conclude that the mass of a body increases in a perfectly definite way with its velocity, and hence in our new mechanics we must define force as equal to the total rate of change of momentum
d(mu) du dm
instead of merely as mass times acceleration m -7- . If we should try
to define force in " relativity mechanics " as merely equal to mass times acceleration, we should find that the application of Newton's third law of motion would then lead to very peculiar results, which would make the mass of a body different in different directions and force us to give up the idea of the conservation of mass.
The Mass of a Moving Particle.
60. In Section 31 we have already obtained in an elementary way an expression for the mass of a moving particle, by considering a collision between elastic particles and calculating how the resulting changes in velocity would appear to different observers who are themselves in relative motion. Since we now have at our command general formula? for the transformation of velocities, we are now in a position to handle this problem much more generally, and in particu lar to show that the expression obtained for the mass of a moving particle is entirely independent of the consideration of any particular type of collision.
61. Transverse Collision. Let us first treat the case of a so-called " transverse " collision. Consider a system of coordinates and two exactly similar elastic particles, each having the mass mQ when at rest, one moving in the X direction with the velocity -f u and the other with the velocity - u. (See figure 11.) Besides the large components of velocity + u and - u which they have in the X direc-
64 Chapter Six.
tion let them also have small components of velocity in the Y direc tion, + v and — v. The experiment is so arranged that the particles will just undergo a glancing collision as they pass each other and
rebound with components
> O _v of velocity in the Y direc-
y +v tion of the same magnitude,
9 **' ~u v, which they originally had,
but in the reverse direction.
(It is evident from the symmetry of the arrangement that the experi ment would actually occur as we have stated.)
We shall now be interested in the way this experiment would appear to an observer who is in motion in the X direction with the velocity V relative to our original system of coordinates.
From equation (14) for the transformation of velocities, it can be seen that this new observer would find for the X component velocities of the two particles the values
u- V -u - V
*P — and u,= -- - (48)
and from equation (15) for the Y component velocities would find the values
(49)
the signs depending on whether the velocities are measured before or after the collision.
Now from Newton's third law of motion (i. e., the principle of the equality of action and reaction) it is evident that on collision the two particles must undergo the same numerical change in momen tum.
For the experiment that we have chosen the only change in mo mentum is in the Y direction, and the observer whose measurements we are considering finds that one particle undergoes the total change
Dynamics of a Particle. 65
in velocity
V~2
and that the other particle undergoes the change in velocity
V2
Since these changes in the velocities of the particles are not equal, it is evident that their masses must also be unequal if the principle of the equality of action and reaction is true for all observers, as we have assumed. This difference in the mass of the particles, each of which has the mass ra0 when at rest, arises from the fact that the mass of a particle is a function of its velocity and for the observer in question the two particles are not moving with the same velocity.
Using the symbols mi and m* for the masses of the particles, we may now write as a mathematical expression of the requirements of the third law of motion
V2 V2
' c2 c2
Simplifying, we obtain by direct algebraic transformation
mi
|
l |
!Z >/ |
1 / " M |
_7y |
|
I \! + |
uV 1 c2 / |
||
|
c2 |
|||
|
1 |
u7 |
I1" ( : |
- r V wF |
66 Chapter Six.
which on the substitution of equations (48) gives us
(50)
ra2
This equation thus shows that the mass of a particle moving with
/ v? the velocity u* is inversely proportional to -\/l — -- , and, denoting
the mass of the particle at rest by ra0, we may write as a general ex pression for the mass of a moving particle
m = -=££= (51)
62. Mass the Same in All Directions. The method of derivation that we have just used to obtain this expression for the mass of a moving particle is based on the consideration of a so-called " trans verse collision," and in fact the expression obtained has often been spoken of as that for the transverse mass of a moving particle, while
a different expression, 7 — _2 X3/2 , has been used for the so-called
longitudinal mass of the particle. These expressions . and
m-,
I are, as a matter of fact, the values of the electric force
necessary to give a charged particle unit acceleration respectively at right angles and in the same direction as its original velocity, and hence such expressions would be proper for the mass of a moving par ticle if we should define force as mass times acceleration. As already
* For simplicity of calculation we consider the case where the components of velocity in the Y direction are small enough to be negligible in their effect on the mass of the particles compared with the large components of velocity u\ and u2 in the X direction. ., / f ,
Dynamics of a Particle. 67
stated, however, it has seemed preferable to retain, for force, Newton's original definition which makes it equal to the rate of change of momentum, and we shall presently see that this more suitable defini tion is in perfect accord with the idea that the mass of a particle is the same in all directions.
Aside from the unnecessary complexity which would be intro duced, the particular reason making it unfortunate to have different expressions for mass in different directions is that under such con ditions it would be impossible to retain or interpret the principle of the conservation of mass. And we shall now proceed to show that by introducing the principle of the conservation of mass, the con sideration of a " longitudinal collision " will also lead to exactly the
?fto .
same expression, — . , for the mass of a moving particle as we
c2 have already obtained from the consideration of a transverse collision.
63. Longitudinal Collision. Consider a system of coordinates and two elastic particles moving in the X direction with the velocities 4- u and — u so that a " longitudinal " (i. e., head-on) collision will occur. Let the particles be exactly alike, each of them having the mass mQ when at rest. On collision the particles will evidently come to rest, and then under the action of the elastic forces developed start up and move back over their original paths with the respective veloci ties — u and -f- u of the same magnitude as before.
Let us now consider how this collision would appear to an observer who is moving past the original system of coordinates with the velocity V in the X direction. Let Ui and w2 be the velocities of the particles as they appear to this new observer before the collision has taken place. Then, from our formula for the transformation of velocities (14), it is evident that we shall have
+ -*=£' and V-'^f <<«>
1 -~$ 1+~?
Since these velocities u\ and u-2 are not of the same magnitude, the two particles which have the same mass when at rest do not have the same mass for this observer. Let us call the masses before col lision mi and w2.
68 Chapter Six.
Now during the collision the velocities of the particles will all the time be changing, but from the principle of the conservation of mass the sum of the two masses must all the time be equal to m\ + m2. When in the course of the collision the particles have come to relative rest, they will be moving past our observer with the velocity — V, and their momentum will be — (mi + m2)F. But, from the principle of the equality of action and reaction, it is evident that this momen tum must be equal to the original momentum before collision occurred. This gives us the equation — (mi + m^V = m\u\ + m2w2. Substi tuting our values (52) for u\ and uz we have
(-7)
and by direct algebraic transformation, as in the previous proof, this can be shown to be identical with
Wl
m-2
/i u^
V1--?-
leading to the same expression that we obtained before for the mass of a moving particle, viz.:
m
64. Collision of Any Type. We have derived this formula for the mass of a moving particle first from the consideration of a transverse and then of a longitudinal collision between particles which are elastic and have the same mass when at rest. It seems to be desirable to show, however, that the consideration of any type of collision between particles of any mass leads to the same formula for the mass of a moving particle.
For the mass m of a particle moving with the velocity u let us write the equation m = m0/r(w2), where F( ) is the function whose form we wish to determine. The mass is written as a function of
Dynamics of a Particle. 69
the square of the velocity, since from the homogeneity of space the mass will be independent of the direction of the velocity, and the mass is made proportional to the mass at rest, since a moving body may evidently be thought of as divided into parts without change in mass. It may be further remarked that the form of the function F( ) must be such that its value approaches unity as the variable approaches zero.
Let us now consider two particles having respectively the masses mo and HQ when at rest, moving with the velocities u and w before collision, and with the velocities U and W after a collision has taken place.
From the principle of the conservation of mass we have
m0F(ux2 + uy2 + u?} + n<>F(wx* + wv2 + wz2)
= mQF(Ux* + U* + U*) + nQF(Wx* + W* + W*), (53)
and from the principle of the equality of action and reaction (i. e., Newton's third law of motion)
mQF(ux1 + u* + uf)uf + noF(w&2 + wy* + wflw*
= mQF(Ux2 + U* + Uf)U, + n,F(WI2 + Wf + W *)W *, (54) mQF(ux2 + uf + u^u, + nQF(wx2 + wy2 + w*)wu
These velocities, uxj uy, uz, wx, wy, wz, Ux, etc., are measured, of course, with respect to some definite system of " space-time " coordi nates. An observer moving past this system of coordinates with the velocity V in the X direction would find for the corresponding com ponent velocities the values
/ y2 / v^
ux-V V1 ""? \ " c2 wx- V
X -. /,! pfp
~^V> u,VU» u,VU" W.T'
-jT ~*~ c2 *
as given by our transformation equations for velocity (14, 15, 16).
70
Chapter Six.
Since the law of the conservation of mass and Newton's third law of motion must also hold for the measurements of the new ob server; we may write the following new relations corresponding to equations 53 to 56:
ux-V
72
uxV
*-:?
(53a)
m<)F{ux
c2
UX-V
1 -
u,v
c2
WX-V
(54a)
mQF{ux
yi
1 -
uxV c2
nQF{w:
I V2
V1 - 72-
1 -
wxV
V2 c2
1 -
uxv
/- p
V1--?
(55a)
i
Dynamics of a Particle. 71
(56a)
It is evident that these equations (53a-56a) must be true no matter what the velocity between the new observer and the original system of coordinates, that is, true for all values of V. The velocities ux, uy, ugf wx, etc., are, however, perfectly definite quantities, measured with reference to a definite system of coordinates and entirely inde pendent of V. If these equations are to be true for perfectly definite values of ux, uy, uz, wx, etc., and for all values of V, it is evident that the function F( ) must be of such a form that the equations are identities in V. As a matter of fact, it is found by trial that V can be cancelled from all the equations if we make F( ) of the form
— jp= ; and we see that the expected relation is a solution of the
/ "I
equations, although perhaps not necessarily a unique solution.
Before proceeding to use our formula for the mass of a moving particle for • the further development of our system of mechanics, we may call attention in passing to the fact that the experiments of Kaufmann, Bucherer, and Hupka have in reality shown that the mass of the electron increases with its velocity according to the formula which we have just obtained. We shall consider the dynamics of the electron more in detail in the chapter devoted to electromagnetic theory. We wish to point out now, however, that in this derivation we have made no reference to any electrical charge which might be carried by the particle whose mass is to be determined. Hence we may reject the possibility of explaining the Kaufmann experiment by assuming that the charge of the electron decreases with its velocity, since the increase in mass is alone sufficient to account for the results of the measurement.
72 Chapter Six.
Transformation Equations for Mass.
65. Since the velocity of a particle depends on the particular system of coordinates chosen for the measurement, it is evident that the mass of the particle will also depend on our reference system of coordinates. For the further development of our system of dynamics, we shall find it desirable to obtain transformation equations for mass similar to those already obtained for velocity, acceleration, etc.
We have
ra0
m =
where the velocity u is measured with respect to some definite system of coordinates, S. Similarly with respect to a system of coordinates S' which is moving relatively to S with the velocity V in the X direc tion we shall have
We have already obtained, however, a transformation equation (17) for the function of the velocity occurring in these equations and on substitution we obtain the desired transformation equation
ra' = f 1 - /U~- } Km, (57)
where K has the customary significance
K =
Z! " c2
By differentiation of (57) with respect to the time and simpli fication, we obtain the following transformation equation for the rate at which the mass of a particle is changing owing to change in velocity
mV
Dynamics of a Particle.
73
Equation for the Force Acting on a Moving Particle.
06. We are now in a position to return to our development of the dynamics of a particle. In the first place, the equation which we have now obtained for the mass of a moving particle will permit us to rewrite the original equation by which we defined force, in a number of ways which will be useful for future reference.
We have our equation of definition (47)
d . du dm
F =-(„„) »„_ + _„,
which, on substitution of the expression for m, gives us mo
m0 du d ^~dt + Jt
I u*
V1 -7J
u (59)
or, carrying out the indicated differentiation,
F =
u du
T -?
(60)
Transformation Equations for Force.
67. We are also in position to obtain transformation equations for force. We have
d '
F = — (mu) = mu + mu
or
Fx = mux Fv = muy F2 = muz
mux,
mu-.
We have transformation equations, however, for all the quantities on the right-hand side of these equations. For the velocities we have equations (14), (15) and (16), for the accelerations (18), (19) and (20), for mass, equation (57) and for rate of change of mass, equation (58). Substituting above we obtain as our transformation
74
Chapter Six.
equations for force
Fx-mV
1 -
uxV c2
= FX-
c2-uxV
c*-uxV
F "
(61)
1 -
uxV
Fy,
(62)
1 -
uxV
F..
(63)
We may now consider a few applications of the principles governing the dynamics of a particle.
The Relation between Force and Acceleration.
68. If we examine our equation (59) for the force acting on a particle
du d
(59)
we see that the force is equal to the sum of two vectors, one of which is in the direction of the acceleration — and the other in the direction
of the existing velocity u, so that in general force and the acceleration
it produces are not in the same di rection. We shall find it interesting to see, however, that if the force which does produce acceleration in a given direction be resolved per pendicular and parallel to the accel eration, the two components will be connected by a definite relation. Consider a particle (fig. 12) in plane space moving with the ve locity
0
FIG. 12. u = uxi + uy].
m <$
Dynamics of a Particle.
75
Let it be accelerated in the X direction by the action of the com ponent forces Fx and Fy.
From our general equation (59) for the force acting on a particle we have for these component forces
Ff =
I u*
V1-^
dui d ~dt ^~ di
mo
Fv ~~ '- ^2 dt + dt
Uy.
(64)
(65)
Introducing the condition that all the acceleration is to be in the Y direction, which makes -~ = 0, and further noting that u? = ux2 + uv2, by the division of equation (64) by (65), we obtain
F,
Fv
Fx =
uxu
C2 _ u^1
U£Uy
r «•
U.
(66)
Hence, in order to accelerate a particle in a given direction, we may apply any force Fy in the desired direction, but must at the same time apply at right angles another force Fx whose magnitude is given by equation (66).
Although at first sight this state of affairs might seem rather unexpected, a simple qualitative consideration will show the necessity of a component of force perpendicular to the desired acceleration. Refer again to figure 12; since the particle is being accelerated in the Y direction, its total velocity and hence its mass are increasing. This increasing mass is accompanied by increasing momentum in the X direction even when the velocity in that direction remains constant. The component force Fx is necessary for the production of this increase in X-momentum.
In a later paragraph we shall show an application of equation (66) in electrical theory.
76 Chapter Six.
Transverse and Longitudinal Acceleration.
69. An examination of equation (66) shows that there are two special cases in which the component force Fx disappears and the force and acceleration are in the same direction. Fx will disappear when either ux or uy is equal to zero, so that force and acceleration will be in the same direction when the force acts exactly at right angles to the line of motion of the particle, or in the direction of the motion (or of course also when ux and uy are both equal to zero and the particle is at rest). It is instructive to obtain simplified ex pressions for force for these two cases of transverse and longitudinal acceleration.
Let us again examine our equation (60) for the force acting on a particle
niQ du mo u du
(-!)'
For the case of a transverse acceleration there is no component of force in the direction of the velocity u and the second term of the equation is equal to zero, giving us
F -
For the case of longitudinal acceleration, the velocity u and the acceleration -r are in the same direction, so that we may rewrite the second term of (60), giving us
ra0 du mo u2 du
~==~
° u* V ~ c*
and on simplification this becomes
m»
«* V" <?)
Dynamics of a Particle. 77
An examination of this expression shows the reason why -j— ° 3/2
is sometimes spoken of as the expression for the longitudinal mass of a particle.
The Force Exerted by a Moving Charge.
70. In a later chapter we shall present a consistent development of the fundamentals of electromagnetic theory based on the Einstein transformation equations for space and time and the four field equa tions. At this point, however, it may not be amiss to point out that the principles of mechanics themselves may sometimes be employed to obtain a simple and direct solution of electrical problems.
Suppose, for example, we wish to calculate the force with which a point charge in uniform motion acts on any other point charge. We can solve this problem by considering a system of coordinates which move with the same velocity as the charge itself. An observer making use of the new system of coordinates could evidently calcu late the force exerted by the charge in question by Coulomb's familiar inverse square law for static charges, and the magnitude of the force as measured in the original system of coordinates can then be deter mined from our transformation equations for force. Let us proceed to the specific solution of the problem.
Consider a system of coordinates S, and a charge e in uniform motion along the X axis with the velocity V. We desire to know the force acting at the time t on any other charge ei which has any desired coordinates x, y, and z and any desired velocity ux, uy and ux.
Assume a system of coordinates, S', moving with the same velocity as the charge e which is taken coincident with the origin. To an observer moving with the system S', the charge e appears to be always at rest and surrounded by a pure electrostatic field. Hence in system S' the force with which e acts on e\ will be, in accordance with Coulomb's law*
ee\r'
F' =
r'3
* It should be noted that in its original form Coulomb's law merely stated that the force between two stationary charges was proportional to the product of the charges and inversely to the distance between them. In the present derivation
78 Chapter Six.
or
)"• (69)
2'
where x', y', and z' are the coordinates of the charge e\ at the time tf* For simplicity let us consider the force at the time t' = 0; then from transformation equations (9), (10), (11), (12) we shall have
x' = K-^-X, y' = y, zf = z.
Substituting in (69), (70), (71) and also using our transformation equations for force (61), (62), (63), we obtain the following equations for the force acting on ei, as it appears to an observer in system S:
'- - (rt??+*)'*(X+ **<*"* t>*)' (?2)
_utV c2
:z2 + ?/2 + z2)3/2'
_^F c2
These equations give the force acting on e\ at the time t. From
V transformation equation (12) we have t = — x, since t' = 0. At this
C
time the charge e, which is moving with the uniform velocity V along
we have extended this law to apply to the instantaneous force exerted by a stationary charge upon any other charge.
The fact that a charge of electricity appears the same to observers in all systems is obviously also necessary for the setting up of equations (69), (70), (71). That such is the case, however, is an evident consequence of the atomic nature of elec tricity. The charge e would appear of the same magnitude to observers both in system S and system S't since they would both count the same number of electrons on the charge. (See Section 157.)
Dynamics of a Particle. 79
the X axis, will evidently have the position 72
x< = ~# x> y* = °> z« = °-
For convenience we may now refer our results to a system of coordinates whose origin coincides with the position of the charge e at the instant under consideration. If X, Y and Z are the coordi nates of ei with respect to this new system, we shall evidently have
the relations
72 X = x - ~x = /c-% Y = y, Z = z,
UX = UX, Uy = Uy, Ug = Ut.
Substituting into (72), (73), (74) we obtain
.
where for simplicity we have placed
5 =
These are the same equations which would be obtained by sub stituting the well-known formula, for the strength of the electric and magnetic field around a moving point charge into the fifth funda mental equation of the Maxwell-Lorentz theory, f=p(e +-[uXh]*J.
They are really obtained in this way more easily, however, and are seen to come directly from Coulomb's law.
The Field around a Moving Charge. Evidently we may also use these considerations to obtain an expression for the electric field produced by a moving charge e, if we consider the particular case that the charge e\ is stationary (i. e., Ux = Uv = Ut = 0) and equal
80 Chapter Six.
to unity. Making these substitutions in (75), (76), (77) we obtain the well-known expression for the electrical field in the neighborhood of a moving point charge
where
r = Xi + Fj + Zk.
71. Application to a Specific Problem. Equations (75), (76), (77)
can also be applied in the solution of a rather interesting specific problem.
Consider a charge e constrained to move in the X direction with the ve locity V and at the instant under con-
y sideration let it coincide with the origin
of a system of stationary coordinates
=v> YeX (fig. 13). Suppose now a second
charge e\, situated at the point X = 0, Y = Y and moving in the X direction with the same velocity V as the charge e, and also having a component velocity in the F direction Uy. Let us predict
the nature of its motion under the influ-
^ X
•p 13 ence of the charge e, it being otherwise
unconstrained.
From the simple qualitative considerations placed at our disposal by the theory of relativity, it seems evident that the charge ei ought merely to increase its component of velocity in the Y direction and retain unchanged its component in the X direction, since from the point of view of an observer moving along with e the phenomenon is merely one of ordinary electrostatic repulsion.
Let us see whether our equations for the force exerted by a moving charge actually lead to this result. By making the obvious sub stitutions in equations (75) and (76) we obtain for the component forces on e\
(79) (80)
Dynamics of a Particle. 81
Now under the action of the component force Fx we might at first sight expect the charge e\ to obtain an acceleration in the X direction, in contradiction to the simple qualitative prediction that we have just made on the basis of the theory of relativity. We remember, however, that equation (66) prescribes a definite ratio between the component forces Fx and Fy if the acceleration is to be in the Y direction, and dividing (79) by (80) we actually "obtain the necessary relation
F,_ VUV
Fy~ c2 -V2'
Other applications of the new principles of dynamics to electrical, magnetic and gravitational problems will be evident to the reader.
Work.
72. Before proceeding with the further development of our theory of dynamics we shall find it desirable to define the quantities work, kinetic, and potential energy.
We have already obtained an expression for the force acting on a particle and shall define the work done on the particle as the integral of the force times the distance through which the particle is dis placed. Thus
(81)
where r is the radius vector determining the position of the particle.
Kinetic Energy.
73. When a particle is brought from a state of rest to the velocity u by the action of an unbalanced force F, we shall define its kinetic energy as numerically equal to the work done in producing the velocity. Thus
K = W =
Since, however, the kinetic energy of a particle turns out to be entirely independent of the particular choice of forces used in pro ducing the final velocity, it is much more useful to have an expression for kinetic energy in terms of the mass and velocity of the particle.
We have
82 Chapter Six.
Substituting the value of F given by the equation of definition (47) we obtain
du
f —
J dt U
= I mu-du -f- I u-udra
= I mudu -f- I u2dm. Introducing the expression (51) for the mass of a moving particle
m,r*
— , we obtain
-du + I ^-7— ^ 0x3/2^
and on integrating and evaluating the constant of integration by placing the kinetic energy equal to zero when the velocity is zero, we easily obtain the desired expression for the kinetic energy of a particle :
K = m0c2
1
- 1
(82)
= c2(m - w0). (83)
It should be noticed, as was stated above, that the kinetic energy of a particle does depend merely on its mass and final velocity and is entirely independent of the particular choice of forces which happened to be used in producing the state of motion.
It will also be noticed, on expansion into a series, that our ex pression (82) for the kinetic energy of a particle approaches at low velocities the form familiar in the older Newtonian mechanics,
K = Potential Energy.
74. When a moving particle is brought to rest by the action of a
Dynamics of a Particle. 83
conservative* force we say that its kinetic energy has been trans formed into potential energy. The increase in the potential energy of the particle is equal to the kinetic energy which has been destroyed and hence equal to the work done by the particle against the force, giving us the equation
= - W = --F-dr. (84)
The Relation between Mass and Energy.
75. We may now consider a very important relation between the mass and energy of a particle, which was first pointed out in our chapter on " Some Elementary Deductions."
When an isolated particle is set in motion, both its mass and energy are increased. For the increase in mass we may write
Am = m — WQ,
and for the increase in energy we have the expression for kinetic energy given in equation (83), giving us
A# = cz(m - W0), or, combining with the previous equation,
AE = c2Aw. (85)
Thus the increase in the kinetic energy of a particle always bears the same definite ratio (the square of the velocity of light) to its increase in mass. Furthermore, when a moving particle is brought to rest and thus loses both its kinetic energy and its extra (" kinetic ") mass, there seems to be every reason for believing that this mass and energy which are associated together when the particle is in motion and leave the particle when it is brought to rest will still remain always associated together. For example, if the particle is brought to rest by collision with another particle, it is an evident
* A conservative force is one such that any work done by displacing a system against it would be completely regained if the motion of the system should be re versed.
Since we believe that the forces which act on the ultimate particles and con stituents of matter are in reality all of them conservative, we shall accept the general principle of the conservation of energy just as in Newtonian mechanics. (For a logical deduction of the principle of the conservation of energy in a system of par ticles, see the next chapter, section 89.)
84 Chapter Six.
consequence of our considerations that the energy and the mass corresponding to it do remain associated together since they are both passed on to the new particle. On the other hand, if the particle is brought to rest by the action of a conservative force, say for example that exerted by an elastic spring, the kinetic energy which leaves the particle will be transformed into the potential energy of the stretched spring, and since the mass which has undoubtedly left the particle must still be in existence, we shall believe that this mass is now asso ciated with the potential energy of the stretched spring.
76. Such considerations have led us to believe that matter and energy may be best regarded as different names for the same funda mental entity: matter, the name which has been applied when we have been interested in the property of mass or inertia possessed by the entity, and energy, the name applied when we have been interested in the part taken by the entity in the production of motion and other changes in the physical universe. We shall find these ideas as to the relations between matter, energy and mass very fruit ful in the simplification of physical reasoning, not only because it identifies the two laws of the conservation of mass and the conser vation of energy, but also for its frequent application in the solution of specific problems.
77. We must call attention to the great difference in size between the two units, the gram and the erg, both of which are used for the measurement of the one fundamental entity, call it matter or energy as we please. Equation (85) gives us the relation
E = czm, (86)
where E is expressed in ergs and m in grams; hence, taking the velocity of light as 3 X 1010 centimeters per second, we shall have
1 gram = 9 X 1020 ergs. (87)
The enormous number of ergs necessary for increasing the mass of a system to the amount of a single gram makes it evident that experi mental proofs of the relation between mass and energy will be hard to find, and outside of the experimental work on electrons at high veloci ties, already mentioned in Section 64 and the well-known relations
Dynamics of a Particle. 85
between the energy and momentum of a beam of light, such evidence has not yet been forthcoming.
As to the possibility of obtaining further direct experimental evidence of the relation between mass and energy, we certainly can not look towards thermal experiments with any degree of confidence, since even on cooling a body down to the absolute zero of temperature it loses but an inappreciable fraction of its mass at ordinary tempera tures.* In the case of some radioactive processes, however, we may find a transfer of energy large enough to bring about measurable differences in mass. And making use of this point of view we might account for the lack of exact relations between the atomic weights of the successive products of radioactive decomposition.!
78. Application to a Specific Problem. We may show an inter esting application of our ideas as to the relation between mass and energy, in the treatment of a specific problem. Consider, just as in Section 63, two elastic particles both of which have the mass m0 at rest, one moving in the X direction with the velocity -f u and the other with the velocity — uy in such a way that a head-on collision between the particles will occur and they will rebound over their original paths with the respective velocities — u and + u of the same magnitude as before.
Let us now consider how this collision would appear to an observer who is moving past the original system of coordinates with the velocity V in the X direction. To this new observer the particles will be moving before the collision with the respective velocities
u-V -u -V
Ul = ~— and u2 = —r,
as given by equation (14) for the transformation of velocities. Fur thermore; when in the course of the collision the particles have come to relative rest they will obviously be moving past our observer with the velocity — V.
*It should be noticed that our theory points to the presence of enormous-. stores of interatomic energy which are still left in substances cooled to the absolute zero.
t See, for example, Comstock, Philosophical Magazine, vol. 15, p. 1 (1908).
86
Chapter Six.
Let us see what the masses of the particles will be both before and during the collision. Before the collision, the mass of the first particle will be
u- V
uV
c2
c2
and the mass of the second particle will be
ra0
F
Adding these two expressions, we obtain for the sum of the masses of the two particles before collision,
Now during the collision, when the two particles have come to relative rest, they will evidently both be moving past our observer with the velocity — V and hence the sum of their masses at the instant of relative rest would appear to be
2m0
a quantity which is smaller than that which we have just found for the sum of the two masses before the collision occurred. This apparent discrepancy between the total mass of the system before and during the collision, is removed, however, if we realize that when the par-
Dynamics of a Particle. 87
tides have come to relative rest an amount of potential energy of elastic deformation has been produced, which is just sufficient to re store them to their original velocities, and the mass corresponding to this potential energy will evidently be just sufficient to make the total mass of the system the same as before collisipn.
In the following chapter on the dynamics of a system of particles we shall make further use of our ideas as to the mass corresponding to potential energy.
CHAPTER VII.
THE DYNAMICS OF A SYSTEM OF PARTICLES.
79. In the preceding chapter we discussed the laws of motion of a particle. With the help of those laws we shall now derive some useful general dynamical principles which describe the motions of a system of particles, and in the following chapter shall consider an application of some of these principles to the kinetic theory of gases.
The general dynamical principles which we shall present in this chapter will be similar in form to principles which are already familiar in the classical Newtonian mechanics. Thus we shall deduce princi ples corresponding to the principles of the conservation of momentum, of the conservation of moment of momentum, of least action and of vis viva, as well as the equations of motion in the Lagrangian and Hamiltonian (canonical) forms. For cases where the velocities of all the particles involved are slow compared with that of light, we shall find, moreover, that our principles become identical in content, as well as in form, with the corresponding principles of the classical mechanics. Where high velocities are involved, however, our new principles will differ from those of Newtonian mechanics. In par ticular we shall find among other differences that in the case of high velocities it will no longer be possible to define the Lagrangian function as the difference between the kinetic and potential energies of the system, nor to define the generalized momenta used in the Hamil tonian equations as the partial differential of the kinetic energy with respect to the generalized velocity.
On the Nature of a System of Particles.
80. Our purpose in this chapter is to treat dynamical systems consisting of a finite number of particles, each obeying the equation of motion which we have already written in the forms,
-I
d_ dt
du dm = m~di + ~dt^ m0 du d
(47)
u. (59)
88
Dynamics of a System of Particles. 89
It is not to be supposed, however, that the total mass of such a system can be taken as located solely in these particles. It is evident rather, since potential energy has mass, that there will in general be mass distributed more or less continuously throughout the space in the neighborhood of the particles. Indeed we have shown at the end of the preceding chapter (Section 78) that unless we take account of the mass corresponding to potential energy we can not maintain the principle of the conservation of mass, and we should also find it impossible to retain the principle of the conservation of momentum unless we included the momentum corresponding to potential energy.
For a continuous distribution of mass we may write for the force acting at any point on the material in a small volume, 5V,
*.
= ~dt
where f is the force per unit volume and g is the density of momentum. This equation is of course merely an equation of definition for the intensity of force at a point. We shall assume, however, that New ton's third law, that is, the principle of the equality of action and reaction, holds for forces of this type as well as for those acting on particles. In later chapters we shall investigate the way in which g depends on velocity, state of strain, etc., but for the purposes of this chapter we shall not need any further information as to the nature of the distributed momentum.
Let us proceed to the solution of our specific problems.
The Conservation of Momentum.
81. We may first show from Newton's third law of motion that the momentum of an isolated system of particles remains constant.
Considering a system of particles of masses mi, mz, ms, etc., we may write in accordance with equation 47,
f[
Fi+1,-1
etc.,
90 Chapter Seven.
where FI, F2, etc., are the external forces impressed on the individual particles from outside the system and Ii, I2; etc., are the internal forces arising from mutual reactions within the interior of the system. Considering the distributed mass in the system, we may also write, in accordance with (47 A) the further equation
(f 4- i)5V = jt(g8V), (90)
where f and i are respectively the external and internal forces acting per unit volume of the distributed mass. Integrating throughout the whole volume of the system V we have
= ^f' (91)
where G is the total distributed momentum in the system. Adding this to our previous equations (89) for the forces acting on the indi vidual particles, we have
VT7 L VT _L f^T7 I f 5x717 ^V I_ ^ 2/Jc i -p 2^1i "J~ I 1$ r "p I Id V — — Z/miUi "j~ .
J J at at
But from Newton's third law of motion (i. e., the principle of the equality of action and reaction) it is evident that the sum of the
internal forces, Sli -f J idV, which arise from mutual reactions within the system must be equal to zero, which leads to the desired equation
of momentum
/j fdv =- 37 (SwiUi -j- G). (92)
at
In words this equation states that at any given instant the vector sum of the external forces acting on the system is equal to the rate at which the total momentum of the system is changing.
For the particular case of an isolated system there are no external forces and our equation becomes a statement of the principle of the conservation of momentum. The Equation of Angular Momentum.
82. We may next obtain an equation for the moment of momentum of a system about a point.
Dynamics of a System of Particles. 91
Consider a particle of mass mi and velocity Ui. Let TI be the radius vector from any given point of reference to the particle. Then for the moment of momentum of the particle about the point we may write
M! = TI X ?HiUi,
and summing up for all the particles of the system we may write
2Mi = Zfo X WiUi). (93)
Similarly, for the moment of momentum of the distributed mass we may write
Mdis, = / (r X g)dF, (94)
where r is the radius vector from our chosen point of reference to a point in space where the density of momentum is g and the inte gration is to be taken throughout the whole volume, F, of the system. Adding these two equations (93) and (94), we obtain for the total amount of momentum of the system about our chosen point
M = ZOTiXrKiuO + / (r X g)dV;
and differentiating with respect to the time we have, for the rate of change of the moment of momentum,
dM dt
or, making the substitutions given by equations (89) and (90), and
dTi writing — = Ui, etc. we have
~ = Z(rL X FO + Z(f! X Ii) + z(Ui X miUi) at
+ / (r X f)dV + f (r X i)dV + / (u X g)dV.
To simplify this equation we may note that the third term is equal to zero because it contains the outer product of a vector by itself. Fur thermore, if we accept the principle of the equality of action and
92 Chapter Seven.
reaction, together with the further requirement that forces are not only equal and opposite but that their points of application be in the same straight line, we may put the moment of all the internal forces equal to zero and thus eliminate the second and fifth terms. We obtain as the equation of angular momentum
-—
l X FO + J (r X f)dV 4- J (u X g)dV. (95)
We may call attention to the inclusion in this equation of the interesting term J (u X g)dV. If density of momentum and velocity should always be in the same direction this term would vanish, since the outer product of a vector by itself is equal to zero. In our con sideration of the " Dynamics of Elastic Bodies/' however, we shall find bodies with a component of momentum at right angles to their direction of motion and hence must include this term in a general treatment. For a completely isolated system it can be shown, how ever, that this term vanishes along with the external forces and we then have the principle of the conservation of moment of momentum. The Function T.
83. We may now proceed to the definition of a function which will be needed in our treatment of the principle of least action.
One of the most valuable properties of the Newtonian expression, %mQu2, for kinetic energy was the fact that its derivative with respect to velocity is evidently the Newtonian expression for momentum, mQu. It is not true, however, that the derivative of our new expression
for kinetic energy (see Section 73), m0c
1
- 1
, with respect
to velocity is equal to momentum, and for that reason in our non- Newtonian mechanics we shall find it desirable to define a new func tion, !F, by the equation,
/ / 1^2 \
(96)
For slow velocities (i. e., small values of u) this reduces to the Newtonian expression for kinetic energy and at all velocities we have
Dynamics of a System of Particles. 93
the relation,
dT
' = = mu, (97)
du du \ c2 / 1/2
c
/, ^ V1-?
showing that the differential of T with respect to velocity is momentum. For a system of particles we shall define T as the summation of the values for the individual particles:
(98)
The Modified Lagrangian Function.
84. In the older mechanics the Lagrangian function for a system of particles was defined as the difference between the kinetic and potential energies of the system. The value of the definition rested, however, on the fact that the differential of the kinetic energy with respect to velocity was equal to momentum, so that we shall now find it advisable to define the Lagrangian function with the help of our new function T in accordance with the equation
L = T - U. (99)
The Principle of Least Action.
85. We are now in a position to derive a principle corresponding to that of least action in the older mechanics. Consider the path by which our dynamical system actually moves from state (1) to state (2). The motion of any 'particle in the system of mass m will be governed by the equation
F=|(mu). (100)
Let us now compare the actual path by which the system moves from state (1) to state (2) with a slightly displaced path in which the laws of motion are not obeyed, and let the displacement of the particle at the instant in question be 5r.
Let us take the inner product of both sides of equation (100) with
94 Chapter Seven.
5r; we have
F-6r = -r(mu)'5r
d . d8r
= _(mu.5r)._mu.__
d = — (rau • 5r) — rau • 5u
(rau-5u + J?-dr)dt = d(wu-5r).
Summing up for all the particles of the system and integrating between the limits ti and t2) we have
(Srau-Su + SF-5r)<i£ = [Srau-5r]'j.
Since ti and t2 are the times when the actual and displaced motions coincide, we have at these times 5r = 0; furthermore we also have u • 5u = udu, so that we may write
(2mu5u + f'8r)dt = 0.
With the help of equation (97), however, we see that 2mudu = dT, giving us
f
•J ti
= 0. (101)
If the forces F are conservative, we may write F • 5r = — dU, where dU is the difference between the potential energies of the displaced and the actual configurations. This gives us
5 f 2 (T - U)dt = 0
J«, or
r
«/«,
Ldt = 0, (102)
which is the modified principle of least action. The principle evi dently requires that for the actual path by which the system goes
Dynamics of a System of Particles. 95
/•H
from state (1) to state (2), the quantity I Ldt shall be a minimum (or
Jtl maximum).
Lagrange's Equations.
86. We may now derive the Lagrangian equations of motion from the above principle of least action. Let us suppose that the position of each particle of the system under consideration is completely deter mined by n independent generalized coordinates 91, <£2, fa • • • </>n and hence that L is some function of 91, 92, 9s • • • 9n, <£i, <fo, 4>z • - • 9n,
where for simplicity we have put <£i = —rr , fa = —rr , etc. From equation (102) we have
/»<j /Vo / n *\T n r\T \
(5L)dt= (Evr-501 + ZvT-^i )dt = 0. (103)
»/<! J<! \ 1 "91 1 "91 /
But
591 = ^ which gives us
tn- ta- dL d
•£ a
or, since at times £1 and ^2, ^91 is zero, the first term in this expression disappears and on substituting in equation (103) we obtain
Since, however, the limits Zi and tz are entirely at our disposal we must have at every instant
Finally, moreover, since the 4>'s are independent parameters, we can assign perfectly arbitrary values to $91, 692, etc., and hence must have
96 Chapter Seven.
the series of equations
<L( <**L\ _ dL dt \ a<£i / d^i " 0>
dt\d<j> etc.
These correspond to Lagrange's equations in the older mechanics, differing only in the definition of L.
Equations of Motion in the Hamiltonian Form.
87. We shall also find it desirable to obtain equations of motion in the Hamiltonian or canonical form.
Let us define the generalized momentum \f/i corresponding to the coordinate </>i by the equation,
It should be noted that the generalized momentum is not as in ordinary mechanics the derivative of the kinetic energy with respect to the generalized velocity but approaches that value at low velocities.
Consider now a function Tr defined by the equation
T' - Vi0i + iW2 + • • • - T. (106)
Differentiating we have
dT' = i<i
dT _ dT .
^**~5£*i"
dT dT
-aT!^1"^^2"
and this, by the introduction of (105), becomes
dTf =
Dynamics of a System of Particles. 97
Examining this equation we have
'£--£• (108)
O(p\ C/<pi
¥r = 4i. (109)
In Lagrange's equations we have
But since U is independent of ^i we may write d(T - U) dT
a ; ~ = VT = rlj
091 001
and furthermore by (108),
Substituting these two expressions in Lagrange's equations we obtain
d^l= _ d(T + U) dt ''
or, writing T' + U = E, we have
--— , (110)
dt d(f>i
and since U is independent of fa we may rewrite equation (109) in the form
^ = 4^. an)
The set of equations corresponding to (110) and (111) for all the coordinates </>i, </>2, fa, • ' • <t>n and the momenta t£i, fa, fa, • - fa are the desired equations of motion in the canonical form.
88. Value of the Function T'. We have given the symbol E to the quantity T' + U, since T' actually turns out to be identical with
8
98 Chapter Seven.
the expression by which we defined kinetic energy, thus making E = T' + U the sum of the kinetic and potential energies of the system.
To show that T' is equal to K, the kinetic energy, we have by the equation of definition (106)
r = <Wi + M* + . - . - T,
. dT . dT
But T by definition, equation (98), is T =
which gives us
— ( _ul\~i'2 ^L
d(j>i \ c2 / d(j>i
du
and substituting we obtain
du . . . du
—- — j_ . . . j£
(112)
We can show, however, that the term in parenthesis is equal to u. If the coordinates x, y, z determine the position of the particle in question, we have,
and differentiating with respect to the <£'s, we obtain, d &( dx dx dx dx dx
etc.,
Dynamics of a System of Particles. 99
Similarly
^L - ^M. dy_ _ dy^
dz dz dz dz
Let us write now
u = Az2 + y2 + z2,
du^ __ 1 / . dx . dy . dz \
dfa" Vx2 + if- + &\dfa ^i 2d*i/>
5x dy or making the substitutions for —r , -r-r- , etc., given above, we have,
= -., -.j - a^i ^V^^i ^i 2
Substituting now in (112) we shall obtain,
or, introducing the value of T given by equation (98), we have
— m0),
which is the expression (83) for kinetic energy.
Hence we see that the Hamiltonian function E = T -f U is the sum of the kinetic and potential energies of the system as in Newtonian mechanics. The Principle of the Conservation of Energy.
89. We may now make use of our equations of motion in the canonical form to show that the total energy of a system of interacting
100 Chapter Seven.
particles remains constant. If such were not the case it is obvious that our definitions of potential and kinetic energy would not be very useful.
Since E = Tr + U is a function of <£i, $2, fo, • • - ti, ^2, ^3, • • •, we may write
dE d# . dE
dE . dE .
+ T-r<£l+TT-'/'2 + ••••
d\l/\ d\l/z
r)JF f)W
Substituting the values of - p , - p , etc., given by the canonical
0<pi 0^1
equations of motion (110) and (111), we have dE
— = — 1//10! — 1//202 -
+ Wz + ' ' '
which gives us the desired proof that just as in the older Newtonian mechanics the total energy of an isolated system of particles is a conservative quantity.
On the Location of Energy in Space.
90. This proof of the conservation of energy in a system of inter acting particles justifies us in the belief that the concept of energy will not fail to retain in the newer mechanics the position of great importance which it gradually acquired in the older systems of physical theory. Indeed, our newer considerations have augmented the important role of energy by adding, to its properties the attribute of mass or inertia, and thus leading to the further belief that matter and energy are in reality different names for the same fundamental entity.
The importance of this entity, energy, makes it very interesting to consider the possibility of ascribing a definite location in space to any given quantity of energy. In the older mechanics we had a hazy notion that the kinetic energy of a moving body was probably located in some way in the moving body itself, and possibly a vague
Dynamics of a System of Particles. 101
idea that the potential energy of a raised weight might be located in the space between the weight and the earth. Our discovery of the relation between mass and energy has made it possible, however, to give a much more definite, although not a complete, answer to inquiries of this kind.
In our discussions of the dynamics of a particle (Chapter VI, Section 61) we saw that an acceptance of Newton's principle of the equality of action and reaction forced us to ascribe an increased mass to a moving particle over that which it has at rest. This increase in the mass of the moving particle is necessarily located either in the particle itself or distributed in the surrounding space in such a way that its center of mass always coincides with the position of the particle, and since the kinetic energy of the particle is the energy corresponding to this increased mass we may say that the kinetic energy of a moving particle is so distributed in space that its center of mass always coincides with the position of the particle.
If now we consider the transformation of kinetic energy into potential energy we can also draw somewhat definite conclusions as to the location of potential energy. By the principle of the conserva tion of mass we shall be able to say that the mass of any potential energy formed is just equal to the " kinetic " mass which has dis appeared, and by the principle of the conservation of momentum we can say that the velocity of this potential energy is just that necessary to keep the total momentum of the system constant. Such con siderations will often permit us to reach a good idea as to the location of potential energy.
Consider, for example, a pair of similar attracting particles which are moving apart from each other with the velocities + u and - u and are gradually coming to rest under the action of their mutual attraction, their kinetic energy thus being gradually changed into potential energy. Since the total momentum of the system must always remain zero, wre may think of the potential energy which is formed as left stationary in the space between the two particles.
CHAPTER VIII. THE CHAOTIC MOTION OF A SYSTEM OF PARTICLES.
The discussions of the previous chapter have placed at our disposal generalized equations of motion for a system of particles similar in form to those familiar in the classical mechanics, and differing only in the definition of the Lagrangian function. With the help of these equations it is possible to carry out investigations parallel to those already developed in the classical mechanics, and in the present chapter we shall discuss the chaotic motion of a system of particles. This problem has received much attention in the classical mechanics because of the close relations between the theoretical behavior of such an ideal system of particles and the actual behavior of a mona- tomic gas. We shall find no more difficulty in handling the problem than was experienced in the older mechanics, and our results will of course reduce to those of Newtonian mechanics in the case of slow velocities. Thus we shall find a distribution law for momenta which reduces to that of Maxwell for slow velocities, and an equipartition law for the average value of a function which at low velocities becomes identical with the kinetic energy of the particles.
91. The Equations of Motion. It has been shown that the Hamil- tonian equations of motion
dE_ _
~ dt
dE _ dfr (113)
dS" dt :=01'
etc.,
will hold in relativity mechanics provided we define the generalized momenta \j/i, fa, etc., not as the differential of the kinetic energy with respect to the generalized velocities <£i, $2, etc., but as the dif ferential with respect to 0i; <£2, etc., of a function
T =
102
Chaotic Motion of a System of Particles. 103
where 7n0 is the mass of a particle having the velocity u and the sum mation 2 extends over all the particles of the system.
92. Representation in Generalized Space. Consider now a system defined by the n generalized coordinates <£i, <£2> <fo, • • • , <t>n, and the corresponding momenta \j/i, i/% fa, ' ' ' > tn- Employing the methods so successfully used by Jeans,* we may think of the state of the system at any instant as determined by the position of a point plotted in a 2n-dimensional space. Suppose now we had a large number of systems of the same structure but differing in state, then for each system we should have at any instant a corresponding point in our 2n-dimensional space, and as the systems changed their state, in the manner required by the laws of motion, the points would describe stream lines in this space.
93. Liouville's Theorem. Suppose now that the points were originally distributed in the generalized space with the uniform density p. Then it can be shown by familiar methods that, just as in the classical mechanics, the density of distribution remains uniform.
Take, for example, some particular cubical element of our gener alized space dcfridfadfa - - • d\f/id\f/zdfa - - •. The density of dis tribution will evidently remain uniform if the number of points entering any such cube per second is equal to the number leaving. Consider now the two parallel bounding surfaces of the cube which are perpendicular to the $1 axis, one cutting the axis at the point 0i and the other at the point 4>i + dfa. The area of each of these surfaces is d^^dfa - • • d^id^zdfa • • - , and hence, if <j>i is the component
of velocity which the points have parallel to the 4>i axis, and T— is
001
the rate at which this component is changing as we move along the axis, we may obviously write the following expression for the differ ence between the number of points leaving and entering per second through these two parallel surfaces
Finally, considering all the pairs of parallel bounding surfaces, we * Jeans, The Dynamical Theory of Gases, Cambridge, 1916.
104 -Chapter Eight.
find for the total decrease per second in the contents of the element
I I I _|_ ,
d0i 302 30s
But the motions of the points are necessarily governed by the Hamil- tonian equations (113) given above, and these obviously lead to the relations
etc.
So that our expression for the change per second in the number of points in the cube becomes equal to zero, the necessary requirement for preserving uniform density.
This maintenance of a uniform distribution means that there is no tendency for the points to crowd into any particular region of the generalized space, and hence if we start some one system going and plot its state in our generalized space, we may assume that, after an indefinite lapse of time, the point is equally likely to be in any one of the little elements dV. In other words, the different states of a system, which we can specify by stating the region d<f>id4>zd(j>3 • • - d\l/idfadfa - - - in which the values of the coordinates and momenta of the system fall, are all equally likely to occur.*
94. A System of Particles. Consider now a system containing Na particles which have the mass ma when at rest, Nb particles which have the mass m^ Nc particles which have the mass mcj etc. If at any given instant we specify the particular differential element dx dy dz d\l/x d\j/y d\j/z which contains the coordinates x, y, z, and the corresponding momenta \f/X) \f/v, \}/z for each particle, we shall thereby completely determine what Planckf has well called the microscopic state of the system, and by the previous paragraph any microscopic
* The criterion here used for determining whether or not the states are equally liable to occur is obviously a necessary requirement, although it is not so evident that it is a sufficient requirement for equal probability.
t Planck, Warmestrahlung, Leipzig, 1913.
Chaotic Motion of a System of Particles. 105
state of the system in which we thus specify the six-dimensional position of each particle is just as likely to occur as any other micro scopic state.
It must be noticed, however, that many of the possible micro scopic states which are determined by specifying the six-dimensional position of each individual particle are in reality completely identical, since if all the particles having a given mass ma are alike among them selves, it makes no difference which particular one of the various available identical particles we pick out to put into a specified range dx dy dz d^x d^y d\I/z.
For this reason we shall usually be interested in specifying the statistical state* of the system, for which purpose we shall merely state the number of particles of a given kind which have coordinates falling in a given range dx dy dz d\l/x d\j/v d\f/z. We see that corre sponding to any given statistical state there will be in general a large number of microscopic states.
95. Probability of a Given Statistical State. We shall now be particularly interested in the probability that the system of particles will actually be in some specified statistical state, and since Liou- ville's theorem has justified our belief that all microscopic states are equally likely to occur, we see that the probability of a given statis tical state will be proportional to the number of microscopic states which correspond to it.
For the system under consideration let a particular statistical state be specified by stating that Na', Na", Nar", • • •, AY, AY', AY", •••, etc., are the number of particles of the corresponding masses ma, mb, etc., which fall in the specified elementary regions dx dy dz d$x dtv d^z, Nos. la, 2a, 3a, • • •, 16, 26, 36, • • •, etc. By familiar methods of calculation it is evident that the number of arrangements by which the particular distribution of particles can be effected, that is, in other words, the number of microscopic states, W, which correspond to the given statistical state, is given by the expression
Na\Nb\Nc...
W
Na'Na"N
* What we have here defined as the statistical state is what Planck calls the macroscopic state of the system. The word macroscopic is unfortunate, however, in implying a less minute observation as to the size of the elements dx dy dz d\f/x d\f/y d^» in which the representative points are found.
106 Chapter Eight.
and this number W is proportional to the probability that the system will be found in the particular statistical state considered.
If now we assume that each of the regions
dx dy dz d\f/x d^y d$z, Nos. la, 2a, 3a, • • •, 16, 26, 36, • • •, etc. is great enough to contain a large number of particles,* we may apply the Stirling formula
\N= J
for evaluating \Naj [N±, etc., and omitting negligible terms, shall obtain for log W the result
'"
(N ' N ' N " N " N -iV a , Jt" a . •*" a , -iV a , -tV a , a
.v: log ]v7 + -Nl log If: + -N7 log Iv7
Nb' Nb" Nb" Nb'" Nb'"
etc.
jya' j\rar/ For simplicity let us denote the ratios -rp- , -r^- , etc., by the
J\l a -i* a
symbols waf, wa", etc. These quantities wa', wa", etc., are evidently the probabilities, in the case of this particular statistical state, that any given particle ma will be found in the respective regions Nos. la, 2a, etc.
We may now write
lOg W = — Na2Wa log Wa — Nb2Wb log Wb ~ , etc.,
where the summation extends over all the regions Nos. la, 2a, • • •, 16, 26, etc.
96. Equilibrium Relations. Let us now suppose that the system of particles is contained in an enclosed space and has the definite energy content E. Let us find the most probable distribution of the particles. For this the necessary condition will be
d lOg W = - Na£(log Wa + l)BWa
- N6Z(log wb + l)dwb • • • =0. (114)
In carrying out our variation, however, the number of particles of
* The idea of successive orders of infinitesimals which permit the differential region dx dy dz d^x d\f/y d\J/t to contain a large number of particles is a familiar one in mathematics.
Chaotic Motion of a System of Particles. 107
each kind must remain constant so that we have the added relations 2dwa = 0, 25wb = 0, etc. (115)
Finally, since the energy is to have a definite value E, it must also remain constant in the variation, which will provide still a further relation. Since the energy of a particle will be a definite function of its position and momentum,* let us write the energy of the system in the form
E = NaZWaEa + N^w^Eb •+•••,
where Ea is the energy of a particle in the region la, etc.
Since in carrying out our variation the energy is to remain con stant, we have the relation
E = Na2EadWa + Nb?EbdWb + '" = 0. (116)
Solving the simultaneous equations (114), (115), (116) by familiar methods we obtain
log Wa + 1+ \Ea + M6 = 0,
log wb + 1 + \Eb + M6 = 0, etc.,
where X, na, M&, etc., are undetermined constants. (It should be specially noticed that X is the same constant in each of the series of equations.)
Transforming we have
wa = aae~hEa,
Wb = abe-hEb, (117)
etc.,
as the expressions which determine the chance that a given particle of mass ma, mb, etc., will fall in a given region dx dy dz d\f/x d\[/v d\l/f, when we have the distribution of maximum probability. It should be noticed that h, which corresponds to the X of the preceding equa tions, is the same constant in all of the equations, while aa, ab, etc., are different constants, depending on the mass of the particles ma, mb, etc.
* We thus exclude from our considerations systems in which the potential energy depends appreciably on the relative positions of the independent particles.
108 Chapter Eight.
97. The Energy as a Function of the Momentum. Ea, Eb, etc., are of course functions of x, y, z, \[/x, \l/y, \f/e. Let us now obtain an expression for Ea in terms of these quantities. If there is no external field of force acting, the energy of a particle Ea will be independent of x, y} and z, and will be determined entirely by its velocity and mass. In accordance with the theory of relativity we shall have*
where ma is the mass of the particle at rest.
Let us now express Ea as a function of \px) \f/y, \j/t.
We have from our equations (105) and (98), which were used for defining momentum
max
Constructing the similar expressions for \j/y and \f/z we may write the relation
2 + f + Z2) mltf
1--
-*• o
which also defines
* This expression is that for the total energy of the particle, including that internal energy m0c2 which, according to relativity theory, the particle has when it is at rest. (See Section 75.) It would be just as correct to substitute for Ea in
equation (117) the value of the kinetic energy mac2 / — — 1 \ instead of the
total energy — a , since the two differ merely by a constant maci which would
11 -?
be taken care of by assigning a suitable value to aa.
Chaotic Motion of a System of Particles. 109
By simple transformations and the introduction of equation (118) we obtain the desired relation
Ea = cV^2 + ma2c2. (120)
98. The Distribution Law. We may now rewrite equations (117) in the form
Wa = aa<
,.— AC^^/Z+OTftSc* flO]}
Wb = &b6 i \*-£L)
etc.
These expressions determine the probability that a given particle of mass ma, mb, etc. will fall in a given region dxdydzd\f/xd\t/yd\j/z, and correspond to Maxwell's distribution law in ordinary mechanics. We see that these probabilities are independent of the position x, y, z* but dependent on the momentum.
aae~Ac^*2+ma2c2 is the probability that a given particle will fall in a particular six-dimensional cube of volume dxdydzd\f/xd\f/yd\f/z. Let us now introduce, for convenience, a new quantity aae~hc *1+m»2c2 which will be the probability per unit volume that a given particle will have the six dimensional location in question, the constants aa and aa standing in the same ratio as the volumes dxdydzd\f/xd\[/yd\f/z and unity.
We may then write
wa = Wb =
etc.
Since every particle must have components of momentum lying between minus and plus infinity, and lie somewhere in the whole volume V occupied by the mixture, we have the relation
£00 /~+oo /»+ I I _j «/_ 00 «/ — oo
(122)
It is further evident that the average value of any quantity A which depends on the momentum of the particles is given by the * This is true only when, as assumed, no external field of force is acting.
110 Chapter Eight.
expression
[A]av. = V + ° r ' r°aae-hc^+^Adtxdtvdtz, (123)
— oo *—
where A is some function of \l/x, \j/y, and \J/Z.
99. Polar Coordinates. We may express relations corresponding to (122) and (123) more simply if we make use of polar coordinates. Consider instead of the elementary volume d\l/xd^yd\^z the volume \f/z $meddd<j>d$ expressed in polar coordinates, where
The probability that a particle raa will fall in the region dxdydz\l/z sm0d6d(j)d\f/ will be
dxdydz\f/2 sin
and since each particle must fall somewhere in the space x y z \l/x \f/v \pz we shall have corresponding to (122) the relation
XTT /»2n- /»» Jo Jo
(124)
Corresponding to equation (123), we also see that the average value of any quantity A, which is dependent on the momentum of the molecules of mass ma, will be given by the expression
[A]av. = 47rF {* aae-****"** Apd*. (125)
Jo
100. The Law of Equipartition. We may now obtain a law which corresponds to that of the equipartition of vis viva in the classical mechanics. Considering equation (124) let us integrate by parts, we obtain
f*^a Jo 3
- he)
Chaotic Motion of a System of Particles. Ill
Substituting the limits into the first term we find that it becomes zero and may write
r Jo
+ m02c2
But by equation (125) the left-hand side of this relation is the
\f/zc average value of . =• for the particles of mass ma. We have
ma2c2
Introducing equation (119) which defines t2, we may transform this expression into
= I • (126)
Since we have shown that h is independent of the mass of the particles, we see that the average value of — .— is the same for particles
of all different masses. This is the principle in relativity mechanics that corresponds to the law of the equipartition of vis viva in the classical mechanics. Indeed, for low velocities the above expression reduces to mQu2, the vis viva of Newtonian mechanics, a fact which affords an illustration of the general principle that the laws of New tonian mechanics are always the limiting form assumed at low veloci ties by the more exact formulations of relativity mechanics.
We may now call attention in passing to the fact that this quantity
, whose value is the same for particles of different masses, is
2 A.
not the relativity expression for kinetic energy, which is given rather
by the formula c5
77*0
. So that in relativity mechanics
112 Chapter Eight.
the principle of the equipartition of energy is merely an approximation. We shall later return to this subject.
101. Criterion for Equality of Temperature. For a system of par ticles of masses ma, w&, etc., enclosed in the volume V, and having the definite energy content E, we have shown that
and
are the respective probabilities that given particles of mass ma or mass w& will have momenta between \f/ and \t/ + d\j/. Suppose now we consider a differently arranged system in which we have Na par ticles of mass ma by themselves in a space of volume Va and Nb particles of mass ra& in a contiguous space of volume Vb, separated from Va by a partition which permits a transfer of energy, and let the total energy of the double system be, as before, a definite quantity E (the energy content of the partition being taken as negligible). Then, by reasoning entirely similar to that just employed, we can obviously show that
and
are now the respective probabilities that given particles of mass ma or mass mb will have momenta between \J/ and \J/ + d\j/, the only changes in the expressions being the substitution of the volumes Va and Vb in the place of the one volume V. Furthermore, this distribution law will evidently lead as before to the equality of the average values of
m,t*J
and
Since, however, the spaces containing the two kinds of particles are in thermal contact, their temperature is the same. Hence we find that
the equality of the average values of — . w the necessary condition for
u2
v1-*
equality of temperature.
Chaotic Motion of a System of Particles. 113
The above distribution law also leads to the important corollary that for any given system of particles at a definite temperature the momenta and hence the total energy content is independent of the volume.
We may now . proceed to the derivation of relations which will
permit us to show that the important quantity . is directly
proportional to the temperature as measured on the absolute ther- modynamic temperature scale.
102. Pressure Exerted by a System of Particles. We first need to obtain an expression for the pressure exerted by a system of N particles enclosed in the volume V. Consider an element of surface dS perpendicular to the X axis, and let the pressure acting on it be p. The total force which the element dS exerts on the particles that impinge will be pdS, and this will be equal to the rate of change of the momenta in the X direction of these particles.*
Now by equation (122) the total number of particles having momenta between \f/x and \l/x + d^x in the positive direction is
/*\l/x+d\!ix /»+» /»
NV I
J\l/x J— oo J —
But xdS gives us the volume which contains the number of particles having momenta between \I/X and \px + d\f/x which will reach dS in a second. Hence the number of such particles which impinge per second will be
xds r****** r+o° r+c
-—
V J^ J-ao J-ao
and their change in momentum, allowing for the effect of the rebound, will be
2NdS
^x+d^z /»+» /» J J-ao J-
Finally, the total change in momentum per second for all particles can be found by integrating for all possible positive values of \f/x.
* The system is considered dilute enough for the mutual attractions of the particles to be negligible in their effect on the external pressure. 9
114 Chapter Eight.
Equating this to the total force pdS we have
Xoo /*+oo /*+oo I I ae-hc^^^ J—fO J— 00
Cancelling dS, multiplying both sides of the equation by the volume V, changing the limits of integration and substituting m0x for \l/x,
we have
/*+°o /•*-}- oo /»-fc t/ — oo *J—ao *J—ao
But this by equation (123) reduces to p7 = N
1 - 1 ~c2
or, since
Wnli2 m0X2 ...^
=== + ~. —
we have from symmetry
N f -m^/2 1
(127)
Since at a given temperature we have seen that the term in parenthesis is independent of the volume and the nature of the particles, we see that the laws of Boyle and Avogadro hold also in relativity mechanics for a system of particles.
For slow velocities equation (127) reduces to the familiar expression
N
pV = — (m0w2)av..
103. The Relativity Expression for Temperature. We are now in a position to derive the relativity expression for temperature. The thermodynamic scale of temperature may be defined in terms of the efficiency of a heat engine. Consider a four-step cycle performed with a working substance contained in a cylinder provided with a piston. In the first step let the substance expand isothermally and
Chaotic Motion of a System of Particles. 115
reversibly, absorbing the heat Q2 from a reservoir at temperature T2; in the second step cool the cylinder down at constant volume to TI', in the third step compress to the original volume, giving out the heat Qi at temperature TI, and in the fourth step heat to the original temperature. Now if the working substance is of such a nature that the heat given out in the second step could be used for the reversible heating of the cylinder in the fourth step, we may define the absolute
Tz Q2
temperatures T2 and T\ by the relation — = 77- .*
1 1 Ui
Consider now such a cycle performed on a cylinder which con tains one of our systems of particles. Since we have shown (Section 101) that at a definite temperature the energy content of such a system is independent of the volume, it is evident that our working substance fulfils the requirement that the heat given out in the second step shall be sufficient for the reversible heating in the last step. Hence, in accordance with the thermodynamic scale, we may measure
T 0
the temperatures of the two heat reservoirs by the relation — = 77-
1 1 vi
and may proceed to obtain expressions for Q2 and Qi.
In order to obtain these expressions we may again make use of the principle that the energy content at a definite temperature is inde pendent of the volume. This being true, we see that Q2 and Qi must be equal to the work done in the changes of volume that take place respectively at T2 and T7!, and we may write the relations
= f
Jv
Q! = pdV(zt rO-
«/F
But equation (127) provides an expression for p in terms of F, leading on integration to the relations
N
* We have used this cycle for defining the thermodynamic temperature scale instead of the familiar Carnot cycle, since it avoids the necessity of obtaining an expression for the relation between pressure and volume in an adiabatic expansion.
116
Chapter Eight.
N
<*-¥
which gives us on division
^2
Qi
We see that the absolute temperature measured on the thermodynamic scale is proportional to the average value of
We may finally express our temperature in the same units custom arily employed by comparing equation (127)
PF=?
with the ordinary form of the gas law
pV = nRT,
where n is the number of mols of gas present. We evidently obtain
(128)
T =
|
N |
mQuz |
||
|
3 |
W-?] |
av. |
|
|
N |
" TOoM2 " |
1 |
" m0w2 |
|
3nR |
VT-f_ |
3/b av. |
U- |
where the quantity — , which may be called the gas constant for a single molecule, has been denoted, as is customary, by the letter k.
Chaotic Motion of a System of Particles. 117
Remembering the relation | — p^ = - , we have
C" J av.
fcr = r. (129)
104. The Partition of Energy. We have seen that our new equi- partition law precludes the possibility of an exact equipartition of energy. It becomes very important to see what the average energy of a particle of a given mass does become at any temperature.
Equation (125) provides a general expression for the average value of any property of the particles. For the average value of the energy 4- mQ2c2 of particles of mass mQ (see equation 120) we shall have
foe ae-hcV**+mWc^^ 4- _
The unknown constant a may be eliminated with the help of the relation (124)
X00
and for h we may substitute the value given by (129), which gives us the desired equation
f " J~5 ~z
[E]&v. = * ^ (130)
Jo
105. Partition of Energy for Zero Mass. Unfortunately, no gen eral method for the evaluation of this expression seems to be available. For the particular case that the mass m0 of the particles approaches zero compared to the momentum, the expression reduces to
fV
Jo
118 Chapter Eight.
in terms of integrals whose values are known. Evaluating, we obtain
[£]„. = 3fcT.
For the total energy of N such particles we obtain
E =-- ZNkT,
nR and introducing the relation k = — by which we defined k we have
E = ZnRT (131)
as the expression for the energy of n mols of particles if their value of m0 is small compared with their momentum.
It is instructive to compare this with the ordinary expression of Newtonian mechanics
E = ~nRTt
2t
which undoubtedly holds when the masses are so large and the veloci ties so small that no appreciable deviations from the laws of New tonian mechanics are to be expected. We see that for particles of very small mass the average kinetic energy at any temperature is twice as large as that for large particles at the same temperature. It is also interesting to note that in accordance with equation (131) a mol of particles which approach zero mass at the absolute zero, would have a mass of
3 X 8.31 X 300
grams at room temperature (300° absolute). This suggests a field of fascinating if profitless speculation.
106. Approximate Partition of Energy for Particles of any Desired Mass. For particles of any desired mass we may obtain an approxi mate idea of the relation between energy and temperature by ex panding the expression for kinetic energy into a series. For the aver age kinetic energy of a particle we have
- mo
Chaotic Motion of a System of Particles. 119
Expanding into a series we obtain for the total kinetic energy of N particles
3 u4 15 u6 105 u8 \
+ -- + ~- + ~-+...), (132)
where u2, u4, etc., are the average values of u2, u4, etc., for the indi vidual particles.
To determine approximately how the value of K varies with the temperature we may also expand our expression (128) for temperature,
into a series; we obtain
Combining expressions (132) and (133) by subtraction and trans position, we obtain
For the case of velocities low enough so that u4 and higher powers
can be neglected, this reduces to the familiar expression of Newtonian
3 mechanics, K = -nRT.
In case we neglect in expression (134) powers higher than u4 we have the approximate relation
8c2 " 2Nm0c2 V 2
the left-hand term really being the larger, since the average square of a quantity is greater than the square of its average. Since ( — ^— J
(3 V
- nRT ] , we may write the approxima-
120 Chapter Eight.
tion
or, noting that Nm0 = M, the total mass of the system at the abso lute zero, we have
K
2 8
If we use the erg as our unit of energy, R will be 8.31 X 107; expressing velocities in centimeters per second, c2 will be 1021, and M will be the mass of the system in grams.
For one mol of a monatomic gas we should have in ergs
K = 12.4 X 107r + ^ 10-6?72. M
In the case of the electron M may be taken as approximately 1/1800. At room temperature the second term of our equation would be entirely negligible, being only 3.5 X 10~6 per cent of the first, and still be only 3.5 X 10~4 per cent in a fixed star having a temperature of 30,000°. Hence at all ordinary temperatures we may expect the law of the equipartition of energy to be substantially exact for par ticles of mass as small as the electron.
Our purpose in carrying through the calculations of this chapter has been to show that a very important and interesting problem in the classical mechanics can be handled just as easily in the newer mechanics, and also to point out the nature of the modifications in existing theory which will have to be introduced if the later develop ments of physics should force us to consider equilibrium relations for particles of mass much smaller than that of the electron.
We may also call attention to the fact that we have here con sidered a system whose equations of motion agree with the principles of dynamics and yet do not lead to the equipartition of energy. This is of particular interest at a time when many scientists have thought that the failure of equipartition in the hohlraum stood in necessary conflict with the principles of dynamics.
CHAPTER IX.
THE PRINCIPLE OF RELATIVITY AND THE PRINCIPLE OF LEAST ACTION.
It has been shown by the work of Helmholtz, J. J. Thomson, Planck and others that the principle of least action is applicable in the most diverse fields of physical science, and is perhaps the most general dynamical principle at our disposal. Indeed, for any system whose future behavior is determined by the instantaneous values of a number of coordinates and their time rate of change, it seems possible to throw the equations describing the behavior of the system into the form prescribed by the principle of least action. This generality of the principle of least action makes it very desirable to develop the relation between it and the principle of relativity, and we shall obtain in this way the most important and most general method for deriving the consequences of the theory of relativity. We have already developed in Chapter VII the particular application of the principle of least action in the case of a system of particles, and with the help of the more general development which we are about to present, we shall be able to apply the principle of relativity to the theories of elasticity, of thermodynamics and of electricity and magnetism.
107. The Principle of Least Action. For our purposes the prin ciple of least action may be most simply stated by the equation
" (BH + W)dt = 0. (135)
This equation applies to any system whose behavior is determined by the values of a number of independent coordinates <Ai4>203 • • • and their rate of change with the time <#>i02<^3 • • •, and the equation describes the path by which the system travels from its configuration at any time ti to its configuration at any subsequent time f2.
H is the so-called kinetic potential of the system and is a func tion of the coordinates and their generalized velocities :
H = F(0i020s • • • Mifa • • •)• (136)
121
122 Chapter Nine.
dH is the variation of H at any instant corresponding to a slightly displaced path by which the system might travel from the same initial to the same final state in the same time interval, and W is the external work corresponding to the variation 5 which would be done on the system by the external forces if at the instant in question the system should be displaced from its actual configuration to its con figuration on the displaced path. Thus
W = $i«0i + $25<£2 + <M</>3 + • • •, (137)
where $1, <£2, etc., are the so-called generalized external forces which act in such a direction as to increase the values of the corresponding coordinates.
The form of the function which determines the kinetic potential H depends on the particular nature of the system to which the principle of least action is being applied, and it is one of the chief tasks of general physics to discover the form of the function in the various fields of mechanical, electrical and thermodynamic investigation. As soon as we have found out experimentally what the form of H is for any particular field of investigation, the principle of least action, as expressed by equation (135), becomes the basic equation for the mathematical development of the field in question, a development which can then be carried out by well-known methods.
The special task for the theory of relativity will be to find a general relation applicable to any kind of a system, which shall connect the value of the kinetic potential H as measured with respect to a set of coordinates S with its value Hf as measured with reference to another set of coordinates Sf which is in motion relative to S. This relation will of course be of such a nature as to agree with the principle of the relativity of motion, and in this way we shall introduce the principle of relativity at the very start into the fundamental equation for all fields of dynamics.
Before proceeding to the solution of that problem we may put the principle of least action into another form which is sometimes more convenient, by obtaining the equations for the motion of a system in the so-called Lagrangian form.
108. The Equations of Motion in the Lagrangian Form. To ob tain the equations of motion in the Lagrangian form we may evidently
Relativity and the Principle of Least Action. 123
rewrite our fundamental equation (135) in the form
^(dH dH dH dH
( TT 5*i + ^T 5^2 + ' ' ' + TT d*1 + ^T 5^2 + ' ' ' Ji! \ofa 0^2 0^1 d02
+ $i5</>i + ^502 + • • • ) dt = 0. We have now, however,
d d
60i = jt (60!), 6<£2 = ^ (602), etc.,
which gives us
or, since 60i, 602, etc., are by hypothesis zero at times £1 and Z2, we obtain
etc. On substituting these expressions in (138) we obtain
and since the variations of 0i, fa, etc., are entirely independent and the limits of integration ti and tz are entirely at our disposal, this equation will be true only when each of the following equations is true. And these are the equations of motion in the desired Lagrangian
124 Chapter Nine.
form,
^d# _d# at d(f)i d(f)\
d_m_BH_ = ^ (139>
dt d(f)2 C/02
etc.
In these equations H is the kinetic potential of a system whose state is determined by the generalized coordinates </>i, $2, etc., and their time derivatives <£i, 02, etc., where 3>i, <£2, etc., are the gener alized external forces acting on the system in such a sense as to tend to increase the values of the corresponding generalized coordinates.
109. Introduction of the Principle of Relativity. Let us now in vestigate the relation between our dynamical principle and the prin ciple of the relativity of motion. To do this we must derive an equa tion for transforming the kinetic potential H for a given system from one set of coordinates to another. In other words, if S and S' are two sets of reference axes, Sf moving past S in the X-direction with the velocity Y, what will be the relation between H and H', the values for the kinetic potential of a given system as measured with reference to S and S'f
It is evident from the theory of relativity that our fundamental equation (135) must hold for the behavior of a given system using either set of coordinates S or £', so that both of the equations
f 2 (dH + W)dt = 0 and f ' (BHf + W')dt' = 0 (140) J t-i Jtif
or
f 2 (dH + W)dt = f 2 (dH' + W')dt' = 0
J t-i vti*
must hold for a given process, where it will be necessary, of course, to choose the limits of integration ti and tz, t\ and t2' wide enough apart so that for both sets of coordinates the varied motion will be completed within the time interval. Since we shall find it possible
now to show that in general f Wdt = f Wdt', we shall be able to obtain from the above equations a simple relation between H and H'.
110. Relation between f W'dt' and / Wdt. To obtain the desired
Relativity and the Principle of Least Action. 125
proof we must call attention in the first place to the fact that all kinds of force which can act at a given point must be governed by the same transformation equations when changing from system S to system S'. This arises because when two forces of a different nature are of such a magnitude as to exactly balance each other and produce no acceleration for measurements made with one set of coordinates they must evidently do so for any set of coordinates (see Chapter IV, Section 42). Since we have already found transformation equations for the force acting at a point, in our consideration of the dynamics of a particle, we may now use these expressions in general for the
evaluation / W'dt'.
W is the work which would be done by the external forces if at any instant t' we should displace our system from its actual con figuration to the simultaneous configuration on the displaced path. Hence it is evident that / W'dt will be equal to a sum of terms of the
type
(Fx'Sxf + Fy'6y' + Fz'8,')dt',
where Fx, Fy', Fz', is the force acting at a given point of the system and 8xf, by', bz' are the displacements necessary to reach the corre sponding point on the displaced path, all these quantities being measured with respect to S'.
Into this expression we may substitute, however, in accordance with equations (61), (62), (63) and (13), the values
c2
F K~I ' = ±V '
C~ (141)
dt' =
126 Chapter Nine.
We may also make substitutions for dx', dy' and dz' in terms of dx, dy and dz, but to obtain transformation equations for these quanti ties is somewhat complicated owing to the fact that positions on the actual and displaced path, which are simultaneous when measured with respect to S', will not be simultaneous with respect to S. We have denoted by t' the time in system S' when the point on the actual path has the position x', yf, z' and simultaneously the point on the displaced path has the position (xr + dx'), (yf + 5yr), (z' + dz')9 when measured in system S', or by our fundamental transformation equations (9), (10) and (11) the positions K(X' + Vt'), y', z' and K([X' + dx'] + Vt'), (y' + dy'), (z' + dz') when measured in system S. If now we denote by tA and tD the corresponding times in system S we shall have, by our fundamental transformation equation (12),
and we see that in system S the point has reached the displaced position at a time later than that of the actual position by the amount
KV b -tA=--—f 8x >
and, since during this time-interval the displaced point would have moved, neglecting higher-order terms, the distances
•jr*. v
KV . . .*V
these quantities must be subtracted from the coordinates of the displaced point in order to obtain a position on the displaced path which will be simultaneous with tA as measured in system S. We obtain for the simultaneous position on the displaced path
xV vV
K([X' + dxf] + Vt') - K -5- 8x'9 y' + dyf - K y— xft
c c
Z' + dZf - K dXf,
Relativity and the Principle of Least Action. 127
and for the corresponding position on the actual path
K(x' + vf), y', *',
and obtain by subtraction
8x =
By = 5y' - K 5x'} (142)
zV
5z = dz' - K — &c'. c2
Substituting now these equations, together with the other trans formation equations (1.41), in our expression we obtain
i
" " 1 "
(143)
-F- te"
We thus see that we must always have the general equality
/ W'dt' = f Wdt. (144)
111. Relation between H' and H. Introducing this equation into our earlier expression (140) we obtain as a general relation between H' and H
f dH'dt' = f dHdt. (145)
Restricting ourselves to systems of such a nature that we can
128 Chapter Nine.
assign them a definite velocity u = xi + yj + ^k, we can rewrite this expression in the following form, where by H& and HA we denote the values of the kinetic potential respectively on the displaced and actual paths
J SH'dt' - J HJdt' - f HA'dt' = J HA ( 1 - (*
+ dt
c -* = HDdt- HAdt,
•and hence obtain for such systems the simple expression
zj/ _ H
( xV\
K(1-^)
I y/2 I ~tf
Noting the relation between \l 1 — — and \/ 1 — — given in equation
» c * c
V
(17), this can be rewritten
TTf TT
£1 £1
(146)
and this is the expression which we shall find most useful for our future development of the consequences of the theory of relativity. Expressing the requirement of the equation in words we may say
TT
that the theory of relativity requires an invariance of • . --- in the
Lorentz transformation.
112. As indicated above, the use of this equation is obviously restricted to systems moving with some perfectly definite velocity u. Systems satisfying this condition would include particles, infinitesimal portions of continuous systems, and larger systems in a steady state.
113. Our general method of procedure in different fields of investi gation will now be to examine the expression for- kinetic potential which is known to hold for the field in question, provided the velocities involved are low and by making slight alterations when necessary,
Relativity and the Principle of Least Action. 129
see if this expression can be made to agree with the requirements of equation (146) without changing its value for low velocities. Thus it is well known, for example, that, in the case of low velocities, for a single particle acted on by external forces the kinetic potential may be taken as the kinetic energy im0w2. For relativity mechanics, as will be seen from the developments of Chapter VII, we may take for
the kinetic potential, — m^~ ^ 1 — — 2 , an expression which, except for
an additive constant, becomes identical with %m0y? at low velocities, and which at all velocities agrees with equation (146).
10
CHAPTER X. THE DYNAMICS OF ELASTIC BODIES.
We shall now treat with the help of the principle of least action the rather complicated problem of the dynamics of continuous elastic media. Our considerations will extend the appreciation of the inti mate relation between mass and energy which we found in our treat ment of the dynamics of a particle. We shall also be able to show that the dynamics of a particle may be regarded as a special case of the dynamics of a continuous elastic medium, and to apply our considerations to a number of other important problems.
114. On the Impossibility of Absolutely Rigid Bodies. In the older treatises on mechanics, after considering the dynamics of a particle it was customary to proceed to a discussion of the dynamics of rigid bodies. These rigid bodies were endowed with definite and nuchangeable size and shape and hence were assigned five degrees of freedom, since it was necessary to state the values of five variables completely to specify their position in space. As pointed out by Laue, however, our newer ideas as to the velocity of light as a limiting value will no longer permit us to conceive of a continuous body as having only a finite number of degrees of freedom. This is evident since it is obvious that we could start disturbances simultaneously at an indefinite number of points in a continuous body, and as these disturbances cannot spread with infinite velocity it will be necessary to give the values of an infinite number of variables in order com pletely to specify the succeeding states of the system. For our newer mechanics the nearest approach to an absolutely rigid body would of course be one in which disturbances are transmitted with the velocity of light. Since, then, the theory of relativity does not permit rigid bo-dies we may proceed at once to the general theory of deformable bodies.
PART I. STRESS AND STRAIN.
115. Definition of Strain. In the more familiar developments of the theory of elasticity it is customary to limit the considerations to
130
Dynamics of Elastic Bodies. 131
the case of strains small enough so that higher powers of the dis placements can be neglected, and this introduces considerable simpli fication into a science which under any circumstances is necessarily one of great complication. Unfortunately for our purposes, we cannot in general introduce such a simplification if we wish to apply the theory of relativity, since in consequence of the Lorentz shortening a body which appears unstrained to one observer may appear tre mendously compressed or elongated to an observer moving with a different velocity. The best that we can do will be arbitrarily to choose our state of zero deformation such that the strains will be small when measured in the particular system of coordinates S in which we are specially interested.
A theory of strains of any magnitude was first attempted by Saint-Venant and has been amplified and excellently presented by Love in his Treatise on the Theory of Elasticity, Appendix to Chapter I. In accordance with this theory, the strain at any point in a body is completely determined by six component strains which can be defined by the following equations, wherein (u, v, w) is the displacement of a point having the unstrained position (z, y, z} :
_ du
€yy=d-V + * <^V • (d"V ' (~ dw
(148)
dw dv du du dv dv dw dw
~dy + 'dz + dy to* dy d* 9y dz*
dw du du du dv_ dv_ dw dw fo+fc+tote+te~d~zdx dz' du du du du dv dv dw dw
It will be seen that these expressions for strain reduce to those familiar in the theory of small strains if such second-order quantities as.
.or — — can be neglected. dx J dy dz
132 Chapter Ten.
116. A physical significance for these strain components will be obtained if we note that it can be shown from geometrical considera tions that lines which are originally parallel to the axes have, when strained, the elongations
ex = VI + 2exx - 1,
(149)
and that the angles between lines originally parallel to the axes are given in the strained condition by the expressions
cos 0u =
cos 9XZ = **-=, (150)
COS VXy =
Geometrical considerations are also sufficient to show that in case the strain is a simple elongation of amount e the following equa tion will be true:
™ €yz €xz €xy — P i I,,* n ci \
~ ~ 2 '
I2 ~ m* ~ n* ~ 2mn 2ln ~ 2lm
where I, m, n are the cosines which determine the direction of the elongation.
117. Definition of Stress. We have just considered the expres sions for the strain at a given point in an elastic medium; we may now define stress in terms of the work done in changing from one .state of strain to another. Considering the material contained in unit volume when the body is unstrained, we may write, for the work done by this material on its surroundings when a change in strain takes place,
Dynamics of Elastic Bodies. 133
dW = — 8E = txx8exx + tvyd€yy + £zz5e*2
(152)
and this equation serves to define the stresses trx, tyv, etc. In case the strain varies from point to point we must consider of course the work done per unit volume of the unstrained material. In case the strains are small it will be noticed that the stresses thus defined are identical with those used in the familiar theories of elasticity.
118. Transformation Equations for Strain. We must now prepare for the introduction of the theory of relativity into our considerations, by determining the way the strain at a given point P appears to ob servers moving with different velocities. Let the point P in question be moving with the velocity u = xi + 2/j + zk as measured in sys tem S. , Since the state of zero deformation from which to measure strains can be chosen perfectly arbitrarily, let us for convenience take the strain as zero as measured in system S, giving us
€xi = fyy = *zz = tyz = *xz = *xy = 0. (153)
What now will be the strains as measured by an observer moving along with the point P in question? Let us call the system of coordi nates used by this observer S°. It is evident now from our considera tions as to the shape of moving systems presented in Chapter V that in system S° the material in the neighborhood of the point in question will appear to have been elongated in the direction of motion in the
/ ~u* ratio of 1 : -x/l — — . Hence in system S° the strain will be an elonga-
» c
tion
« = - 4= - 1 (154)
in the line determined by the direction cosines
m = — , u
(155)
We may now calculate from this elongation the components of strain by using equation (151). We obtain
134
Chapter Ten.
vz
2c2
c2
(156)
xy
1 -
and these are the desired equations for the strains at the point P, the accent ° indicating that they are measured with reference to a system of coordinates S° moving along with the point itself.
119. Variation in the Strain. We shall be particularly interested in the variation in the strain as measured in S° when the velocity experiences a small variation Su, the strains remaining zero as mea sured in S. For the sake of simplicity let us choose our coordinates in such a way that the X-axis is parallel to the original velocity, so that our change in velocity will be from u = xi to
u + <5u = (x + dx)i + dyj
Taking Su small enough so that higher orders can be neglected, and noting that y = z = 0, we shall then have, from equations (156),
Dynamics of Elastic Bodies. 135
6e°xx = 7 o\, ~5 &c, 6e vu = 0,
5e%2 = 0, 5e%, = 0, (157)
5e°" = 7~ ~7^\7zd^ 8e°*v = 7 ^\~28V-
We shall also be interested in the variation in the strain as measured in S° produced by a variation in the strain as measured in S. Con sidering again for simplicity that the X-axis is parallel to the motion of the point, we must calculate the variation produced in e°xx, e°vv, etc., by changing the values of exx, eyy, etc., from zero to 5exx, 8eyy, etc.
The variation dexx will produce a variation in e°xx whose amount can be calculated as follows: By equations (149) a line which has unit length and is parallel to the X-axis in the unstrained condition will have when strained the length A/1 + 2exx when measured in system S and Vl + 2e°xx when measured in system S°. Since the strain in system S is small, the line remains sensibly parallel to the X-axis, which is also the direction of motion, and these quantities will be connected in accordance with the Lorentz shortening by the equation
Vl + 2exx = A/1 - -J VT+ 2c°xx. (158)
* c
Carrying out now our variation 5exx, neglecting exx in comparison with larger quantities and noting that except for second order quanti ties,
1
tif VA^^;
^ c^" we obtain
s o gc«'
(160)
Since the variations deyv, 5ezz, 8ey, affect only lines which are at right angles to the direction of motion, we may evidently write
8e°yy = 8€VV) de°zz = 5ezz, 5e%2 = &„,. (161)
136 Chapter Ten.
To calculate 8e°xz we may note that in accordance with equations (150) we must have
cos 6XZ =
Vl + 2«M Vl + 2e,~, '
C°S9°"=Vl + 2eCvi + 2e°,-
where 0X2 is the angle between lines which in the unstrained condition are parallel to the X and Z axes respectively. In accordance with the Lorentz shortening, however, we shall have
cos exz = - - cos X2.
Introducing this relation, remembering that exx = e22 = t°zz = 0, and noting equation (159), we obtain
(162)
(>-*)
and similarly
('-!')'
We may now combine these equations (160), (161), (162) and (163) with those for the variation in strain with velocity and obtain the final set which we desire:
1 x 1
—„ 5x
de yy — deyy, 8e°zz = 5ezz,
dt°yZ = 5tyZ,
"o-?r (»-«
Dynamics of Elastic Bodies. 137
These equations give the variation in the strain measured in system S° at a point P moving in the X direction with velocity u, provided the strains are negligibly small as measured in S.
PART II. INTRODUCTION OF THE PRINCIPLE OF LEAST ACTION.
120. The Kinetic Potential for an Elastic Body. We are now in a position to develop the mechanics of an elastic body with the help of the principle of least action. In Newtonian mechanics, as is well known, the kinetic potential for unit volume of material at a given point P in an elastic body may be put equal to the density of kinetic energy minus the density of potential energy, and it is obvious that our choice for kinetic potential must reduce to that value at low velocities. Our choice of an expression for kinetic potential is further more limited by the fundamental transformation equation for kinetic potential which we found in the last chapter
(146)
Taking these requirements into consideration, we may write for the kinetic potential per unit volume of the material at a point P moving with the velocity u the expression
H = -
where E° is the energy as measured in system S° of the amount of material which in the unstrained condition (i. e., as measured in system S) is contained in unit volume.
The above expression obviously satisfies our fundamental trans formation equation (146) and at low velocities reduces in accordance with the requirements of Newtonian mechanics to
H = JroV - E°, provided we introduce the substitution made familiar by our previous
ElO
work, m° = — .
138
Chapter Ten.
121. Lagrange's Equations. Making use of this expression for the kinetic potential in an elastic body, we may now obtain the equations of motion and stress for an elastic body by substituting into Lagrange's equations (139) Chapter IX.
Considering the material at the point P contained in unit volume in the unstrained condition, we may choose as our generalized co ordinates the six component strains exx, eyy, etc., with the corre sponding stresses — txxj — tyy, etc., as generalized forces, and the three coordinates x, y, z which give the position of the point with the corresponding forces Fx, Fy and Fz.
It is evident that the kinetic potential will be independent of the time derivatives of the strains, and if we consider cases in which E° is independent of position, the kinetic potential will also be inde pendent of the absolute magnitudes of the coordinates x, y and 2. Substituting in Lagrange's equations (139), we then obtain
-£(-
«
(165)
dtdx\
dtd$\
d d dt'dl
1 - -„ =
1 - —„ =
(166)
Dynamics of Elastic Bodies. 139
We may simplify these equations, however; by performing the indicated differentiations and making suitable substitutions, we have
dE°xx dE°xxde
dexx de x£ dexx
But in accordance with equation (152) we may write
dE°
_ _ _ /
-j O l XX
0€ xx
and from equations (164) we may put
de°xx
dexx
2 '
Making the substitutions in the first of the Lagrangian equations we obtain
-
V1 - *
122. Transformation Equations for Stress. Similar substitutions can be made in all the equations of stress, and we obtain as our set of transformation equations
(167)
123. Value of E°. With the help of these transformation equations for stress we may calculate the value of E0, the energy content, as measured in system S°, of material which in the unstrained condition is contained in unit volume.
Consider unit volume of the material in the unstrained condition and call its energy content w°°. Give it now the velocity u = x, keeping its state of strain unchanged in system S. Since the strain
140
Chapter Ten.
is not changing in system S, the stresses txx, etc., will also be constant in system S. In system S°, however, the component strain will change in accordance with equations (156) from zero to
x* 1
xx 2c2/ _w_2V
V1 ~c2;
and the corresponding stress will be given at any instant by the expression just derived,
txx being, as we have just seen, a constant. We may then write for E° the expression
1 xz
~2c2
E
--^d
Noting that it = x we obtain on integration,
po _
(168)
as the desired expression for the energy as measured in system S° contained in the material which in system S is unstrained and has unit volume.
124. The Equations of Motion in the Lagrangian Form. We are now in a position to simplify the three Lagrangian equations (166) for FXJ Fy and Fz. Carrying out the indicated differentiation we have
u* dE° c2 dx
and introducing the value of E° given by equation (168) we obtain
w°° + t,
(169)
Dynamics of Elastic Bodies.
141
Simple calculations will also give us values for Fv and Fg. We have from (166)
_y_ I _^2aE
~ 2 c2 ' \ c2 a?>
But since we have adapted our considerations to cases in which the direction of motion is along the X-axis, we have y = 0; furthermore we may substitute, in accordance with equations (152), (157) and (167),
dE
dy
1
- t:
dy
We thus obtain as our three equations of motion
w°°-t-txxx
Fy=dt(txy^)>
(170)
In these equations the quantities Fx, Fy and F2 are the components of force acting on a particular system, namely that quantity of material which at the instant in question has unit volume. Since the volume of this material will in general be changing, Fx, Fv and F, do not give us the force per unit volume as usually defined. If we represent, however, by fx, fv and fg the components of force per unit volume, we may rewrite these equations in the form
V - -
^x ~ dt
(171)
142
Chapter Ten.
where by 67 we mean a small element of volume at the point in question.
125. Density of Momentum. Since we customarily define force as equal to the time rate of change of momentum, we may now write for the density of momentum g at a point in an elastic body which is moving in the X direction with the velocity u = x
txx X
(172)
It is interesting to point out that there are components of momen tum in the Y and Z directions in spite of the fact that the material at the point in question is moving in the X direction. We shall later see the important significance of this discovery.
126. Density of Energy. It will be remembered that the forces whose equations we have just obtained are those acting on unit volume of the material as measured in system S, and hence we are now in a position to calculate the energy density of our material. Let us start out with unit volume of our material at rest, with the energy content w°° and determine the work necessary to give it the velocity u = x without change in stress or strain. Since the only component of force which suffers displacement is Fx, we have
+ (w°° +
|
' ' W°° + txx X |
j |
xdt, 1 x ' |
|
1 u* c IV1-* •txx) r xd Jo |
||
|
v/i-p"1. |
(173)
as an expression for the energy density of the elastic material.
127. Summary of Results Obtained from the Principle of Least Action. We may now tabulate for future reference the results ob tained from the principle of least action.
Dynamics of Elastic Bodies. 143
At a given point in an elastic medium which is moving in the X direction with the velocity u = x, we have for the components of stress
t\* i *
u- ,---2, *„-
V1-^
(167) i° t°
<> XZ I XV
I I/'
V1-^
c* \ c2
For the density of energy at the point in question we have
w = W r-^-^f - tx*. (173)
For the density of momentum we have
g* = — 7== ~2» Sv = **v-2, g* = ^V' (172>
y1--?
PART III. SOME MATHEMATICAL RELATIONS.
Before proceeding to the applications of these results which we have obtained from the principle of least action, we shall find it de sirable to present a number of mathematical relations which will later prove useful.
128. The Unsymmetrical Stress Tensor t. We have defined the components of stress acting at a point by equation (152)
where 5W is the work which accompanies a change in strain and is performed on the surroundings by the amount of material which was contained in unit volume in the unstrained state. Since for con venience we have taken as our state of zero strain the condition of the body as measured in system S, it is evident that the components ^*x, tvv, etc., may be taken as the forces acting on the faces of a unit cube of material at the point in question, the first letter of the sub-
144 . Chapter Ten.
script indicating the direction of the force and the second subscript the direction of the normal to the face in question.
Interpreting the components of stress in this fashion, we may now add three further components and obtain a complete tensor
i.
t = -\tyx tyy tyg, (174)
The three new components tyx, tzx, tzy are forces acting on the unit cube, in the directions and on the faces indicated by the sub scripts. A knowledge of their value was not necessary for our develop ments of the consequences of the principle of least action, since it was possible to obtain an expression for the work accompanying a change in strain without their introduction. We shall find them quite im portant for our later considerations, however, and may proceed to determine their value.
tyx is the force acting in the Y direction tangentially to a face of the cube perpendicular to the X-axis, and measured with a system of coordinates S. Using a system of coordinates S° which is stationary with respect to the point in question, we should obtain, for the measure ment of this force,
.o t yx -
in accordance with our transformation equation for force (62), Chapter VI. Similarly we shall have the relation
t°Xy = txy.
In accordance with the elementary theory of elasticity, however, the forces t°yx and t°xy which are measured by an observer moving with the body will be connected by the relation
/° ^ vx
* xy •
t°xy being larger than t°yx in the ratio of the areas of face upon which they act. Combining these three equations, and using similar methods
Dynamics of Elastic Bodies. 145
for the other quantities, we can obtain the desired relations
^*- I1 -?)*»" ^ = (l-^r)u U-<f» (175)
We see that t is an unsymmetrical tensor.
129. The Symmetrical Tensor p. Besides this unsymmetrical ten sor t we shall find it desirable to define a further tensor p by the
equation
p = t + gu. (176)
We shall call gu the tensor product of g and u and may indicate tensor products in general by a simple juxtaposition of vectors, gu is itself a tensor with components as indicated below:
f gxux gxuy gxuz,
gu = J gvux gyuy gyuz, (177)
I gzux gzuy gzuz.
Unlike t, p will be a symmetrical tensor, since we may show, by substitution of the values for g and u already obtained, that
Pyx = Pxy, Pzx = Pxz, Pzy = Pvz- (178)
Consider for example the value of pyx; we have from our definition
Pyx = tyx + QyUx,
and by equations (175) and (172) we have
Ux
Qy ~ xy '
and hence by substitution obtain
Pyx = *
We also have, however, by definition
and since for the case we are considering uy = 0, we arrive at the equality
Pxy = Pyx-
The other equalities may be shown in a similar way. 11
146 Chapter Ten.
130. Relation between div t and tn. At a given point P in our elastic body we shall define the divergence of the tensor t by the equa tion
dtzx dtzy dtzz
^"~*T"~dF/ '
where i, j and k are unit vectors parallel to the axes, div t thus being an ordinary vector. It will be seen that div t is the elastic force acting per unit volume of material at the point P.
Considering an element of surface dS, we shall define a further vector tn by the equation
tn = (txx COS a + txy COS ]8 + txz COS 7)!
+ (tyx cos a + tyy cos )8 + tyg cos 7)j (180)
+ (tzx cos a + tzy cos j8 + tzz cos 7)k,
where cos a, cos /3 and cos 7 are the direction cosines of the inward- pointing normal to the element of surface dS.
Considering now a definite volume V enclosed by the surface S, it is evident that div t and tn will be connected by the relation
• JdivW = ftndS, (181)
where the symbol 0 indicates that the integration is to be taken over the whole surface which encloses the volume V. This equation is of course merely a direct application of Gauss's formula, which states in general the equality
= I (P cos a + Q cos |8 + R cos y)dS, J%
where P, Q and R may be any functions of x, y and z.
Dynamics of Elastic Bodies. 147
We shall also find use for a further relation between div t and tn. Consider a given point of reference 0, and let r be the radius vector to any point P in the elastic body; we can then show with the help of Gauss's Formula (182) that
- J (r X div t)dV = J" (r X tJdS
• f[(ty. - UJk + (**. - Uik + (txv - tyjc)ij]dV,
where X signifies as usual the outer product. Taking account of equations (172) and (175) this can be rewritten
(183)
131. The Equations of Motion in the Eulerian Form. We saw in
sections (124) and (125) that the equations of motion in the Lagran- gian form might be written
where f is the density of force acting at any point and g is the density of momentum.
Provided that there are no external forces acting and f is pro duced solely by the elastic forces, our definition of the divergence of a tensor will now permit us to put
f = - div t, and write for our equation of motion
dg Expressing — in terms of partial differentials, and putting
d(5V)
— ,— = 57 div u dt
we obtain
148 Chapter Ten.
Our symmetrical tensor p, however, was defined by the equation (176)
p = t + gu,
and hence we may now write our equations of motion in the very beautiful Eulerian form
- div p - |5 . (184)
We shall find this simple form for the equations of motion very interesting in connection with our considerations in the last chapter.
PART IV. APPLICATIONS OF THE RESULTS.
We may now use the results which we have obtained from the principle of least action to elucidate various problems concerning the behavior of elastic bodies.
132. Relation between Energy and Momentum. In our work on the dynamics of a particle we found that the mass of a particle was equal to its energy divided by the square of the velocity of light, and hence have come to expect in general a necessary relation between the existence of momentum in any particular direction and the trans fer of energy in that same direction. We find, however, in the case of elastically stressed bodies a somewhat more complicated state of affairs than in the case of particles, since besides the energy which is transported bodily by the motion of the medium an additional quan tity of energy may be transferred through the medium by the action of the forces which hold it in its state of strain. Thus, for example, in the case of a longitudinally compressed rod moving parallel to its length, the forces holding it in its state of longitudinal compression will be doing work at the rear end of the rod and delivering an equal quantity of energy at the front end, and this additional transfer of energy must be included in the calculation of the momentum of the bar.
As a matter of fact, an examination of the expressions for momen tum which we obtained from the principle of least action will show the justice of these considerations. For the density of momentum in the X direction we obtained the expression
/v«
Qx = (w + txx) — , c
Dynamics of Elastic Bodies. 149
and we see that in order to calculate the momentum in the X direc tion we must consider not merely the energy w which is being bodily carried along in that direction with the velocity z, but also must take into account the additional flow of energy which arises from the stress txx. .As we have already seen in Section 128, this stress txx can be thought of as resulting from forces which act on the front and rear faces of a centimeter cube of our material. Since the cube is moving with the velocity x, the force on the rear face will do the work txxx per second and this will be given up at the forward face. We thus have an additional density of energy-flow in the X direction of the magnitude txxx and hence a corresponding density of momen-
txxx turn — — .
c2
Similar considerations explain the interesting occurrence of com ponents of momentum in the Y and Z directions,
in spite of the fact that the material involved is moving in the X direction. The stress txy, for example, can be thought of as resulting from forces which act tangentially in the X direction on the pair of faces of our unit cube which are perpendicular to the Y axis. Since the cube is moving in the X direction with the velocity x, we shall have the work txyx done at one surface per second and transferred to the other, and the resulting flow of energy in the X direction is ac-
txvx companied by the corresponding momentum — — .
133. The Conservation of Momentum. It is evident from our previous discussions that we may write the equation of motion for an elastic medium in the form
fBV =
where g is the density of momentum at any given point and f is the force acting per unit volume of material. We have already obtained , from the principle of least action, expressions (172) which permit the calculation of g in terms of the energy density, stress and velocity at the point in question, and our present problem is to discuss some what further the nature of the force f .
150 Chapter Ten.
We shall find it convenient to analyze the total force per unit volume of material f into those external forces iexl. like gravity, which are produced by agencies outside of the elastic body and the internal force tint- which arises from the elastic interaction of the parts of the strained body itself. It is evident from the way in which we have defined the divergence of a tensor (179) that for this latter we may
write
fin*. = - div t. (185)
Our equation of motion then becomes
(iext. -divt)67 = ~^, (186)
(MI or, integrating over the total volume of the elastic body,
ft~..dV- fdivtdV = ~fgdV = ^j-, (187)
where G is the total momentum of the body. With the help of the purely analytical relation (181) we may transform the above equation into
s-f, (188)
where tn is defined in accordance with (180) so that the integral
XtndS becomes the force exerted by the surroundings on the sur face of the elastic body.
In the case of an isolated system both fext. and tn would evidently be equal to zero and we have the principle of the conservation of momentum.
134. The Conservation of Angular Momentum. Consider the radius vector r from a point of reference O to any point P in an elastic body; then the angular momentum of the body about O will be
M = / (r X g)dV, and its rate of change will be
Dynamics of Elastic Bodies. 151
Substituting equation (186), this may be written
X f«t.W -- (r X div t)dV + (u X
or, introducing the purely mathematical relation (183) we have,
C
X fea)dV + (r X t.)dS. (190)
r/TVT C
-# = J
We see from this equation that the rate of change of the angular momentum of an elastic body is equal to the moment of the external forces acting on the body plus the moment of the surface forces.
In the case of an isolated system this reduces to the important principle of the conservation of angular momentum.
135. Relation between Angular Momentum and the Unsymmetrical Stress Tensor. The fact that at a point in a strained elastic medium there may be components of momentum at right angles to the motion of the point itself, leads to the interesting conclusion that even in a state of steady motion the angular momentum of a strained body will in general be changing.
This is evident from equation (189), in the preceding section, which may be written
In the older mechanics velocity u and momentum g were always in the same direction so that the last term of this equation became zero. In our newer mechanics, however, we have found (172) components of momentum at right angles to the velocity and hence even for a body moving in a straight line with unchanging stresses and velocity we find that the angular momentum is increasing at the rate
(192)
and in order to maintain the body in its state of uniform motion we must apply external forces with a turning moment of this same amount.
The presence of this increasing angular momentum in a strained body arises from the unsymmetrical nature of the stress tensor, the inte gral j (u X g)dV being as a matter of fact exactly equal to the integral
152
Chapter Ten.
over the same volume of the turning moments of the unsymmetrical components of the stress. Thus, for example, if we have a body mov ing in the X direction with the velocity u = xi we can easily see from equations (172) and (175) the truth of the equality
(u X g) = [(tv, -
(txg -
(txy - tyt)ij].
136. The Right-Angled Lever. An interesting example of the
principle that in general a turning
_^ 7 *^
1 moment is needed for the uniform translatory motion of a strained body is seen in the apparently paradoxical case of the right-angled lever.
Consider the right-angled lever shown in figure 14. This lever is sta tionary with respect to a system of coordinates S°. Referred to S° the two lever arms are equal in length :
FIG. 14.
and the lever is in equilibrium under the action of the equal forces
Let us now consider the equilibrium as it appears, using a system of coordinates S with reference to which the lever is moving in X direction with the velocity V. Referred to this new system of co ordinates the length li of the arm which lies in the Y direction will be the same as in system S°, giving us
Zi = Zi°.
But for the other arm which lies in the direction of motion we shall have, in accordance with the Lorentz shortening,
For the forces Ft and F2 we shall have, in accordance with our equa-
Dynamics of Elastic Bodies. 153
tions for the transformation of force (61) and (62),
771 771
r 2 = r We thus obtain for the moment of the forces around the pivot B
V*\ Po,ol/2_F,^
• <? ) ~Fltl c> Flh~*>
and are led to the remarkable conclusion that such a moving lever will be in equilibrium only if the external forces have a definite turning moment of the magnitude given above.
The explanation of this apparent paradox is obvious, however, in the light of our previous discussion. In spite of the fact that the lever is in uniform motion in a straight line, its angular momentum is continually increasing owing to the fact that it is elastically strained, and it can be shown by carrying out the integration indicated in equation (192) that the rate of change of angular momentum is as a
V2 matter of fact just equal to the turning moment FA ~ •
0
v2
This necessity for a turning moment FA — can also be showi:
directly from a consideration of the energy flow in the lever. Since the force FI is doing the work F\V per second at the point A, a stream of energy of this amount is continually flowing through the lever from A to the pivot B. In accordance with our ideas as to the rela tion between energy and mass, this new energy which enters at A each
second has the mass — ^- , and hence each second the angular mo mentum of the system around the point B is increased by the amount
FiV V2
~& l ~ rl c2 '
We have already found, however, exactly this same expression for the moment of the forces around the pivot B and hence see that they are of just the magnitude necessary to keep the lever from turning, thus solving completely our apparent paradox.
154
Chapter Ten.
137. Isolated Systems in a Steady State. Our considerations have shown that the density of momentum is equal to the density of energy flow divided by the square of the velocity of light. If we have a system which is in a steady internal state, and is. either isolated or merely subjected to an external pressure with no components of force tangential to the bounding surface, it is evident that the resultant flow of energy for the whole body must be in the direction of motion, and hence for these systems momentum and velocity will be in the same direction without the complications introduced by a trans verse energy flow.
Thus for an isolated system in a steady internal state we may write,
(193)
'-5
138. The Dynamics of a Particle. It is important to note that particles are interesting examples of systems in which there will obviously be no transverse component of energy flow since their infinitesimal size precludes the action of tangential surface forces. We thus see that the dynamics of a particle may be regarded as a special case of the more general dynamics which we have developed in this chapter, the equation of motion for a particle being
m
dt
in agreement with the work of Chapter VI.
139. Conclusion. We may now point out in conclusion the chief results of this chapter. With the help of Einstein's equations for spatial and temporal considerations, we have developed a set of transformation equations for the strain in an elastic body. Using the components of strain and velocity as generalized coordinates, we then introduced the principle of least action, choosing a form of function
Dynamics of Elastic Bodies. 155
for kinetic potential which agrees at low velocities with the choice made in the older theories of elasticity and at all velocities agrees with the requirements of the principle of relativity. Using the Lagrangian equations, we were then able to develop all that is neces sary for a complete theory of elasticity.
The most important consequence of these considerations is an extension in our ideas as to the relation between momentum and energy. We find that the density of momentum in any direction must be placed equal to the total density of energy flow in that same direction divided by the square of the velocity of light; and we find that we must include in our density of energy flow that transferred through the elastic body by the forces which hold it in its state of strain and suffer displacement as the body moves. This involves in general a flow of energy and hence momentum at right angles to the motion of the body itself.
At present we have no experiments of sufficient accuracy so that we can investigate the differences between this new theory of elasticity and the older ones, and hence of course have found no experimental contradiction to the new theory. It will be seen, however, from the expressions for momentum that even at low velocities the conse quences of this new theory will become important as soon as we run across elastic systems in which very large stresses are involved. It is also important to show that a theory of elasticity can be de veloped which agrees with the requirements of the theory of relativity. In fairness, it must, however, be pointed out in conclusion that since our expression for kinetic potential was not absolutely uniquely deter mined there may also be other theories of elasticity which will agree with the principle of relativity and with all the facts as now known.
CHAPTER XI. THE DYNAMICS OF A THERMODYNAMIC SYSTEM.
We may now use our conclusions as to the relation between the principle of least action and the theory of relativity to obtain informa tion as to the behavior of thermodynamic systems in motion.
140. The Generalized Coordinates and Forces. Let us consider a thermodynamic system whose state is defined by the generalized coordinates volume v, entropy S and the values of x, y and z which determine its position. Corresponding to these coordinates we shall have the generalized external forces, the negative of the pressure, — p, temperature, T, and the components of force, Fx, Fy and Fz. These generalized coordinates and forces are related to the energy change 8E accompanying a small displacement 6, in accordance with the equation
5E = - 5W = - pdv + TSS + Fxdx + FySy + F,5z. (194)
141. Transformation Equation for Volume. Before we can apply the principle of least action we shall need to have transformation equations for the generalized coordinates, volume and entropy.
In accordance with the Lorentz shortening, we may write the following expression for the volume v of the system in terms of v° as measured with a set of axes $° with respect to which the system is stationary :
x2 + y2 + z2
c2
where u is the velocity of the system.
By differentiation we may obtain expressions which we shall find useful,
5 = -^=, (195)
dV° V
~Q^ == 7 I7«r\l To = 7 r9~\ ~^> '
156
Dynamics of a Thermodynamic System. 157
142. Transformation Equation for Entropy. As for the entropy of a thermodynamic system, this is a quantity which must appear the same to all observers regardless of their motion. This invariance of entropy is a direct consequence of the close relation between the entropy of a system in a given state and the probability of that state. Let us write, in accordance with the Boltzmann-Planck ideas as to the interdependence of these quantities,
S = k log W,
where S is the entropy of the system in the state in question, fc is a universal constant, and W the probability of having a microscopic arrangement of molecules or other elementary constituent parts which corresponds to the desired thermodynamic state. Since this prob ability is evidently independent of the relative motion of the observer and the system we see that the entropy of a system S must be an invariant and may write
S = S°. '(197)
143. Introduction of the Principle of Least Action. The Kinetic Potential. We are now in a position to introduce the principle of least action into our considerations by choosing a form of function for the kinetic potential which will agree at low velocities with the familiar principles of thermodynamics and will agree at all velocities with the requirements of the theory of relativity.
If we use volume and entropy as our generalized coordinates, these conditions are met by taking for kinetic potential the expression
H = - E°\l --. (198)
This expression agrees with the requirements of the theory of
IT
relativity that . shall be an invariant (see Section 111) and
at low velocities reduces to H = — E, which with our choice of coordinates is the familiar form for the kinetic potential of a thermo dynamic system.
158 Chapter Eleven.
It should be noted that this expression for the kinetic potential of a thermodynamic system applies of course only provided we pick out volume v and entropy S as generalized coordinates." If, following Helmholtz, we should think it more rational to take v as one coordinate and a quantity 0 whose time derivative is equal to temperature, 0 = T7, as the other coordinate, we should obtain of course a different expression for the kinetic potential; in fact should have under those circumstances
H = (E° - T°S
Using this value of kinetic potential, however, with the corresponding coordinates we should obtain results exactly the same as those which we are now going to work out with the help of the other set of coordi nates.
144. The Lagrangian Equations. Having chosen a form for the kinetic potential we may now substitute into the Lagrangian equa tions (139) and obtain the desired information with regard to the behavior of thermodynamic systems.
Since we shall consider cases in which the energy of the system is independent of the position in space, the kinetic potential will be independent of the coordinates x, y and 2, depending only on their time derivatives. Noting also that the kinetic potential is inde pendent of the time derivatives of volume and entropy, we shall obtain the Lagrangian equations in the simple form
Dynamics of a Thermodynamic System. 159
145. Transformation Equation for Pressure. We may use the first of these equations to show that the pressure is a quantity which appears the same to all observers regardless of their relative motion. We have
U*\ Ji U23E° Ji u*
--? = - V1 -7«"ST := " V1 -7*
dv° dv '
dE°
But, in accordance with equation (194), p° = — —5- . and in ac-
ov
cordance with equation (195),
dv° 1
dv
which gives us the desired relation
p = P°. (200)
Defining pressure as force per unit area, this result will be seen to be identical with that which is obtained from the transformation equations for force and area which result from our earliest considera tions.
146. Transformation Equation for Temperature. The second of the Lagrangian equations (199) will provide us information as to measurements of temperature made by observers moving with different velocities. We have
~ dE° dS° dS° dS '
But, in accordance with equation (194), — ^ = T° and in accordance
r) Si°
with (197) -r^r = 1. We obtain as our transformation equation, oo
~7> (201)
T
and see that the quantity . is an invariant for the Lorentz
transformation
160
Chapter Eleven.
147. The Equations of Motion for Quasistationary Adiabatic Acceleration. Let us now turn our attention to the last three of the Lagrangian equations. These are the equations for the motion of a thermodynamic system under the action of external force. It is evident, however, that these equations will necessarily apply only to cases of quasistationary acceleration, since our development of the principle of least action gave us an equation for kinetic potential which was true only for systems of infinitesimal extent or large systems in a steady internal state. It is also evident that we must confine our considerations to cases of adiabatic acceleration, since otherwise the value of E° which occurs in the expression for kinetic potential might be varying in a perfectly unknown manner.
The Lagrangian equations for force may be advantageously trans formed. We have
= - — (-
1 ~ dtdx\~
E
x__ I _ it? 3EC c2 V1 ~c2 dx
But by equations (194), (196) and (197) we have
8E
We obtain
dx
and
dx
= 0.
E
(202)
Similar equations may be obtained for the components of force in the Y and Z directions and these combined to give the vector equation
(203)
Dynamics of a Thermodynamic System. 161
This is the fundamental equation of motion for the dynamics of a thermodynamic system.
148. The Energy of a Moving Thermodynamic System. We may use this equation to obtain an expression for the energy of a moving thermodynamic system. If we adiabatically accelerate a thermo dynamic system in the direction of its motion, its energy will increase both because of the work done by the force
which produces the acceleration and because of the work done by the pressure p = p° which acts on a volume which is continually dimin ishing as the velocity u increases, in accordance with the expression
/ u2
v° A/I — — .
Hence we may write for the total energy
E = E° + -IT
E° °° ° °°
V1
pv E = -== - P°v° I - - = — - P1'- (204)
149. The Momentum of a Moving Thermodynamic System. We may compare this expression for the energy of a thermodynamic system with the following expression for momentum which is evident from the equation (203) for force:
c
We find again, as in our treatment of elastic bodies presented in the last chapter, that the momentum of a moving system may be calculated by taking the total flow of energy in the desired direction
12
162
Chapter Eleven.
and dividing by c2. Thus, comparing equations (204) and (205), we have
E
pv
(206)
E
where the term -^ u takes care of the energy transported bodily along c
pv
by the system and the term — u takes care of the energy transferred
c
in the u direction by the action of the external pressure on the rear and front end of the moving system.
150. The Dynamics of a Hohlraum. As an application of our con siderations we may consider the dynamics of a hohlraum, since a hohlraum in thermodynamic equilibrium is of course merely a special example of the general dynamics which we have just developed. The simplicity of the hohlraum and its importance from a theoretical point of view make it interesting to obtain by the present method the same expression for momentum that can be obtained directly but with less ease of calculation from electromagnetic considerations.
As is well known from the work of Stefan and Boltzmann, the energy content E° and pressure p° of a hohlraum at rest and in thermo dynamic equilibrium are completely determined by the temperature T° and volume v° in accordance with the equations
where a is the so-called Stefan's constant.
Substituting these values of E° and p° in the equation for the motion of a thermodynamic system (203), we obtain
dt
|
'4 av°T°* u' |
d |
"4 avT* u" |
|
I U2C { V1-^ J |
dt |
°(>-$r. |
(207)
as the equation for the quasistationary adiabatic acceleration of a
Dynamics of a Thermodynamic System. 163
hohlraum. In view of this equation we may write for the momentum of a hohlraum the expression
It is a fact of significance that our dynamics leads to a result for the momentum of a hohlraum which had been adopted on the ground of electromagnetic considerations even without the express intro duction of relativity theory.
CHAPTER XII. ELECTROMAGNETIC THEORY.
The Einstein theory of relativity proves to be of the greatest significance for electromagnetics. On the one hand, the new electro magnetic theory based on the first postulate of relativity obviously accounts in a direct and straightforward manner for the results of the Michelson-Morley experiment and other unsuccessful attempts to detect an ether drift, and on the other hand also accounts just as simply for the phenomena of moving dielectrics as did the older theory of a stationary ether. Furthermore, the theory of relativity provides considerably simplified methods for deriving a great many theorems which were already known on the basis of the ether theory, and gives us in general a clarified insight into the nature of electro magnetic action.
151. The Form of the Kinetic Potential. In Chapter IX we in vestigated the general relation between the principle of least action and the theory of the relativity of motion. We saw that the develop ment of any branch of dynamics would agree with the requirements of relativity provided only that the kinetic potential H has such a form
TT
that the quantity , is an invariant for the Lorentz transfor
mation. Making use of this discovery we have seen the possibility of developing the dynamics of a particle, the dynamics of an elastic body, and the dynamics of a thermodynamic system, all of them in forms which agree with the theory of relativity by merely introducing slight modifications into the older expressions for kinetic potential in
TT
such a way as to obtain the necessary invariance for
In the case of electrodynamics, however, on account of the closely interwoven historical development of the theories of electricity and relativity, we shall not find it necessary to introduce any modification
164
Electromagnetic Theory. 165
in the form of the kinetic potential, but may take for H the following expression, which is known to lead to the familiar equations of the Lorentz electron theory
(curl <t>)2 /e u\ 1
^_? _<)> . (?_|_PHU f (209)
where the integration is to extend over the whole volume of the system 7, e is the intensity of the electric field at the point in question, <(> is the value of the vector potential, p the density of charge and u its velocity.*
Let us now show that the expression which we have chosen for kinetic potential does lead to the familiar equations of the electron theory.
152. The Principle of Least Action. If now we denote by f the force per unit volume of material exerted by the electromagnetic action it is evident that we may write in accordance with the principle of least action (135)
= 0, (210)
where 5r is the variation in the radius vector to the particle under consideration, and where the integration is to be taken over the whole volume occupied by the system and between two instants of time ti and tz at which the actual and displaced configurations of the system coincide.
153. The Partial Integrations. In order to simplify this equation, we shall need to make use of two results which can be obtained by partial integrations with respect to time and space respectively.
Thus we may write
£* dt(abb} = f' ad(bb) = [a56]g - J dt\-^
* Strictly speaking this expression for kinetic potential is not quite correct, since kinetic potential must have the dimensions of energy. To complete the equa tion and give all the terms their correct dimensions, we could multiply the first term by the dielectric inductivity of free space e, and the last two terms by the magnetic permeability i*. Since, however, e and n have the numerical value unity with the usual choice of units, we shall not be led into error in our particular considerations if we omit these factors.
166 Chapter Twelve.
or, since the displaced and actual motions coincide at ti and £2,
fdt(adb) = ~ We may also write
fdV(a^) = f dydz(adb) =
or, since we are to carry out our integrations over the whole volume occupied by the system, we shall take our functions as zero at the limits of integration and may write
/"(•£ )--/"(•=)•
Since similar considerations apply to derivatives with respect to the other variables y and z, we can also obtain
/ dV a div b = - / dV b - grad a, (213)
fdVa.- curl b = / dV b - curl a. (214)
154. Derivation of the Fundamental Equations of Electromagnetic Theory. Making use of these purely mathematical relationships we are now in a position to develop our fundamental equation (210).
Carrying out the indicated variation, noting that Su = , and making use of (211) and (214) we easily obtain
+f
1 5r = 0.
In developing the consequences of this equation, it should be noted, however, that the variations are not all of them independent; thus, since we shall define the density of charge by the equation
p = div e, (216)
it is evident that it will be necessary to preserve the truth of this equation in any variation that we carry out. This can evidently be
Electromagnetic Theory. 167
done if we add to our equation (215) the expression / dtdVifr[&p - div 5e] = 0,
where ^ is an undetermined scalar multiplier. We then obtain with the help of (213)
I dtdV [ e + - ~ + grad ^ | • 5e + | curl curl <|>
(217) = 0,
and may now treat the variations 5e and 6<(> as entirely independent of the others; we must then have the following equations true
e= -i|£-grad*, (218)
curl curl <(>=- + — , (219)
c c
and have thus derived from the principle of least action the funda mental equations of modern electron theory. We may put these in their familiar form by denning the magnetic field strength h by the equation
h = curl <|> (220)
We then obtain from (219)
and, noting the mathematical identity curl grad ^ = 0, we obtain from (218)
curle= --§. (222)
C dt
We have furthermore by definition (216)
div e = p, (223)
and noting equation (220) may write the mathematical identity
div h = 0. (224)
168 Chapter Twelve.
These four equations (221—4) are the familiar expressions which have been made the foundation of modern electron theory. They differ from Maxwell's original four field equations only by the intro duction in (221) and (223) of terms which arise from the density of charge p of the electrons, and reduce to Maxwell's set in free space.
155. We have not yet made use of the last three terms in the fundamental equation (217) which results from the principle of least action. As a matter of fact, it can be shown that these terms can be transformed into the expression
fdtdV [f * - f [uX curl <H* + p grad * + f J -Sr;
(225)
and hence lead to the familiar fifth fundamental equation of modern electron theory,
(226)
f = p{e+ ["xh]*J.
The transformation of the last three terms of (217) into the form given above (225) is a complicated one and it has not seemed neces sary to present it here since in a later paragraph we shall show the possibility of deriving the fifth fundamental equation of the electron theory (226) by combining the four field equations (221-4) with the transformation equations for force already obtained from the principle of relativity. The reader may carry out the transformation himself, however, if he makes use of the partial integrations which we have already obtained, notes that in accordance with the principle of the conservation of electricity we must have dp = — divp 6r and notes
that Su = — — , where the differentiation — indicates that we are at at
following some particular particle in its motion, while the differentia-
d d<|>
tion — occurring in — indicates that we intend the rate of change
at some particular stationary point.
156. The Transformation Equations for e, h and p. We have thus shown the possibility of deriving the fundamental equations of modern
Electromagnetic Theory. 169
electron theory from the principle of least action. We now wish to introduce the theory of relativity into our discussions by presenting a set of equations for transforming measurements of e, h and p from one set of space-time coordinates S to another set S' moving past S in the X-direction with the velocity V. This set of equations is as follows :
ex = ex,
(227)
z' = K ( ez +— hy J ,
ez = hx' = h hy' = «
&•/ « i
. ' - (229)
where K has its customary significance
As a matter of fact, this set of transformation equations fulfills all the requirements imposed by the theory of relativity. Thus, in the first place, it will be seen, on development, that these equations are themselves perfectly symmetrical with respect to the primed and unprimed quantities except for the necessary change from + V to — V. In the second place, it will be found that the substitution of these equations into our five fundamental equations for electro magnetic theory (221-2-3-4-6) will successfully transform them into an entirely similar set with primed quantities replacing the unprimed ones. And finally it can be shown that these equations agree with the general requirement derived in Chapter IX that the
170 Chapter Twelve.
TT
quantity — . shall be an invariant for the Lorentz trans-
1 -
formation.
rr
To demonstrate this important invariance of — . we may
point out that by introducing equations (220), (221) and (214), our original expression for kinetic potential
2 (curl «t»)2 +--Y- ->
can easily be shown equal to
and, noting that our fundamental equations for space and time pro vide us with the relation
we can easily show that our transformation equations for e and h do lead to the equality
H Hr
tf I u>*
1-^ \l—3
We thus know that our development of the fundamental equations for electromagnetic theory from the principle of least action is indeed in complete accordance with the theory of relativity, since it conforms with the general requirement which was found in Chapter IX to be imposed by the theory of relativity on all dynamical considerations.
157. The Invariance of Electric Charge. As to the significance of the transformation equations which we have presented for e, h and p, we may first show, in accordance with the last of these equations, that a given electric charge will appear the same to all observers no matter what their relative motion.
Electromagnetic Theory. 171
To demonstrate this we merely have to point out that, by intro ducing equation (17), we may write our transformation equation for p (229) in the form
which shows at once that the two measurements of density of charge made by 0 and 0' are in exactly the same ratio as the corresponding measurements for the Lorentz shortening of the charged body, so that the total charge will evidently measure the same for the two observers.
We might express this invariance of electric charge by writing the equation
Q' = Q. (231)
It should be noted in passing that this result is in entire accord with the whole modern development of electrical theory, which lays increasing stress on the fundamentally and indivisibility of. the electron as the natural unit quantity of electricity. On this basis the most direct method of determining the charge on an electrified body would be to count the number of electrons present and this number must obviously appear the same both to observer 0 and observer 0'.*
158. The Relativity of Magnetic and Electric Fields. As to the significance of equations (227) and (228) for transforming the values of the electric and magnetic field strengths from one system to another, we see that at a given point in space we may distinguish between the electric vector e = ex\ + evj + ezk as measured by our original observer 0 and the vector e' = ex'i -f- evfj -f ez'k as measured in units of his own system by an observer 0' who is moving past 0 with the velocity V in the .XT-direction. Thus if 0 finds in an unvarying electromagnetic field that Qe is the force on a small test charge Q which is stationary with respect to his system, 0' will find experi-
* A similar invariance of electric charge has been made fundamental in the author's development of the theory of similitude (i. e., the theory of the relativity of size). See for example Phys. Rev., vol. 3, p. 244 (1914).
172 Chapter Twelve.
mentally for a similar test charge that moves along with him a value for the force Qef, where e' can be calculated from with the help of these equations (227). Similar remarks would apply to the forces which would act on magnetic poles.
These considerations show us that we should now use caution in speaking of a pure electrostatic or pure magnetic field, since the description of an electromagnetic field is determined by the particular choice of coordinates with reference to which the field is measured.
159. Nature of Electromotive Force. We also see that the " elec tromotive " force which acts on a charge moving through a magnetic field finds its interpretation as an " electric " force provided we make use of a system of coordinates which are themselves stationary with respect to the charge. Such considerations throw light on such ques tions, for example, as to the seat of the " electromotive " forces in " homopolar " electric dynamos where there is relative motion of a conductor and a magnetic field.
Derivation of the Fifth Fundamental Equation.
160. We may now make use of this fact that the forces acting on a moving charge of electricity may be treated as purely electrostatic, by using a set of coordinates which are themselves moving along with the charge, to derive the fifth fundamental equation of electromagnetic theory.
Consider an electromagnetic field having the values e and h for the electric and magnetic field strengths at some particular point. What will be the value of the electromagnetic force f acting per unit volume on a charge of density p which is passing through the point in question with the velocity u?
To solve the problem take a system of coordinates Sf which itself moves with the same velocity as the charge, for convenience letting the X-axis coincide with the direction of the motion of the charge. Since the charge of electricity is stationary with respect to this system, the force acting on it as measured in units of this system will be by definition equal to the product of the charge by the strength of the electric field as it appears to an observer in this system, so that we may write
F' = Q'e',
Electromagnetic Theory. 173
or
Fx' = Q'ef', Fyf = Q'evf, Fzf = Q'e,'.
For the components of the electrical field ex, ey', ez', we have just obtained the transformation equations (227), while in our earlier dynamical considerations in Chapter VI we obtained transformation equations (61), (62), and (63) for the components of force. Sub stituting above and bearing in mind that ux = V, uy = uz = 0, and that Q' = Q, we obtain on simplification
Fx = Qex,
Fy = Q ( ey - y hz J ;
r. -«(•• + 7 *i)i
which in vectorial form gives us the equation
F = or for the force per unit volume
f = pe + [uxh]*. (226)
This is the well-known fifth fundamental equation of the Maxwell- Lorentz theory of electromagnetism. We have already indicated the method by which it could be derived from the principle of least action. This derivation, however, from the transformation equations, provided by the theory of relativity, is particularly simple and attractive.
Difference between the Ether and the Relativity Theories of Electro-
magnetism.
161. In spite of the fact that we have now found five equations which can be used as a basis for electromagnetic theory which agree with the requirements of relativity and also have exactly the same form as the five fundamental equations used by Lorentz in building up the stationary ether theory, it must not be supposed that the relativity and ether theories of electromagnetism are identical. Al though the older equations have exactly the same form as the ones which we shall henceforth use, they have a different interpretation, since our equations are true for measurements made with the help of any non-accelerated set of coordinates, while the equations of
174 Chapter Twelve.
Lorentz were, in the first instance, supposed to be true only for mea surements which were referred to a set of coordinates which were stationary with respect to the assumed luminiferous ether. Suppose, for example, we desire to calculate with the help of equation (226),
the force acting on a charged body which is moving with the velocity u; we must note that for the stationary ether theory, u must be the velocity of the charged body through the ether, while for us u may be taken as the velocity past any set of unaccelerated coordinates, pro vided e and h are measured with reference to the same set of co ordinates. It will be readily seen that such an extension in the mean ing of the fundamental equations is an important simplification.
162. A word about the development from the theory of a stationary ether to our present theory will not be out of place. When it was found that the theory of a stationary ether led to incorrect con clusions in the case of the Michelson-Morley experiment, the hypo thesis was advanced by Lorentz and Fitzgerald that the failure of that experiment to show any motion through the ether was due to a con traction of the apparatus in the direction of its motion through the
/ 172
ether in the ratio 1 : A/I — — . Lorentz then showed that if all sys- \ c2
terns should be thus contracted in the line of their motion through the ether, and observers moving with such system make use of suitably contracted meter sticks and clocks adjusted to give what Lorentz called the " local time," their measurements of electromagnetic phenomena could be described by a set of equations which have nearly the same form as the original four field equations which would be used by a stationary observer. It will be seen that Lorentz was thus making important progress towards our present idea of the com plete relativity of motion. The final step could not be taken, however, without abandoning our older ideas of space and time and giving up the Galilean transformation equations as the basis of kinematics. It was Einstein who, with clearness and boldness of vision, pointed out that the failure of the Michelson-Morley experiment, and all other attempts to detect motion through the ether, is not due to a
Electromagnetic Theory. 175
fortuitous compensation of effects but is the expression of an important general principle, and the new transformation equations for kinematics to which he was led have not only provided the basis for an exact transformation of the field equations but have so completely revo lutionized our ideas of space and time that hardly a branch of science remains unaffected.
163. With regard to the present status of the ether in scientific theory, it must be definitely stated that this concept has certainly lost both its fundamentality and the greater part of its usefulness, and this has been brought about by a gradual process which has only found its culmination in the work of Einstein. Since the earliest days of the luminiferous ether, the attempts of science to increase the substantiality of this medium have met with little success. Thus we have had solid elastic ethers of most extreme tenuity, and ethers with a density of a thousand tons per cubic millimeter; we have had quasi-material tubes of force and lines of force ; we have had vibratory gyrostatic others and perfect gases of zero atomic weight; but after every debauch of model-making, science has recognized anew that a correct mathematical description of the actual phenomena of light propagation is superior to any of these sublimated material media. Already for Lorentz the ether had been reduced to the bare function of providing a stationary system of reference for the measurement of positions and velocities, and now even this function has been taken from it by the work of Einstein, which has shown that any unaccel- erated system of reference is just as good as any other.
To give up the notion of an ether will be very hard for many physicists, in particular since the phenomena of the interference and polarization of light are so easily correlated with familiar experience with wave motions in material elastic media. Consideration will show us, however, that by giving up the ether we have done nothing to destroy the periodic or polarizable nature of a light disturbance. When a plane polarized beam of light is passing through a given point in space we merely find that the electric and magnetic fields at that point lie on perpendiculars to the direction of propagation and undergo regular periodic changes in magnitude. There is no need of going beyond these actual experimental facts and introducing any hypothetical medium. It is just as simple, indeed simpler, to say
176 Chapter Twelve.
that the electric or magnetic field has a certain intensity at a given point in space as to speak of a complicated sort of strain at a given point in an assumed ether.
Applications to Electromagnetic Theory.
164. The significant fact that the fundamental equations of the new electromagnetic theory have the same form as those of Lorentz makes it of course possible to retain in the structure of modern elec trical theory nearly all the results of his important researches, care being taken to give his mathematical equations an interpretation in accordance with the fundamental ideas of the theory of relativity. It is, however, entirely beyond our present scope to make any presenta tion of electromagnetic theory as a whole, and in the following para graphs we shall confine ourselves to the proof of a few theorems which can be handled with special ease and directness by the methods intro duced by the theory of relativity.
165. The Electric and Magnetic Fields around a Moving Charge. Our transformation equations for the electromagnetic field make it very easy to derive expressions for the field around a point charge in uniform motion. Consider a point charge Q moving with the velocity V. For convenience consider a system of reference S such that Q is moving along the X-axis and at the instant in question, t = 0, let the charge coincide with the origin of coordinates 0. We desire now to calculate the values of electric field e and the magnetic field h at any point in space xt y, z.
Consider another system of reference, S', which moves along with the same velocity as the charge Q, the origin of coordinates 0' ', and the charge always coinciding in position. Since the charge is sta tionary with respect to their new system of reference, we shall have the electric field at any point x', y', z' in this system given by the equations
Qx'
ex' =
e' =
(V2 + y'2 + z'1)31* ' Qy'
(x* + y'2 + z'2)"* ' Qz'
'**?*'
Electromagnetic Theory. 177
while the magnetic field will obviously be zero for measurements made in system £', giving us
hx' = 0,
A/ = 0,
A,' = 0.
Introducing our transformation equations (9), (10) and (11) for xf, y' and z' and our transformation equations (227) and (228) for the electric and magnetic fields and substituting i = 0, we obtain for the values of e and h in system S at the instant when the charge passes through the point 0,
Or/
hx = 0,
V v c
hz = - ey,
or, putting s for the important quantity ^ x2 + ( 1 - -^ j (?/2 + s2) and writing the equations in the vectorial form where we put
r = (x\ + yj + zk),
we obtain the familiar equations for the field around a point charge 13
178 Chapter Twelve.
in uniform motion with the velocity u = V in the X-direction
(232)
h = - [u X e].* (233)
c
166. The Energy of a Moving Electromagnetic System. Our
transformation equations will permit us to obtain a very important expression for the energy of an isolated electromagnetic system in terms of the velocity of the system and the energy of the same system as it appears to an observer who is moving along with it.
Consider a physical system surrounded by a shell which is im permeable to electromagnetic radiation. This system is to be thought of as consisting of the various mechanical parts, electric charges and electromagnetic fields which are inside of the impermeable shell. The system is free in space, except that it may be acted on by external electromagnetic fields, and its energy content thus be changed.
Let us now equate the increase in the energy of the system to the work done by the action of the external field on the electric charges in the system. Since the force which a magnetic field exerts on a charge is at right angles to the motion of the charge it does no work and we need to consider only the work done by the external electric field and may write for the increase in the energy of the system
A# = //// p(exux + eyuy + ezuz)dx dy dz dt, (234)
where the integration is to be taken over the total volume of the system and over any time interval in which we may be interested.
Let us now transform this expression with the help of our trans formation equations for the electric field (227) for electric charge (229), and for velocities (14-15-16). Noting that our fundamental equations for kinematic quantities give us dx dy dz dt = dxf dyf dz' dt' , we obtain
A# = K ffff P'(ex'uxf + ev'uv' + ez'us')dx' dy' dzf dt'
+ «V ffff P' (••' + ^ h,' - ^f hy' ) dx' dy' dz' dt'.
Electromagnetic Theory. 179
Consider now a system which both at the beginning and end of our time interval is free from the action of external forces; we may then rewrite the above equation for this special case in the form
A# = K&E' + KV f 2Fx'dt',
where, in accordance with our earlier equation (234), kE' is the increase in the energy of the system as it appears to observer 0' and ZFZ' is the total force acting on the system in X-direction as measured byO'.
The restriction that the system shall be unacted on by external forces both at the beginning and end of our time interval is necessary because it is only under those circumstances that an integration between two values of t can be considered as an integration between two definite values of t', simultaneity in different parts of the system not being the same for observers 0 and 0' '.
We may now apply this equation to a specially interesting case. Let the system be of such a nature that we can speak of it as being at rest with respect to S', meaning thereby that all the mechanical parts have low velocities with respect to S' and that their center of gravity moves permanently along with S'. Under these circum stances we may evidently put j *LFt'dt' = 0 and may write the above equation in the form
/—*
V1-?
or
dE
where u is the velocity of the system, and E° is its energy as measured by an observer moving along with it. The energy of a system which is unacted on by external forces is thus a function of two variables, its energy E0 as measured by an observer moving along with the system and its velocity u.
180 Chapter Twelve.
We may now write
E = EQ + (f>(u) + const.,
where <j>(u) represents the energy of the system which depends solely on the velocity of the system and not on the changes in its EQ values. <j>(u) will thus evidently be the kinetic energy of the mechanical masses in the system which we have already found (82) to have the value
0 — m0c2 where ra0 is to be taken as the total mass of the
c2 mechanical part of our system when at rest. We may now write
E = — (m0c2 + EQ) — mQc2 + const.
u2
Or, assuming as before that the constant is equal to ra0c2, which will be equivalent to making a system which has zero energy also have zero mass, we obtain
=- (».* + *«). (235)
which is the desired expression for the energy of an isolated system which may contain both electrical and mechanical parts.
167. Relation between Mass and Energy. This expression for the energy of a system that contains electrical parts permits us to show that the same relation which we found between mass and energy for mechanical systems also holds in the case of electromagnetic energy. Consider a system containing electromagnetic energy and enclosed by a shell which is impermeable to radiation. Let us apply a force F to the system in such a way as to change the velocity of the system without changing its EQ value. We can then equate the work done per second by the force to the rate of increase of the energy of the system. We have
dE
F-u - ,. . dt
Electromagnetic Theory.
181
But from equation (235) we can obtain a value for the rate of increase
dE . . of energy —- , giving us
F-u = Fxux + Fyiiy + FzUg = ( m0 +-7 )
and solving this equation for F we obtain
E0
du dt
IT
(236)
which for low velocities assumes the form
(237)
Examination of these expressions shows that our system which contains electromagnetic energy behaves like an ordinary mechanical
Tjl
^ ) at low velocities or ra0 + —^
— =5*
system with the mass
any desired velocity u. To the energy of the system E0, part of which
is electromagnetic, we must ascribe the mass — just as we found in
the case of mechanical energy. We realize again that" matter and energy are but different names for the same fundamental entity, 1021 ergs of energy having the mass 1 gram.
The Theory of Moving Dielectrics.
168. The principle of relativity proves to be very useful for the development of the theory of moving dielectrics.
It was first shown by Maxwell that a theory of electromagnetic phenomena in material media can be based on a set of field equations, similar in form to those for free space, provided we introduce besides the electric and magnetic field strengths, E and F, two new field vectors.
182 Chapter Twelve.
the dielectric displacement D and the magnetic induction B, and also the density of electric current in the medium i. These quantities are found to be connected by the four following equations similar in form to the four field equations for free space:
(238)
curl E= -if, (239)
div D = p, (240)
div B = 0. (241)
For stationary homogeneous media, the dielectric displacement, magnetic induction and electric current are connected with the electric and magnetic field strengths by the following equations:
D = eE, (242)
B - MH, (243)
i = o-E, (244)
where e is the dielectric constant, fj, the magnetic permeability and a the electrical conductivity of the medium in question.
169. Relation between Field Equations for Material Media and Electron Theory. It must not be supposed that the four field equa tions (238-241) for electromagnetic phenomena in material media are in any sense contradictory to the four equations (221-224) for free space which we took as the fundamental basis for our development of electromagnetic theory. As a matter of fact, one of the main achieve ments of modern electron theory has been to show that the electro magnetic behavior of material media can be explained in terms of the behavior of the individual electrons and ions which they contain, these electrons and ions acting in accordance with the four fundamental field equations for free space. Thus our new equations for material media merely express from a macroscopic point of view the statistical result of the behavior of the individual electrons in the material in question. E and H in these new equations are to be looked upon as the average values of e and h which arise from the action of the individual electrons in the material, the process of averaging being so
Electromagnetic Theory. 183
carried out that the results give the values which a macroscopic ob server would actually find for the electric and magnetic forces acting respectively on a unit charge and a unit pole at the point in question. These average values, E and H, will thus pay no attention to the rapid fluctuations of e and h which arise from the action and motion of the individual electrons, the macroscopic observer using in fact differentials for time, dt, and space, dx, which would be large from a microscopic or molecular viewpoint.
Since from a microscopic point of view E and H are not really the instantaneous values of the field strength at an actual point in space, it has been found necessary to introduce two new vectors, electric displacement, D, and magnetic induction, B, whose time rate of change will determine the curl of E and H respectively. It will evidently be possible, however, to relate D and B to the actual electric and magnetic fields e and h produced by the individual electrons, and this relation has been one of the problems solved by modern electron theory, and the field equations (238-241) for material media have thus been shown to stand in complete agreement with the most modern views as to the structure of matter and electricity. For the purposes of the rest of our discussion we shall merely take these equations as expressing the experimental facts in stationary or in moving media.
170. Transformation Equations for Moving Media. Since equa tions (238 to 241) are assumed to give a correct description of electro magnetic phenomena in media whether stationary or moving with respect to our reference system S, it is evident that the equations must be unchanged in form if we refer our measurements to a new system of coordinates Sf moving past S, say, with the velocity V in the X-direction.
As a matter of fact, equations (238 to 241) can be transformed into an entirely similar set
curl E = — — TT7 ; div D' = P', div B' = 0,
184 Chapter Twelve.
provided we substitute for x, y, z and t the values of x', y', z' and tf given by the fundamental transformation equations for space and time (9 to 12), and substitute for the other quantities in question the relations
Ev' =
Ezf = Dx' = Dx,
Dz' = Hx' = Hx,
B,' = B
(246)
ix = K(IX - Vp), (247)
t,' = lz.
It will be noted that for free space these equations will reduce to the same form as our earlier transformation equations (227 to 229) since we shall have the simplifications D = E, B = H and i = pu.
We may also call attention at this point to the fact that our funda-
Electromagnetic Theory. 185
mental equations for electromagnetic phenomena (238-241) in di electric media might have been derived from the principle of least action, making use of an expression for kinetic potential which could
//E-D H-B\ dV( — - --- — 1, and it will be noticed
that our transformation equations for these quantities are such as to
IT
preserve that necessary in variance for —7= = which we found in
Chapter IX to be the general requirement for any dynamical develop ment which agrees with the theory of relativity.
171. We are now in a position to handle the theory of moving media. Consider a homogeneous medium moving past a system of coordinates S in the Jf -direction with the velocity F; our problem is to discover relations between the various electric and magnetic vectors in this medium. To do this, consider a new system of co ordinates Sf also moving past our original system with the velocity V. Since the medium is stationary with respect to this new system S' we may write for measurements referred to S' in accordance with equa tions (242 to 244) the relations
D' = eE', B' - jeH',
i' = <rE',
which, as we have already pointed out, are known experimentally to be true in the case of stationary, homogeneous media. €, ^ and a- are evidently the values of dielectric constant, permeability and con ductivity of the material in question, which would be found by an experimenter with respect to whom the medium is stationary.
Making use of our transformation equations (245 to 247) we can obtain by obvious substitutions the following set of relations for measurements made with respect to the original system of coordi nates S:
D, = eEx,
(248)
186
Chapter Twelve.
>z +—Hy = e( E2 + — By }
c \ c V
Bx =
V
BV+-ES = v,
V Bz --Ev -
(249)
(250)
172. Theory of the Wilson Experiment. The equations which we have just developed for moving media are, as a matter of fact, in complete accord with the celebrated experiment of H. A. Wilson on moving dielectrics and indeed all other experiments that have been performed on moving media.
Wilson's experiment consisted in the rotation of a hollow cylinder of dielectric, in a magnetic field which was parallel to the axis of the cylinder. The inner and outer surfaces of the cylinder were covered with a thin metal coating, and arrangements made with the help of wire brushes so that electrical contact could be made from these coatings to the pairs of quadrants of an electrometer. By reversing the magnetic field while the apparatus was in rotation it was possible to measure with the electrometer the charge produced by the electrical displacement in the dielectric. We may make use of our equations to compute the quantitative size of the effect.
FIG. 15.
Electromagnetic Theory. 187
Let figure 15 represent a cross-section of the rotating cylinder. Consider a section of the dielectric AA which is moving perpendicularly to the plane of the paper in the X-direction with the velocity V. Let the magnetic field be in the F-direction parallel to the axis of rotation. The problem is to calculate dielectric displacement Dz in the Z- direction.
Referring to equations (248) we have
and, substituting the value of Bv given by equations (249),
V „
we obtain
V
V
or, neglecting terms of orders higher than — , we have
c
Dz = eE,+- (efjL - l)Hv. (251)
For a substance whose permeability is practically unity such as Wilson actually used the equation reduces to
D, = eEz + -j (e - !)#„,
and this was found to fit the experimental facts, since measurements with the electrometer show the surface charge actually to have the magnitude Dz per square centimeter in accordance with our equation div D = p.
It would be a matter of great interest to repeat the Wilson experi ment with a dielectric of high permeability so that we could test the complete equation (251). This is of some importance since the original Lorentz theory led to a different equation,
Dz = eE2 + (e -
CHAPTER XIII. FOUR-DIMENSIONAL ANALYSIS.
173. In the present chapter we shall present a four-dimensional method of expressing the results of the Einstein theory of relativity, a method which was first introduced by Minkowski, and in the form which we shall use, principally developed by Wilson and Lewis. The point of view adopted, consists essentially in considering the properties of an assumed four-dimensional space in which intervals of time are thought of as plotted along an axis perpendicular to the three Car tesian axes of ordinary space, the science of kinematics thus becoming the geometry of this new four-dimensional space.
The method often has very great advantages not only because it sometimes leads to considerable simplification of the mathematical form in which the results of the theory of relativity are expressed, but also because the analogies between ordinary geometry and the geometry of this imaginary space often suggest valuable modes of attack. On the other hand, in order to carry out actual numerical calculations and often in order to appreciate the physical significance of the conclusions arrived at, it is necessary to retranslate the results obtained by this four-dimensional method into the language of ordinary kinematics. It must further t>e noted, moreover, that many im portant results of the theory of relativity can be more easily obtained if we do not try to employ this four-dimensional geometry. The reader should also be on his guard against the fallacy of thinking that extension in time is of the same nature as extension in space merely because intervals of space and time can both be represented by plotting along axes drawn on the same piece of paper.
174. Idea of a Time Axis. In order to grasp the method let us consider a particle constrained to move along a single axis, say OX, and let us consider a time axis OT perpendicular to OX. Then the position of the particle at any instant of time can be represented by a point in the XT plane, and its motion as time progresses by a line in the plane. If, for example, the particle were stationary, its behavior
188
Four Dimensional Analysis.
189
in time and space could be represented by a line parallel to the time axis OT as shown for example by the line ab in figure 16. A particle
A
\ Ax
FIG. 16.
dx
moving with the uniform velocity u = -r- could be represented by a
straight line etc making an angle with the time axes, and the kine- matical behavior of an accelerated particle could be represented by a curved line.
By conceiving of a /tmr-dimensional space we can extend this method which we have just outlined to include motion parallel to all three space axes, and in accordance with the nomenclature of Minkowski might call such a geometrical representation of the space- time manifold " the world," and speak of the points and lines which represent the instantaneous positions and the motions of particles as " world-points " and " world-lines."
175. Non-Euclidean Character of the Space. It will be at once evident that the graphical method of representing kinematical events which is shown by Figure 16 still leaves something to be desired. One of the most important conclusions drawn from the theory of relativity was the fact that it is impossible for a particle to move with a velocity greater than that of light, and it is evident that there is nothing in our plot to indicate that fact, since we could draw a line making any desired angle with the time axis, up to perpendicularity, and thus
190
Chapter Thirteen.
represent particles moving with any velocity up to infinity,
Ax
u
= co.
It is also evident that there is nothing in our plot to correspond to that invariance in the velocity of light which is a cornerstone of the theory of relativity. Suppose, for example, the line OC, in figure 17,
/Q
T'l
|
/ / / |
/ / |
/ ! 1 1 |
|
/ / I / |
1 1 1 1 1 1 1 1 1 1 i 1 |
FIG. 17. represents the trajectory of a beam of light with the velocity —7 = c;
there is then nothing so far introduced into our method of plotting to indicate the fact that we could not equally well make use of another set of axes OX'T' ', inclined to the first and thus giving quite a different
Ax' value, —7 , to the velocity of the beam of light.
There are a number of methods of meeting this difficulty and obtaining the invariance for the four-dimensional expression x2 + y2 + z2 — c2t2 (see Chapter IV) which must characterize our system of kinematics. One of these is to conceive of a four-dimensional Eu-
Four Dimensional Analysis. 191
clidean space with an imaginary time axis, such that instead of plotting real instants in time along this axis we should plot the quantity I = ict where i = V — 1. In this way we should obtain invariance for the quantity x2 + y2 + z2 -+- I2 = x2 + y2 + z2 — c2t2, since it may be regarded as the square of the magnitude of an imaginary four- dimensional radius vector. This method of treatment has been especially developed by Minkowski, Laue, and Sommerfeld. Another method of attack, which has been developed by Wilson and Lewis and is the one which we shall adopt in this chapter, is to use a real time axis, for plotting the real quantity ct, but to make use of a non- Euclidean four-dimensional space in which the quantity (x2 + y2 + z2 — c2t2) is itself taken as the square of the magnitude of a radius vector. This latter method has of course the disadvantages that come from using a non-Euclidean space; we shall find, however, that these reduce largely to the introduction of certain rules as to signs. The method has the considerable advantage of retaining a real time axis which is of some importance, if we wish to visualize the methods of attack and to represent them graphically.
We may now proceed to develop an analysis for this non-Euclidean space. We shall find this to be quite a lengthy process but at its completion we shall have a very valuable instrument for expressing in condensed language the results of the theory of relativity. Our method of treatment will be almost wholly analytical, and the geo metrical analogies may be regarded merely as furnishing convenient names for useful analytical expressions. A more geometrical method of attack will be found in the original work of Wilson and Lewis.
PART I. VECTOR ANALYSIS OF THE NON-EUCLIDEAN FOUR- DIMENSIONAL MANIFOLD.
176. Consider a four-dimensional manifold in which the position of a point is determined by a radius vector
r =
where ki, k2, k3 and k4 may be regarded as unit vectors along four mutually perpendicular axes and Xi, xz, xs, and x^. as the magnitudes of the four components of r along these four axes. We may identify Xi, Xz, and #3 with the three spatial coordinates of a point x, y and z
192 Chapter Thirteen.
with reference to an ordinary set of space axes and consider x± as a coordinate which specifies the time (multiplied by the velocity of light) when the occurrence in question takes place at the point xyz. We have
Xi = x, x2 = y, x3 = z, £4 = ct, (252)
and from time to time we shall make these substitutions when we wish to interpret our results in the language of ordinary kinematics. We shall retain the symbols Xi, x2, x3, and x.i throughout our develop ment, however, for the sake of symmetry.
177. Space, Time and Singular Vectors. Our space will differ in an important way from Euclidean space since we shall consider three classes of one-vector, space, time and singular vectors. Considering the coordinates Xi, xz, #3 and x± which determine the end of a radius vector, Space or y-vectors will have components such that
and we shall put for their magnitude
Time or d-vectors will have components such that
Z42 > (X? + X? + Z32),
and we shall put for their magnitude
s = z42 - x? - x22 - x32. . (254)
Singular or a-vectors will have components such that
and their magnitude will be zero.
178. Invariance of x2 + y2 + z2 — cH2. Since we shall naturally consider the magnitude of a vector to be independent of any particular choice of axes we have obtained at once by our definition of magnitude for any rotation of axes that invariance for the expression
Four Dimensional Analysis. 193
which is characteristic of the Lorentz transformation, and have thus evidently set up an imaginary space which will be suitable for plotting kinematical events in accordance with the requirements of the theory of the relativity of motion.
179. Inner Product of One-Vectors. We shall define the inner product of two one-vectors with the help of the following rules for the multiplication of unit vectors along the axes
ki-kj = k2-k2 - k3-k3 = 1, k4-k4 = -- 1, kn-km = 0. (255)
It should be noted, of course, that there is no particular sig nificance in picking out the product k4-ki as the one which is nega tive; it would be equally possible to develop a system in which the products kL • ki, k2 • k2, and k3 • k3 should be negative and k4 • k4 positive.
The above rules for unit vectors are sufficient to define completely the inner product provided we include the further requirements that this product shall obey the associative law for a scalar factor and the distributive and commutative laws, namely
= n(a-b) = (a-b)(n), a-(b + c) = a-b + a-c, (256)
a-b = b-a.
For the inner product of a one-vector by itself we shall have, in accordance with these rules,
r-r = (ziki + £2k2 + z3k3 + z4k4) • (ziki + z2k2 + z3k3 -f z4k4)
= (Zl2 + Z2* + X.2 - Z42)
and hence may use the following expressions for the magnitudes of vectors in terms of inner product
s = A/r-r for ^-vectors, s = V— r-r for 5-vectors. (258)
For curved lines we shall define interval along the curve by the equations
f ds = f Vdr • dr for ^-curves,
J _ (259)
J" ds = f V — dr - dr for 6-curves. 14
194 Chapter Thirteen.
Our rules further show us that we may obtain the space components of any one vector by taking its inner product with a unit vector along the desired axis and may obtain the time component by taking the negative of the corresponding product. Thus
r-k2 = (ziki + z2k2 + £3k3 + Z4k4)-k2 = x2,
(260) r-k3
r-k4 = (ziki + z2k2 + z3k3 + £4k4) -k4 = — z4.
We see finally moreover in general that the inner product of any pair of vectors will be numerically equal to the product of the mag nitude of either by the projection of the other upon it, the sign de pending on the nature of the vectors involved.
180. Non-Euclidean Angle. We shall define the non-Euclidean angle 6 between two vectors rx and r2 in terms of their magnitudes Si and s2 by the expressions
db TI-TZ = (si X projection s2) = SiS2 cosh 0, (261)
the sign depending on the nature of the vectors in the way indicated in the preceding section. We note the analogy between this equation and those familiar in Euclidean vector-analysis, the hyperbolic trigonometeric functions taking the place of the circular functions used in the more familiar analysis.
For the angle between unit vectors k and k' we shall have
coshfl = ± k-k', (262)
where the sign must be chosen so as to make cosh 6 positive, the plus sign holding if both are 7-vectors and the minus sign if both are 5- vectors.
181. Kinematical Interpretation of Angle in Terms of Velocity. At this point we may temporarily interrupt the development of our four-dimensional analysis to consider a kinematical interpretation of non-Euclidean angles in terms of velocity. It will be evident from our introduction that the behavior of a moving particle can be repre sented in our four-dimensional space by a 5-curve,* each point on
* It is to be noted that the actual trajectories of particles are all of them repre sented by 5-curves since as we shall see 7-curves would correspond to velocities greater than that of light.
Four Dimensional Analysis. 195
this curve denoting the position of the particle at a given instant of time, and it is evident that the velocity of the particle will be deter mined by the angle which this curve makes with the axes.
Let r be the radius vector to a given point on the curve and con sider the derivative of r with respect to the interval s along the curve; we have
dr dxi dx2 dx3 dx4
and this may be regarded as a unit vector tangent to the curve at the point in question.
If 0 is the angle between the k4 axis and the tangent to the curve at the point in question, we have by equation (262)
cosh 4> = — w • k4 = ~ ; as
making the substitutions for xlf x2, x3) and x4, in terms of x, y, z and t we may write, however,
ds = Vdz42 - dx? - dxf - dxf ~ 1 ~~ cdt>
which gives us
cosh 0 = . - = (265)
2
and by the principles of hyperbolic trigonometry we may write the further relations
u
sinh 0 = -7=^== , (266)
(267)
VECTORS OF HIGHER DIMENSIONS
182. Outer Products. We shall define the outer product of two one-vectors so that it obeys the associative law for a scalar factor, the
196 Chapter Thirteen.
distributive law and the anti-commutative law, namely,
(na) X b = w(a X b) = a X (wb),
aX(b + c)=aXb+aXc (a + b)Xc = aXc + bXc; (268) a X b = -bXa.
From a geometrical point of view, we shall consider the outer product of two one-vectors to be itself a two-vector, namely the paral lelogram, or more generally, the area which they determine. The sign of the two-vector may be taken to indicate the direction of pro gression clockwise or anti-clockwise around the periphery. In order to accord with the requirement that the area of a parallelogram deter mined by two lines becomes zero when they are rotated into the same direction, we may complete our definition of outer product by adding the requirement that the outer product of a vector by itself shall be
zero.
a X a = 0. (269)
We may represent the outer products of unit vectors along the chosen axes as follows:
ki X ki = k2 X k2 = k3 X k3 - k4 X k4 = 0,
ki X k2 = - k2 X ki = kia = - k21, (270)
ki X k3 = — k3 X ki = k^ = — k3i, etc.,
where we may regard ki2, for example, as a unit parallelogram in the plane XfiX*.
We shall continue to use small letters in Clarendon type for one- vectors and shall use capital letters in Clarendon type for two-vectors. The components of a two-vector along the six mutually perpendicular planes XiOX2, XiOX3, etc., may be obtained by expressing the one- vectors involved in terms of their components along the axes and carrying out the indicated multiplication, thus:
A = a X b = (aiki + a2k2 -f a3k3 +
X (6iki + 62k2 + 63k3 + 64k4) = (a i62 - a2&i)ki2 + (ai&3 - as&i)ki3 + (0164 - a46i)ki4
-f (a264 — a462)k24 + (a364 — a463)k34,
Four Dimensional Analysis. 197
or, calling the quantities (ai&2 — a2fri), etc., the component magni tudes of A, A 12, etc., we may write
A = A12k12 + A13k13 + 4i4k,4 + A23k23 + A24k24 + A34k,4. (272)
The concept of outer product may be extended to include the idea of vectors of higher number of dimensions than two. Thus the outer product of three one-vectors, or of a one-vector and a two-vector will be a three-vector which may be regarded as a directed parallelo- piped in our four-dimensional space. The outer product of four one- vectors will lead to a four-dimensional solid which would have direction only in a space of more than four dimensions and hence in our case will be called a pseudo-scalar. The outer product of vectors the sum of whose dimensions is greater than that of the space considered will vanish.
The results which may be obtained from different types of outer multiplication are tabulated below, where one-vectors are denoted by small Clarendon type, two-vectors by capital Clarendon type, three-vectors by Tudor black capitals, and pseudo-scalars by bold face Greek letters.
A = a X b = -bXa= (alb2 - «2&i)k12 + (a^ - a3&i)k13 + (a 164 — a46i) ku + (a263 — a362)k23 + (a264 — a462)k2i + (a364 — a463)k34,
H = c X A = (ciA23 - c2Ai3 4- c3Ai2)k123
(273) c4A23)k234,
a = A X B =
The signs in these expressions are determined by the general rule that the sign of any unit vector knmo will be reversed by each transposition of the order of a pair of adjacent subscripts, thus:
kabcd = — kbacd = kbcad, etc., ' ' ' . (274)
198 Chapter Thirteen.
183. Inner Product of Vectors in General. We have previously defined the inner product for the special case of a pair of one-vectors, in order to bring out some of the important characteristics of our non-Euclidean space. We may now give a general rule for the inner product of vectors of any number of dimensions.
The inner product of any pair of vectors follows the associative law for scalar factors, and follows the distributive and commutative laws.
Since we can express any vector in terms of its components, the above rules will completely determine the inner product of any pair of vectors provided that we also have a rule for obtaining the inner products of the unit vectors determined by the mutually perpendicular axes. This rule is as follows: Transpose the subscripts of the unit vectors involved so that the common subscripts occur at the end and in the same order and cancel these common subscripts. If both the unit vectors still have subscripts the product is zero; if neither vector has subscripts the product is unity, and if one of the vectors still has subscripts that itself will be the product. The sign is to be taken as that resulting from the transposition of the subscripts (see equa tion (274)), unless the subscript 4 has been cancelled, when the sign will be changed.
For example:
ki24-k34 = ki2-k3 = 0,
k132-ki23 = - ki23-k123 = - 1, (275)
ki24-k42 = - ki24-k24 = ki.
It is evident from these rules that we may obtain the magnitude of any desired component of a vector by taking the inner product of the vector by the corresponding unit vector, it being noticed, of course, that when the unit vector involved contains the subscript 4 we obtain the negative of the desired component. For example, we may obtain the ki2 component of a two-vector as follows:
A]2 = A-ki2 = (Ai2ki2 + A13k13 + A]4ki4
(276) + A23k23 + Asiksi + A34k34) -ki2.
184. The Complement of a Vector. In an n-dimensional space any m-dimensional vector will uniquely determine a ~new vector of
Four Dimensional Analysis. 199
dimensions (n— m) which may be called the complement of the original vector. The complement of a vector may be exactly defined as the inner product of the original vector with the unit pseudo-scalar ki23 ... n. In general, we may denote the complement of a vector by placing an asterisk * after the symbol. As an example we may write as the complement of a two-vector A in our non-Euclidean four- dimensional space:
A* = A-ki234 = (Ai2k12 + Ai3ki3 + Auku
+ A23k23 + A24k24 + Awkrf) • k1234 (277)
185. The Vector Operator, <0 or Quad. Analogous to the familiar three-dimensional vector-operator del,
v-k'i + *£, + *£;• (278)
we may define the four-dimensional vector-operator quad,
o'**£+*£+*5-*5;- (279)
If we have a scalar or a vector field we may apply these operators by regarding them formally as one-vectors and applying the rules for inner and outer multiplication which we have already given.
Thus if we have a scalar function F which varies continuously from point to point we can obtain a one-vector which we may call the four-dimensional gradient of F at the point in question by simple multiplication; we have
GradF=OF = k1g + k2g+k,g-k,g;. (280)
If we have a one-vector field, with a vector f whose value varies from point to point we may obtain by inner multiplication a scalar quantity which we may call the four-dimensional divergence of f we have
^-f + + + -. (281)
OX\ OXz 0X3 0X4
Taking the outer product with quad we may obtain a two-vector, the
200 Chapter Thirteen.
four-dimensional curl of f,
k24 +
By similar methods we could apply quad to a two-vector function F and obtain the one-vector function 0 • F and the three-vector func tion 0 X F.
186. Still regarding 0 as a one-vector we may obtain a number of important expressions containing 0 more than once; we have:
0 X (OF) = 0, (283) 0 X (0 X f) = 0, (286)
0-(0-F)=0, (284) OX(OXF)=0, (287)
O'(0-lf)=0, (285)
0:(0 xf) = O(O-f) - (O-O)f, (288)
0-(0 X F) = 0 X (0-F) + (O'O)F, (289)
0-(0 xf)= 0 x(O-JT)- (0-0)3f- (290)
The operator 0 ' 0 or O2 nas l°ng been known under the name of the D'Alembertian,
a2 d2 a2 d2 a2
(291)
From the definition of the complement of a vector given in the previous section it may be shown by carrying out the proper expansions that
(0 X0)* = 0-0*, (292)
where 0 is a vector of any number of dimensions.
187. Tensors. In analogy to three-dimensional tensors we may define a four-dimensional tensor as a quantity with sixteen components as given in the following table:
rn /TT /TT rn
JL 11 J- 12 1 13 * 14,
rji rri rn /TT
-i 21 J 22 Ji 23 J 24j
r r r r (293)
J- 31 * 32 -L 33 * 34>
Four Dimensional Analysis. 201
with the additional requirement that the divergence of the tensor, defined as follows, shall itself be a one-vector.
}
(294)
•MS*--
188. The Rotation of Axes. Before proceeding to the application of our four-dimensional analysis to the actual problems of relativity theory we may finally consider the changes in the components of a vector which would be produced by a rotation of the axes. We have already pointed out that the quantity (xi2 -f x22 + £s2 — #42) is an invariant in our space for any set of rectangular coordinates having the same origin since it is the square of the magnitude of a radius vector, and have noted that in this way we have obtained for the quantity (x2 + y2 + z2 — c2t2) the desired invariance which is charac teristic of the Lorentz transformation. In fact we may look upon the Lorentz transformation as a rotation from a given set of axes to a new set, with a corresponding re-expression of quantities in terms of the new components. The particular form of Lorentz transformation, familiar in preceding chapters, in which the new set of spatial axes has a velocity component relative to the original set, in the X-direction alone, will be found to correspond to a rotation of the axes in which only the directions of the Xi and X4 axes are changed, the X% and X3 axes remaining unchanged in direction.
Let us consider a one-vector
a = (a-iki + a2k2 + a3k3 + a4ki) = (a/k/ + a2'k2' + a/k3' + a/k/),
where ab a2, a3 and a4 are the component magnitudes, using a set of axes which have kx, k2, k3 and k4 as unit vectors and a/, a2'> as' and a/ the corresponding magnitudes using another set of mutually per pendicular axes with the unit vectors k/, k/, k/ and k/. Our problem,
202
Chapter Thirteen.
now, is to find relations between the magnitudes «i, a2, a3 and a4 and a/, a2', as' and a/.
We have already seen sections, (179) and (183), that we may obtain any desired component magnitude of a vector by taking its inner product with a unit vector in the desired direction, reversing the sign if the subscript 4 is involved. We may obtain in this way an expression for a\ in terms of a/, a/, a/ and a/. We have
Ol = a-ka = (a/k/ + a2'k2' + a3'k3' + a/k^-kx
= ai'ki' • ki + a2'k2' • ki + a/k/ • ki + a/k/ • ki. (295)
By similar multiplications with k2, k3 and k4 we may obtain expres sions for a2, a3, and — a4. The results can be tabulated in the con venient form
|
a/ |
a,' |
V |
a*' |
|
|
ai |
k/'k! |
k/.ki |
k/'ki |
fc'-k, |
|
a2 |
k/.k2 |
k/-k2 |
ka'-k, |
ki'vk, |
|
a3 |
k/-k3 |
k2'-k3 |
ks'-k. |
k/-k3 |
|
a4 |
- k/.k4 |
- k/.k4 |
- k/-k4 |
-k/-k4 |
(296)
Since the square of the magnitude of the vector, (ai2 + a22 + a23 — a42), is a quantity which is to be independent of the choice of axes, we shall have certain relations holding between the quantities k/-ki, kir-k2, etc. These relations, which are analogous to the familiar
Four Dimensional Analysis.
203
conditions of orthogonality in Euclidean space, can easily be shown to be
and
+ (k/.k,)2 + (k/.k,)2 - (k/-k4)2 = 1, + (k2'-k2)2 + (k2'-k3)2 - (k/-k4)2 = 1, + (k3'-k2)2 + (k/-k3)2 - (k/-k4)2 - 1, + (k^-ko)2 + (k/-k3)2 - (k/-k4)2 = -
H (k/.k2)(k2'.k2) + (k/-k3)(k/-k3)
(297)
=0,
etc., for each of the six pairs of vertical columns in table (296).
Since we shall often be interested in a simple rotation in which the directions of the X2 and Xs axes are not changed, we shall be able to simplify this table for that particular case by writing
k2' = k2, k3' = k3,
and noting the simplifications thus introduced in the products of the unit vectors, we shall obtain
|
ai1 |
a,' |
a,' |
04' |
|
|
a\ |
k^-k, |
0 |
0 |
ki'-k, |
|
a2 as |
0 |
1 0 |
0 |
0 |
|
0 |
1 |
0 |
||
|
0| |
-ki'-ki |
0 |
0 |
- k/-k4 |
(298)
204 Chapter Thirteen.
If now we call 0 the angle of rotation between the two time axes ' and OX4} we may write, in accordance with equation (262),
- k/-k4 = cosh </>.
Since we must preserve the orthogonal relations (297) and may also make use of the well-known expression of hyperbolic trigonometry
cosh2 <f> — sinh2
1,
we may now rewrite our transformation table in the form
|
Oi' |
a,' |
a,' |
a/ |
|
|
hi |
cosh (f) |
0 |
0 |
sinh 0 |
|
a2 |
0 |
1 |
0 |
0 |
|
«3 |
0 |
0 |
1 |
0 |
|
a4 |
sinh </> |
0 |
0 |
cosh </> |
(299)
By a similar process we may obtain transformation tables for the components of a two-vector A. Expressing A in terms of the unit vectors ki2', ki3', kH', etc., and taking successive inner products with the unit vectors ki2, ki3, ki4, etc., we may obtain transformation equations which can be expressed by the tabulation (300) shown on the following page.
Four Dimensional Analysis.
205
A 12 ki2'-ki2
k12'-k13
A14
ki3'-k12 ki4'-k12
k13'-k13 k14'-k13
A24'
k24'-k12
k24'-k13
-k12'-ki4 ;-ki/-k14 -ki4'-ki4 -k22/-k14 -k24/-ki4
k13' • k23 k14'
13 -K23 KM 'K23
-k12'-ko4
w
-k13'-k24-k14'.k24! -k2/-k24 -k2/
k3/-k12
k3/-k13
-k3/-k14; (300)
/ • k34 — ki4' • k34 — k23' • k34 — k24' • k34 — k?4r • k24
For the particular case of a rotation in which the direction of the X2 and X3 axes are not changed we shall have
k2' = k2, k3' = k3,
and very considerable simplification will be introduced. We shall have, for example,
k12'-k12 = (k/ X k^-Ckx X k2) = (k/ X k2)-(k! X k2) = k/.k^ kw'-kn = (k/ X kaO-Cki X k2) = (k/ X ks)-(ki X k2) = 0, etc. Making these and similar substitutions and introducing, as before,
206
Chapter Thirteen.
the relation — k'4-k4 = cosh 0 where 0 is the non-Euclidean angle between the two time axes, we may write our transformation table in the form
|
A12' |
Ar; |
Au' |
A,,' |
A*' |
Au' |
|
|
A 12 |
cosh 0 |
0 |
0 |
0 |
sinh 0 |
0 |
|
Aa |
0 |
cosh 0 |
0 |
0 |
0 |
sinh 0 |
|
Au |
0 |
0 |
1 |
0 |
0 |
0 |
|
A 23 |
0 |
0 |
0 |
1 |
0 |
0 |
|
A^ |
— sinh 0 |
0 |
0 |
0 |
cosh 0 |
0 |
|
AM |
0 |
— sinh 4> |
0 |
0 |
0 |
cosh 0 |
(301)
189. Interpretation of the Lorentz Transformation as a Rotation of Axes. We may now show that the Lorentz transformation may be looked upon as a change from a given set of axes to a rotated set.
Since the angle 0 which occurs in our transformation tables is that between the k4 axis and the new k/ axis, we may write, in ac cordance with equations (265) and (266),
cosh 0 =
sinh 0 = — p=
where V is the velocity between the two sets of space axes which correspond to the original and the rotated set of four-dimensional axes. This will permit us to rewrite our transformation table for the
Four Dimensional Analysis.
207
components of a one-vector in the forms
Vic
a,
a,
a
F/c
72
_
oV
_Z!
0 0
V/c
72
(302)
Consider now any point P(XI, x2, x3, ar4). The radius vector from the origin to this point will be r = (ziki + a:2k2 + x3k3 + x4k4), or, making use of the relations between xi, x2, a:3, x4 and x, y, z, t given by equations (252), we may write
r =
Applying our transformation table to the components of this one- vector, we obtain the familiar equations for the Lorentz transformation
x - VI
208 Chapter Thirteen.
y = y,
z' = z,
V2
We thus see that the Lorentz transformation is to be interpreted in our four-dimensional analysis as a rotation of axes.
190. Graphical Representation. Although we have purposely re stricted ourselves in the foregoing treatment to methods of attack which are almost purely analytical rather than geometrical in nature, the importance of a graphical representation of our four-dimensional manifold should not be neglected. The difficulty of representing all four axes on a single piece of two-dimensional paper is not essentially different from that encountered in the graphical representation of the facts of ordinary three-dimensional solid geometry, and these diffi culties can often be solved by considering only one pair of axes at a time, say OXi and OX4) and plotting the occurrences in the XiOX^ plane. The fact that the geometry of this plane is a non-Euclidean one presents a more serious complication since the figures that we draw on our sheet of paper will obviously be Euclidean in nature, but this difficulty also can be met if we make certain conventions as to the significance of the lines we draw, conventions which are funda mentally not so very unlike the conventions by which we interpret as solid, a figure drawn in ordinary perspective.
Consider for example the diagram shown in figure 18, where we have drawn a pair of perpendicular axes, OXi, and OX4, and the two unit hyperbolae given by the equations
x? - x? = 1,
(303) xf -x? = - 1,
together with their asymptotes, OA and OB, given by the equation
xf - x? = 0. (304)
This purely Euclidean figure permits, as a matter of fact, a fairly satisfactory representation of the non-Euclidean properties of the manifold with which we have been dealing.
Four Dimensional Analysis.
209
OX i and OX> may be considered as perpendicular axes in the non-Euclidean XiOX^ plane. Radius vectors lying in the quadrant A OB, will have a greater component along the X4 than along the Xi axis and hence will be S-vectors with the magnitude s = Vx42 — £i2, where x\ and x\ are the coordinates of the terminal of the vector.
D
FIG. 18.
7-radius-vectors will lie in the quadrant BOC and will have the mag nitude s = V#i2 — £42. Radius vectors lying along the asymptotes OA and OB will have zero magnitudes (s = Vzr — x42 = 0) and hence will be singular vectors.
Since the two hyperbola? have the equations x-f — x,f = 1 and Xi~ — #42 = — 1, rays such as Oa, Oaf, Ob, etc., starting from the origin and terminating on the hyperbola?, will all have unit magnitude. Hence we may consider the hyperbolae as representing unit pseudo- circles in our non-Euclidean plane and consider the rays as repre senting the radii of these pseudo-circles.
A non-Euclidean rotation of axes will then be represented by changing from the axes OXi and OX± to OAY and 0AY, and taking
Oaf and Ob' as unit distances along the axes instead of Oa and Ob. 15
210 Chapter Thirteen.
It is easy to show, as a matter of fact, that such a change of axes and units does correspond to the Lorentz transformation. Let Xi and Xi be the coordinates of any point with respect to the original axes OXi and OX4, and Xi" and x*" the coordinates of the same point referred to the oblique axes OXi and OX/, no change having yet been made in the actual lengths of the units of measurement. Then, by familiar equations of analytical geometry, we shall have
Xi = Xi" cos 6 + #4" sin 6, (
(305) £4 = Xi" sin 0 -f x" cos 0,
where 0 is the angle XiOX\.
We have, moreover, from the properties of the hyperbola,
Oa Ob " Vcos2 e - sin20 '
and hence if we represent by x\ and x<{ the coordinates of the point with respect to the oblique axes and use Oa' and Ob' as unit distances instead of Oa and Ob, we shall obtain
cos 0 sin B
Vcos2 0 — sin2 0 Vcos2 0 — sin2 0 '
, sin 0 , ; cos 0
Vcos2 0 - sin2 0 ! Vcos2 0 - sin2 0 ' It is evident, however, that we may write
sin 0 dxi V
—i = tan 0 = — - = — , cos 0 cte4 c
where V may be regarded as the relative velocity of our two sets of space axes. Introducing this into the above equations and also writing Xi = x, x4 = ct, x^ = x', x4' = ctf, we may obtain the familiar equations
V1-.-?
Four Dimensional Analysis. 211
We thus see that our diagrammatic representation of non-Euclidean rotation in the XiOX^ plane does as a matter of fact correspond to the Lorentz transformation.
Diagrams of this kind can now be used to study various kine- matical events. 5-curves can be drawn in the quadrant A OB to repre sent the space-time trajectories of particles, their form can be in vestigated using different sets of rotated axes, and the equations for the transformation of velocities and accelerations thus studied. 7-lines perpendicular to the particular time axis used can be drawn to correspond to the instantaneous positions of actual lines in ordinary space and studies made of the Lorentz shortening. Singular vectors along the asymptote OB can be used to represent the trajectory of a ray of light and it can be shown that our rotation of axes is so devised as to leave unaltered, the angle between such singular vectors and the OX4 axis, corresponding to the fact that the velocity of light must appear the same to all observers. Further development of the possi bilities of graphical representation of the properties of our non- Euclidean space may be left to the reader.
PART II. APPLICATIONS OF THE FOUR-DIMENSIONAL ANALYSIS.
191. We may now apply our four-dimensional methods to a number of problems in the fields of kinematics, mechanics and electro magnetics. Our general plan will be to express the laws of the par ticular field in question in four-dimensional language, making use of four-dimensional vector quantities of a kinematical, mechanical, or electromagnetic nature. Since the components of these vectors along the three spatial axes and the temporal axis will be closely related to the ordinary quantities familiar in kinematical, mechanical, and electrical discussions, there will always be an easy transition from our four-dimensional language to that ordinarily used in such dis cussions, and necessarily used when actual numerical computations are to be made. We shall find, however, that our four-dimensional language introduces an extraordinary brevity into the statement of a number of important laws of physics.
KINEMATICS.
192. Extended Position. The position of a particle and the par ticular instant at which it occupies that position can both be indi-
212 Chapter Thirteen.
cated by a point in our four-dimensional space. We can call this the extended position of the particle and determine it by stating the value of a four-dimensional radius vector
r = (xiki + z2k2 + z3k3 + x4k4). (306)
193. Extended Velocity. Since the velocity of a real particle can never exceed that of light, its changing position in space and time will be represented by a 6-curve.
The equation for a unit vector tangent to this 5-curve will be
dr dxi dxz dx3 dx*
(307)
where ds indicates interval along the 5-curve; and this important vector w may be called the extended velocity of the particle. Remembering that for a 5-curve
ds = Vcfo42 - dx? - dxz2 - dx32 = cdt-yl - — , (308)
we may rewrite our expression for extended velocity in the form
w- ,—- l-.+ kJ. (309)
where u is evidently the ordinary three-dimensional velocity of the particle.
Since w is a four-dimensional vector in our imaginary space, we may use our tables for transforming the components of w from one set of axes to another. We shall find that we may thus obtain trans formation equations for velocity identical with those already familiar in Chapter IV.
The four components of w are
u,
k
V'-S
and with the help of table (302) we may easily obtain, by making simple algebraic substitutions, the following familiar transformation
Four Dimensional Analysis.
213
equations :
1 -
c2
yj
' c2 L/
This is a good example of the ease with which we can derive our familiar transformation equations with the help of the four-dimensional method.
194. Extended Acceleration. We may define the extended accel eration of a particle as the rate of curvature of the 6-line which deter mines its four-dimensional position. We have
c =
d2! dw _ d ~ ds ~ ds
Or, introducing as before the relation ds = cdl-\J 1 — — , we may write 1 du 1 u du
(310)
u du
(311)
214 Chapter Thirteen.
where u is evidently the ordinary three-dimensional velocity, and -j-
CtL
the three-dimensional acceleration; and we might now use our trans formation table to determine the transformation equations for accel eration which we originally obtained in Chapter IV.
195. The Velocity of Light. As an interesting illustration of the application to kinematics of our four-dimensional methods, we may point out that the trajectory of a ray of light will be represented by a singular line. Since the magnitude of all singular vectors is zero by definition, we have for any singular line
or, since the magnitude will be independent of any particular choice of axes, we may also write
Transforming the first of these equations we may write
dxj + dx22 + dx^ _ dx* + dy2 + dz2 _
dxt2 cW
or
dl
dt = C'
Similarly we could obtain from the second equation
dl'
dt = C'
We thus see that a singular line does as a matter of fact correspond to the four-dimensional trajectory of a ray of light having the velocity c, and that our four-dimensional analysis corresponds to the require ments of the second postulate of relativity that a ray of light shall have the same velocity for all reference systems.
THE DYNAMICS OF A PARTICLE.
196. Extended Momentum. We may define the extended momen tum of a material particle as equal to the product ra0w of its mass mQ, measured when at rest, and its extended velocity w. In accordance
Four Dimensional Analysis. 215
with equation (309) for extended velocity, we may write then, for the extended momentum,
(312)
Or, if in accordance with our considerations of Chapter VI we put for the mass of the particle at the velocity u
we may write
m =
u
m0w = m- + mk4. (313)
c
We note that the space component of this vector is ordinary momen tum and the time component has the magnitude of mass, and by applying our transformation table (302) we can derive very simply the transformation equations for mass and momentum already obtained in Chapter VI.
197. The Conservation Laws. We may now express the laws for the dynamics of a system of particles in a very simple form by stating the principle that the extended momentum of a system of particles is a quantity which remains constant in all interactions of the particles, we have then
/ wu \
;m0w = 2 I — + wk4 ) = a constant,
(314)
where the summation 2 extends over all the particles of the system. It is evident that this one principle really includes the three principles of the conservation of momentum, mass, and energy. This is true because in order for the vector 2m Ow to be a constant quantity, its components along each of the four axes must be con stant, and as will be seen from the above equation this necessitates the constancy of the momentum 2rau, of the total mass 2m, and of
m the total energy 2 — .
216 Chapter Thirteen.
THE DYNAMICS OF AN ELASTIC BODY.
Our four-dimensional methods may also be used to present the results of our theory of elasticity in a very compact form.
198. The Tensor of Extended Stress. In order to do this we shall first need to define an expression which may be called the four-dimen sional stress in the elastic medium. For this purpose we may take the symmetrical tensor Tm defined by the following table:
xx Pxy Pxz CQX) v^ Pvy Pyz
rp
"* Pzx Pzy Pzz CQZ, (315)
So o
X o y o z
w,
1C C C
where the spatial components of Tm are equal to the components of the symmetrical tensor p which we have already defined in Chapter X and the time components are related to the density of momentum g, density of energy flow s and energy density w, as shown in the tabu lation.
From the symmetry of this tensor we may infer at once the simple relation between density of momentum and density of energy flow:
g = ^ i (316)
with which we have already become familiar in Section 132.
199. The Equation of Motion. We may, moreover, express the equation of motion for an elastic medium unacted on by external forces in the very simple form
div Tm = 0. (317)
It will be seen from our definition of the divergence of a four- dimensional tensor, Section 187, that this one equation is in reality equivalent to the two equations
div p + ~ = 0 (318)
and
div
div s + — = 0. ot
Four Dimensional Analysis. 217
The first of these equations is identical with (184) of Chapter X, which we found to be the equation for the motion of an elastic medium in the absence of external forces, and the second of these equations expresses the principle of the conservation of energy.
The elegance and simplicity of this four-dimensional method of expressing the results of our laborious calculations in Chapter X can not fail to be appreciated.
ELECTROMAGNETICS.
We also find it possible to express the laws of the electromagnetic field very simply in our four-dimensional language.
200. Extended Current. We may first define the extended current, a simple but important one-vector, whose value at any point will de pend on the density and velocity of charge at that point. We shall take as the equation of definition
q = Pow = p + k4 , (319)
where
Po
is the density of charge at the point in question.
201. The Electromagnetic Vector M. We may further define a two-vector M which will be directly related to the familiar vectors strength of electric field e and strength of magnetic field h by the equation of definition
M = (fcikw + /?2k3i + h3kn - e&u - e2k24 - e3k34)
or (320)
M* = (eik23 + e2k3i -f eski2 + /hkH + /i2k24 + /?3k34),
where e\, e->, e3, and hi, hZj h3 are the components of e and h.
202. The Field Equations. We may now state the laws of the electromagnetic field in the extremely simple form
0-M = q, (321)
0 X M = 0. (322)
218 Chapter Thirteen.
These two simple equations are, as a matter of fact, completely equivalent to the four field equations which we made fundamental for our treatment of electromagnetic theory in Chapter XII. Indeed if we treat {> formally as a one-vector
/ d
( k! —
\ dxL
d d d
o - + k3 — - k4 --
dx2 dx3 dx4
and apply it to the electromagnetic vector M expressed in the extended form given in the equation of definition (320) we shall obtain from (321) the two equations
1 de u
div e = p, and from (322)
div h = 0,
1 dh
where we have made the substitution #4 = ct. These are of course the familiar field equations for the Maxwell-Lorentz theory of electro- magnetism.
203. The Conservation of Electricity. We may also obtain very easily an equation for the conservation of electric charge. In accord ance with equation (284) we may write as a necessary mathematical identity
0-(0'M)=0. (323)
Noting that 0 • M = q, this may be expanded to give us the equation of continuity.
div Pu + ^ = 0. (324)
ot
204. The Product M • q. We have thus shown the form taken by the four field equations when they are expressed in four dimensional language. Let us now consider with the help of our four-dimensional methods what can be said about the forces which determine the motion of electricity under the action of the electromagnetic field.
Consider the inner product of the electromagnetic vector and
Four Dimensional Analysis. 219
the extended current:
+ k4
(325)
f [uXh]*\ e-u = P(e+- — -j + P— k,
We see that the space component of this vector is equal to the ex pression which we have already found in Chapter XII as the force acting on the charge contained in unit volume, and the time com ponent is proportional to the work done by this force on the moving charge; hence we may write the equation
(326)
an expression which contains the same information as that given by the so-called fifth fundamental equation of electromagnetic theory, f being the force exerted by the electromagnetic field per unit volume of charged material.
205. The Extended Tensor of Electromagnetic Stress. We may now show the possibility of defining a four-dimensional tensor Te such that the important quantity M-q shall be equal to — div Te. This will be valuable since we shall then be able to express the equation of motion for a combined mechanical and electrical system in a very simple and beautiful form.
Consider the symmetrical tensor
Tfji rri
12 ^13 1 14,
31 -t 32 * 33 -t 34, T rn rn rri
41 1 42 1 43 1 44,
defined by the expression
(328) where j, k = 1, 2, 3, 4.
220 Chapter Thirteen.
It can then readily be shown by expansion that
- div Te = M-(O-M) + M*-(0-M*).
But, in accordance with equations (321), (326), (292) and (322), this is equivalent to
Since in free space the value of the force f is zero, we may write for free space the equation
div Te = 0. (330)
This one equation is equivalent, as a matter of fact, to two im portant and well-known equations of electromagnetic theory. If we develop the components Tu, T12, etc., of our tensor in accordance with equations (328) and (320) we find that we can write
Yxx Yxy *//xz — , C
T, = 1
'vv tyz ~ , C
(331)
So o
x oy e>2
7 7 ~c w'
where we shall have
*« = " \(e? txv = - (exev
etc.
sx = c(ejiz — ezhv),
etc.
- ej - e? + h* - hj - h*
(332)
^ thus being equivalent to the well-known Maxwell three-dimensional stress tensor, sxj sv, etc., being the components of the Poynting vector c[e X h]*, and w being the familiar expression for density of electro-
Four Dimensional Analysis. 221
magnetic energy — | — . We thus see that equation (330) is equiva lent to the two equations
1 as
div* + -2~ = 0,
dw div s + — = 0.
ot
The first of these is the so-called equation of electromagnetic momen tum, and the second, Poynting's equation for the flow of electromag netic energy.
206. Combined Electrical and Mechanical Systems. For a point not in free space where mechanical and electrical systems are both involved, taking into account our previous considerations, we may now write the equation of motion for a combined electrical and mechanical system in the very simple form
div Tm + div Te = 0.
And we may point out in closing that we may reasonably expect all forces to be of such a nature that our most general equation of motion for any continuous system can be written in the form
div T7! + div T2 + • • • = 0.
APPENDIX I.— SYMBOLS FOR QUANTITIES.
Scalar Quantities. (Indicated by Italic type.)
c speed of light.
e electric charge.
E energy.
H kinetic potential.
K kinetic energy.
I, m, n direction cosines.
L Lagrangian function.
p pressure.
Q quantity of electricity.
S entropy.
t time.
/ ; 2 \
T temperature, function 2m0c2 1 1 — -\! 1 -- - 1
U potential energy.
v volume.
V relative speed of coordinate systems, volume.
w energy density.
W work.
e dielectric constant.
V
ju index of refraction, magnetic permeability.
v frequency.
p density of charge.
a electrical conductivity.
</> non-Euclidean angle between time axes.
• • generalized coordinates.
\l/ scalar potential.
\J/i\f/z\l/3- - • generalized momenta.
222
Appendix I. 223
Vector Quantities. (Indicated by Clarendon type.)
B magnetic induction.
c extended acceleration.
D dielectric displacement,
e electric field strength in free space.
E electric field strength in a medium.
f force per unit volume.
F force acting on a particle.
g density of momentum.
h magnetic field strength in free space.
H magnetic field strength in a medium.
i density of electric current.
M angular momentum, electromagnetic vector.
p symmetrical elastic stress tensor.
q extended current.
r radius vector
s density of energy flow.
t unsymmetrical elastic stress tensor.
u velocity.
w extended velocity.
<(> vector potential.
APPENDIX II.— VECTOR NOTATION. Three Dimensional Space. Unit Vectors, i j k Radius Vector, r = x\ + yj -f zk Velocity,
dr
U = ~ = Xi + yj + 'Zk
= uxi + uvj + uzk Acceleration,
= uxi + iyj + wzk Inner Product,
a-b = axbx + atf6tf + aebz Outer Product,
a X b = (axbv - aj>x)ij + (ay6, - a,6v)jk + (a A - az62)ki Complement of Outer Product,
[a X b]* = (aybz - a,6y)i + (aJ>, - axbz)j + (axby - aybx)k The Vector Operator Del or V,
d d d
dy dz
curia = [V X a]* = _
dy dz dz dx
224
/day _dOx\ \dx ' dy J
Appendix II. 225
Non-Euclidean Four Dimensional Space.
Unit Vectors, ki k2 k3 k4 Radius Vector,
r = ziki + z2k2 + z3k3 4- = xi + yj + 2k + dk4
One Vector,
a = Qiki + a2k2 + a3k3 + Two Vector,
A = ^i2k12 + Ai3k13 + Auku + A23k23 + A24k24 + A34k34 Three Vector,
H = 5Ii23ki23 + 5li24ki24 + Hi34ki34 + 5I234k234
Pseudo Scalar,
a = aki234
Transposition of Subscripts,
Inner Product of One Vectors,
(See Section 183). Outer Product of One Vectors,
ka&... X knm... =
Complement of a Vector,
0* = <£-k
The Vector Operator Quad or 0>
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