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Full text of "An Introduction to Tensor Calculus & Relativity"

AN INTRODUCTION 

Tensor Calculus 
and Relativity 

DEREK F. LAWDEN 

From Methuen's Monographs on Physical Subjects 



4> 



AN INTRODUCTION TO 
TENSOR CALCULUS AND RELATIVITY 



An Introduction to 
Tensor Calculus and Relativity 

(methuen's monographs 
on physical subjects) 



Derek F. Lawden, 

SC.D., F.R.S.N.Z. 

Professor of Mathematics 
University of Canterbury, N.Z. 



METHUEN & CO LTD 

and 
SCIENCE PAPERBACKS 

11 NEW FETTER LANE, LONDON EC4P 4EE 



First published by Methuen & Co Ltd 1962 

Second edition 1967 

Reprinted 1968 

Reprinted 1971 

S.B.N. 416 43140 2 

2.3 

First published as a Science Paperback 1967 

Reprinted 1968 

Reprinted 1971 

S.B.N. 412 20370 7 

© 1967 Ay ZtereA: F. Lawden 

Reproduced and printed in Great Britain by 

Redwood Press Limited, Trowbridge & London 

This book is available in both hardbound and paperback 
editions. The paperback edition is sold subject to the 
condition that it shall not, by way of trade or otherwise, 
be lent, resold, hired out, or otherwise circulated without 
the publisher' 's prior consent in any form of binding or 
cover other than that in which it is published and without 
a similar condition including this condition being imposed 
on the subsequent purchaser. 



Distributed in the U.S.A. 
by Barnes & Noble Inc. 



Contents 

Preface page ix 

1. Special Principle of Relativity . Lorentz Transformations 1 

1. Newton's laws of motion 1 

2. Covariance of the laws of motion 4 

3. Special principle of relativity 5 

4. Lorentz transformations. Minkowski space-time 7 

5. The special Lorentz transformation 11 

6. Fitzgerald contraction. Time dilation 14 

7. Spacelike and timelike intervals. Light cone 17 
Exercises 1 20 



2. Orthogonal Transformations. Cartesian Tensors 23 

8. Orthogonal transformations 23 

9. Repeated index summation convention 25 

10. Rectangular Cartesian tensors 26 

11. Invariants. Gradients. Derivatives of tensors 29 

12. Contraction. Scalar product. Divergence 30 

13. Tensor densities 32 

14. Vector products. Curl 33 
Exercises 2 35 



3. Special Relativity Mechanics 38 

15. The velocity vector 38 

16. Mass and momentum 41 

17. The force vector. Energy 43 

18. Lorentz transformation equations for force 47 

19. Motion with variable proper mass 48 

20. Lagrange's and Hamilton's equations 50 
Exercises 3 51 



Vi CONTENTS 

4. Special Relativity Electrodynamics page 59 

21. 4-Current density 59 

22. 4- Vector potential 60 

23. The field tensor 62 

24. Lorentz transformations of electric and magnetic 

intensities 65 

25. The Lorentz force 67 

26. Force density 68 

27. The energy-momentum tensor for an electromagnetic 

field 69 

28. Equations of motion of a charge flow 74 
Exercises 4 77 

5. General Tensor Calculus, Riemannian Space 81 

29. Generalized JV-dimensional spaces 81 

30. Contravariant and co variant tensors 85 

31. The quotient theorem. Conjugate tensors 92 

32. Relative tensors and tensor densities 94 

33. Covariant derivatives. Parallel displacement. Affine 

connection 96 

34. Transformation of an affinity 99 

35. Covariant derivatives of tensors 101 

36. Covariant differentiation of relative tensors 104 

37. The Riemann-Christoffel curvature tensor 108 

38. Geodesic coordinates. The Bianchi identities 1 1 3 

39. Metrical connection. Raising and lowering of indices 115 

40. Scalar products. Magnitudes of vectors 117 

41 . The Christoffel symbols. Metric affinity 1 18 

42. The covariant curvature tensor 121 

43. Divergence. The Laplacian. Einstein's tensor 123 

44. Geodesies 125 
Exercises 5 129 

6. General Theory of Relativity 137 

45. Principle of equivalence 137 

46. Metric in a gravitational field 141 



CONTENTS VU 

47. Motion of a free particle in a gravitational field page 145 

48. Einstein's law of gravitation 147 

49. Acceleration of a particle in a weak gravitational field 1 50 

50. Newton's law of gravitation 153 

51. Metrics with spherical symmetry 155 

52. Schwarzschild's solution 158 

53. Planetary orbits 160 

54. Gravitational deflection of a light ray 166 

55. Gravitational displacement of spectral lines 168 

56. Maxwell's equations in a gravitational field 171 
Exercises 6 173 

Appendix Bibliography 182 

Index 183 



Preface 

Now that half a century has elapsed since the special and general 
theories of relativity were constructed, it is possible to perceive more 
clearly their true significance for the development of mathematical 
physics as a whole. Although these theories possessed a decidedly 
revolutionary appearance when they were announced, it has now be- 
come clear that they represented the natural termination for the classi- 
cal theories of mechanics and electromagnetism, rather than a break 
with these systems of ideas and the inception of a new line of thought. 
In this respect, the relativity theories are to be contrasted with that 
other great achievement of modern theoretical physics, namely 
quantum mechanics, based as this is upon principles which are com- 
pletely at variance with those which were fundamental for Newtonian 
mechanics. That relativity theory has proved to be somewhat sterile 
by comparison with the enormous fertility of the new ideas introduced 
in quantum theory may, perhaps, be partly accounted for by this 
difference. However, the circumstance that the applications of rela- 
tivity theory are chiefly to problems in the fields of astronomy and 
cosmology, fields which until recently have not received great atten- 
tion from physicists due to the difficulty of experimentation, whereas 
quantum mechanics is associated with phenomena of atomic dimen- 
sions more easily investigated in the laboratory, may also have a 
bearing upon the matter. Now that interplanetary and even inter- 
stellar instrumented probes are approaching realization, it may be 
that interest in cosmological problems will be stimulated and that 
significant advances from the position reached by relativity theory 
some decades ago will result. 

However, relativity theory's present status is as the culmination of 
the classical theories of mechanics and electromagnetism and, since 
much time is normally spent during undergraduate courses giving a 
detailed account of these classical theories, it has always seemed to me 
improper that such courses should be terminated without a descrip- 
tion being given of the very natural and illuminating form into which 
it is possible to cast these theories by use of the device, basic for 

viii 



PREFACE IX 

relativity theory, of mapping events on a four-dimensional space-time 
manifold. Such an introductory course of lectures on relativity theory 
helps to clarify the principles upon which the classical theories are 
based and is far more rewarding from the student's point of view than 
is a course devoted to the solution of intricate problems of statics and 
dynamics. By some sacrifice of the time usually devoted to conven- 
tional techniques of classical mechanics and electromagnetism, it is 
possible to provide an introductory course of relativity theory suitable 
for students in their third honours year at the university who, judging 
from our experience at Canterbury, are at this stage sufficiently 
mature mathematically to appreciate the great beauty of this system 
of ideas. I have delivered such a course of lectures at Canterbury over 
a number of years and this book has grown out of the lecture notes I 
have prepared. Although there is no shortage of excellent accounts of 
the subject of this book (many are listed in the bibliography at the end) 
the great majority of these are intended for study by post-graduate 
students who are specializing in this or related fields and are not suit- 
able as adjuncts to an introductory course of lectures at the under- 
graduate level. I hope, therefore, that there is a place for this book and 
that university teachers who are responsible for courses of this type 
and their students will find it helpful. 

The plan of the book is as follows : The basic idea of the covariance 
of physical laws with respect to transformations between reference 
frames is introduced in Chapter 1 and is illustrated by showing that 
the laws of Newtonian mechanics are covariant with respect to trans- 
formations between inertial frames. This leads to the special principle 
of relativity and its verification for electrodynamics by the Michelson- 
Morley experiment. The details of this crucial experiment are not dis- 
cussed and its aftermath is only briefly alluded to, since I feel that, 
though of great historical interest, now that more than half a century 
has elapsed since it was performed, its significance for the special 
theory, so strongly supported by numerous other experimental results, 
is not now so great as formerly. The remainder of the first chapter is 
devoted to the Lorentz transformation which, from the outset, is 
treated as a rotation of rectangular Cartesian axes in Minkowski 
space-time. I have thought it desirable to relieve the undergraduate 
reader, for whom this book is primarily intended, of the necessity for 



X PREFACE 

coping immediately with the full rigours of general tensor calculus and 
I have therefore treated the Minkowski space-time of the special 
theory as a Euclidean manifold, with the formalism of which he will 
already be familiar. The time components of vectors are then, of 
course, purely imaginary, but this is salutary in one respect for it 
serves to emphasize the basic physical distinction which exists between 
space and time measurements and so to check the impression that 
relativity theory implies that space and time are basically of the same 
nature. The theory of tensors relative to rectangular Cartesian co- 
ordinate frames in an iV-dimensional Euclidean space is developed in 
Chapter 2 and this is made use of in the succeeding two chapters to 
define the principal vectors and tensors of special relativity physics in 
the Minkowski space-time continuum. All such vectors and tensors 
are denoted by bold roman capital letters and their components by 
the corresponding italic characters with appropriate subscripts (e.g. 
F, F t for the 4-force vector and its components). The corresponding 
3-vectors defined with respect to the rectangular axes employed by an 
inertial observer are denoted by bold roman lower case letters and 
their components by the corresponding italic lower case symbols with 
subscripts (e.g. f,f { for the 3-force vector and its components). All sets 
of transformation equations relating the components of a 3-vector 
relative to different inertial frames are consistently obtained from the 
corresponding 4-vector transformation equations with respect to a 
change of axes in space-time. The reader will accordingly become very 
familiar with this important technique. The laws of mechanics are all 
expressed in covariant four-dimensional form in Chapter 3 and the 
electrodynamic laws are exhibited in this form in Chapter 4. A student 
of physics, for whom the general theory of relativity is of less interest, 
should proceed no further than Chapter 4 except, perhaps, to read 
sections 45 and 46 in which the approach to the general principle is 
outlined. 

The techniques of general tensor calculus, which are employed when 
expounding the general theory of relativity, are explained in Chapter 5. 
The algebra and analysis of tensors and relative tensors are first con- 
structed in an affinely connected space, the special case of a Rieman- 
nian space where the affinity is conveniently related to the metric being 
studied later. The coordinates of a point are here denoted by x* (rather 



PREFACE XI 

than xi), for the obvious reason that the dx { are the components of a 
contravariant vector and that I have never found the apologies for 
employing subscripts at all acceptable. I am certain that this practice 
magnifies, quite unnecessarily, the difficulties of the beginner. 

The reason why a theory, which accepts as basic the general prin- 
ciple of relativity, must necessarily be a theory of gravitation is first 
explained in Chapter 6. This leads to Einstein's law of gravitation, the 
Schwarzschild spherically symmetric metric for empty space with a 
point singularity and the three standard physical tests of the theory. 
It will, perhaps, prevent confusion if it is here remarked that, the 
interval ds between two neighbouring points of space-time is defined 
in such a way that, if (x,y,z) are rectangular Cartesian coordinates 
relative to an inertial frame falling freely in the gravitational field and 
t is time in this frame, then 

ds 2 = dx 2 + dy 2 + d£-(?dt 2 . 

For the mode of development chosen in this book, this proves to be 
the most convenient definition of ds. 

When giving consideration to the manner in which this material 
should be presented to the reader, I have greatly profited by referring 
to the original memoirs of the subject and to the books listed in the 
bibliography. All these works have influenced my own presentation to 
some extent, but I have found the accounts by Einstein, Moller and 
Schrodinger especially stimulating. I am happy to acknowledge my 
indebtedness to the authors of all books which I have consulted. Also, 
to my colleague Professor W. R. Andress, who has read the manuscript 
and made many valuable suggestions, I tender my grateful thanks. As 
a result of our discussions a number of errors and obscurities have 
been removed. 

Certain of the exercises have been taken from examination 
papers set at the Universities of Cambridge, London and Liverpool. 
These have been indicated as follows: M.T. (Mathematics Tripos), 
L.U. (London) and Li. U. (Liverpool). I am grateful to the author- 
ities concerned for permission to make use of these. 

D. F. LAWDEN 

Mathematics Department University of Canterbury 
Christchurch, N.Z. September 1960 



Preface to the Second Edition 

A small number of typographical errors have been corrected, but 
the chief improvement has been the addition of a large number of 
new exercises, many of which have been taken from examination 
papers set at the University of Canterbury. There are now 1 14 of 
these exercises and it is hoped that these will provide the student with 
an adequate collection for the purpose of testing his understanding 
of the text. 

D. F. LAWDEN 

Mathematics Department University of Canterbury, Christchurch, 
N.Z. September, 1966. 



CHAPTER 1 

Special Principle of Relativity. Lorentz 
Transformations 

1. Newton's laws of motion 

A proper appreciation of the physical content of Newton's three laws 
of motion is an essential prerequisite for any study of the special 
theory of relativity. It will be shown that these laws are in accordance 
with the fundamental principle upon which the theory is based and 
thus they will also serve as a convenient introduction to this principle. 

The first law states that any particle which is not subjected to forces 
moves along a straight line at constant speed. Since the motion of a 
particle can only be specified relative to some coordinate frame of 
reference, this statement has meaning only when the reference frame 
to be employed when observing the particle's motion has been indi- 
cated. Also, since the concept of force has not, at this point, received a 
definition, it will be necessary to explain how we are to judge when a 
particle is * not subjected to forces '. It will be taken as an observed fact 
that if rectangular axes are taken with their origin at the centre of the 
sun and these axes do not rotate relative to the most distant objects 
known to astronomy, viz. the extra-galactic nebulae, then the motions 
of the neighbouring stars relative to this frame are very nearly uniform. 
The departure from uniformity can reasonably be accounted for as 
due to the influence of the stars upon one another and the evidence 
available suggests very strongly that if the motion of a body in a 
region infinitely remote from all other bodies could be observed, 
then its motion would always prove to be uniform relative to our 
reference frame irrespective of the manner in which the motion was 
initiated. 

We shall accordingly regard the first law as asserting that, in a region 
of space remote from all other matter and empty save for a single test 
particle, a reference frame can be defined relative to which the particle 
will always have a uniform motion. Such a frame will be referred to as 

1 



2 TENSOR CALCULUS AND RELATIVITY 

an inertial frame. An example of such an inertial frame which is con- 
veniently employed when discussing the motions of bodies within the 
solar system has been described above. However, if S is any inertial 
frame and S is another frame whose axes are always parallel to those of 
S but whose origin moves with a constant velocity u relative to S, then 
S also is inertial. For, if v, v are the velocities of the test particle relative 
to S, S respectively, then 

v = v-u (1.1) 

and, since v is always constant, so is v. It follows, therefore, that a 
frame whose origin is at the earth's centre and whose axes do not 
rotate relative to the stars can, for most practical purposes, be looked 
upon as an inertial frame, for the motion of the earth relative to the 
sun is very nearly uniform over periods of time which are normally 
the subject of dynamical calculations. In fact, since the earth's rota- 
tion is slow by ordinary standards, a frame which is fixed in this body 
can also be treated as approximately inertial and this assumption will 
only lead to appreciable errors when motions over relatively long 
periods of time are being investigated, e.g. Foucault's pendulum, long 
range gunnery calculations. 

Having established an inertial frame, if it is found by observation 
that a particle does not have a uniform motion relative to the frame, 
the lack of uniformity is attributed to the action of a. force which is 
exerted upon the particle by some agency. For example, the orbits of 
the planets are considered to be curved on account of the force of 
gravitational attraction exerted upon these bodies by the sun and when 
a beam of charged particles is observed to be deflected when a bar 
magnet is brought into the vicinity, this phenomenon is understood to 
be due to the magnetic forces which are supposed to act upon the 
particles. If v is the particle's velocity relative to the frame at any 
instant t, its acceleration a = dvjdt will be non-zero if the particle's 
motion is not uniform and this quantity is accordingly a convenient 
measure of the applied force f. We take, therefore, 

fa a, 
or f = wza, (1.2) 

where m is a constant of proportionality which depends upon the 
particle and is termed its mass. The definition of the mass of a particle 



SPECIAL PRINCIPLE OF RELATIVITY 3 

will be given almost immediately when it arises quite naturally out of 
the third law of motion. Equation (1.2) is essentially a definition of 
force relative to an inertial frame and is referred to as the second law 
of motion. It is sometimes convenient to employ a non-inertial frame 
in dynamical calculations, in which case a body which is in uniform 
motion relative to an inertial frame and is therefore subject to no 
forces, will nonetheless have an acceleration in the non-inertial frame. 
By equation (1.2), to this acceleration there corresponds a force, but 
this will not be attributable to any obvious agency and is therefore 
usually referred to as a * fictitious ' force. Well-known examples of such 
forces are the centrifugal and Coriolis forces associated with frames 
which are in uniform rotation relative to an inertial frame, e.g. a frame 
rotating with the earth. By introducing such 'fictitious' forces, the 
second law of motion becomes applicable in all reference frames. 

According to the third law of motion, when two particles P and Q 
interact so as to influence one another's motion, the force exerted by P 
on Q is equal to that exerted by Q on P but is in the opposite sense. 
Defining the momentum of a particle relative to a reference frame as 
the product of its mass and its velocity, it is proved in elementary text- 
books that the second and third laws taken together imply that the 
sum of the momenta of any two particles involved in a collision is 
conserved. Thus, if m^ m 2 are the masses of two such particles and 
Ui, u 2 are their respective velocities immediately before the collision 
and v u y 2 are their respective velocities immediately afterwards, then 

miu 1 + m 2 u 2 = w 1 v 1 + m 2 v 2 (1.3) 

i.e. ^ (u2 _ V2) = Vi _ Uif (1 . 4) 

mi 

This last equation implies that the vectors u 2 - v 2 , V! - u t are parallel, 
a result which has been checked experimentally and which constitutes 
the physical content of the third law. However, equation (1.4) shows 
that the third law is also, in part, a specification of how the mass of a 
particle is to be measured and hence provides a definition for this 
quantity. For 

nh = l?i-"il (15) 

Wj |u 2 -v 2 l 



4 TENSOR CALCULUS AND RELATIVITY 

and hence the ratio of the masses of two particles can be found from 
the results of a collision experiment. If, then, one particular particle is 
chosen to have unit mass (e.g. the standard gramme, pound, etc.), the 
masses of all other particles can, in principle, be determined by per- 
mitting them to collide with this standard and then employing 
equation (1.5). 

2. Covariance of the laws of motion 

It has been shown in the previous section that the second and third 
laws are essentially definitions of the physical quantities force and 
mass relative to a given reference frame. In this section, we shall 
examine whether these definitions lead to different results when differ- 
ent inertial frames are employed. 

Consider first the definition of mass. If the collision between the 
particles m it m 2 is observed from the inertial frame S, let u,, u 2 be the 
particle velocities before the collision and v t , v 2 the corresponding 
velocities after the collision. By equation (1.1), 

fii = U!-u,etc. (2.1) 

and hence 

vi-Qj = vi-ii!, u 2 -v 2 = u 2 -v 2 . (2.2) 

It follows that if the vectors \i -u u u 2 - v 2 are parallel, so are the 
vectors f i - u t , u 2 - v 2 and consequently that, in so far as the third law 
is experimentally verifiable, it is valid in all inertial frames if it is valid 
in one. Now let rh\ , fh 2 be the particle masses as measured in S. Then, 
by equation (1.5), 

/wi |u 2 -v 2 | |u 2 -v 2 | mi ' 

But, if the first particle is the unit standard, then m t = m t = 1 and 
hence 

m 2 = m 2 , (2.4) 

i.e. the mass of a particle has the same value in all inertial frames. We 
can express this by saying that mass is an invariant relative to trans- 
formations between inertial frames. 



SPECIAL PRINCIPLE OF RELATIVITY 5 

By differentiating equation (1.1) with respect to the time t, since u 
is constant it is found that 

a = a, (2.5) 

where a, a are the accelerations of a particle relative to S, S respec- 
tively. Hence, by the second law (1.2), since m = m, it follows that 

f = f, (2.6) 

i.e. the force acting upon a particle is independent of the inertial 
frame in which it is measured. 

It has therefore been shown that equations (1.2), (1 .4) take precisely 
the same form in the two frames, S, S, it being understood that mass, 
acceleration and force are independent of the frame and that velocity 
is transformed in accordance with equation (1.1). When equations 
preserve their form upon transformation from one reference frame to 
another, they are said to be covariant with respect to such a transfor- 
mation. Newton's laws of motion are covariant with respect to a 
transformation between inertial frames. 

3. Special principle of relativity 

The special principle of relativity asserts that all physical laws are 
covariant with respect to a transformation between inertial frames. This 
implies that all observers moving uniformly relative to one another 
and employing inertial frames will be in agreement concerning the 
statement of physical laws. No such observer, therefore, can regard 
himself as being in a special relationship to the universe not shared by 
any other observer employing an inertial frame; there are no privi- 
leged observers. When man believed himself to be at the centre of 
creation both physically and spiritually, a principle such as that we 
have just enunciated would have been rejected as absurd. However, 
the revolution in attitude to our physical environment initiated by 
Copernicus has proceeded so far that today the principle is accepted 
as eminently reasonable and very strong evidence contradicting the 
principle would have to be discovered to disturb it as a foundation 
upon which theoretical physics is based. 

It has been shown already that Newton's laws of motion obey the 
principle. Let us now transfer our attention to another set of funda- 
mental laws governing non-mechanical phenomena, viz. Maxwell's 



6 TENSOR CALCULUS AND RELATIVITY 

laws of electrodynamics. These are more complex than the laws of 
Newton and are most conveniently expressed by the equations 

13H 

curlE — , (3.1) 

c ot 



1 (a • 8E \ 



curlH = -|4ttj+— I, (3.2) 

divE = 4tt/>, (3.3) 

divH = 0, (3.4) 

where E, H are the electric and magnetic field intensities respectively, 
j is the current density, p is the charge density and the region of space 
being considered is assumed to be empty save for the presence of the 
electric charge. Units have been taken to be Gaussian, so that c is the 
ratio of the electromagnetic unit of charge to the electrostatic unit 
(= 3 x 10 10 cm/sec). Experiment confirms that these equations are 
valid when any inertial frame is employed. The most famous such 
experiment was that carried out by Michelson and Morley, who veri- 
fied that the velocity of propagation of light waves in any direction is 
always measured to be c relative to an apparatus stationary on the 
earth. As is well-known, light has an electromagnetic character and 
this result is predicted by the equations (3.1)-(3.4). However, the 
velocity of the earth in its orbit at any time differs from its velocity six 
months later by twice the orbital velocity, viz. 60 km/sec and thus, by 
taking measurements of the velocity of light relative to the earth on 
two days separated by this period of time and showing them to be 
equal, it is possible to confirm that Maxwell's equations conform to 
the special principle of relativity. This is effectively what Michelson 
and Morley did. However, this interpretation of the results of their 
experiment was not accepted immediately, since it was thought that 
electromagnetic phenomena were supported by a medium called the 
aether and that Maxwell's equations would prove to be valid only in 
an inertial frame stationary in this medium, i.e. the special principle 
of relativity was denied for electromagnetic phenomena. The con- 
troversy which ensued is of great historical interest, but will not be 
recounted in this book. The special principle is now firmly established 



SPECIAL PRINCIPLE OF RELATIVITY / 

and is accepted on the grounds that the conclusions which may be 
deduced from it are everywhere found to be in conformity with experi- 
ment and also because it is felt to possess a priori a high degree of 
plausibility. A description of the steps by which it ultimately came to 
be appreciated that the principle was of quite general application 
would therefore be superfluous in an introductory text. It is, however, 
essential for our future development of the theory to understand the 
prime difficulty preventing an early acceptance of the idea that the 
electromagnetic laws are in conformity with the special principle. 

Consider the two inertial frames S, S. Suppose that an observer 
employing S measures the velocity of a light pulse and finds it to be c. 
If the velocity of the same light pulse is measured by an observer 
employing the frame S, let this be c. Then, by equation (1.1), 

c = c-u (3.5) 

and it is clear that, in general, the magnitudes of the vectors c, c will 
be different. It appears to follow, therefore, that either Maxwell's 
equations (3.1)-(3.4) must be modified, or the special principle of 
relativity abandoned for electromagnetic phenomena. Attempts were 
made (e.g. by Ritz) to modify Maxwell's equations, but certain conse- 
quences of the modified equations could not be confirmed experi- 
mentally. Since the special principle was always found to be valid, the 
only remaining alternative was to reject equation (1.1) and to replace 
it by another in conformity with the experimental result that the speed 
of light is the same in all inertial frames. As will be shown in the next 
section, this can only be done at the expense of a radical revision of our 
intuitive ideas concerning the nature of space and time and this was 
very understandably strongly resisted. 

4. Lorentz transformations. Minkowski space-time 

Let the reference frame S comprise rectangular Cartesian axes Oxyz. 
We shall assume that the coordinates of a point relative to this frame 
are measured by the usual procedure and employing a measuring 
scale which is stationary in S (it is necessary to state this precaution, 
since it will be shown later that the length of a bar is not independent 
of its motion). It will also be supposed that clocks, stationary relative 
to S, are distributed throughout space and are all synchronized with a 



8 TENSOR CALCULUS AND RELATIVITY 

master clock at O. A satisfactory synchronization procedure would be 
as follows: Warn observers at all clocks that a light source at O will 
commence radiating at t = t . When an observer at a point P first 
receives light from this source, he is to set the clock at P to read 
to+OP/c, i.e. it is assumed that light travels with a speed c relative to 
S, as found by experiment. The position and time of an event can now 
be specified relative to S by four coordinates (x, y, z,i),t being the time 
shown on the clock which is contiguous to the event. We shall often 
refer to the four numbers (x,y,z, t) as an event. 

Let Oxyi be rectangular Cartesian axes determining the frame S 
(to be precise, these are rectangular as seen by an observer stationary 
in S) and suppose that clocks at rest relative to this frame are synchro- 
nized with a mastei* at O. Any event can now be fixed relative to S by 
four coordinates (x, y,z, i), the space coordinates being measured by 
scales which are at rest in S and the time coordinate by the contiguous 
clock at rest in S. If (x,y,z, t), (x,y,z, t) relate to the same event, in this 
section we are concerned to find the equations relating these corre- 
sponding coordinates. 

The possibility that the length of a scale and the rate of a clock may 
be affected by uniform motion relative to a reference frame was 
ignored in early physical theories. Velocity measurements were agreed 
to be dependent upon the reference frame, but lengths and time 
measurements were thought to be absolute. We shall make no such 
assumption, but will choose the equations relating the coordinates of 
an event in the two frames to be of such a form that (i) a particle which 
has uniform motion relative to one frame, has uniform motion relative 
to the other and (ii) the velocity of propagation of light is the same 
constant c in both frames. Unless (i) is true, Newton's first law must 
be abandoned and, with it, the very concept of an inertial frame. 
Experimental results force us to accept (ii). 

To comply with requirement (i), we shall assume that each of the 
coordinates (x,y,z, f) is a linear function of the coordinates (x,y, z, t). 
The inverse relationship is then of the same type. A particle moving 
uniformly in S with velocity (v x ,v y ,v z ) will have space coordinates 
(x,y,z) such that 

x = x +v x t, y = y +v y t, z = z +v z t. (4.1) 



SPECIAL PRINCIPLE OF RELATIVITY 9 

If linear expressions in the coordinates (x,y,z, i) are now substituted 
for (x,y, z, t), it will be found on solving for (x, y, z) that these quantities 
are linear in f and hence that the particle's motion is uniform relative 
to S. In fact, it may be proved that only a linear transformation can 
satisfy the requirement (i). 

Now suppose that at the instant t = t a light source situated at the 
point P (xo,yo,z ) in S radiates a pulse of short duration. At any 
later instant t, the wavefront will occupy the sphere whose centre is 
P and radius c(t—t ). This has equation 

(x-x ) 2 + (y-y ) 2 + (z-x ) 2 = c\t-t ) 2 . (4.2) 

Let (x , y ,z ) be the coordinates of the light source as observed from 
S at the instant i=i the short pulse is radiated. At any later instant /, 
in accordance with requirement (ii), the wavefront must also appear 
from S to occupy a sphere of radius c(i — i ) and centre (xo,y ,z ). This 
has equation 

(x-x ) 2 +ty-yo) 2 + (.z-z ) 2 = cHi-i Q f. (4.3) 

Equations (4.2), (4.3) describe the same set of events in languages 
appropriate to S, S respectively. It follows that the equations relating 
the coordinates (x,y,z, t), (x,y,z, i) must be so chosen that, upon sub- 
stitution for the 'barred' quantities appearing in equation (4.3) the 
appropriate linear expressions in the 'unbarred' quantities, equation 
(4.2) results. 

A mathematical device due to Minkowski will now be employed. 
We shall replace the time coordinate / of any event observed in S by 
a purely imaginary coordinate x 4 = ict (/= V~ !)• The space co- 
ordinates (x,y,z) of the event will be replaced by (^1,^2,^3) so that 

x = x it y = x 2 , z = x 3 , ict = x 4 (4.4) 

and any event is then determined by four coordinates x t (1 = 1 , 2, 3, 4). 
A similar transformation to coordinates Jc,- will be carried out in S. 
Equations (4.2), (4.3) can then be written 

S (*,-*/o) 2 = 0, (4.5) 

/=i 

S (x -Xiof = 0. (4.6) 

=1 



10 TENSOR CALCULUS AND RELATIVITY 

The Jc f are to be linear functions of the x { and such as to transform 
equation (4.6) into equation (4.5) and hence such that 

2 (*«-*,o) 2 -* k 2 ( Xi -x i0 ) 2 . (4.7) 

/=i »=i 

k can only depend upon the relative velocity of S and S. It is reasonable 
to assume that the relationship between the two frames is a reciprocal 
one, so that, when the inverse transformation is made from S to S, then 

2 (Xi-x i0 ) 2 -* k 2 (Jc.-^o) 2 . (4.8) 

But the transformation followed by its inverse must leave any function 
of the coordinates x { unaltered and hence k 2 = 1. In the limit, as the 
relative motion of S and S is reduced to zero, it is clear that k -> + 1 . 
Hence k ^ - 1 and we conclude that k is identically unity. 

The x t will now be interpreted as rectangular Cartesian coordinates 
in a four-dimensional Euclidean space which we shall refer to as «^ 4 . 
This space is termed Minkowski space-time. The left-hand member of 
equation (4.5) is then the square of the 'distance' between two points 
having coordinates x h x i0 . It is now clear that the x t can be interpreted 
as the coordinates of the point x t referred to some other rectangular 
Cartesian axes in <? 4 . For such an interpretation will certainly enable 
us to satisfy the requirement (4.7) (with k-l). Also, the x h x t will 
then be related by equations of the form 

4 

^ = 2 ayxj+bi, (4.9) 

}=\ 

where / = 1 , 2, 3, 4 and the ay, b t are constants and this relationship is 
linear. The b t are the coordinates of the origin of the first set of rect- 
angular axes relative to the second set. The a i} will be shown to satisfy 
certain identities in Chapter 2 (equations (8.14), (8.15)). It is proved 
in algebra texts that the relationship between the x t and x t must be of 
the form we are assuming, if it is (i) linear and (ii) such as to satisfy 
the requirement (4.7). 

Changing back from the x h x t to the original coordinates of an 
event by equations (4.4), the equations (4.9) provide a means of re- 
lating space and time measurements in S with the corresponding 



SPECIAL PRINCIPLE OF RELATIVITY 



11 



measurements in S. Subject to certain provisos (e.g. an event which 
has real coordinates in S, must have real coordinates in S), this trans- 
formation will be referred to as the General Lorentz Transformation. 

5. The special Lorentz transformation 

We shall now investigate the special Lorentz transformation obtained 
by supposing that the jc r axes in ^ 4 are obtained from the x r axes by a 
rotation through an angle a parallel to the x t x 4 -plane. The origin and 




Fig. 



the * 2 , x 3 -axes are unaffected by the rotation and it will be clear after 
consideration of Fig. 1 therefore that 

jci = xi cos a +x 4 sin a, x 2 = 

jc 4 = -Jtisina+xicosa, x 3 
Employing equations (4.4), these transformation equations may be 
written 

x = jtcos<x+/cfsina, y = y, 

id = — xsina+fc/cosa, z = z. 



= *2>1 
= X 3 .j 



(5.1) 



(5.2) 



12 TENSOR CALCULUS AND RELATIVITY 

To interpret the equations (5.2), consider a plane which is stationary 
relative to the S frame and has equation 

dx+By+cz+d = (5.3) 

for all f . Its equation relative to the S frame will be 

(a cos a) x + By + cz+d+ictd sin a = 0, (5.4) 

at any fixed instant t. In particular, if a — B = d= 0, this is the co- 
ordinate plane Oxy and its equation relative to S is z = 0, i.e. it is the 




Fig. 2 

plane Oxy. Again, if B = c - d= 0, the plane is Oyz and its equation 
in S is 

x = — /cftana, (5.5) 

i.e. it is a plane parallel to Oyz displaced a distance - /c/tan« along 
Ox. Finally, if a = c = *7= 0, the plane is OiJc and its equation with 
respect to S is y = 0, i.e. it is the plane Ozx. We conclude, therefore, 
that the Lorentz transformation equations (5.2) correspond to the 
particular case when the coordinate planes comprising S are obtained 
from those comprising S at any instant / by a translation along Ox a 
distance - fc/tana (Fig. 2). Thus, if u is the speed of translation of S 
relative to S 

u = — /ctana. (5.6) 



SPECIAL PRINCIPLE OF RELATIVITY 13 

It should also be noted that the events 

x = y = z=t = 0, x = y = z = i = 

correspond and hence that, at the instant O and 6 coincide, the S and 
S clocks at these points are supposed set to have zero readings; all 
other clocks are then synchronized with these. 

Equation (5.6) indicates that a is imaginary and is directly related 
to the speed of translation. We have tana = iu/c and hence 

1 . (ju/c) 

va-«W sina_ vd-« 2 /c 2 ) 



cos a = — — iTjk* sina = ~~ 77\ 27~2>" (•*''' 



Substituting in the equations (5.2), the special Lorentz transformation 
is obtained in its final form, viz. 



x — ut _ 

^VO-kV)' y " y * 

_ t-(ux/c 2 ) 

'"Vd-^/c 2 )' Z ~ Z ' 



(5.8) 



If u is small by comparison with c, as is generally the case, these 
equations may evidently be approximated by the equations 

x = x-ut, y = yA (59) 

i = t, z = z.\ 

This set of equations, called the Special Galilean Transformation 
equations, is, of course, the set which was assumed to relate space and 
time measurements in the two frames in classical physical theory. 
However, the equation i = t was rarely stated explicitly, since it was 
taken as self-evident that time measurements were absolute, i.e. quite 
independent of the observer. It appears from equations (5.8) that this 
view of the nature of time can no longer be maintained and that, in 
fact, time and space measurements are related, as is shown by the 
dependence of i upon both t and jc. This revolutionary idea is also 
suggested by the manner in which the special Lorentz transformation 
has been derived, viz. by a rotation of axes in a manifold which has 
both spacelike and timelike characteristics. However, this does not 



14 TENSOR CALCULUS AND RELATIVITY 

imply that space and time are now to be regarded as basically similar 
physical quantities, for it has only been possible to place the time co- 
ordinate on the same footing as the space coordinates in 84 by multi- 
plying the former by /. Since x 4 must always be imaginary, whereas 
x it x 2 , x 3 are real, the fundamentally different nature of space and 
time measurements is still maintained in the new theory. 

If u > c, both Jc and / as given by equations (5.8) are imaginary. We 
conclude that no observer can possess a velocity greater than that of 
light relative to any other observer. 

If equations (5.8) are solved for (x,y,z,t) in terms of (x,y,z,i), it 
will be found that the inverse transformation is identical with the 
original transformation, except that the sign of w is reversed. This also 
follows from the fact that the inverse transformation corresponds to 
a rotation of axes through an angle — a in space-time. Thus, the frame 
S has velocity — u when observed from S. 

6. Fitzgerald contraction. Time dilation 

In the next two sections, we shall explore some of the more elementary 
physical consequences of the transformation equations (5.8). 

Consider first a rigid rod stationary in S and lying along the x-axis. 
Let x = x x , x = x 2 at the two ends of the bar so that its length as 
measured in S is given by 

l = x 2 -x x . (6.1) 

At the instant / in S, suppose these ends occupy the positions x = x lt 
x = x 2 . Then, by equations (5.8), 

*» - va-^/c 2 )' 2 ~ va-u 2 jc 2 ) c } 

But x 2 — x 1 = / is the length of the bar as measured in S and it follows 
by subtraction of equations (6.3) that 

/^MI-kV). (6.3) 

The length of a bar accordingly suffers contraction when it is moved 
longitudinally relative to an inertial frame. This is the Fitzgerald 
contraction. 



SPECIAL PRINCIPLE OF RELATIVITY 15 

This contraction is not to be thought of as the physical reaction of 
the rod to its motion and as belonging to the same category of physical 
effects as the contraction of a metal rod when it is cooled. It is due to a 
changed relationship between the rod and the instruments measuring 
its length, /is a measurement carried out by scales which are stationary 
relative to the bar, whereas / is the result of a measuring operation with 
scales which are moving with respect to the bar. Also, the first opera- 
tion can be carried out without the assistance of a clock, but the second 
operation involves simultaneous observation of the two ends of the 
bar and hence clocks must be employed. In classical physics, it was 
assumed that these two measurement procedures would yield the same 
result, since it was supposed that a rigid bar possessed intrinsically an 
attribute called its length and that this could in no way be affected by 
the procedure employed to measure it. It is now understood that 
length, like every other physical quantity, is defined by the procedure 
employed for its measurement and that it possesses no meaning apart 
from being the result of this procedure. From this point of view, it is 
not surprising that, when the procedure must be altered to suit the 
circumstances, the result will also be changed. It may assist the reader 
to adopt the modern view of the Fitzgerald contraction if we remark 
that the length of the rod considered above can be altered at any 
instant by simply changing our minds and commencing to employ the 
5" frame rather than the S frame. Clearly, such a change of mathe- 
matical description can have no physical consequences. 

Now consider two events which have coordinates (x\,yi,z u t), 
(x 2 ,y2,Z2>t) in S and hence occur simultaneously at different points. 
These events are not necessarily simultaneous in S. For, if i = 1 1 at 
the first mentioned event and / = f 2 at the second, then 

h-h = "(*i-*2)/V0-"V) (6-4) 

c 

and h * f ! , unless x x = x 2 . The concept of simultaneity is accordingly, 
also, a relative one and has no absolute meaning as was previously 
thought. 

The registration by the clock moving with O of the times i i , i 2 , con- 
stitutes two events having coordinates (0,0,0,^), (0,0,0,/ 2 ) respec- 
tively in S. Employing the inverse transformation to (5.8), it follows 



16 TENSOR CALCULUS AND RELATIVITY 

that the times t lt t 2 of these events as measured in S are given by 

'i = hi V(l - «V), h = i 2 \ V(l - u 2 /c 2 ), (6.5) 

and hence that 

h-h = (.h-t 2 W(\-u 2 l<?). (6.6) 

This equation shows that the clock moving with 6 will appear from S 
to have its. rate reduced by a factor V(l - ^/c 2 ). This is the time 
dilation effect. 

Since any cyclic physical process, i.e. one which returns to some 
initial state after the lapse of a period of time, can be employed as a 
clock, the result just obtained implies that all physical processes will 
evolve more slowly when observed from a frame relative to which 
they are moving. Thus, the rate of decay of radioactive particles pres- 
ent in cosmic rays and moving with high velocities relative to the earth, 
has been observed to be reduced by exactly the factor predicted by 
equation (6.6). 

It may also be deduced that, if a human passenger were to be 
launched from the earth in a rocket which attained a speed approach- 
ing that of light and after proceeding to a great distance returned to 
the earth with the same high speed, suitable observations made from 
the earth would indicate that all physical processes occurring within 
the rocket, including the metabolic and physiological processes taking 
place inside the passenger's body, would suffer a retardation. Since all 
physical processes would be affected equally, the passenger would be 
unaware of this effect. Nonetheless, upon return to the earth he would 
find that his estimate of the duration of the flight was less than the 
terrestrial estimate. It may be objected that the passenger is entitled to 
regard himself as having been at rest and the earth as having suffered 
the displacement and therefore that the terrestrial estimate should be 
less than his own. This is the clock paradox. The paradox is resolved by 
observing that a frame moving with the rocket is subject to an acceler- 
ation relative to an inertial frame and consequently cannot be treated 
as inertial. The results of special relativity only apply to inertial frames 
and the rocket passenger is accordingly not entitled to make use of 
them in his own frame. As will be shown later, the methods of general 
relativity theory are applicable in any frame and it may be proved 



SPECIAL PRINCIPLE OF RELATIVITY 17 

that, if the passenger employs these methods, his calculations will yield 
results in agreement with those obtained by the terrestrial observer. 

7. Spacelike and timelike intervals. Light cone 

We have proved in section 4 that if x h x i0 are the coordinates in 
Minkowski space-time of two events, then 

S (*,-*/o) 2 (7.D 

is invariant, i.e. has the same value for all observers employing inertial 
frames and thus rectangular axes in space-time. Reverting by equa- 
tions (4.4) to the ordinary space and time coordinates employed in an 
inertial frame, it follows that 

(x-x ) 2 + (y-y ) 2 + (z-z ) 2 -c 2 (t-t ) 2 , (7.2) 

is invariant for all inertial observers. 

Thus, if (x,y,z,t), (x ,y ,ZQ,t ) are the coordinates of two events 
relative to any inertial frame S and we define the proper time interval r 
between the events by the equation 

r 2 = {t-t ) 2 -±{(x-x ) 2 + (y-y ) 2 +(.z-z ) 2 }, (7.3) 

then t is an invariant for the two events. Two observers employing 
different inertial frames may attribute different coordinates to the 
events, but they will be in agreement concerning the value of t. 

Denoting the time interval between the events by A t and the distance 
between them by Ad, both relative to the same frame S and positive, 
it follows from equation (7.3) that 

r 2 = At 2 -^Ad 2 . (7.4) 

cr 

Suppose that a new inertial frame S is now defined, moving in the 
direction of the line joining the events with speed Ad/ A t. This will only 
be possible if Ad/ At < c. Relative to this frame the events will occur 
at the same point and hence Ad = 0. By equation (7.4), therefore, 

t 2 = At 2 , (7.5) 

i.e. the proper time interval between two events is the ordinary time 



18 TENSOR CALCULUS AND RELATIVITY 

interval measured in a frame (if such exists) in which the events occur 
at the same space point. In this case, it is clear that t 2 > and the 
proper time interval between the events is said to be timelike. 

Suppose, if possible, that a frame S can be chosen relative to which 
the events are simultaneous. In this frame At = and 

r 2 = -^Ad 2 . (7.6) 

(T 

Thus t 2 < 0, and, in any frame, Ad/ At > c. t is then purely imaginary 
and the interval between the events is said to be spacelike. 

If the interval is timelike, Ad/ At < c and it is possible for a material 
body to be present at both events. On the other hand, if the interval is 
spacelike, Ad/At>c and it is not possible for such a body to be present 
at both events. The intermediate case is when Ad/ At = c and then 
t = 0. Only a light pulse can be present at both events. It also follows 
that the proper time interval between the transmission and reception 
of a light signal is zero. 

We shall now represent the event (x,y,z,t) by a point having these 
coordinates in a four-dimensional space. This space is also often 
referred to as Minkowski space-time but, unlike the space-time con- 
tinuum introduced in section 4, it is not Euclidean. However, this 
representation has the advantage that the coordinates all take real 
values and it is therefore more satisfactory when diagrams are to be 
drawn. Suppose a particle is at the origin O of S at t = and com- 
mences to move along Ox with constant speed u. Its y- and z-coordin- 
ates will always be zero and the representation of its motion in 
space-time will be confined to the xf-plane. In this plane, its motion 
will appear as the straight line QP, Q being the point x = y = z=t = 
(Fig. 3). QP is called the world-line of the particle. If LPQt = 6, 
tan0 = u. But \u\^c and hence the world-line of the particle must lie 
in the sector AQB, where LAQB = 2a and tana = c. Similarly, the 
world-line of a particle which arrives at O at t = after moving along 
Ox, must lie in the sector A' QB'. It follows that any event in either of 
these sectors must be separated from the event Q by a timelike inter- 
val, since a particle can be present at both events. Events in the sectors 
AQB', A' QB are separated from Q by spacelike intervals. A' A, B'B 
are the world-lines of light signals passing through O at t = and 



SPECIAL PRINCIPLE OF RELATIVITY 19 

being propagated in the directions of the positive and negative x-axis 
respectively. 
For any event in A QB, f > 0, i.e. it is in the future with respect to the 




Fig. 3 



event Q when the frame S is being employed. However, by no choice 
of frame can it be made simultaneous with Q, for this would imply a 
spacelike interval. A fortiori, in no frame can it occur prior to Q. The 
sector AQB accordingly contains events which are in the absolute 



20 TENSOR CALCULUS AND RELATIVITY 

future with respect to the event Q. Similarly all events in the sector 
A' QB' are in the absolute past with respect to Q. On the other hand, 
events lying in the sectors AQB',A' QB are separated from Qby space- 
like intervals and can all be made simultaneous with Q by proper 
choice of inertial frame. These events may occur before or after Q 
depending upon the frame being used. These two sectors define a 
region of space-time which will be termed the conditional present. 

Since no physical signal can have a speed greater than c, the world- 
line of any such signal emanating from Q must lie in the sector AQB. 
It follows that the event Q can be the physical cause of only those 
events which are in the absolute future with respect to Q. Similarly, Q 
can be the effect of only those events in its absolute past. Q cannot be 
causally related to events in its conditional present. 

This state of affairs should be contrasted with the essentially simpler 
situation of classical physics where there is no upper limit to the signal 
velocity and AA', BB' coincide along the x-axis. Past and future are 
then separated by a perfectly precise present in which events all have 
the time coordinate t = for all observers. 

In the four-dimensional space Qxyzt, the three regions of absolute 
past, absolute future and conditional present are separated from one 
another by the hyper-cone 

x 2 +y 2 +z 2 -c 2 t 2 = 0. (7.7) 

A light pulse transmitted from Q will have its world-line on this sur- 
face, which is accordingly called the light cone at Q. Since any arbitrary 
event can be selected to be Q, any event is the apex of a light cone 
which separates the space-time continuum in an absolute manner into 
three distinct regions relative to the event. 



Exercises 1 

1. A particle of mass m is moving in the plane of axes Oxy under 
the action of a force f . Oxy is an inertial frame. Ox'y' is rotating rela- 
tive to the inertial frame so that Ax' Ox = tf/ and tf> = co. (r, &) are polar 
coordinates of the particle relative to the rotating frame. If (f r , fe) 
are the polar components off, (a r ,ae) are the polar components of the 



SPECIAL PRINCIPLE OF RELATIVITY 21 

particle's acceleration relative to Ox'y', v is the particle's speed rela- 
tive to this frame and ^ is the angle its direction of motion makes with 
the radius vector in this frame, obtain the equations of motion in the 
form 

ma r = f r + 2maw sin <f> + mrca 2 , 

mag = fg — 2mciw cos <f> — mrw. 

Deduce that the motion relative to the rotating frame is in accordance 
with the second law if, in addition to f, the following forces are also 
taken to act on the particle : (i) mio 2 r radially outwards (the centrifugal 
force), (ii) 2m<nv at right angles to the direction of motion (the Coriolis 
force), (iii) mrw transversely. (The latter force vanishes if the rotation 
is uniform.) 

2. A bar lies along Ox and is stationary in S. Show that if the po- 
sitions of its ends are observed in S at instants which are simultaneous 
in S, its length deduced from these observations will be greater than 
its length in S by a factor (1 - J/c 2 ) ~ 1/2 . 

3. Suppose that the bar referred to in Exercise 2 takes a time 7" to 
pass a fixed point on the jc-axis, Toeing measured by a clock stationary 
at the fixed point. Defining the length of the bar in the S-frame to be 
uT, deduce the Fitzgerald contraction. 

4. The measuring rod employed by S will appear from S to be 
shortened by a factor (1 - i^/c 2 ) l /2 . Hence, when S measures the length 
of the bar fixed in S he might be expected to obtain the result 

/=//(l-« 2 /c 2 ) ,/2 . 

This contradicts equation (6.3). Resolve the contradiction. [Hint: It 
will be observed from S that S fixes the position of the forward end of 
the bar first and the position of the rear end a time hZ/c 2 later.] 

5. A bar lies stationary along the *-axis of S, Show that the 
world-lines of the particles of the bar occupy a certain 'band' in 
the xi jc 4 -plane. By measuring the width of this 'band' parallel to the 
xi-axis, deduce the Fitzgerald contraction. 

6. Verify that the transformation equations (5.8) are such that 



22 TENSOR CALCULUS AND RELATIVITY 

7. Two light pulses are moving in the positive direction along the 
x-axis of the frame S, the distance between them being d. Show that, 
as measured in S, the distance between the pulses is 



•m 



8. A and B are two points of an inertial frame S a distance d apart. 
An event occurs at B a time T (relative to clocks in S) after another 
event occurs at A. Relative to another inertial frame S, the events are 
simultaneous. If AP is a displacement vector in S representing the 
velocity of S relative to S, prove that P lies in a plane perpendicular 
to AB, distance c 2 T/d from A. 

9. S, S are the inertial frames considered in section 5. The length 
of a moving rod, which remains parallel to the x and x axes, is 
measured as a in the frame S and a in the frame S. By consideration 
of a Minkowski diagram for the rod, or otherwise, show that the 
proper length of the rod is 

adfiu/c 



V(2Pad-a 2 -a 2 ) 



where jS = (l-«2/ c 2)-i/2 > 

10. If the position vectors r = (x,y,z), t = (x,y,z) of an event as 
determined by the observers in the parallel inertial frames S, S 
respectively are mapped in the same independent «^ 3 , prove that 



f = r+ufe r (j3-l) + jS*j, 
i = £(/+u.r/c 2 ), 



where jS = (1 — u 2 /c 2 ) 1/2 and u is the velocity of S as measured from 
S. 

11. The centroid and axis of a right circular cylinder are fixed in 
the inertial frame S relative to which the cylinder rotates about its 
axis with uniform velocity a>. Prove that, when observed from S, the 
cylinder will appear to be twisted about its axis through an angle 
uw/c 2 per unit rest-length of the cylinder. 



CHAPTER 2 

Orthogonal Transformations. Cartesian 
Tensors 

8. Orthogonal transformations 

In section 4 events have been represented by points in a space <? 4 . The 
resulting distribution of points was described in terms of their co- 
ordinates relative to a set of rectangular Cartesian axes. Each such set 
of axes was shown to correspond to an observer employing a rect- 
angular Cartesian inertial frame in ordinary «? 3 -space and clocks 
which are stationary in this frame. In this representation, the descrip- 
tions of physical phenomena given by two such inertial observers are 
related by a transformation in <^ 4 from one set of rectangular axes to 
another. Such a transformation has been given at equation (4.9) and 
is called an orthogonal transformation. In general, if x h x t (/= 1, 
2,...,N) are two sets of N quantities which are related by a linear 
transformation 

N 

Xi = 2 ayXj+b h (8.1) 

Mi 

and, if the coefficients % of this transformation are such that 

is an identity for all corresponding sets x h x t and y u y h then the trans- 
formation is said to be orthogonal. It is clear that the x h x t hiay be 
thought of as the coordinates of a point in & N referred to two different 
sets of rectangular Cartesian axes and then equation (8.2) states that 
the square of the distance between two points is an invariant, inde- 
pendent of the Cartesian frame. 
Writing z, = x t -y h z t = x t -p h it follows from equation (8.1) that 

N 
*i = 2 a ij z i' (8*3) 

[23] 



24 TENSOR CALCULUS AND RELATIVITY 

Let z denote the column matrix with elements z h 2 the column matrix 
with elements 2 , and A the N x JVmatrix with elements ay. Then the set 
of equations (8.3) is equivalent to the matrix equation 

2 = Az. (8.4) 

Also, if z' is the transpose of z, 

z'z = 2 z? (8.5) 

/=i 

and thus the identity (8.2) may be written 

2' 2 = z'z. (8.6) 

But, from equation (8.4), 

z' = z'A'. (8.7) 

Substituting in the left-hand member of equation (8.6) from equations 
(8.4), (8.7), it will be found that 

z'A'Az = z'z. (8.8) 

This can only be true for all z if 

A' A = I, (8.9) 

where / is the unit N x N matrix. 

Taking determinants of both members of the matrix equation (8.9), 
we find that \A\ 2 — 1 and hence 

\A\ = ±1. (8.10) 

A is accordingly regular. Let A ~ i be its inverse. Multiplication on the 
right by A~ l of both members of equation (8.9) then yields 

A' = A~\ (8.11) 

It now follows that 

AA' = AA~ l = /. (8.12) 

Let 8fj be the if* element of /, so that 



= 0, ijtj.j 



(8.13) 



ORTHOGONAL TRANSFORMATIONS 25 

The symbols 8y are referred to as the Kronecker deltas. Equations 
(8.9), (8.12) are now seen to be equivalent to 

N 

1=1 

2 a,ia u = 8 Jk (8.15) 

/=i 

respectively. These conditions are necessarily satisfied by the co- 
efficients a u of the transformation (8. 1) if it is orthogonal. Conversely, 
if either of these conditions is satisfied, it is easy to prove that equation 
(8.6) follows and hence that the transformation is orthogonal. 

9. Repeated index summation convention 

At this point it is convenient to introduce a notation which will greatly 
abbreviate future manipulative work. It will be understood that, 
wherever in any term of an expression a literal index occurs twice, this 
term is to be summed over all possible values of the index. For 
example, we shall abbreviate by writing 

2 a r b r = a r b r . (9.1) 

r=l 

The index must be a literal one and we shall further stipulate that it 
must be a small letter. Thus a 2 b 2 , a N b N are individual terms of the 
expression a r b r> and no summation is intended in these cases. 

Employing this convention, equations (8.14) and (8.15) can be 
written 

OffOtk = $jk> a Jt a ki = S Jk (9.2) 

respectively. Again, with z t = x t - y h equation (8.2) may be written 

ZiZi = z t Zi. (9.3) 

More than one index may be repeated in the same term, in which 
case more than one summation is intended. Thus 

N N 

aijbj k c k = 2 S a u b Jk e k . (9.4) 

y-i *=i 

It is permissible to replace a repeated index by any other small 



26 TENSOR CALCULUS AND RELATIVITY 

letter, provided the replacement index does not occur elsewhere in 
the same term. Thus 

a t bt = ajbj = a k b k , (9.5) 

but 

«/./«/* ^ ajja Jkt (9.6) 

irrespective of whether the right-hand member is summed with respect 
toy or not. A repeated index shares this property with the variable of 
integration in a definite integral. Thus 

b b 

j f(x)dx - j f(y)dy. (9.7) 

a a 

A repeated index is accordingly referred to as a dummy index. Any 
other index will be called a free index. 

It will be assumed, in future, that any equation remains true when 
the free indices assume all possible values. Thus, equations (9.2) are 
true for ally = 1,2, . . .,Na.nd all k = 1,2,. . ,,N. It is clear that the free 
indices on the two sides of an equation will be identical. 

The reader should note carefully the identity 

$(/«; = «/, (9.8) 

for it will be of frequent application. 8y is often called a substitution 
operator, since when it multiplies a symbol such as a Jt its effect is to 
replace the index j by /. 

10. Rectangular Cartesian tensors 

Let x h y t be rectangular Cartesian coordinates of two points Q, P 
respectively in S N . Writing z t = x t — y h the z t are termed the compon- 
ents of the displacement vector PQ relative to the axes being used. If 
xt, y ( are the coordinates of Q, P with respect to another set of rect- 
angular axes, the new coordinates will be related to the old by the 
transformation equations (8.1). Then, iff, are the components of PQ 
in the new frame, it follows (equation (8.3)) that 

2t = oyzj. (10.1) 

Any set of N quantities which take the values A t when the first co- 
ordinate frame is being employed and which transform in the same 



ORTHOGONAL TRANSFORMATIONS 27 

manner as the z t when referred to a new coordinate frame, i.e. are 
such that 

A ( = a i} A h (10.2) 

are said to be the components of a vector in £ N relative to rectangular 
Cartesian reference frames. We shall frequently abbreviate ' the vector 
whose components are A? to 'the vector A t \ We shall also denote the 
vector by A. 

If A h B t are two vectors, consider the N 2 quantities AiBj. Upon 
transformation of axes, these quantities transform thus: 

A^j = a lkajl A k B h (10.3) 

Any set of N 2 quantities Cy which transform in this manner, i.e. which 
are such that 

Cy = a ik a n C kU (10.4) 

are said to be the components of a tensor of the second rank. We shall 
speak of 'the tensor Cy. Such a tensor is not, necessarily, represent- 
able as the product of two vectors. 

A set of N 3 quantities Dy k which transform in the same manner as 
the product of three vectors A t Bj C k , form a tensor of the third rank. 
The transformation law is 

^Uk- a il a jmOkn D lmn' (10-5) 

The generalization to a tensor of any rank should now be obvious. 
Vectors are, of course, tensors of the first rank. 

If Ay, By are tensors, the sums Ay+By are N 2 quantities which 
transform according to the same law as the Ay and By. The sum of two 
tensors of the second rank is accordingly also a tensor of this rank. 
This result can be generalized immediately to the sum of any two 
tensors of identical rank. Similarly, the difference of two tensors of 
the same rank is also a tensor. 

Our method of introducing a tensor implies that the product of any 
number of vectors is a tensor. Quite generally, if Ay._, JB y ... are 
tensors of any ranks (which may be different), then the product 
Ay . . B M . . . is a tensor whose rank is the sum of the ranks of the two 
factors. The reader should prove this formally for a product such as 



28 TENSOR CALCULUS AND RELATIVITY 

AyB k i m , by writing down the transformation equations. (N.B. the 
indices in the two factors must be kept distinct, for otherwise a sum- 
mation is implied and this complicates matters; see section 12.) 

The components of a tensor may be chosen arbitrarily relative to 
any one set of axes. The components of the tensor relative to any other 
set are then fixed by the transformation equations. Consider the tensor 
of the second rank whose components relative to the x,-axes are the 
Kronecker deltas 8y. In the ^,-frame, the components are 

8y = aik<*ji8 kl = a ik a Jk = 8 y , (10.6) 

by equations (9.2). Thus this tensor has the same components relative 
to all sets of axes. It is termed the fundamental tensor of the second 
rank. 
If, to take the particular case of a third rank tensor as an example, 

A ijk = A jik (10.7) 

for all values of i,y, k, A ijk is said to be symmetric with respect to its 
indices i,j. Symmetry may be with respect to any pair of indices. If 
A iJk is a tensor, its property of symmetry with respect to two indices is 
preserved upon transformation, for 

Ajik = a jl a im a knAi m m 
~ a im a )l a knA m \n, 

- *ijk, (10.8) 

where, in the second line, we have rearranged and put A imn = A m i„. 
Unless a property is preserved upon transformation, it will be of little 
importance to us, for we shall later employ tensors to express relation- 
ships which are valid for all observers and a chance relationship, true 
in one frame alone, will be of no fundamental significance. 
Similarly, if 

A m = -A M (10.9) 

for all values of i, j, k, A ijk is said to be skew-symmetric or anti- 
symmetric with respect to its first two indices. This property also is 
preserved upon transformation. Since A nk = —A nk , A nk = 0. All 
components of A uk with the first two indices the same are clearly zero. 



ORTHOGONAL TRANSFORMATIONS 29 

A tensor whose components are all zero in one frame, has zero 
components in every frame. A corollary to this result is that if Ay ,, ., 
By , . . are two tensors of the same rank whose corresponding compon- 
ents are equal in one frame, then they are equal in every frame. This 
follows because Ay, ..— By,,, is a tensor whose components are all 
zero in the first frame and hence in every frame. Thus, a tensor equation 

Ay... = By... (10.10) 

is valid for all choices of axes. This explains the importance of tensors 
for our purpose. By expressing a physical law as a tensor equation, we 
shall guarantee its covariance with respect to a change of inertial 
frame. 

11. Invariants. Gradients. Derivatives of tensors 

Suppose that V is a quantity which is unaffected by any change of 
axes. Then V is called a scalar invariant or simply an invariant. Its 
transformation equation is simply 

V= V. (11.1) 

As will be proved later (section 21), the charge of an electron is 
independent of the inertial frame from which it is measured and is, 
therefore, the type of quantity we are considering. 

If a value of V is associated with each point of a region of & N , an 
invariant field 'is defined over this region. In this case Fwill be a func- 
tion of the coordinates x t . Upon transformation to new axes, V will 
be expressed in terms of the new coordinates x t ; when so expressed, it 
is denoted by V. Thus 

P(*i,*2 x N ) = V(xi,x 2 ,...,x N ) (11.2) 

is an identity. The reader should, perhaps, be warned that Vis not, 
necessarily, the same function of the x t that Kis of the x t . 

If Ay is a tensor, it is obvious that VAy is also a tensor of the second 
rank. It is therefore convenient to regard an invariant as a tensor of 
zero rank. 

Consider the N partial derivatives 8V/dx t . These transform as a 
vector. To prove this it will be necessary to examine the transformation 



30 TENSOR CALCULUS AND RELATIVITY 

inverse to (8.1). In the matrix notation of section 8, this may be written 

x = A~\x-b) = A'(x-b), (11.3) 

having made use of equation (8.11). Equation (11.3) is equivalent to 

x i = a' u (x J -bj), (11.4) 

where a^ is the if h element of A'. But a'y = a, 7 and hence 

xi = ajiixj-bj). (11.5) 

It now follows that 

dx 



and hence that 



«*;-"* (,L6) 

dp BVdxj 8V __ 

proving that dV/dx t is a vector. It is called the gradient of Fand is 
denoted by grad V or V V. 

If a tensor Ay . . . is defined at every point of some region of £ N , the 
result is a tensor field. The partial derivatives &4,y. /Sx r can now be 
formed and constitute a tensor whose rank is one greater than that of 
Ay .... We shall prove this for a second rank tensor field Ay. The 
argument is easily made general. We have 

8A U d < . x 

tt- = 7T \fli r a js A rs \ 

OXfc OXjt 



by equation (11.6). 






= a ir a js a kt ^, (11.8) 



12. Contraction. Scalar product. Divergence 

If two indices are made identical, a summation is implied. Thus, 
consider Ay k . Then 

A® = A in + A m + ... + A lNN . (12.1) 



ORTHOGONAL TRANSFORMATIONS 31 

There are JV 3 quantities A uk . However, of the indices in Ayj, only i 

remains free to range over the integers 1,2 N, and hence there are 

but JV quantities Ayj and we could put B t = Ayj. The rank has been 
reduced by two and the process is accordingly referred to as 
contraction. 

Contraction of a tensor yields another tensor. For example, if 
Bj = Ay then, employing equations (9.2), 

Si = A VJ = a iq a Jr a js A qrs = a iq 8 rs A qrx = a iq A qrr = a iq B q . (12.2) 

Thus J5,is a vector. The argument is easily generalized. 

In the special case of a tensor of rank two, e.g. Ay, it follows that 
Ay = A ih i.e. Ay is an invariant. Now, if A h B t are vectors, A t Bj is a 
tensor. Hence, A { Bi is an invariant. This contracted product is called 
the inner product or the scalar product of the two vectors. We shall 
write 

AiBi = A-B. (12.3) 

In particular, the scalar product of a vector with itself is an invariant. 
The positive square root of this invariant will be called the magnitude 
of the vector. Thus, if A is the magnitude of A h then 

A 2 = A,A, = AA = A 2 . (12.4) 

In <? 3 , if 6 is the angle between two vectors A and B, then 

ABcos9 = A>B. (12.5) 

In 6 Nt this equation is used to define 9. Hence, if 

A-B = 0, (12.6) 

then 8 = \tt and the vectors A, B are said to be orthogonal. 

If A t is a vector field, dAi/dxj is a tensor. By contraction it follows 
that dAi/dXi is an invariant. This invariant is called the divergence of 
A and is denoted by div A. Thus 

divA = ^'- (12.7) 

8xi 

More generally, if Ay„, is a tensor field, 8Ay„Jdx r is a tensor. This 
tensor derivative can now be contracted with respect to the index r 



32 TENSOR CALCULUS AND RELATIVITY 

and any other index to yield another tensor, e.g. 8A U . . . /dxj. This con- 
traction is also referred to as the divergence of Ay ^ with respect to 
the index j and we shall write 

-^ = div^y.... (12.8) 

dxj 

13. Tensor densities 

%j is a tensor density if, when the coordinates are subjected to the 
transformation (8.1), its components transform according to the law 

% J =\A\a ik a Jl * kh (13.1) 

\A\ being the determinant of the transformation matrix A. Since for 
orthogonal transformations \A\= ±1 (equation (8.10)), relative to 
rectangular Cartesian frames, tensors and tensor densities are identi- 
cal except that, for certain changes of axes, all the components of a 
density will be reversed in sign. For example, if in # 3 a change is made 
from the right-handed system of axes to a left-handed system, the 
determinant of the transformation will be — 1 and the components of 
a density will then be subject to this additional sign change. 

Let *u . . . n be a density of the N th rank which is skew-symmetric with 
respect to every pair of indices. Then all its components are zero, 
except those for which the indices ij, . . ., n are all different and form a 
permutation of the numbers 1, 2, . . ., N. The effect of transposing any 
pair of indices in e y ..„ is to change its sign. It follows that if the 

arrangement i, j, ...,n can be obtained from 1,2 N by an even 

number of transpositions, then e y ... B = + e 12 ...Ar> whereas if it can 
be obtained by an odd number c I y. --n = — c 12 ...jv. Relative to the x r 
axes let c 12 . . . at = 1 • Then, in this frame, t tj . . . „ is if ij, . . ., n is not a 

permutation of 1 , 2 N, is + 1 if it is an even permutation and is — 1 

if it is an odd permutation. Transforming to the Jc r axes, we find that 

*12...N = \Aaua2j-'-aNn*ij...n = Ml 2 = I- (13.2) 

But ly . . „ is also skew-symmetric with respect to all its indices, since 
this is a property preserved by the transformation. Its components 
are also 0, ± 1 therefore and ey.,.„ is a density with the same com- 
ponents in all frames. It is called the Levi-Civita tensor density. 



ORTHOGONAL TRANSFORMATIONS 33 

It may be shown without difficulty that: 

(i) the sum or difference of two densities of the same rank is a 

density, 
(ii) the product of a tensor and a density is a density, 
(iii) the product of a density and a density is a tensor, 
(iv) the partial derivative of a density with respect to x t is a 

density, 
(v) a contracted density is a density. 
Thus, to prove (iii), let %, 33/ be two vector densities. Then 

%®j = \A\ 2 a lk aj,* k ®, = a**!//**®/. (13.3) 

The method is clearly quite general. The remaining results will be left 
as exercises for the reader. 

14. Vector products. Curl 

Throughout this section we shall be assuming that N = 3, i.e. the space 
will be ordinary Euclidean space. 
Let A tJ be a skew-symmetric tensor. Denning 

% = hti} k A jkt (14.1) 

21/ is a vector density. Its components are 

«1 = h{A 23 -A 32 ) = A 23 ,' 

«2 = £0*31-^13) = 4n, ' (14.2) 

% = UAi2-A 2l ) = Ai 2 , 

i.e. are the three distinct non-zero components of Ay. We have proved, 
therefore, that these three components of any skew-symmetric tensor 
of the second rank may be regarded as the components of a vector 
density. 
It is easy to verify that the inverse relationship to (14.1) is 

Ay = t ijk K k . (14.3) 

Let A h B t be two vectors. From these we can form a skew-symmetric 
tensor of the second rank Cy such that 

Cy = AiBj-AjB,. (14.4) 



34 TENSOR CALCULUS AND RELATIVITY 

From Cy we can then form a vector density, viz. 

£/ = frukCjk = l*ijk(AjB k -A k Bj) = t uk AjB k , (14.5) 
whose components are 

Ci = A 2 B 3 -A 3 B 2i ' 

(£2 = AiBi-AiB}, - (14.6) 

Provided we employ only right-handed systems of axes or only left- 
handed systems in <^ 3 , £,- is indistinguishable from a vector. If, 
however, a change is made from a left-handed system to a right-handed 
system, or vice versa, the components of (£/ are multiplied by — 1 in 
addition to the usual vector transformation. Since it is usual to employ 
only right-handed frames, (£,- is often referred to as a vector (or an 
axial vector) and treated as such. It is then called the vector product of 
A and B and we write 

£ = AXB. (14.7) 

Vector multiplication is non-commutative, for 

BXA = t uk BjA k = -t ik jA k Bj = -AXB, (14.8) 

having made use of t ikj = -<t ijk . However, vector multiplication 
obeys the distributive law, for 

A X (B + C) = e uk Aj(B k + C k ) = t uk Aj B k + t ijk A } C k 

= AxB+AxC. (14.9) 

Again, if A t is a vector field, we can construct the skew-symmetric 
tensor of the second rank 

Rv = ^~ = Aj.t-Au- 04.10) 

dx t oxj 

We have introduced here the abbreviated notation A u = dArfdXj for 
partial derivatives with respect to the coordinates. This will be made 
use of in many later arguments. Corresponding to Ry, there is the 
vector density 5R, where 

»/ = frijkRjk = \t uk {A kJ -A hk ) = t uk A kJ . (14.11) 



ORTHOGONAL TRANSFORMATIONS 

This has components 

dA$ dA 2 

dx 2 fo 3 

dAj dA 3 

dx 3 dx t ' 

8A 2 &A\ 



35 



<R 1= r^_^f 



% 



*, = — -r-^ 



(14.12) 



3*1 dx 2 

and is denoted by curl A. It, also, is an axial vector. 

Equation (14.5) can still be employed to define a vector product 
when either or both of the vectors A, B are replaced by vector densities. 
If only one is replaced by a vector density, the right-hand member of 
equation (14.5) will involve the product of two densities and a vector. 
The resulting vector product will then be a vector. Similarly, by re- 
placing A in equation (14. 1 1) by a vector density, the curl of a vector 
density is defined as an ordinary vector. 

Exercises 2 
1. Show that, in two dimensions, the general orthogonal trans- 
formation has matrix A given by 



(cos 6 sin 6\ 
— sin0 cos 8 J 



Verify that \A\ = 1 and that A~* = A'. Ty is a tensor in this space. 
Write down in full the transformation equations for all its com- 
ponents and deduce that T u is an invariant. 

2. x = Ax, X = Bx are two successive orthogonal transformations 
relative to each of which T v transforms as a tensor. Show that the 
resultant transformation j? = BAx is orthogonal and that T tj trans- 
forms as a tensor with respect to it. 

3. Show that contraction of the Levi-Civita density results in the 
zero tensor density. 

4. In ^ 3 , prove that 



curlgrad V = 0, divcurlA = 0. 



36 TENSOR CALCULUS AND RELATIVITY 

5. In <^ 3 , prove that 

(0 *ikl*imn = §km%ln — &kn&lm> 

(") *ikl*ikm = 23/ m . 

6. In ^ 3 , show that 



„, etv d 2 v Pv 



dxj dxi 



7. In ^ 3 , prove that 

curl curl A = graddivA— V 2 A. 
[Hint: Employ Exercise 5 (i).] 

8. In ^ 3 , prove that 

(i) AX(BXC) = A-CB-A-BC, 



(ii) 



ABXC = 



A, 


Bi 


c, 


A 2 


B 2 


C 2 


A 3 


Bi 


c 3 



9. In £ N , prove that 

divFA = KdivA+A'gradK 

10. In ^ 3 , prove that 

(i) curl VA = Fcurl A - A X grad V, 

(ii) div(AxB) = B«curlA-A«curlB, 
(iii) curl(AxB) = B-VA-A-VB+AdivB- B divA, 
(iv) grad(A«B) = B-VA+A-VB+AXcurlB+BxcurlA, 
where A-VB = AjB u . 

1 1 . If Ay is a tensor and By = A jh prove that By is a tensor. Deduce 
that if Ay is symmetric in one frame, it is so in all. 



ORTHOGONAL TRANSFORMATIONS 37 

12. Prove that 8 v 8 ik = 8 Jk 

and that t ijk t lmn has the value + lifi,j,k are all different and (Jmri) 
is an even permutation of (ijk), — 1 if *, y, A: are all different and (//wi) 
is an odd permutation of (ijk), and otherwise. Deduce that 



*Vk*imn = S, 7 8 /m 8 A .„+8 /OT S /rt S ifc/ +S jn S // 8 A . OT 
Hence prove that 



- &in 8/w 8*/ - 8// 8 ya 8 Aw - 8^ 8ji 8^,. 



13. In ^ 3 , prove that 

(i) (aXbMcXd) = a*cb*d-a>db*c, 
(ii) (aXb)X(cXd) = [acd]b-[bcd]a 
= [abd]c-[abc]d, 

where [abc] = a*bxc. 



(M.T.) 



CHAPTER 3 

Special Relativity Mechanics 

15. The velocity vector 

Suppose that a point P is in motion relative to an inertial frame 5". 
Let ds be the distance between successive positions of P which it 
occupies at times t,t+dt respectively. Then, by equation (7.4), if dr 
is the proper time interval between these two events, 

/ 1 V 2 
dr = LdP—^ds 1 ] = (l-v 2 /(^) ll2 dt, (15.1) 

where v = ds/dt is the speed of P as measured in S. Now, as shown in 
section 7, dr is the time interval between the two events as measured 
in a frame for which the events occur at the same point. Thus dr is 
the time interval measured by a clock moving with P. dt is the time 
interval measured by clocks stationary in 5*. Equation (15.1) indicates 
that, as observed from S, the rate of the clock moving with P is slow 
by a factor (1— v 2 /*?) 112 . This is the phenomenon of time dilation 
already commented upon in section 6. If P leaves a point AaXt=t i 
and arrives at a point B at / = t 2 , the time of transit as registered by a 

clock moving with P will be 

ft 

T 2 -T! = j (\-1?l<?) XI2 dt. (15.2) 

h 
The successive positions of P together with the times it occupies 
these positions constitute a series of events which will lie on the point's 
world-line in Minkowski space-time. Erecting rectangular axes in 
space-time corresponding to the rectangular Cartesian frame S, let 
x t , Xt+dxi be the coordinates of adjacent points on the world-line. 
These points will represent the events (x,y,z,t), (x+dx,y+dy,z+dz y 
t+ dt) in S. If (v x , v y , v 2 ) are the components of the velocity vector v 
of P relative to S, then 

dx dy dz 

V * = Jt> V » = dl> ">=*' 05 - 3) 

[38] 



SPECIAL RELATIVITY MECHANICS 39 

v does not possess the transformation properties of a vector relative 
to orthogonal transformations (i.e. Lorentz transformations) in space- 
time. It is a vector relative to rectangular axes stationary in S only. 
However, we can define a 4-velocity vector which does possess such 
properties as follows: dx t is a displacement vector relative to rect- 
angular axes in space-time and dr is an invariant. It follows that dxi/dr 
is a vector relative to Lorentz transformations expressed as orthogonal 
transformations in space-time. It is called the 4-velocity vector of P 
and will be denoted by V. 
V can be expressed in terms of v thus : 

dXi = *, A = _ u2/c 2 r l/2^ (15 4) 

dr dt dr 

by equation (15.1). Also, from equations (4.4) we obtain 

xi = v x , x 2 = v y , jc 3 = v z , x+ = ic. (15.5) 

It now follows from these equations that 

V = (l-«V)- 1/2 C^,^,^,/c) = (l-v 2 l^r xi2 (yjc), (15.6) 

where the notation should be clear without further explanation. 

Knowing the manner in which the components of V transform when 
new axes are chosen in space-time, equation (15.6) enables us to calcu- 
late how the components of v transform when 5" is replaced by a new 
inertial frame S. Thus, consider the orthogonal transformation (5.1) 
which has been interpreted as a change from an inertial frame S to 
another S related to the first as shown in Fig. 2. The corresponding 
transformation equations for V are 

V x = Kxcosa+J^sina, V 2 = V 2 ,\ ^ ?) 

p 4 = - K,sina+ Kjcosa, V 3 = V 3 . J 

By equation (15.6), these equations are equivalent to 

(l-v 2 /c 2 )~ 1,2 v x = (l-v 2 /c 2 )~ li2 (v x cosa+icsin<x), 

(l-v 2 /c 2 r ll2 * y = (l-*Vr ,/: S, I 

{\-&ic 2 r xl2 * z = <\-v 2 i<?r m vz> 

(1 - &/<?) - 1J2 ic = (1 - i^/c 2 ) ~ ,/2 ( - v x sin a + ic cos a), 



40 TENSOR CALCULUS AND RELATIVITY 

where v is the velocity of the point as measured in the frame S. Substi- 
tuting for cosa, sina from equations (5.7), equations (15.8) can be 
written 

v x = Q(v x -u), 

v y = ea-«V) 1/2 ^, 

«,= e(l-«V) 1/2 w„ 
1 = Q(l-uv x fc\ 

;2/„2 



(15.9) 



where 



„ r \-v 2 ic 2 v> 2 
H (i-*V)(i-«V) J (1510) 



Dividing the first three equations (15.9) by the fourth, we obtain the 
special Lorentz transformation equations for v in their final form, viz. 



v v = 



v r = 



1 — uvjc 2 

(\-u 2 lc 2 ) m v y 
l—uv x /c 2 

(1-kV) 1 ^ 
1 — uvjc 2 



(15.11) 



If u and v are small by comparison with c, equations (15.1 1) can be 
replaced by the approximate equations 



v x = v x -u, v y = v y , v z = v z 



(15.12) 



These are equivalent to the vector equation (1.1) relating velocity 
measurements in two inertial frames according to Newtonian 
mechanics. 

Since, by the fourth of equations (15.9) Q must be real, equation 
(15.10) implies that if v < c then v < c. Thus, if a point is moving with 
a velocity approaching c in S and S is moving relative to S with a 
velocity of the same order, the point's velocity relative to S will still 
be less than c. Such a result is, of course, completely at variance with 
classical ideas. In particular, if a light pulse is being propagated along 
Ox so that v x = c, Vy = v z = 0, then it will be found that v x = c, 



SPECIAL RELATIVITY MECHANICS 41 

v y = v z = 0. This confirms that light is propagated with speed c in all 
inertial frames. 

The transformation inverse to (15.1 1) can be found by exchanging 
barred' and 'unbarred' velocity components and replacing u by — u. 



16. Mass and momentum 

In section 2 it was shown that Newton's laws of motion conform to 
the special principle of relativity. However, the argument involved 
classical ideas concerning space-time relationships between two 
inertial frames and these have since been replaced by relationships 
based upon the Lorentz transformation. The whole question must 
therefore be re-examined and this we shall do in this and the following 
section. 

We shall begin by considering the conservation of momentum 
equation (1 .3) for the impact of two particles by which mass is defined 
in classical mechanics. Since the velocity vectors Ui, etc. are not vec- 
tors relative to orthogonal transformations in space-time and indeed 
transform between inertial frames in a very complex manner, it is at 
once evident that equation (1 .3) is not covariant with respect to trans- 
formations between inertial frames. It will accordingly be replaced, 
tentatively, by another equation, viz. 

Mi U, + M 2 U 2 = Mj V 2 + M 2 V 2 , (16.1) 

where Uj, etc. are the 4-velocities of the particles and M u M 2 are 
invariants associated with the particles which will correspond to their 
classical masses. This is a vector equation and hence is covariant with 
respect to orthogonal transformations in space-time as we require. 
Equation (16.1) will be abbreviated to the statement 

2 AfV is conserved (16.2) 

and then, by equation (15.6), this implies that 

2 w(v, id) is conserved, (16.3) 

M 
where m = p.^ ifr' < 16 - 4 ) 



42 TENSOR CALCULUS AND RELATIVITY 

By consideration of the first three (or space) components of (16.3), it 
will be clear that 

2 mv is conserved, (16.5) 

and, by consideration of the fourth (or time) component that 

2 m is conserved. (16.6) 

If, therefore, m is identified as the quantity which will play the role of 
the Newtonian mass in special relativity mechanics, our tentative 
conservation law (16.1) is seen to incorporate both the principles of 
conservation of momentum and of mass from Newtonian mechanics. 
The principle (16.1) is accordingly eminently reasonable. However, 
our ultimate justification for accepting it is, of course, that its conse- 
quences are verified experimentally. We shall refer to such checks at 
appropriate points in the later development. 

It appears from equation (1 6.4) that the mass of a particle must now 
be regarded as being dependent upon its speed v . If v = 0, then m = M. 
Thus Mis the mass of the particle when measured in an inertial frame 
in which it is stationary. M will be referred to as the rest mass or proper 
mass and will, in future, be denoted by wq. Then 

m = o-„W)" 2 ' (16 - 7) 

Clearly m -*■ oo as v -*- c, implying that inertia effects become increas- 
ingly serious as the velocity of light is approached and prevent this 
velocity being attained by any material particle. This is in agreement 
with our earlier observations. Formula (16.7) has been verified by 
observation of collisions between atomic nuclei and cosmic ray par- 
ticles (e.g. see Exercise 14 at the end of this chapter). 

We shall define the 4-momentum vector P of a particle whose proper 
mass is mo and whose 4-velocity is V, by the equation 

P = /mqV. (16.8) 

Since wo is an invariant and V is a vector in space-time, P is a vector. 
By equation (15.6), 

P = mo(l-v 2 /c 2 )- ll2 (y,ic) = (rmjmc) = (p,/wc), (16.9) 

where p = mv is the classical momentum. 



SPECIAL RELATIVITY MECHANICS 43 

Relative to the special orthogonal transformation (5.1), the trans- 
formation equations for the components of P are 

P t = Pi cos a +P 4 sin a, P 2 = 
J* 4 = — P 1 sina+P 4 cosa, P 3 

Substituting for the components of P from equation (16.9) and 
similarly for P, and employing equations (5.7), it will be found that 

.1 



:£} (i&uo 



p x -mu 
Px = 



(16.11) 



(l-^/c 2 ) 1 ' 2 

Py = Py* 
Pz = />« 

m = (l-«w (1612) 

Equations (1 6. 1 1 ) constitute the special Lorentz transformation equa- 
tions for the components of the momentum p and equation (1 6. 1 2) the 
corresponding transformation equation for mass. Since p x = mv x , 
this equation can also be written 

1 —UVJ(T 

* = n — 2fkm m - < 16 - 13 > 

(1 — ir/<r) ' 

This reduces to the classical form of equation (2.4) if u, v x are neg- 
ligible by comparison with c. 

17. The force vector. Energy 

We have seen that in classical mechanics, when the mass of a particle 
has been determined, the force acting upon it at any instant is specified 
by Newton's second law. Force receives a similar definition in special 
relativity mechanics. The mass of a particle with a given velocity can 
be determined by permitting it to collide with a standard particle and 
applying the principle of momentum conservation. Equation (16.7) 
then gives its mass at any velocity. The force f acting upon a particle 
having mass m and velocity v relative to some inertial frame is then 
defined by the equation, 



44 TENSOR CALCULUS AND RELATIVITY 

where p is the particle's momentum. Clearly f will be dependent upon 
the inertial frame employed, a departure from classical mechanics. 

Definition (17. 1) implies that, if equal and opposite forces act upon 
two colliding particles, momentum is conserved. However, although 
experiment confirms that momentum is indeed conserved, Newton's 
third law cannot be incorporated in the new mechanics, for it will 
appear later that, if the forces are equal and opposite for one inertial 
observer, in general they are not so for all such observers. Equation 
(16.1) therefore replaces this law in the new mechanics. 

f is not a vector with respect to Lorentz transformations in space- 
time. However, a 4-force F can be defined which has this property. 
The natural definition is clearly 

„ dP dV 

F = * = "*>;*;• < 17 - 2) 

P being the 4-momentum and t the proper time for the particle. F is 
immediately expressible in terms of f for, by equation (16.9), 

F = -rfaimc), 
dr 

d dt 

= dt^ imC) dr 

= (i-i>Vr 1/2 (P,wic), 

= (1 - t; 2 /c 2 r 1/2 (f, imc). (17.3) 

The vectors V, F are orthogonal. This is proved as follows: From 
equation (15.6) 

V 2 = -c 2 . (17.4) 

Differentiating with respect to t, 

dV 
V'Tr = °' 

i.e. V-F = 0, (17.5) 

as stated. This result has very important consequences. Substituting 



SPECIAL RELATIVITY MECHANICS 45 

for V and F from equations (15.6) and (17.3) respectively, it is clear 
that 

(1 -t^/c 2 ) -1 (v, ic) -(f, imc) = 0. (17.6) 

This is equivalent to 

vf-c 2 ro = 0. (17.7) 

But, by definition, vf is the rate at which f is doing work. It follows 

that the work done by the force acting on the particle during a time 

interval (,t u t 2 ) is 

'• 

f <?mdt = m 2 c 2 -m l c 2 . (17.8) 

h 

The classical equation of work is 

work done = increase in kinetic energy, (17.9) 

where T= \rm? is the kinetic energy. Equation (17.8) indicates that 
in special relativity mechanics we must define T by a formula of the 
type 

T = me 1 + constant. (1 7. 1 0) 

When v = 0, T= and this determines the unknown constant to be 
- mo c 2 . Thus 

r - o-?/^ ""* A (17 - n) 

If v/c is small, (l-w 2 /c 2 ) _1/2 = l+v 2 /^ approximately and the 
above equation reduces to T= \m^iP , i in agreement with classical 
theory. 

According to equation (17.10), any increase in the kinetic energy 
of a particle will result in a proportional increase in its mass. Thus, if 
a body is heated so that the thermal agitation of its molecules is 
increased, the masses of these particles, and hence the total body mass, 
will increase in proportion to the heat energy which has been 
communicated. 

Again, suppose two equal elastic particles approach one another 



46 TENSOR CALCULUS AND RELATIVITY 

along the same straight line with equal speeds v. If their proper masses 
are both mo, the net mass in the system before collision is 

2mo/(l-t>V) 1/2 . 

It has been accepted as a fundamental principle that this mass will 
be conserved during the collision. However, from considerations of 
symmetry, it is obvious that at some instant during the impact both 
particles will be brought to rest and their masses at this instant will 
be proper masses m^. By our principle, 

2m « = d-^W 2 * (17,12) 

It follows, therefore, that, at this instant, the proper mass of each 
particle has increased by 

"* mo = T/c 2 , (17.13) 



(l-^/c 2 ) 1 ' 2 

where T is the original KE of the particle and use has been made of 
equation (17. 11). Now, in losing this KE, the particle has had an equal 
amount of work done upon it by the force of interaction and this has 
resulted in a distortion of the elastic material of which it is made. 
At the instant each particle is brought to rest, this distortion is at a 
maximum and the elastic potential energy as measured by the work 
done will be exactly T. If we assume that this increase in the internal 
energy of the particle leads to a proportional increase in mass, the 
increment of rest mass (17.13) is explained. If the particles are not 
perfectly elastic, the work done in bringing them to rest will not only 
increase the internal elastic energy, but will also generate heat. Both 
forms of energy will then contribute to increase the proper masses. 

Such considerations as these suggest very strongly that mass and 
energy are equivalent, being two different measures of the same 
physical quantity. Thus, the distinction between mass and energy 
which was maintained in classical physical theories, has now been 
abandoned. All forms of energy E, mechanical, thermal, electromag- 
netic, are now taken to possess inertia of mass m, according to 
Einstein's Equation, viz. 

E = mc 2 . (17.14) 



SPECIAL RELATIVITY MECHANICS 47 

Conversely, any particle whose mass is m, has associated energy E 
and, by equation (17.11), 

E=T+mo<?. (17.15) 

mo c 2 is interpreted as the internal energy of the particle when station- 
ary. If the particle were converted completely into electro-magnetic 
radiation, m c 2 would be the energy released. This is the source of the 
energy released in an atomic explosion. The mass of the material 
products of the explosion is slightly less than the net mass present 
before the explosion, the difference being accounted for by the mass 
of the energy released. Even a small mass deficiency implies that an 
immense quantity of energy has been released. Thus, if m = 1 gm, 
c = 3 x 10 10 cm/sec and hence E = 9 x 10 20 ergs = 2-5 x 10 7 kilowatt 
hours. 

The principle of conservation of mass, which has been incorporated 
into the new mechanics, is now seen to be identical with the principle 
of conservation of energy, which is accordingly also regarded as valid 
in the new mechanics. However, the distinction between the two 
principles, which was a feature of the older mechanics, has 
disappeared. 

18. Lorentz transformation equations for force 
By equation (17.7), 

irhc = -f«v. (18.1) 

c 

Referring to equation (17.3), F can now be completely expressed in 
terms off thus: 

F = (1 -i> 2 /c 2 r 1/2 (f,^f.v) • (18.2) 

Relative to the special Lorentz transformation, the transformation 
equations for the components of F are 



', = F!COSa + F 4 sina, F 2 = F 2 , 1 „ g „ 

\ = -Fxsina + i^cosa, F 3 = F 3 . J 



A- 



48 



TENSOR CALCULUS AND RELATIVITY 



Substituting from equation (1 8.2) into the first three of these equations 
and employing equations (5.7), it follows that 






(18.4) 



where Q is given by equation (15.10). Substituting for Q from the 
fourth of equations (15.9), it will be found that 



/*=/*- 



U (fyVy+f 2 V z ) 

c 2 l—uvjc 2 



(18.5) 



, (1-kW% 

Jy i / 2 Jy* 

\—uv x l<r y 

, (1-kW)'/% 



These are the special Lorentz transformation equations for f . If u, v 
are negligible by comparison with c, these equations reduce to the 
classical form of equation (2.6). 

It is clear from equations (18.5) that, if equal and opposite forces 
are observed from S to act upon two particles, the forces observed 
from S will not be so related unless the particles' velocities are the 
same. 



19. Motion with variable proper mass 

In section 17 it has been assumed that the proper mass thq of the 
particle which is moving under the action of the force f, is constant 
throughout the motion. If, however, the particle is being heated or 
cooled during its motion, or if any non-mechanical forms of energy 
are being communicated to it from an external source, its proper mass 
will vary and our equations must be modified to take account of this 
variation. 
Thus, consider equation (17.2). The 4-momentum P is still defined 



SPECIAL RELATIVITY MECHANICS 49 

by equation (16.8) but, since thq is variable, the 4-force is given by 

F = U m V) = m„^+V^- (19.1 

or ar dr 

Equation (17.4) remains valid and hence, differentiating, 

V-^ = 0. (19.2) 

ar 

Substituting for dV/dr from equation (19.1), we obtain 

V. F =-c 2 ^. (19.3) 

dr 

We conclude that V and F are no longer orthogonal vectors. Substi- 
tuting in equation (19.3) for V and F from equations (15.6) and (17.3) 
respectively, it will be found that 

f =f . v+(c 2_.2 ) ^0. (194) 

dt dr 

This is the modified equation of work. Its physical interpretation is 
clearly: 

rate of increase in particle's energy 

= rate of doing work by applied force 

+ rate of energy input from the external source. (19.5) 

We deduce that energy is being taken from the external source at a 
rate 

R = (c 2 -» 2 )^° (19.6) 

dr 

as measured in the inertial frame being employed. 
Equation (19.4) can therefore be written 

£= c 2 m = f-v+i? ( (19.7) 

and hence, by equation (17.3), 

F = (l-* 2 /c 2 )" 1/2 k^v+*)l- (19.8) 

This is the modified form of equation (18.2). 



50 TENSOR CALCULUS AND RELATIVITY 

20. Lagrange's and Hamilton's equations 

Suppose that a particle having constant proper mass m is in motion 
relative to an inertial frame under the action of a force derivable from 
a potential V. Then its equations of motion are 

' ' - ,etc. (20.1) 



.)- 



dtXil-v^c 2 ) 1 ' 2 } dx 

Expressed in Lagrange form, these equations must be 
d_(dL\ _ 8L 



\e*J 



*ia-i * ' etc - (20 - 2) 

dt\dx) dx 

and hence L must be a function of x, y, 2, x, y, z, such that 

dL _ m x 8L _ 8V 

^- ( l_„2 /c2) i/2> 8x ~ ejc .etc. (20.3) 

Since v 2 = x 2 +y 2 +z 2 , these equations can be validated by taking 

L = -moc 2 (l-* 2 /c 2 ) 1/2 - V, (20.4) 

which is accordingly the Lagrangian for the particle. 
Now 

~= Px ,etc. (20.5) 

dx 

and it follows exactly as in classical theory that, if the Hamiltonian H 
is defined by the equation 

H = p x v x +p y v y +p 2 v z - L, (20.6) 

and is then expressed as a function of the quantities x, y, z, p x , p y , p z 
alone, the Lagrange equations (20.2) are equivalent to Hamilton's 
equations 

* = —, P X = -£-.ete. ( 20 - 7 > 

8p x dx 

Now 

m ° v2 (20.8) 



p x v x +p y v y +p z v z = , l _ v zj < ayi2 



and hence 



SPECIAL RELATIVITY MECHANICS 51 

*** +v 9 



" (l-vV) 1 ' 2 

= E+ V (20.9) 

the total energy, precisely as for classical theory. 
But 

2 2 
2 2 2 m v 

P X +Py+P: = JZ^P* 

= -nftf + E 2 /*?, (20.10) 

and it follows that 

£ 2 = <?<j>l+p 2 y +p 2 z +n%<?). (20.11) 

Substituting in equation (20.9) 

H = cG£+*J+ J p5+«S«*> l/2 + K> ( 20.12) 

expressing H as a function of x, y, z, p x , p y , p z . The reader is now left 
to verify that Hamilton's equations are equivalent to the equations 
of motion (20.1). 

Exercises 3 

1 . Obtain the transformation equations for v by differentiating the 
Lorentz transformation. 

2. Obtain the transformation equations for the acceleration a by 
differentiating the transformation equations for v and express them 
in the form 

(1-kV) 3 ' 2 



52 TENSOR CALCULUS AND RELATIVITY 

- 1-kW / v y ul<? \ 
a >~ {\-v x ul<z?\ a >+T^i? ax )> 

- 1~«/ 2 C 2 / VzU/c 2 \ 

Deduce that a point which has uniform acceleration in one inertial 
frame has not, in general, uniform acceleration in another. 

3. If S has velocity c relative to S, show that all points moving 
relative to S with velocities less than c have a velocity c as observed 
from S. 

4. Two points are moving in opposite directions with speeds c 
relative to some inertial frame. Show that their relative velocity is c. 

5. Show that the 4-velocity V is of constant magnitude ic. 

6. A beam of light is being propagated in the jc^-plane of S at an 
angle a to the *-axis. Relative to S it is observed to make an angle a 
with Ox. Prove the aberration of light formula, viz. 

cot a — (u/c) cosec a 
COta= (I-kW 2 
Deduce that, if u ^ c, then 

A U • 

Zla = a — a = -sina, 
c 

approximately. 

7. A particle of proper mass ttiq is moving under the action of a 
force f with velocity v. Show that 

thq ds mQVv/c 2, 
(l-v 2 /c 2 ) l ' 2 Jt + (.l-v 2 /c 2 ) 3 l 2V ' 

Hence, if the acceleration dv/dt is parallel to v, show that 
f - "*o dv 

~ (i-i>V) 3/2 dt' 

and if the acceleration is perpendicular to v, then 

f ntp dv 

(l-J/W 2 *' 



SPECIAL RELATIVITY MECHANICS 53 

8. Show that P = (p, iE/c) and deduce that 

p 2 -E 2 /<? 

is an invariant -moc 2 with respect to Lorentz transformations. 

9. Show that i? transforms under a special Lorentz transformation 
according to the equation 

£ = E-p x ii 



(l-« 2 /c 2 ) 1/2 

10. A rocket moves along the jc-axis in 5", commencing its motion 
with velocity v and ending it with velocity v\. If w is the jet velocity 
as measured by the crew (assumed constant), show that the mass ratio 
of the manoeuvre (i.e. initial mass/final mass) as measured by the crew 
is 



l(c-vi)(c+v )\ 



What does this reduce to as c-»- oo? Deduce that, if the rocket starts 
from rest in 5" and its jet is a stream of photons, the mass ratio to 
velocity v is 



m 



Show that, with a mass ratio of 6, the rocket can attain 35/37 of the 
velocity of light in S. 

11. S, S, S are inertial frames with their axes parallel. S has a 
velocity u relative to S and S has a velocity v relative to S, both velo- 
cities being parallel to the x-axes. If transformation from S to S 
involves a rotation through an angle a of the axes in space-time and 
transformation from S to S a rotation jS, a transformation from S to 
S involves a rotation y where y — a+ jS. Deduce from this equation 
the relativistic law for the composition of velocities, viz. 

u+v 

1 + uv c 2 



54 TENSOR CALCULUS AND RELATIVITY 

12. A force f acts upon a particle of mass m whose velocity is v. 
Show that 

dv f«v 

f = m — + -^=-v. 
dt c 2 

1 3. An electrified particle having charge e and rest mass /wq moves 
in a uniform electric field of intensity E parallel to the jc-axis. If it is 
initially at rest at the origin, show that it moves along the x-axis so 
that at time t 



-U4')- 



where k = eE/m . Show that this motion approaches that predicted 
by classical mechanics as c-> a>. [It may be assumed that the force 
acting upon the particle is eE in the direction of the field at all times.] 
14. A particle is moving with velocity u when it collides with a 
stationary particle having the same rest mass. After the collision the 
particles are moving at angles 9, <f> with the direction of motion of the 
first particle before collision. Show that 

tanfltan^ = 



y+1 

where y = (1 - i//c 2 ) ~ 1/2 . 0f c -> oo, y -> 1 and 6+ <f> = frr. This is the 
prediction of classical mechanics. However, if the particles are elec- 
trons and u is near to c in value, 0+tf> < Jw. This effect has been 
observed in a Wilson cloud chamber.) [Hint : Refer the collision to an 
inertial frame in which both particles have equal and opposite 
velocities prior to collision.] 

15. A body of mass M disintegrates while at rest into two parts of 
rest masses M x and M 2 . Show that the energies E^E^of the parts are 
given by 

_ ^ M 2 + M 2 -Ml F ,M 2 -M\ + M 2 2 

1 6. A particle of proper mass Wq moves under the action of a central 
force, (r, 6) are its polar coordinates in its plane of motion relative to 
the force centre as pole. V(r) is its potential energy when at a distance r 



SPECIAL RELATIVITY MECHANICS 55 

from the centre. Obtain Lagrange's equations for the motion in the 
form 

£(yf)-yrP+ - V = 0, j(yr 2 6) = 0, 
at m at 

where y = [1 - (r 2 + r 2 ^ 2 )/^] ~ 1/2 . Write down the energy equation for 
the motion and obtain the differential equation for the orbit in the 
form 



, 2 2 [d 2 u \ C-V „, 



where u=\jr and h, C are constants. In the inverse square law case 
when V= —\i\r t deduce that the polar equation of the orbit can be 
written 

lu = \ + ecosr)d t 

where rj 1 = l—^/n^h 2 ^. If fx/mohc is small, show that the orbit is 
approximately an ellipse whose major axis rotates through an angle 
Tr^/mo^c 2 per revolution. 

17. A photon having energy E collides with a stationary electron 
whose rest mass is rriQ. As a result of the collision the direction of the 
photon's motion is deflected through an angle 6 and its energy is 
reduced to E'. Prove that 



moi 



(H)=- 



C 2 \4r.-i\ = 1-COS0. 



(It may be assumed that the momentum of a photon having energy E 
is Etc.) 
Deduce that the wavelength A of the photon is increased by 

JA = — sin 2 i0, 

UIqC 

where h is Planck's constant. (This is the Compton Effect. For a 
photon, take A = hc/E.) 

18. A particle of rest mass m\ and speed v collides with a particle 
of rest mass m 2 which is stationary. After collision the two particles 



56 TENSOR CALCULUS AND RELATIVITY 

coalesce. Assuming that there is no radiation of energy, show that 
the rest mass of the combined particle is M, where 

-> i 2/Mi m 2 

and find its speed. 

19. A luminous disc of radius a has its centre fixed at the point 
(£,0,0) of the 5-frame and its plane is perpendicular to the *-axis. It 
is observed from the origin in the .S-frame at the instant the origins 
of the two frames coincide and is measured to subtend an angle 2a. 
Prove that, if a < x, then 



tan 



« = '- /( C -±-"Y 
x*J \c — u) 



(Hint : employ the aberration of light formula, exercise 6 above.) 

20. Two particles having proper masses m u m 2 are moving with 
velocities u x , u 2 respectively, when they collide and cohere. If a is the 
angle between their lines of motion before collision, show that the 
proper mass of the combined particle is m, where 



2 2 2/wi m 2 (c 2 — u\ u 2 cos a) 
V{(c 2 -«?)(c 2 -«l)} ' 



m 2 = mi + miH- ■ ,-. „ , 

' f ' "2 — i/f 



Show that, for all values of a, m > m 1 + m 2 and explain the increase 
in proper mass. 

21. v, v are the velocities of a point relative to the inert ial frames 
S, S respectively. Representing these vectors as position vectors in an 
independent <^ 3 , show that 



jSv= e r ¥ +u{^<j8-l)+j8j] f 



where p = (1 - «2/c 2 )~ 1/2 and 

Q = 1/0+u.v/c 2 ). 
Show further that 

M 2j8v = Q[(l-j3)ux(vxu) + j8«2( u + v)], 



SPECIAL RELATIVITY MECHANICS 57 

and hence verify that 

t;2 = Q2[( U + v )2 _ ( v x U )2/ C 2]. 

22. A particle moves along the x-axis of the frame S with velocity 
v and acceleration a. Show that the particle's acceleration in S is 

-_ 0-" 2 /c 2 ) 3/2 

a ~ (i-uv/c 2 y a ' 

If the particle always has constant acceleration a relative to an 
inertial frame in which it is instantaneously at rest, prove that 

>> - - 

where j3 = (1 — u 2 /c 2 ) -1 / 2 and t is time in S. 

Assuming that the particle is at rest at the origin of S at / = 0, 
show that its x-coordinate at time t is given by 

OJC = C 2 [(l + a2/2/ c 2)l/2_l]. 

23. Three rectangular cartesian inertial frames S, S, S are initially 
coincident. As seen from S, S moves with velocity u parallel to Ox 
and, as seen from S, S moves with velocity v parallel to Oy. If the 
direction of S 's motion as seen from £ makes an angle 8 with Ox and 
the direction of S"s motion as seen from S makes an angle <f> with Ox, 
prove that 



vl J/2\l/2 „/ v 2\~ 



1/2 



Deduce that, if u, v <^ c, then 

<f>-d = uv/2c 2 

approximately. 

24. A particle is moving with velocity v when it disintegrates into 
two photons having energies E u E 2 , moving in directions making 
angles a, j3 with the original direction of motion and on opposite 



58 TENSOR CALCULUS AND RELATIVITY 

sides of this direction. Show that 

c—v 
taniatanip = • 

c+v 

Deduce that, if a photon disintegrates into two photons, they must 
both move in the same direction as the original photon. (The momen- 
tum of a photon having energy E is Ejc.) 



CHAPTER 4 

Special Relativity Electrodynamics 

21. 4-Current density 

In this chapter we shall study the electromagnetic field due to a flow 
of charge which will be assumed known. Relative to an inertial frame 
S, let p be the charge density and v its velocity of flow. Then, if j is 
the current density, 

j = pv. (21.1) 

Assuming that charge can neither be created nor destroyed, the 
equation of continuity 

divj + ^ = (21.2) 

will be valid for the charge flow in S. This equation must be valid in 
every inertial frame and hence must be expressible in a form which is 
covariant with respect to orthogonal transformations in space-time. 
Introducing the coordinates x t by equations (4.4) and employing 
equation (21.1), equation (21.2) is seen to be equivalent to 

1 ?-(pv x ) + ^-(pv y ) + ^-(pv z )+ ir (icp) = 0. (21.3) 

OXi 0X2 8x 3 8x 4 

This equation is covariant as required if (pv x , pv yt pv z , icp) are the 
four components of a vector in space-time. For, if J is this vector, 
equation (21.3) can be written 

//,/ = 0, (21.4) 

and this is covariant with respect to orthogonal transformations. 
Now, by equation (15.6), 

J = (py,icp) = p(l -«V) 1/2 V, (21.5) 

[59] 



60 TENSOR CALCULUS AND RELATIVITY 

where V is the 4-velocity of flow and hence J is a vector if p(l - « 2 /c 2 ) ' /2 
is an invariant. Denoting the invariant by p , we have 

/> = (i-J°W 2 ' (2L6) 

It follows that p = p if v = and hence that p is the" charge density 
as measured from an inertial frame relative to which the charge being 
considered is instantaneously at rest. p is called the proper charge 
density. 

J is called the 4-current density and it is clear from equation (21.5) 
that 

J = ft>V = afc/>). (21.7) 

It is now clear that, when J has been specified throughout space-time, 
the charge flow is completely determined, for the space components 
of J fix the current density and the time component fixes the charge 
density. Hence, given J, the electromagnetic field must be calculable. 
The equations which form the basis for this calculation will be derived 
in the next two sections. 

Let d(o be the volume of a small element of charge as measured 
from an inertial frame So relative to which the charge is instantane- 
ously at rest. The total charge within the element is pod<t) . Due to the 
Fitzgerald contraction, the volume of this element as measured from 
S will be da>, where 

dto = (1 - t>V) 1/2 da> . (21 .8) 

The total charge within the element as measured from S is therefore 

pdai = p(l-v 2 /c 2 ) ll2 dw = p Q doi Q , (21.9) 

by equation (21.6). It follows that the electric charge on a body is 
invariant for all inertial observers. 

22. 4- Vector potential 

In classical theory, the equations determining the electromagnetic 
field due to a given charge flow are Maxwell's equations (3.1)— (3.4). 
To ensure covariance of the laws of mechanics with respect to Lorentz 
transformations, it proved necessary to modify classical Newtonian 
theory slightly. However, it will be shown that Maxwell's equations 



SPECIAL RELATIVITY ELECTRODYNAMICS 61 

are covariant without any adjustment being necessary. Indeed, the 
Lorentz transformation equations were first noticed as the transfor- 
mation equations which leave Maxwell's equations unaltered in form. 
To prove this, it will be convenient to introduce the scalar and vector 
potentials, <p and A respectively, of the field. It is proved in textbooks 
devoted to the classical theoryt that A satisfies the equations 

ld<f> 
divA+-^ = 0, (22.1) 

c dt 

_, A 1 B*A An. 

V 2 A- ?1? =- 7 i, (22.2) 



and <f> satisfies the equation 



1 d 2 ^ 



V2 *--2-$ = -*" P . (22.3) 

We now define a 4-vector potential SI in any inertial frame S by the 
equation 

Sl = (A,i<f>). (22.4) 

It is easily verified that equations (22.2), (22.3) are together equivalent 
to the equation 

D 2 « = -—J, (22.5) 

c 

where the operator D 2 is defined by 

a 2 d 2 e 2 a 2 

dx\ a*| 8x1 dx l 

The space components of equation (22.5) yield equation (22.2) and 
the time component, equation (22.3). If £,, /,- are the components of 
SI and J respectively, equation (22.5) can be written 

Q UJ= ~^ J i> (22.7) 



D 2 = ^2 + ^2 + 7-2+^-2- (22.6) 



in which form it is clearly covariant with respect to Lorentz transfor- 
mations provided SI is a vector. This confirms that equation (22.4) 

t See, e.g., A Course in Applied Mathematics by D. F. Lawden, p. 527. 
English Universities Press. 



62 



TENSOR CALCULUS AND RELATIVITY 



does, in fact, define a quantity with the transformation properties of 
a vector in space-time. 

Next, it is necessary to show that equation (22.1) is also covariant 
with respect to orthogonal transformations in space-time. It is clearly 
equivalent to the equation 

divft = Q iti = 0, (22.8) 

which is in the required form. 
J being given, SI is now determined by equations (22.7) and (22.8). 



23. The field tensor 

When A and $ are known in an inertial frame, the electric and magnetic 
intensities E and H respectively at any point in the electromagnetic 
field follow from the equations 



a± laA 
E = -grad^-- — , 

H = curlA. 



(23.1) 
(23.2) 



Making use of equations (4.4) and (22.4), these equations are easily 



shown to be equivalent to the set 



-iE x 



8Q 4 8Q X 
dx\ dx 4 ' 



y 8x 2 te 4 



-iE,= 



H r = 



H y= * 

ox 



(23.3) 



H,= 



8Q A 8D 3 

8x 3 8x4 » 

8iJ 3 8Q2 
8x 2 8x 3 

8Q Y 8Q 3 
3 d *i 

di?2 d&\ 
8x x 8x2 



(23.4) 



SPECIAL RELATIVITY ELECTRODYNAMICS 



63 



Equations (23.3), (23.4) indicate that the six components of the vec- 
tors - iE, H with respect to the rectangular Cartesian inertial frame S 
are the six distinct non-zero components in space-time of the skew- 
symmetric tensor Q jti —Qj. We have proved, therefore, that equa- 
tions (23.1), (23.2) are valid in all inertial frames if 



W = 




(23.5) 



is assumed to transform as a tensor with respect to orthogonal trans- 
formations in space-time. The equations (23.3), (23.4) can then be 
summarized in the tensor equation 



Fy = Qj ti —Qij. 



(23.6) 



Fy is called the electromagnetic field tensor. The close relationship 
between the electric and magnetic aspects of an electromagnetic field 
is now revealed as being due to their both contributing as components 
to the field tensor which serves to unite them. 

Consider now equations (3.2), (3.3). Employing the field tensor 
defined by equation (23.5) and the current density given by equation 
(21.7), these equations are seen to be equivalent to 

8F l2 8F n 8F U An 
8x 2 8x 3 ox 4 c 



or, in short, 



dF 2l 8F 23 8F 24 An 

-7. 1--7 *--£ — = — J 2> 

OX 1 OX3 CX4 c 

8F 31 8F 32 8F 34 4tt 
8x1 8x 2 8x4 c 

8F AX dF4 2 dF 43 An 

■7, r— H— — = — J4, 

8x\ 8x 2 8x 3 c 



An 
F ijJ ~ — J i> 



(23.7) 



(23.8) 



64 



TENSOR CALCULUS AND RELATIVITY 



an equation which is covariant with respect to Lorentz trans- 
formations. 
Finally, consider equations (3.1) and (3.4). These can be written 

8x 2 8x 3 8x 4 



(23.9) 



8F 4l ( 8F n | 8F 34 = ^ 
8x 3 8x 4 8x\ 

8F l2 8F 24 8F 4l 

— - + — - +— - = 0, 

dx 4 8x1 8x 2 

8F 23 | 8F 3l t eF 12 = Q 

3*1 9jC2 #*3 

These equations are summarized thus: 

Fi],k+Fjk,i+Fku = 0. (23.10) 

If any pair from i, j, k are equal, since F /y is skew-symmetric, the left- 
hand member of this equation is identically zero and the equation is 
trivial. The four possible cases when ij, k are distinct are the equations 
(23.9). Equation (23.10) is a tensor equation and is therefore also 
covariant with respect to Lorentz transformations. 

To sum up, Maxwell's equations in 4-dimensional covariant form 
are: 



4tt 



(23.11) 



Fy,k + Fjk,i + F ki j — 0. 

Given // at all points in space-time, these equations determine the 
field tensor F u . The solution can be found in tei 
tial Q t which satisfies the following equations: 

Q ul = 0, 

~ 47T 



(23.12) 



SPECIAL RELATIVITY ELECTRODYNAMICS 65 

Qi being determined, F u follows from the equation 

F v = Qj.,-Qij. (23.13) 



24. Lorentz transformations of electric and magnetic intensities 

Since F y is a tensor, relative to the special Lorentz transformation 
(5.1) its non-zero components transform thus: 



F 23 — ^23> 

F 31 = F 3 iCOSa+F 34 sina, 
F\2 = Fi 2 cosa+F 42 sina, 

Fi4 = F 14 , 

F 24 = -F 2 isina+F 24 cosa, 

F 34 = — F 3 iSina + F 34 COSa. 



(24.1) 



(24.2) 



Substituting for the components of F y from equation (23.5) and for 
sin a, cos a from equations (5.7), the above equations (24.1) yield the 
special Lorentz transformation equations for H, viz. 



H v = H\ 



H y + (u/c)E z 
"'"(I-kV) 1 ' 2 ' 



#z = 



H z -(u/c)E y 
(I-kV) 1 ' 2 



(24.3) 



Similarly, equations (24.2) yield the transformation equations for E, 
viz. 



p - E E - 



E y -(u/c)H z ,, _ E z +(ulc)H y (244) 



y ~ a-i/V) 1 ' 2 



- E = 

2) % 



(l-^/c 2 ) 1 ' 2 



As an example of the use to which these transformation formulae 
may be put, consider the electromagnetic field of a point charge e 
moving uniformly with speed « relative to the observer. Let S be the 
inertial frame employed by the observer and suppose e moves along 
its x-axis. S will be a parallel inertial frame with the point charge 
stationary at its origin. Consider the instant / = in S, when the point 
charge is at the origin of S and the origins of S and S coincide. For an 
observer employing S, the electromagnetic field is that due to a 



66 



TENSOR CALCULUS AND RELATIVITY 



stationary point charge and is accordingly specified at the point 
(x,y,z) for all i by the equations 



£ = -%(x,y,z), fl = 0, 



(24.5) 



where r 2 = x 2 + y 2 + z 2 . The field in S can now be calculated from the 
transformation equations inverse to (24.3), (24.4) (replace u by — «) 
and proves to be given by 



E 



vy^'a-aWT 



]■ 



_£["- : 

~ f 3 [ X, (l-u 

h _ 1 \n u2 I° *oV c 1 

But, putting / = in equations (5.8), it will be found that 



(24.6) 



Whence, equations (24.6) are equivalent to 

e 



E = 



r'\\-u z l(ry l ' L 



eu/c 
H = r>\l-u 2 /<?)^ >- Z > y) > 



(24.7) 



(24.8) 



where 



,'2 = 



(i-«V) 



+ / + Z 2 . 



If r is the position vector of the point {x,y,z) with respect to the origin 
of S, equations (24.8) can be written 



E 



H = 



r, 



r' 3 (l-u 2 /c 2 ) 1 ' 2 

1 e 

cr' 3 (l-«V) 1/2 



uxr. 



(24.9) 



SPECIAL RELATIVITY ELECTRODYNAMICS 67 

These equations indicate that, at this instant in S, the E-lines are 
straight lines radiating from O and the H-lines are circles whose 
centres lie on Ox and whose planes are parallel to Oyz. 
If («/c) 2 is considered to be negligible, equations (24.9) reduce to 

E = -U H = --,uXr. (24.10) 

r 3 cr 3 

The first equation shows that the electric field is, to this order of 
approximation, identical with the field of a stationary charge and the 
second equation is the Biot-Savart Law. 

25. The Lorentz force 

We shall now calculate the force exerted upon a point charge e in 
motion in an electromagnetic field. 

At any instant, we can choose an inertial frame relative to which the 
point charge is instantaneously at rest. Let Eq be the electric intensity 
at the point charge relative to this frame. Then, by the physical 
definition of electric intensity as the force exerted upon unit stationary 
charge, the force exerted upon e will be eEo- It follows from equation 
(18.2) that the 4-force acting upon the charge in this frame is given by 

F = 0?Eo,0). (25.1) 

The 4-velocity of the charge in this frame is also given by 

V = (0,/c) (25.2) 

and hence, by equation (23.5), 

- c Fy Vj = eiE^E^E^O) = (eE ,0). (25.3) 

It has accordingly been shown that, in an inertial frame relative to 
which the charge is instantaneously stationary, 

Ft^-FijVj. (25.4) 

c 

But this is an equation between tensors and is therefore true for all 
inertial frames. 



68 TENSOR CALCULUS AND RELATIVITY 

Substituting in equation (25.4) for the components F h F u , V } from 
equations (18.2), (23.5) and (1 5.6) respectively, the followingequations 
are obtained: 

fx = c (.H z v y -H y v z + cE x ), 



fy = - (H x v z -H z v x + cE y ), 



e 
fz = ~ (H y v x -H x v y + cE z ). 



\ (25.5) 



l X Vy 

These equations are equivalent to the 3-vector equation 

C=eE + -vXH. (25.6) 

c 

f is called the Lorentz force acting upon the charged particle. 

26. Force density 

Consider a continuous distribution of matter in motion under the 
action of some field of force. Let dco be the proper volume of any 
small element of the distribution and let F be the 4-force exerted upon 
the element by the field. Writing 

F = D dco Q , (26.1) 

it follows that, since F is a vector and d<o is an invariant, then D is also 
a vector in space-time. It is termed the 4-force density vector for the 
field.. 

When measured from an inertial frame S, let dm be the volume of 
the element and let f be the 3-force exerted by the field upon it. We 
shall define the 3-force density vector d in S by the equation 

f=ddco. (26.2) 

da> is an invariant with respect to transformations between Cartesian 
frames stationary in S and hence d is a vector relative to such 
frames. 



SPECIAL RELATIVITY ELECTRODYNAMICS 69 

Substituting for F, f from equations (26.1), (26.2) respectively into 
equation (18.2), we obtain 

Dda) = U,-d-v\da>(l-v 2 /c 2 r l l 2 . (26.3) 

By virtue of the relationship (21.8), this reduces immediately to the 
equation 

(26.4) 



D = (d,^d.v), 



relating the 3- and 4-force density vectors. 

27. The energy-momentum tensor for an electromagnetic field 

Suppose that a charge distribution is specified by a 4-current density 
vector J. KdoiQ is the proper volume of any small element of the distri- 
bution and po is the proper density of the charge, the charge within 
the element will be podw . It follows from equation (25.4) that the 
4-force exerted upon the element by the electromagnetic field is given 
by 

F^^FyVjdtuo, (27.1) 

c 

V being the 4-velocity of flow for the element. Employing equation 
(21.7), this last equation can be written 

■ F t = -FyJjdcoo (27.2) 

c 

and it follows from the definition given in the last section that the 
4-force density for the electromagnetic field is given by 

D t = -F u Jj. (27.3) 

Substituting for J } from the first of equations (23. 1 1 ), we can express 
D ( in terms of the field tensor thus: 

D, = ^F u F Jktk . (XI A) 



70 TENSOR CALCULUS AND RELATIVITY 

We will now prove that the right-hand member of this equation is, 
apart from sign, the divergence of a certain symmetric tensor Sy given 
by the equation 

s » = ^F ik F jk - j^M*i**f (27.5) 

and called the energy-momentum tensor of the electromagnetic field. 
Taking the divergence of Sy, we have 

S UJ = ^F ik jFj k +—F ik F Jk j-—8yF k iF k ij. (27.6) 

Now 

FikjFjk = Fy.kFkj ~ Fji.kFjk* (27.7) 

since F u is skew-symmetric. Thus 

FikjFj k = h(F ikJ +Fj itk )F jk . (27.8) 

Also 

8(jF kl F k ij — F k iF k i ti = —Fj k F kJti (27.9) 

and it follows from these results that the first and last terms of the 
right-hand member of equation (27.6) can be combined to yield 

^(Fikj+Fj^k+F^dFjk (27.10) 

and this is zero by the second of equations (23.11). 
Hence 

S «J = ^ F ac F JkJ = -^FikFwj = - A- (27.11) 

Substituting for the components of the field tensor from equation 
(23.5), the components of Sy are calculable from equation (27.5) as 
follows : If /, j take any of the values 1 , 2, 3, then writing E t for E x , 
E 2 for E y , etc. 

Sy= - — (EiEj+HiHj), i*j. (27.12) 



SPECIAL RELATIVITY ELECTRODYNAMICS 71 

If 1=7=1, 

*u = ^(#!+iJ 2 3-E?)-i(ff 2 -* 2 ), 

= -±{E\ + H\)+±-(E 2 + H 2 ). (27.13) 

An 07r 

522, £33 may be expressed similarly and therefore, in general, if 
/,/=l,2, 3, 

Sy = -^(EiEj+HiHjH^ByiEt + H 2 ). (27.14) 

Apart from sign, this is Maxwell's stress tensor ty. ty is only a tensor 
with respect to rectangular frames stationary in the inertial frame 
being employed. 
Also, if /= 1,2, 3, 

S/4 = S 4i = ^-(E 2 H 3 -E 3 H 2 ,E i Hi-E 1 H 3 ,EiH 2 -E 2 H l ) 

An 

= /exH = -S, (27.15) 

Air c 

where S is Pointing's vector. 

Finally, 

5^ = - ^- (E 2 + H 2 ) = - U, (27.16) 

07T 

where U is the energy density in the electromagnetic field. 

These results may be summarized conveniently by exhibiting the 
components of Sy in a matrix thus: 

/ ty S/ic) 
(Sy) = 



[S/ic 



(27.17) 



U 



Returning to the set of four equations (27.11), each can now be 
expressed in classical three-dimensional form. Thus, if i = 1, 2, 3, by 



72 TENSOR CALCULUS AND RELATIVITY 

reference to the equations (26.4) and (27.17) the corresponding equa- 
tion is seen to be equivalent to 

1 dS i _, 
tvj ~?li = dh (27,18) 

If / = 4, the corresponding equation is equivalent to 

8U 
divS + — = -d-v. (27.19) 

Equations (27.18), (27.19) can be given simple physical interpreta- 
tions in the case when the motion of a cloud of charged particles which 
do not interact mechanically is being studied. Each particle is then 
subjected to electromagnetic forces only and it will be assumed that 
its rest mass is constant throughout the motion, i.e. no heat is gener- 
ated. Let 2 be a closed surface stationary in the frame S and let JP be 
the region of space it encloses. Integrating equation (27.19) over f 
and employing Green's theorem, it will be found that 

I S n da = I d-vrfco+- J Udw, (27.20) 

e r r 

where da is a surface element of 2, dot is a volume element of r and 
S„ is the component of S along the inwards normal to 2. Now d • \d<o 
is the rate at which the force applied to the charge in dm is doing work 
and is therefore equal, by equation (17.7), to the rate of increase of the 
mechanical energy of this charge. The first term of the right-hand 
member of equation (27.20) accordingly gives the total rate at which 
the mechanical energy of the charge which is inside 2 at the instant 
under consideration is increasing. Since 2 is a fixed surface and the 
charge is moving, some of this mechanical energy will be lost by 
transport of charge across 2. We have, therefore, 

J d'\d<o = rate of increase of mechanical energy inside 2 
r 

+ rate of loss of mechanical energy across 2. (27.21) 

The second term of the right-hand member of equation (27.20) 



SPECIAL RELATIVITY ELECTRODYNAMICS 73 

measures the rate of increase of electromagnetic field energy inside 27. 
Equation (27.20) accordingly asserts that 

\ S n da+ rate of gain of mechanical energy across 27 

E 

= total rate of energy increase within 27. (27.22) 

For the law of conservation of energy to be valid, it is clearly necessary 
to interpret the inward flux of S across 27 as the rate of flow of electro- 
magnetic energy across this surface. Thus S is the energy current 
density vector. 

Taking 27 to be a surface whose elements are all at a great distance 
from the charge flow being considered, so that E = 0, H = over 27 
and integrating the i' th equation (27.18) over T, the contribution of the 
term t tj j will be zero. This follows since tyj is the ordinary divergence 
of a vector having components (t n ,t i2 , t i3 ) and its volume integral, by 
Green's theorem, can be expressed as a surface integral of this vector 
over 27 and the vector is everywhere zero on 27. There results, therefore, 

f d,da> + j ( ^Sidw = 0. (27.23) 

r r 

Now d t dot is the I th component of the force exerted upon the charge in 
dw and therefore gives the rate at which its momentum is increasing. 
It follows from equation (27.23) that the net / th component of momen- 
tum of an isolated system of charges is conserved only if the electro- 
magnetic field is supposed to contribute momentum whose density is 
S/c 2 . This electromagnetic momentum density vector will be denoted 
by g and thus 

g = ^S. (27.24) 

If w is the velocity of propagation of electromagnetic energy, then 
S = £/w (27.25) 

and hence, by equation (27.24), 

g = ^w. (27.26) 

cr 

But, according to equation (17.14), U/c 2 is the mass density associated 



74 TENSOR CALCULUS AND RELATIVITY 

with an energy density U. It is consistent with earlier theory therefore, 
that such a mass density flowing with velocity w should generate a 
momentum density g. 

28. Equations of motion of a charge flow 

In this section, we shall continue to restrict our attention to a system 
comprising a cloud of charged particles which do not interact mechan- 
ically and whose proper masses remain constant during their motions. 
Since the proper masses of the particles are conserved during the 
motion, an equationof continuity for proper mass can be found. Let 
2 be any closed surface bounding a region r. Then the rate at which 
the net proper mass in r is decreasing must equal the rate at Which 
proper mass is being lost by outwards flow across 2. Let dto be the 
volume of an element of the charge distribution as measured in the 
inertial frame S being employed and let ftoda> be the proper mass of 
the element. /u is the density of proper mass in S. ftQ dm is an invariant, 
but dm is not and hence neither is n . Then the mathematical expres- 
sion of the statement we have just made is 

~dt ^ d<ii = V* v " da > ( 28,1 ) 

r z 

where v n is the component of the velocity of flow v along the outward 
normal to 27. Employing Green's theorem, equation (28.1) can be 
written 

J {^+divO*ov)j da> - 0, (28.2) 

r 
and, since P is arbitrary, this implies the equation of continuity 

^+divOxov) = 0. (28.3) 

ot 

Let da> be the proper volume of a charge element and let /*oo dw be 
the proper mass of this element. Then fiQo is called the proper density 
of proper mass. Since yt-oodcoQ and doi are invariants, so is /t*oo- Clearly 

[XoQdtx> Q = n d<*> (28.4) 



SPECIAL RELATIVITY ELECTRODYNAMICS 75 

and hence, by equation (21.8), 

/%> = (1-*V) 1/: W (28.5) 

It now follows that equation (28.3) can be written in the covariant 
form 

^0*00 Vi) = 0, (28.6) 

OXi 

where V t is the 4-velocity of flow. 

The 4-force exerted by the field upon the charge element of proper 
volume da>o is, by equation (26.1), D,cfa> . Since /xoo^o is the proper 
mass of this element, its equation of motion (see equation (17.2)) is 
therefore 

A = vJ^r- (28.7) 

dr 

But V { can be expressed as a function of the x { and, as the charge 
element moves, its coordinates x { vary as functions of its proper time 
t. It follows that 

dr oxj dr oxj 
Hence 

dV t 

= D h (28.9) 

equations (28.6), (28.8) and (28.7) having been employed to effect the 
reduction. 
We now define a symmetric tensor Oy by the equation 

9ll = n)oViVj, (28.10) 



76 TENSOR CALCULUS AND RELATIVITY 

so that D { can be written as the divergence of this tensor thus 

D t = Oyj. (28.11) 

Consider the components of 0y. We have 

& « " " j _ ?;2/c 2 " (1 _ V 2 /C 2?I2 " C * (28 ' 12 ) 

where /u is the mass density in £. Hence, apart from sign, ® 44 is the 
mechanical energy density W. 

If /= 1,2, 3, then 

®«-4 = ^p^ = /C W = **'» (28J3) 

where g = /*v is the momentum density. 
Finally, if /,y = 1, 2, 3, 

®u = f=9^ 2 = ^ = giVj - (28 ' 14) 

Now g,v is the current density of the / th component of the momentum 
and thus the i tb row of the 3 x 3 matrix (0y) can be so interpreted. 
To summarize, we have shown that 






• (28.15) 



-W) 



This representation should be compared with the representation 
(27. 17) of the electromagnetic energy-momentum tensor Sy. Since, by 
equation (27.24), S = c 2 g, it is clear that ©y is the counterpart for the 
mass distribution of Sy for the electromagnetic field. By is called the 
kinetic energy-momentum tensor. 

Substituting in the equation of motion (28.1 1) for D t from equation 
(27.11), we find that 

-Su,j=®ij,r ( 28 - 16 > 

Writing 

Ty = Sy+@y, (28.17) 

we have finally 

Tyj = 0. (28.18) 



SPECIAL RELATIVITY ELECTRODYNAMICS 77 

Ty is the overall energy-momentum tensor including contributions 
from an energy distribution in an electromagnetic form and from an 
energy distribution in a material form. Equation (28.18) indicates 
that the flow is determined by the statement that the divergence of the 
net energy-momentum tensor vanishes. This result can be proved to 
be true quite generally, i.e. corresponding to any energy (matter) 
distribution, there is an energy-momentum tensor Ty whose diver- 
gence vanishes. It will be shown later in section 48 that this tensor also 
determines the gravitational field of the distribution. 

As shown in the previous section, equation (28.18) with / = 1, 2, 3, 
expresses that the net linear momentum of the system is conserved 
and, with i = 4, that the total energy is conserved. 

Exercises 4 

1 . Write down the special Lorentz transformation equations for J 
and deduce the transformation equations for j, p, viz. 

h = (l-J/(?r ll2 (j x -pu), J y = jy, 

p = (\-Jlc 2 r xl2 (j>-j x ulJ\ ] s = A- 

2. Deduce from the Maxwell equation 

Att 
p.. . = J. 

thatdivJ = 0. 

3. Verify that the field tensor defined in terms of the 4-potential 
Q t by equation (23.13) satisfies Maxwell's equations (23.1 1) provided 
Q-, satisfies the equations (23.12). 

4. (i) Prove that 

FyFij = 2(H 2 -E 2 ) 

and deduce that H 2 - E 2 is invariant with respect to Lorentz trans- 
formations, 
(ii) Prove that 

*ijkiFyF kl = -8/E'H 

and deduce that E«H is an invariant density with respect to Lorentz 
transformations. 



78 TENSOR CALCULUS AND RELATIVITY 

5. An infinite unif brm line charge lies along the x-axis of the inertial 
frame 5" and has longitudinal velocity u. As measured in S, the charge 
per unit length is e. A point P is at distance r from the x-axis and a is 
the unit vector along the perpendicular to this axis through P. Show 
that the electric and magnetic field intensities at P are given by 

E = — a, H = — uxa. 

r cr 

6. An observer O at rest in an inertial frame Oxyzt finds himself to 
be in an electric field E = (0,E,0), with no magnetic field. Show that 
an observer O' moving according to O with uniform velocity V at 
right angles to E, finds electric and magnetic fields E', H' connected 
by the relation 

cH'+VxE' = 0. 

(L,U.) 

7. Oxyz are rectangular axes. An electron moves in the xy-plane 
under the action of a uniform magnetic field parallel to Oz. Prove that 
its path is a circle. 

8. A plane monochromatic electromagnetic wave is being propa- 
gated in a direction parallel to the x-axis in the inertial frame S. Its 
electric and magnetic field components are given by 

E = [0,asina>(/-x/c),0], 

H = [0,0,asina>(f--jc/c)]. 

Show that, when observed from the inertial frame S, it appears as the 
plane monochromatic wave 

E = [O,\asm\to(i-x/c),0], 

fl = [0, 0, Xa sin Ao>(f - jc/c)], 



where 



J\l + u/c) 



u being the velocity of S relative to 5". (I.e. both the amplitude and 
frequency are reduced by a factor A. The reduction in frequency is the 
Doppler effect) 



SPECIAL RELATIVITY ELECTRODYNAMICS 79 

9. Show that the Hamiltonian for the motion of a particle with 
charge e and mass m in an electromagnetic field (A, <f>) is 

H= c (p--A) + m 2 cA +e<f> 

and express this equation in the covariant form 
P.ft = -m 2 c 2 . 

10. Verify that, in a region devoid of charge, equations (23. 12) are 
satisfied by 

Q { = A i e ik ' x ' t 

provided A h k p are constants such that 

A t k t = 0, k p k p = 0. 

By considering the 4-vector property of Q h deduce that A t must 
transform as a 4-vector under Lorentz transformations. Deduce also 
that kpXp is a scalar under such transformations and hence that k p is 
a 4-vector. 

A plane electromagnetic wave, whose direction of propagation is 
parallel to the plane Oxy and makes an angle a with Ox, is given by 

Q — A g 2niv(.x cos a+y sin a— ct)/c 

where v is the frequency. The same wave observed from a parallel 
frame Oxyz moving with velocity u along Ox, has frequency v and 
direction of propagation making an angle a with Ox. By writing 
down the transformation equations for the vector k p , prove that 

u u 

1 — cosa cosa — 

c c 

v = — ^— — v, cos a = 



(l-v 2 lc 2 y 2 l--cosa 

c 

1 1. A particle of mass m and charge e moves freely in a magnetic 
field with components (0,0, H/z) and there is no electric field. Show 
that m is constant during the motion and, by a suitable choice of the 



80 TENSOR CALCULUS AND RELATIVITY 

initial conditions, prove that the motion of the particle is given by 

x = at sin (A log 0. 
y = a/ cos (A log/), 
z = kt, 

where A = eHlmck and hence that the particle moves on the surface of 
the cone 

k 2 (x 2 +y 2 ) = a 2 z 2 . 



CHAPTER 5 

General Tensor Calculus. Riemannian Space 

29. Generalized N-dimensional spaces 

In Chapter 2 the theory of tensors was developed in an JV-dimensional 
Euclidean space on the understanding that the coordinate frame being 
employed was always rectangular Cartesian. If*;, x t +dx t are the co- 
ordinates of two neighbouring points relative to such a frame, the 
'distance' ds between them is given by the equation 

ds 2 = dx,dx,. (29.1) 

If x it Xi+dx t a*"e the coordinates of the same points with respect to 
another rectangular Cartesian frame, then 

ds 2 = dXidxi (29.2) 

and it follows that the expression dx t dxi is invariant with respect to a 
transformation of coordinates from one rectangular Cartesian frame 
to another. Such a transformation was termed orthogonal. 

Now, even in ^ 3 , it is very often convenient to employ a coordinate 
frame which is not Cartesian. For example, spherical polar coordin- 
ates (r,6,<f>) are frequently introduced, these being related to rect- 
angular Cartesian coordinates (x,y,z) by the equations 

x = rsmdcos<f>, y = rsin0sin<£, z = rcosd. (29.3) 

In such coordinates, the expression for ds 2 will be found to be 

ds 2 = dx 2 + dy 2 + dz 2 , 

= dr 2 + r 2 dd 2 + r 2 sm 2 ed<l> 2 , (29.4) 

and this is no longer of the simple form of equation (29.1). The co- 
ordinate transformation (29.3) is accordingly not orthogonal. In fact, 
it is not even linear, as was the most general coordinate transforma- 
tion (8.1) considered in Chapter 2. 
The spherical polar coordinate system is an example of a curvilinear 

81 



82 TENSOR CALCULUS AND RELATIVITY 

coordinate frame in ^ 3 . Let (u, v, w) be quantities related to rectangular 
Cartesian coordinates (x,y,z) by equations 

u = u(x,y,z), v = v(x,y,z), w = w(x,y,z), (29.5) 

such that, to each point there corresponds a unique triad of values of 
(u,v, w) and to each such triad there corresponds a unique point. Then 
a set of values of (u, v, w) will serve to identify a point in ^ 3 and (u, v, w) 
can be employed as coordinates. Such generalized coordinates are 
called curvilinear coordinates. 
The equation 

u(x,y,z) = u , (29.6) 

where u is some constant, demies a surface in £ 3 over which u takes 
the constant value ii . Similarly, the equations 

v = v Q , w - w (29.7) 

define a pair of surfaces on which v takes the value v and w the value 
w respectively. These three surfaces will all pass through the point Po 
having coordinates (u ,v , w ) as shown in Fig. 4. They are called the 
coordinate surfaces through P . The surfaces v = v ,w = w will inter- 
sect in a curve P U along which v and w will be constant in value and 
only u will vary. P Uis a coordinate line through P . Altogether, three 
coordinate lines pass throughP ' The equations u = constant, v = con- 
stant, w = constant define three families of coordinate surfaces corre- 
sponding to the three families of planes parallel to the coordinate 
planes x = 0, y = 0, z = of a rectangular Cartesian frame. Pairs of 
these surfaces intersect in coordinate lines which correspond to the 
parallels to the coordinate axes in a Cartesian frame. 

Solving equations (29.5) for (x,y,z) in terms of (u,v,w), we obtain 
the inverse transformation 

x = x(u,v,w), y - y(u,v,w), z = z(u,v,w). (29.8) 

Let (x,y,z), (x+dx,y+dy,z+dz) be the rectangular Cartesian co- 
ordinates of two neighbouring points and let (u,v,w), (u+du,v+dv, 
w+dw) be their respective curvilinear coordinates. Differentiating 
equations (29.8), we obtain 

dx = -£ du + -^ do + -^ dw, etc. (29.9) 

du dv ow 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 83 

Thus, if ds is the distance between these points, 

ds 2 = dx?+df+dz 2 , 

= Adif + Bcht + Cdy^+lFdvdw+lGdwdu+lHdudv, (29.10) 

giving the appropriate expression for ds 2 in curvilinear coordinates. It 
will be noted that the coefficients A, B, etc. are, in general, functions 
of (u,v,w). 




Fig. 4 



If, therefore, curvilinear coordinate frames are to be permitted, the 
theory of tensors developed in Chapter 2 must be modified to make it 
independent of the special orthogonal transformations for which ds 2 
is always expressible in the simple form of equation (29.1). The 
necessary modifications will be described in the later sections of this 
chapter. However, these modifications prove to be of such a nature 



84 TENSOR CALCULUS AND RELATIVITY 

that the amended theory makes no appeal to the special metrical 
properties of Euclidean space, i.e. the theory proves to be applicable 
in more general spaces for which Euclidean space is a particular case. 
This we shall now explain further. 

Let (jc 1 ,* 2 ,...,**") be curvilinear coordinates in ^.f Then, by 
analogy with equation (29.10), if ds is the distance between two 
neighbouring points, it can be shown that 

ds 2 = gijdx l dx J , (29.11) 

where the coefficients gy of the quadratic form in the x' will, in general, 
be functions of these coordinates. Since the space is Euclidean, it is 
possible to transform from the curvilinear coordinates x l to Cartesian 
coordinates y' so that 

ds 2 = dy'dy 1 . (29.12) 

Clearly, the reduction of ds 1 to this simple form is only possible 
because the functions gy satisfy certain conditions. Conversely, the 
satisfaction of these conditions by the gy will guarantee that coordin- 
ates y exist for which ds 2 takes the simple form (29. 1 2) and hence that 
the space is Euclidean. However, in extending the theory of tensors to 
be applicable to curvilinear coordinate frames, we shall, at a certain 
stage, make use of the fact that ds 2 is expressible in the form (29.1 1), 
but no use will be made of the conditions satisfied by the coefficients 
gy which are a consequence of the space being Euclidean. It follows 
that the extended theory will be applicable in a hypothetical TV-dimen- 
sional space for which the ' distance' ds between neighbouring points 
x', x' + dx' is given by an equation (29. 1 1) in which the gy are arbitrary 
functions of the x 1 .* Such a space is said to be Riemannian and will be 
denoted by @n. S n is a particular 0t N for whictithe gy satisfy certain 
conditions. The right-hand member of equation (29.1 1) is termed the 
metric of the Riemannian space. 
The surface of the Earth provides an example of an 3t 2 - If is the 

t The coordinates are here distinguished by superscripts instead of sub- 
scripts for a reason which will be given later. 

* Except that partial derivatives of the g tl will be assumed to exist and to 
be continuous to any order required by the theory. 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 85 

co-latitude and ^ is the longitude of any point on the Earth's surface, 
the distance ds between the points (6, <f>), (d+dd, <f> + d<f>) is given by 

ds 2 = R\dd 2 + sin 2 6d<f> 2 ), (29.13) 

where R is the earth's radius. For this space and coordinate frame, the 
gu take the form 

gn = R\ Si2 = £21 = 0, g 22 = R 2 sm 2 d. (29.14) 

It is not possible to define other coordinates (x,y) in terms of which 

ds 2 = dx 2 +dy 2 (29.15) 

over the whole surface, i.e. this St 2 is not Euclidean. However, the 
surfaces of a right circular cylinder and cone are Euclidean ; the proof 
is left as an exercise for the reader. 

It will be proved in Chapter 6 that, in the presence of a gravitational 
field, space-time ceases to be Euclidean in Minkowski's sense and 
becomes an St^. This is our chief reason for considering such spaces. 
However, we can generalize the concept of the space in which our 
tensors are to be defined yet further. Until section 39 is reached, we 
shall make no further reference to the metric. This implies that the 
theory of tensors, as developed thus far, is applicable in a very general 
iV-dimensional space in which it is assumed it is possible to set up a 
coordinate frame but which is not assumed to possess a metric. In such 
a hypothetical space, the distance between two points is not even 
defined. It will be referred to as £f N . &t N is a particular SP N for which 
a metric is specified. 

30. Contravariant and covariant tensors 

Let x' be the coordinates of a point P in Sf N relative to a coordinate 
frame which is specified in some manner which does not concern us 
here. Let x l be the coordinates of the same point with respect to 
another reference frame and let these two systems of coordinates be 
related by equations 

*' = Jc'Oc 1 ,* 2 ,...,*"). (30.1) 

Consider the neighbouring point P ' having coordinates jc' + dx* in the 



86 TENSOR CALCULUS AND RELATIVITY 

first frame. Its coordinates in the second frame will be x' + dx', where 

dx l = —,dx } , (30.2) 

dx J 

and summation with respect to the index j is understood. The JV 
quantities dx i are taken to be the components of the displacement 
vector PP' referred to the first frame. The components of this vector 
referred to the second frame are, correspondingly, the dx l and these 
are related to the components in the first frame by the transformation 
equation (30.2). Such a displacement vector is taken to be the proto- 
type for all contravariant vectors. 

Thus, A* are said to be the components of a contravariant vector 
located at the point x l , if the components of the vector in the ' barred ' 
frame are given by the equation 

dx 1 . 
A'=-^jA>. (30.3) 

ox 1 

It is important to observe that, whereas in Chapter 2 the coefficients 
a tj occurring in the transformation equation (10.2) were not functions 
of the Cartesian coordinates x t so that the vector A was not, neces- 
sarily, located at a definite point in & N , the coefficients dx l /dx J in the 
corresponding equation (30.3) are functions of the x l and the precise 
location of the vector A 1 must be known before its transformation 
equations are determinate. This can be expressed by saying that there 
are no free vectors in S? N . 

The form of the transformation equation (30.3) should be studied 
carefully. It will be observed that the dummy index y occurs once as a 
superscript and once as a subscript (i.e. in the denominator of the 
partial derivative). Dummy indices will invariably occupy such 
positions in all expressions with which we shall be concerned in the 
sequel. Again, the free index / occurs as a superscript on both sides of 
the equation. This rule will be followed in all later developments, i.e. 
a free index will always occur in the same position (upper or lower) in 
each term of an equation. Finally, it will assist the reader to memorize 
this transformation if he notes that the free index is associated with 
the 'barred' symbol on both sides of the equation. 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 87 

A contravariant vector A 1 may be defined at one point of S? N only. 
However, if it is denned at every point of a certain region, so that the 
A' are functions of the x*, a contravariant vector field is said to exist in 
the region. 

If V is a quantity which is unaltered in value when the reference 
frame is changed, it is said to be a scalar or an invariant in S? N . Its 
transformation equation is simply 

?= V. (30.4) 

Since this equation involves no coefficients dependent upon the x', 
the possibility that Fmay be a free invariant exists. However, Fis more 
often associated with a specific point in Sf N and may be defined at all 
points of a region of S^ N , in which case it defines an invariant field. In 
the latter case 

V= V(x\x 2 ,...,x N ). (30.5) 

Pwill then, in general, be a quite distinct function of the Jc*. If, however, 
in this function we substitute for the x' in terms of the x 1 from equation 
(30.1), by equation (30.4) the right-hand member of equation (30.5) 
must result. Thus 

V(x l ,x 2 , . . ., x N ) = V(xW, • • ., x N ). (30.6) 

V being an invariant field, consider the N derivatives dVJdx*. In the 
x'-frame, the corresponding quantities are dV/dx' and we have 

8f _ 8V dx^_ Bx^dV 

since, by equation (30.6), when Vis expressed as a function of the x 
it reduces to V. As in Chapter 2, the 8V/dx' are taken to be the com- 
ponents of a vector called the gradient of Fand denoted by grad V. 
However, its transformation law (30.7) is not the same as that for a 
contravariant vector, viz. (30.3) and it is taken to be the prototype for 
another species of vectors called covariant vectors. 
Thus, Bj is a covariant vector if 

8x J 
Si = — t Bj. (30.8) 



88 TENSOR CALCULUS AND RELATIVITY 

Covariant vectors will be distinguished from contravariant vectors by 
writing their components with subscripts instead of superscripts. This 
notation is appropriate, for d VJdx' is a covariant vector and the index i 
occurs in the denominator of this partial derivative. The vector dx ! , 
on the other hand, has been shown to be contravariant in its transfor- 
mation properties and this is correctly indicated by the upper position 
of the index. This is the reason for denoting the coordinates by x' 
instead of x h although it must be clearly understood that the x' alone 
are not the components of a vector at all. 

The reader should check that the three rules formulated above in 
relation to the transformation equation (30.3), apply equally to the 
equation (30.8). 

The generalization from vectors to tensors now proceeds along the 
same lines as in section 10. Thus, if A 1 , B J are two contravariant 
vectors, the N 2 quantities A l B } are taken as the components of a 
contravariant tensor of the second rank. Its transformation equation 
is found to be 

dx k dx l 

Any set ofN 2 quantities C iJ transforming in this way is a contravariant 
tensor. 

Again, if A 1 , Bj are vectors, the first contravariant and the second 
covariant, then the N 2 quantities A'Bj transform thus: 

**> = %w A " B - (3010) 

Any set of N 2 quantities Cj transforming in this fashion is a mixed 
tensor, i.e. it possesses both contravariant and covariant properties 
as is indicated by the two positions of its indices. 

Similarly, the transformation law for a covariant tensor of rank 2 
can be assembled from the law for covariant vectors. 

The further generalization to tensors of higher rank should now be 
an obvious step. It will be sufficient to give one example. Aj k is a 
mixed tensor of rank 3, having both the covariant and contravariant 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 89 

properties indicated by the positions of its indices, if it transforms 
according to the equation 

7i dx i dx s dx t Ar 

The components of a tensor can be given arbitrary values in any 
one frame and their values in any other frame are then uniquely de- 
termined by the transformation law. Consider the mixed second rank 
tensor whose components in the jt'-frame are 8j, the Kronecker deltas 
(Sj = 0, / &j and Sj = 1, / =j). The components in the Jc'-frame are Bj, 
where 



SJ 


VJi, 

= dx* 






8x l 
~ dx k 


Bx k 
dx J ' 




_ 3 *' 

~ dx J 


> 



= Sj. (30.12) 

Thus this tensor has the same components in all frames and is called 
the fundamental mixed tensor. However, a second rank covariant ten- 
sor whose components in the x'-frame are the Kronecker deltas (in 
this case denoted by 8 /y ), has different components in other frames and 
is accordingly of no special interest. 

It is reasonable to enquire at this stage why the distinction between 
covariant and contravariant tensors did not arise when the coordinate 
transformations were restricted to be orthogonal. Thus, suppose that 
A 1 , B t are contravariant and covariant vectors with respect to the 
orthogonal transformation (8.1). The inverse transformation has been 
shown to be equation (1 1.5) and it follows from these two equations 
that 

dx t 8X; 

^ = %, s -*. (30.13) 



90 TENSOR CALCULUS AND RELATIVITY 

For the particular case of orthogonal transformations, therefore, 
equations (30.3), (30.8) take the form 

A'^ayA', B i = a ij B j . (30.14) 

It is clear that both types of vector transform in an identical manner 
and the distinction between them cannot, therefore, be maintained. 

As in the case of the Cartesian tensors of Chapter 2, new tensors 
may be formed from known tensors by addition (or subtraction) and 
multiplication. Only tensors of the same rank and type may be added 
to yield new tensors. Thus, if AJ kt BJ k are components of tensors and 
we define the quantities CJ k by the equation 

C) k = Aj k + Bj k , (30.15) 

then Cj k are the components of a tensor having the covariant and 
contravariant properties indicated by the position of its indices. How- 
ever, A}, B u cannot be added in this way to yield a tensor. Any two 
tensors may be multiplied to yield a new tensor. Thus, if Aj, B^ m are 
tensors and we define N 5 quantities Cfc by the equation 

cjL = ajbL ( 30 - 16 > 

these are the components of a fifth rank tensor having the covariant 
and contravariant properties indicated by its indices. The proofs of 
these statements are left for the reader to provide. 

If a tensor is symmetric (or skew-symmetric) with respect to two 
of its superscripts or to two of its subscripts in any one frame, then it 
possesses this property in every frame. The method of proof is identical 
with that of the corresponding statement for Cartesian tensors given 
in section 10. However, if Aj = A{ is true for all i,j when one reference 
frame is being employed, this equation will not, in general, be valid 
in any other frame. Thus, symmetry (or skew-symmetry) of a tensor 
with respect to a superscript and a subscript is not, in general, a 
covariant property. The tensor 8} is exceptional in this respect. 

Another result of great importance which may be established by the 
same argument we employed in the particular case of Cartesian ten- 
sors, is that an equation between tensors of the same type and rank is 
valid in all frames if it is valid in one. This implies that such tensor 
equations are covariant (i.e. are of invariable form) with respect to 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 91 

transformations between reference frames. The usefulness of tensors 
for our later work will be found to depend chiefly upon this property. 
A symbol such as Aj k can be contracted by setting a superscript and 
a subscript to be the same letter. Thus A) h A\ k are the possible con- 
tractions of Aj k and each, by the repeated index summation conven- 
tion, represents a sum. Since in the symbol A) u j alone is a free index, 
this entity has only N components. Similarly A\ k has TV components. 
It will now be proved that, if A) k is a tensor, its contractions are also 
tensors. Specifically, we shall prove that Bj = A} t is a covariant vector. 
For 

- _ dx'dx'dx* 
Bj ~ Aji ~ e*W& A " 

~\dx i dx r )dx J s " 
&dx° 

dx~ r W 
r dx J 



8x^ 
8x J 



~-iA st , 



8x° 

establishing the result. This argument can obviously be generalized 
to yield the result that any contracted tensor is itself a tensor of rank 
two less than the tensor from which it has been derived and of the type 
indicated by the positions of its remaining free indices. In this con- 
nection it should be noted that, if Aj k is a tensor, A)j is not, in general, 
a tensor ; it is essential that the contraction be with respect to a super- 
script and a subscript and not with respect to two indices of the same 
kind. 

If Aj k , B r s are tensors, the tensor Aj k B r s is called the outer product 
of these two tensors. If this product is now contracted with respect to 



92 TENSOR CALCULUS AND RELATIVITY 

a superscript of one factor and a subscript of the other, e.g. Aj k B[, 
the result is a tensor called an inner product. 

31. The quotient theorem. Conjugate tensors 

It has been remarked in the previous section that both the outer and 
inner products of two tensors are themselves tensors. Suppose, how- 
ever, that it is known that a product of two factors is a tensor and that 
one of the factors is a tensor, can it be concluded that the other factor 
is also a tensor? We shall prove the following quotient theorem: If 
the result of taking the product {outer or inner) of a given set of elements 
with a tensor of any specified type and arbitrary components is known 
to be a tensor, then the given elements are the components of a tensor. 
It will be sufficient to prove the theorem true for a particular case, 
since the argument will easily be seen to be of general application. 
Thus, suppose the Aj k are iV 3 quantities and it is to be established that 
these are the components of a tensor of the type indicated by the 
positions of the indices. Let B r s be a mixed tensor of rank 2 whose 
components can be chosen arbitrarily (in any one frame only of course) 
and suppose it is given that the inner product 

A} k B* = C) s (31.1) 

is a tensor for all such B r s . All components have been assumed calcu- 
lated with respect to the x-frame. Transforming to the *-frame, the 
inner product is given to transform as a tensor and hence we have 

A%B k s = Cj„ (31.2) 

where Aj k are the actual components replacing the Aj k when the 
reference frame is changed. Let A) k be a set of elements defined in the 
jc-frame by equation (30.11). Since this is a tensor transformation 
equation, we know that the elements so defined will satisfy 

4fc£* = Cj s . (31.3) 

Subtracting equation (31.3) from (31.2), we obtain 

(A$-A} k )B? = 0. (31.4) 

Since B r s has arbitrary components in the jc-frame, its components in 
the x-frame are also arbitrary and the components B* can assume any 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 93 

convenient values. Thus, taking B k = 1 when k = K and B k = 
otherwise, equation (31.4) yields 

A?K-%K = 0, 

or 41 = A) K . (31.5) 

This being true for K = 1 , 2, . . ., N, we have quite generally 

4* = 4fc- (31.6) 

This implies that A] k does transform as a tensor. 

We will first give a very simple example of the application of this 
theorem. Let A 1 be an arbitrary contravariant vector. Then 

8)A } = A 1 (31.7) 

and since the right-hand member of this equation is certainly a vector, 
by the quotient theorem Sj- is a tensor (as we have proved earlier). 

As a second example, let g i} be a symmetric covariant tensor and 
let g = \gij\ be the determinant whose elements are the tensor's com- 
ponents. We shall denote by G iJ the co-factor in this determinant of 
the element gy. Then, although G u is not a tensor, if g ± 0, G iJ /g = g u 
is a symmetric contravariant tensor. To prove this, we first observe 
that 

gyG kJ = g 8?, giJ G ik = g8}, (31.8) 

and hence, dividing by g, 

gijg ki = 8l gij g ik = %. (31.9) 

Now let A 1 be an arbitrary contravariant vector and define the co- 
variant vector B t by the equation 

Bi = g ik A k . (31.10) 

Since g ^ 0, when the components of B t are chosen arbitrarily, the 
corresponding components of A 1 can always be calculated from this 
last equation, i.e. B t is arbitrary with A*. But 

iPB, = g iJ g ik A k = 8{A k = A j , (31.11) 

having employed the second identity (31.9). It now follows by the 



94 TENSOR CALCULUS AND RELATIVITY 

quotient theorem that g iJ is a contravariant tensor. That it is sym- 
metric follows from the circumstance that G u possesses this property. 
gy, gP are said to be conjugate to one another. 

32. Relative tensors and tensor densities 

The coordinates x 1 being expressed in terms of the coordinates x 1 by 
the equations (30.1), we shall define the determinant D of the trans- 
formation to have dx'/8x J for its y th element. Thus 

dx 1 



D == 



d X J 



(32.1) 



Then 9lj is a relative tensor of weight W(Wsl positive or negative 
integer) having both covariant and contravariant characteristics as 
are indicated by its indices, if its components transform thus: 

This statement can be generalized in the obvious way to relative 
tensors of any rank. 

If fv = 0, a relative tensor reduces to an ordinary tensor. In the 
particular case W= 1, the relative tensor is referred to as a tensor 
density (cf. section 13). 

The following statements follow immediately from the definition of 
a relative tensor and the reader is left to devise formal proofs : (i) Rela- 
tive tensors of the same type and weight can be added or subtracted to 
yield new tensors of the same type and weight ; (ii) two relative tensors 
can be combined into an outer product and this will be a relative 
tensor whose weight is the sum of the weights of its factors; (iii) if a 
relative tensor is contracted with respect to a contravariant and a co- 
variant index, its weight is unaltered but its rank is reduced by two. 

e' 7 • • " denotes a contravariant tensor density of the N th rank which 
is skew-symmetric with respect to every pair of indices. It then follows, 
as in section 13, that its components are determined as follows: 

e »y.../» _ q^ jf two indices are the same, 

_ +c i2...A^ y ij t ... t n is an even permutation of 1,2, ...,N, 
_ _ c i2 . . . n ^ i j^ ^ n j s an fa permutation of 1 , 2, . . ., N. 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 95 

j2...N = i.xhen,inthejc-frame, 



In particular, in the *-frame, we take c 



- t n...N =D & d J?. 



dx" 



dx* dx J 

= DE, (32.3) 

where E is the determinant whose y th element is dx'/dx', i.e. 



E = 



dx J 



(32.4) 



Employing the usual rule for the multiplication of determinants, from 
equations (32.1), (32.4) it follows that 



DE = 



dx^dx^ 

dx r dx J 



dx' 



8x J 



= |Sj| = 1. 



Hence 



■A2...N 



= 1, 



(32.5) 



(32.6) 



proving that the components in the Jc-frame are identical with those 
in the x-frame. The density e' 7- •" accordingly possesses components 
+ 1,-1 and in all frames. 

It is left as an exercise for the reader to prove in a similar way that 
the skew-symmetric covariant relative tensor c y ...„ of the iV th rank 
and weight — 1 , whose component c J2 . . . n is unity in the jc-frame, has 
the same components + 1,-1 and in all frames. 

If Ay is a covariant tenso* 1 , the determinant \Ay\ provides an example 
of a relative invariant. For its transformation law is 



\M = 



= 


dx r 
dx 1 


A rs d -j 


» 


= 


dx r 
dx' 


\A„\ 


dx J 


= 


D 2 \. 


4rsl 





(32.7) 



Its weight is seen to be 2. 



96 TENSOR CALCULUS AND RELATIVITY 

Similarly, if A iJ is a contravariant tensor, the determinant \A iJ \ will 
be found to transform according to the law 

\A V \ = E 2 \A U \ = D~ 2 \A U \ (32.8) 

and is therefore a relative invariant of weight — 2. 

Lastly, if A) is a mixed tensor, the determinant \A]\ is an invariant, 
for its transformation law is: 

\A)\ = DE\AJ\ = \A}\. (32.9) 

33. Covariant derivatives. Parallel displacement. Affine connection 

In the earlier sections of this chapter, the algebra of tensors was estab- 
lished and it is now time to explain how the concepts of analysis can 
be introduced into the theory. Our space SP N has AT dimensions, but 
is otherwise almost devoid of special characteristics. Nonetheless, it 
has so far been able to provide all the facilities required of a stage upon 
which the tensors are to play their roles. It will now be demonstrated, 
however, that additional features must be built into the structure of 
S? Nt before it can function as a suitable environment for the operations 
of tensor analysis. 

It has been proved that, if <f> is an invariant field, dfydx* is a covariant 
vector. But, if a covariant vector is differentiated, the result is not a 
tensor. For, let A t be such a vector, so that 

dx k 
*i = 7=rA k . (33.1) 

dx' 

Differentiating both sides of this equation with respect to x J , we 
obtain 

dAj _ d^cbSdAk a 2 ** 

8x J ~ dx 1 8x J dx 1 + 8x'dx j Ak ' (33,2) 

The presence of the second term of the right-hand member of this 
equation reveals that dAi/dx J does not transform as a tensor. However, 
this fact can be arrived at in a more revealing manner as follows: 

LetP, P' be the neighbouring points x l , x'+dx' and let A h Ai+dA t 
be the vectors of a covariant vector field associated with these points 
respectively. The transformation laws for these two vectors will be 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 97 

different, since the coefficients of a tensor transformation law vary 
from point to point in Sf N . It follows that the difference of these two 
vectors, namely dA h is not a vector. However, 

dAi . 
dAi = —]dx } (33.3) 

ox J 

and, since dx J is a vector, if Aij were a tensor, dA { would be a vector. 
A it j cannot be a tensor, therefore. The source of the difficulty is now 
apparent. To define A ifj , it is necessary to compare the values assumed 
by the vector field A t at two neighbouring, but distinct, points and 
such a comparison cannot lead to a tensor. If, however, this procedure 
could be replaced by another, involving the comparison of two vectors 
defined at the same point, the modified equation (33.3) would be 
expected to be a tensor equation featuring a new form of derivative 
which is a tensor. This leads us quite naturally to the concept of 
parallel displacement. 

Suppose that the vector A t is displaced from the point P, at which 
it is defined, to the neighbouring point.?', without change in magnitude 
or direction, so that it may be thought of as being the same vector now 
defined at the neighbouring point. The phrase in italics has no precise 
meaning in S? N as yet, for we have not defined the magnitude or the 
direction of a vector in this space. However, in the particular case 
when Sf N is Euclidean and rectangular axes are being employed, this 
phrase is, of course, interpreted as requiring that the displaced vector 
shall possess the same components as the original vector. But even in 
S Ni if curvilinear coordinates are being used, the directions of the 
curvilinear axes at the point P' will, in general, be different from their 
directions at P and, as a consequence, the components of the displaced 
vector will not be identical with its components before the displace- 
ment. In S? Nt therefore, components of the displaced vector will be 
denoted by A t + §A t . This vector can now be compared with the field 
vector A t + dA t at the same point P '. Since the two vectors are defined 
at the same point, their difference is a vector at this point, i.e. dA t — 8A t 
is a vector. The modified equation (33.3) is accordingly expected to be 
of the form 

dA t -8A i = A ilJ dx h (33.4) 



98 TENSOR CALCULUS AND RELATIVITY 

where A i; j is the appropriate replacement for A itJ . Since dx J is an 
arbitrary vector and the left-hand member of equation (33.4) is known 
to bea vector, A t .j will, by the quotient theorem, be a covariant tensor. 
It will be termed the covariant derivative of A t . Thus, the problem of 
defining a tensor derivative has been re-expressed as the problem of 
defining parallel displacement (infinitesimal) of a vector. 

We are at liberty to define the parallel displacement of A t from P to 
P' in any way we shall find convenient. However, to avoid confusion, 
it is necessary that the definition we accept shall be in conformity with 
that adopted in £ N , which is a special case of S? N . Suppose, therefore, 
that our S? N is Euclidean and that y are rectangular Cartesian co- 
ordinates in this space. Let B t be the components of the vector field A t 
with respect to these rectangular axes. Then 

If the parallel displacement of the vector A t to the point P' is now 
carried out, its Cartesian components B t will not change, i.e. SB = 0. 
Hence, from the first of equations (33.5), we obtain 



8 „, = S gU) = S g)a, 



= —/— k dx k Bj. (33.6) 

8x i dx k J 

Substituting for Bj into this equation from the second of equations 
(33.5), we find that 

8Af = rlkAjdx'', (33.7) 

d 2 y j dx l 
8x i dx k 8y j ' 

This shows that, in f N , the 8A t are bilinear forms in the A t and dx k . 
In Sf Ni we shall accordingly define the hA t by the equation (33.7), 
determining the JV 3 quantities I* tk arbitrarily at every point of Sf N .\ 

t Subject to the requirement that the F\ k are continuous functions of the 
x* and possess continuous partial derivatives to the order necessary to 
validate all later arguments. 



where P ik = nr?- • (33.8) 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 99 

This set of quantities r* ik is called an affinity and specifies an affine 
connection between the points of Sf N . A space which is affinely con- 
nected possesses sufficient structure to permit the operations of tensor 
analysis to be carried out within it. 
For we can now write 

8A- 
dAt-SA, = -^.dx } -r\A k dxK 



= 0-r?jA k }dxJ. (33.9) 



But, as we have already explained, the left-hand member of this 
equation is a vector for arbitrary dx j and hence it follows that 

dA- 
A i;J = -^r F i A k 03.10) 

is a covariant tensor, the covariant derivative of A t . 

It will be observed from equation (33.10) that, if the components of 
the affinity all vanish over some region of SP N> the covariant and 
partial derivatives are identical over this region. However, this will 
only be the case in the particular reference frame being employed. In 
any other frame the components of the affinity will, in general, be 
non-zero and the distinction between the two derivatives will be main- 
tained. In tensor equations which are to be valid in every frame, 
therefore, only covariant derivatives may appear, even if it is possible 
to find a frame relative to which the affinity vanishes. 

We have stated earlier that, when defining an affine connection, the 
components of an affinity may be chosen arbitrarily. To be precise, a 
coordinate frame must firsi be selected in Sf N and the choice of the 
components of the affinity is then arbitrary within this frame. How- 
ever, when these have been determined, the components of the affinity 
with respect to any other frame are, as for tensors, completely fixed by 
a transformation law. This transformation law for affinities, we now 
proceed to obtain. 

34. Transformation of an Affinity 

The manner in which each of the quantities occurring in equation 
(33.10) transforms is known, with the exception of the affinity 1%. The 



100 TENSOR CALCULUS AND RELATIVITY 

transformation law for this affinity can accordingly be deduced by 
transformation of this equation. Relative to the Jc-frame, the equation 
is written 

A,j = -£-T%A k . (34.1) 

Since A u A t .j are tensors, 

A t = %A„ (34.2) 

dx 

1 8xS8xt A KAVX 

Substituting in equation (34.1), we obtain 

?^^ A -»£?£ d JL + *?L A -I*¥A (344) 
dx { dx jAs;t ~ dx'dx^dx^dx'dx 3 r lJ dx k r ' K5 *' } 

Employing equation (33. 10) to substitute for A s ; , and cancelling a pair 
of identical terms from the two members of equation (34.4), this 
equation reduces to 

J**t.r u A. = ^A,-n%A r . (34.5) 

dx'8x J sl dx'dx J r ,J 8x k r V 

Since A r is an arbitrary vector, we can equate coefficients of A r from 
the two members of this equation to obtain 

v dx k " dx\dxi st+ dx { 8x J l ' 

Multiplying both sides of this equation by 8y}\8x r and using the result 

dx'dx" dx 1 , ..... 

a?«? -«*-** (34J) 

yields finally 

u ~ ax r dx'dx 1 ""We*'^' K } 

which is the transformation law for an affinity. 
It should be noted that, were it not for the presence of the second 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 101 

term in the right-hand member of equation (34.8), ly would trans- 
form as a tensor of the third rank having the covariant and contra- 
variant characteristics suggested by the positions of its indices. Thus, 
the transformation law is linear in the components of an affinity but 
is not homogeneous like a tensor transformation law. This has the 
consequence that, if all the components of an affinity are zero relative 
to one frame, they are not necessarily zero relative to another frame. 
However, in general, there will be no frame in which the components 
of an affinity vanish over a region of S? N , though it will be proved that, 
provided the affinity is symmetric, it is always possible to find a frame 
in which the components all vanish at some particular point (see 
section 38). 

Suppose r$, Jy* are two affinities defined over a region of S? N . 
Writing down their transformation laws and subtracting one from 
the other, it is immediate that 

**-** - ftPMTO^- 7 ^ (34.9) 

i.e. the difference of two affinities is a tensor. However, the sum of two 
affinities is neither a tensor nor an affinity. It is left as an exercise for 
the reader to show, similarly, that the sum of an affinity Jy and a 
tensor Ay is an affinity. 

If /y is symmetric with respect to its subscripts in one frame, it is 
symmetric in every frame. For, from equation (34.8), 

n = £^f^£i' r r 8 * k &x r 
Ji dx r dx J dx l st+ dx r dx j dx r 

dx k dx f dx* r dx k &x r 



dx'dx'dx' ts dx r dx'dx J ' 
= rfj, (34.10) 

where, at the first step, we have put I^ s = r r st . 

35. Covariant derivatives of tensors 

In this section, we shall extend the process of covariant differentiation 
to tensors of all ranks and types. 



102 TENSOR CALCULUS AND RELATIVITY 

Consider first an invariant field V. When Fsuffers parallel displace- 
ment from P to P\ its value will be taken to be unaltered, i.e. 8 V = 
in all frames. Hence 

oV , 
dV-hV=— i dx i (35.1) 

ox 

is the counterpart for an invariant of equation (33.4). It follows that 

V. tl = V h (35.2) 

i.e. the covariant derivative of an invariant is identical with its partial 
derivative or gradient. 

Now let B* be a contravariant vector field and A t an arbitrary co- 
variant vector. Then A t B ' is an invariant and, when parallel displaced 
from P to P\ remains unchanged in value. Thus 

8(A t B') = 0, 

or SAiB'+Ai SB 1 = 0, 

and hence, by equation (33.7), 

A k 8B k = -I%A k dx?B l . (35.3) 

But, since the A k are arbitrary, their coefficients in the two members 
of this equation can be equated to yield 

8B k = -riB i dx J . (35.4) 

This equation defines the parallel displacement of a contravariant 
vector. The covariant derivative of the vector is now deduced as 
before: Thus 

dB k -8B k = (~ + I*B t \dx i (35.5) 

and since dx J is an arbitrary vector and dB k —8B k \s then known to be 
a vector, 



t k 

dx 

is a tensor called the covariant derivative of B k 



8B 

B k j = ^ 1 + rf J B i (35.6) 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 103 

Similarly, if A) is a tensor field, we consider the parallel displacement 
of the invariant AJB t C J t where B u C J are arbitrary vectors. Then, from 

8(A}BiC J ) = (35.7) 

and equations (33.7), (35.4), we deduce that 

SA} = rj k A\dx k - r\ k A l jdx k . (35.8) 

It now follows that 

Aj-.k-fp-llkAt+rlkA} (35.9) 

is the covariant derivative required. 

The rule for finding the covariant derivative of any tensor will now 
be plain from examination of equation (35.9), viz., the appropriate 
partial derivative is first written down and this is then followed by 
'affinity terms'; the 'affinity terms' are obtained by writing down an 
inner product of the affinity and the tensor with respect to each of its 
indices in turn, prefixing a positive sign when the index is contra- 
variant and a negative sign when it is covariant. 

Applying this rule to the tensor field whose components at every 
point are those of the fundamental tensor Sj, it will be found that 

% ;k = r^-rj.sl = rk-1% = o. (35.10) 

Thus, the fundamental tensor behaves like a constant in covariant 
differentiation. 

Finally, in this section, we shall demonstrate that the ordinary rules 
for the differentiation of sums and products apply to the process of 
covariant differentiation. 

The right-hand member of equation (35.9) being linear in the tensor 
Aj, it follows immediately that if 

C} = A}+B}, (35.11) 

then Cj ik = AJ ik +BJ. k . (35.12) 

Now suppose that C* = A^B 1 . (35.13) 



104 TENSOR CALCULUS AND RELATIVITY 

Then 

ac"* 



c i ,k = T - k +n k c% 



= A} ;k B J +B J . k AJ, (35.14) 

which is the ordinary rule for the differentiation of a product. 

36. Covariant differentiation of relative tensors 

We consider first the parallel displacement of an invariant density 91 
from the point P to the neighbouring point P ', at which it will be taken 
to become 91+ 891. It cannot be assumed that 891 = for, even if the 
two invariant densities are identical in one frame, since the transfor- 
mation law at P is different from that at P', they will cease to be 
identical when referred to another frame. If, however, Sf N is Euclidean 
and we restrict the choice of frames to those which are rectangular 
Cartesian, then the transformation law for the density will be 

S = m, (36.1) 

where D is given by equation (32.1) and, in the case of orthogonal 
transformations, can take the values + 1 and - 1 only. In this special 
case, therefore, D does not depend upon the point at which 91 is defined 
and 891 = can be valid for all rectangular Cartesian frames. We shall 
accordingly define the parallel displacement of an invariant density in 
£ N , relative to rectangular Cartesian frames, to be such that the 
density suffers no change. 

Let y, y + dy l be the coordinates of pointsP, P ' relative to rectangu- 
lar Cartesian axes in S N and let 93 be an invariant density calculated 
in this frame and associated with the point P. If this density is parallel 
displaced to P\ its value remains unchanged at 95, i.e. 893 = 0. Let 
x { , *'+<&' be the coordinates of P, P' respectively referred to any 
other coordinate frame (not necessarily Cartesian) and let 91 be the 



But 


dx* 


and thus 


h% = K{Mx\ 


where 


18D 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 105 

same invariant density at P calculated in this new frame. Referred to 
the new frame, let 51+831 be this density after it has been parallel 
displaced to P'. Then 

21 = D93, (36.2) 

where D = \dtflbx!\. Hence 

82t = S(Z>95) = 8D.% = ^8Z>SI. (36.3) 

(36.4) 

(36.5) 

(36.6) 

Equation (36.5) is valid for the parallel displacement of an invariant 
density in a space S? N which is Euclidean. If Sfjf is not Euclidean, the 
parallel displacement of an invariant density will be defined by the 
equation (36.5). Since Cartesian coordinates are not available in such 
a space, the coefficients K t cannot be derived from equation (36.6) and 
must, instead, be imposed upon the space by specification at every 
point (cf. an affinity). This specification will, for the moment, be 
assumed to be performed in an arbitrary manner in any one frame and, 
when this has been done, the K t will be determined in all frames. 

Suppose now that % is an invariant density field and let 21, %+d% 
denote the actual values taken by the density at the points x\ x'+dx*. 
Let Ul be displaced to the point x'+dx*, where it assumes the value 
W.+ 831. Then the difference between the two densities defined at the 
point x^dx', is itself a density and equals 

cM-8% = I— f-KiWldx'. (36.7) 

But dx' being an arbitrary Vector, it follows that 

aui 

* ;l = jn-Kt* (36.8) 

ox' 

is a covariant vector density. This will be termed the covariant 
derivative of %. 



106 TENSOR CALCULUS AND RELATIVITY 

Consider now 93/, a tensor density of the second rank. Let % be any 
invariant density. Then %~ 1 is a relative invariant of weight - 1 and 
hence 

n- l SBJ = BJ (36.9) 

is a tensor. Rewriting this equation thus, 

»j = «flf, (36.10) 

and subjecting it to parallel displacement, we find 
SSBJ = 8(2LBj), 

= %8^j+^j8% 

= %(r Jk B t r -P rk ff])dx k +^K k 'Mx k t 
= <J7ft»/-J?*»5+jr*»yVc*. (36.11) 

where equations (35.8) and (36.5) have been employed in the third line 
of this argument. Equation (36.11) indicates that the law of parallel 
displacement for a tensor density is identical with that for a tensor 
except for the presence of an additional term involving the coefficient 
K k . 

If 93y is a tensor density field, its covariant derivative can now be 
found in the usual way. It is easily shown that this derivative is given 
by 

*J;k = -^ji+ r rkWj-r r jM-K k <8) (36.12) 

and is a tensor density. The rule for differentiating any tensor density 
should now be clear after inspection of the last equation. 

Consider, in particular, the tensor density field whose components 
at every point are t u "". Since these components are all constant, 
their partial derivatives are zero and, by the rule, the covariant 
derivative is 

e y - •"., = r , rt t rJ - a +rJ„t*-"+...+r?,tV-'-K,t if '~ a . 06.13) 

If 1, j, . . ., n is an even permutation of 1, 2, . . ., N, then 



I% a t rJ '" n = rj s (not summed over 
etc. and hence 



JV,e"-" = rj s (not summed overy)| J (36-14) 

e ,y -" : , = r rs -K Si (36.15) 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 107 

where summation with respect to r is intended. If i,j, . . ., n is an odd 
permutation, then 

e* •••".,= -(.r rs -K s ). (36.16) 

If a pair of i, j, . . . n are the same, e.g. i=j = P, then 

_ pP-rP...n_pP-rP...n 

■*■ rs *■ ■*■ rs* > 

= 0, (36.17) 

where summation with respect to P is not intended. Also, the remain- 
ing terms of the right-hand member of equation (36.13) are then zero 
and it follows that, in this case, 

t v - ■ n ;s = 0. (36.18) 

Equations (36.15), (36.16), (36.18) can be summarized in the form 

t«- H it = (r„-Kdt«- H . (36.19) 

c tf...n having the same components in every frame and at every 
point, it is natural to expect its covariant derivative to be identically 
zero. Such a result would certainly facilitate calculations into which 
this density enters, since it could then be treated as a constant with 
respect to covariant differentiation (cf. Sj). Equation (36.19) shows 
that this desirable state of affairs can easily be achieved by taking 

K s = r rs (36.20) 

in the law defining parallel displacement of densities. This we shall 
accordingly do in all future developments. The covariant derivative 
of a tensor density as determined by equation (36.12), will then read 

®j;k = ^ + J*,*»5-J7*8r l -J7*»J. (36.21) 

The parallel displacement of any relative tensor of weight W can 
now be deduced. Let C be a relative invariant of weight W and % an 
invariant density. Then (£/% w is an invariant. Denoting this by V, we 
have 

d = % w V. (36.22) 



108 TENSOR CALCULUS AND RELATIVITY 

Performing a parallel displacement to a neighbouring point, we obtain 

8<Z = 8Q& W V) = W% w ~ l h%V t (36.23) 

since 8V=0. Substituting for B% from equation (36.5) (using the 
above form for Kj), it is found that 

8£ = wr\ k % w Vdx k = wr i ik <£dx k . (36.24) 

This is the law for parallel displacement of a relative invariant of 
weight Wand it will be noted that it is identical with the law (36.5) for 
the parallel displacement of a density, except for the occurrence of 
the numerical factor W. 

The derivation of the covariant derivatives of relative tensors now 
proceeds in the same manner as for densities. The covariant derivative 
of an invariant (E of weight W proves to be 

C;* = ^E-^n*«. (36.25) 

(cf. equation (36.8)), and 

*>* = -^+ r W-r r Jk ^ r - wr rk Vj (36.26) 

is the covariant derivative of a relative tensor (£j of weight W (cf . 
equation (36.21)). 

The covariant relative tensor of weight — 1, which has earlier been 
denoted by e ; y...„, can be regarded as a field and its derivative now 
found. It will be left as an exercise for the reader to show that this 
derivative is identically zero. The method used to derive equation 
(36.19) should be employed. 

37. The Riemann-Christoffel curvature tensor 

If a rectangular Cartesian coordinate frame is chosen in a Euclidean 
space ^Vand if A* are the components of a vector defined at a point Q 
with respect to this frame, then 8 A 1 = for an arbitrary small parallel 
displacement of the vector from Q. This being true for arbitrary A 1 , it 
follows from equation (35.4) that rj k = with respect to this frame at 
every point of ^. Suppose C is a closed curve passing through Q and 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 109 

that A' makes one complete circuit of C, being parallel displaced over 
each element of the path. Then its components remain unchanged 
throughout the motion and hence, if A' + A A 1 denotes the vector upon 
its return to Q, 

A A 1 = 0. (37.1) 

Since A A 1 is the difference between two vectors both defined at Q, it 
is itself a vector and equation ("37. 1) will therefore be a vector equation 
true in all frames. Thus, in & Nt parallel displacement of a vector 
through one circuit of a closed curve leaves the vector unchanged. 

If, however, A' is defined at a point Q in an affinely connected space 
y N , not necessarily Euclidean, it will no longer be possible, in gen- 
eral, to choose a coordinate frame for which the components of the 
affinity vanish at every point. As a consequence, if A' is parallel dis- 
placed around C, its components will vary and it is no longer per- 
missible to suppose that upon its return to Q it will be unchanged, 
i.e. A A 1 ^ 0. We shall now calculate A A' when A 1 is parallel displaced 
around a small circuit C enclosing the point P having coordinates jc 1 
(Fig. 5) at which it is initially defined. 

Let U be any point on this curve and let x'+ £' be its coordinates, 
the £' being small quantities. Fis a point on C near to U and having 
coordinates x'+ tj' + di;'. When A 1 is displaced from U to V, its com- 
ponents undergo a change 

8 A* = -Tj k A J d$ k , (37.2) 

where r} k and A j are to be computed at U. Considering the small 
displacement from P to C/and employing Taylor's theorem, the value 
of rj k at U is seen to be 



dx l 



r}k + -Ae, (37.3) 



to the first order in the £'. In this expression, the affinity and its deriva- 
tive are to be computed at P. A J in equation (37.2) represents the 
vector after its parallel displacement from P to U, i.e. it is 

A J -ri,A r $ l , (37.4) 



110 



TENSOR CALCULUS AND RELATIVITY 



where A J , A r and r j rl are all to be calculated at the point P. To the first 
order in £ ! therefore, equation (37.2) may be written 

8A l = -[jJt^+^^-Jlr/M^jdf*. (37.5) 




Fig. 5 



Integrating around C, it will be found that 

aa'= -rj k AJ jd^+lri.rz-^AJ J£'di k , (37.6) 

where the dummy indices j, r have been interchanged in the final term 
of the right-hand member of equation (37.5). 



Now 



d£ k = A£ k = 0. 



(37.7) 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 111 

Also jd(i l i k ) = A($ ! £ k ) = 0, (37.8) 

c 

so that j> &d? = -j £ k dg l , (37.9) 

c c 

implying that the left-hand member of this equation is skew-sym- 
metric in / and k. Since £', df- k are vectors, it is also a tensor. Denoting 
it by a*', we have 

a*' = I j (£'#*-£*#') (37.10) 

c 

and equation (37.6) then reduces to the form 

j^ = (n*r;,-^y«*. (37.il) 

Apart from its property of skew-symmetry, ct kl is arbitrary. None- 
theless, since it is not completely arbitrary, the quotient theorem 
(section 31) cannot be applied directly to deduce that the contents of 
the bracket in equation (37. 11) constitute a tensor. In fact, this expres- 
sion is not a tensor. However, it is easy to prove that, if A$ is skew- 
symmetric with respect to k, I and if Y y , defined by the equation 

Y iJ = Xg<x kl t (37.12) 

is a tensor for arbitrary skew-symmetric tensors <x w , then Xft is also 
a tensor. 

To prove this, let fl kl be an arbitrary symmetric tensor. Then the 
components of the tensor 

/' = «"+£* (37.13) 

are completely arbitrary, for, assuming k < /, 

/' = «*'+/3" f yf k = - a kl +p kl (37.14) 

and it follows that the values of y kl , y lk can be chosen arbitrarily and 
then 

«*' = i(/'-y'*). P kl = h(y kl +v lk ). (37.15) 

I.e., it is only necessary to fix the values of a, k! , fl kl in the cases k < I in 



112 TENSOR CALCULUS AND RELATIVITY 

order that the y kl shall assume any specified values over the complete 
range of its superscripts, with the exception of the cases when two 
superscripts are equal. If the superscripts are equal, * kl = and 
y kl = p kl . But these j8 w are also arbitrary and hence so again are the 
y kl with equal superscripts. 
Since j8 w is symmetric and X\li is skew-symmetric, 

*#£*' = 0. (37.16) 

Adding equations (37.12) and (37.16), we obtain therefore 

Xk J iV kl = 7°. (37.17) 

But y kl is an arbitrary tensor and hence, by the quotient theorem, X$ 
is a tensor. 

The multiplier of a kl in equation (37.1 1) is not skew-symmetric in 
k> I. However, it can be made so as follows: Interchange the dummy 
indices k, I in this equation to obtain 

AA< = (r^-^AU*. (37.18) 

Adding equations (37.11) and (37.18) and noting that a. kl - -af k t it 
will be found that 

AA' = ifart-nilJk+^-^A'** 1 . (37.19) 

The bracketed expression is now skew-symmetric in k, I and hence 

(n^-I^+g*-!*)^ (37.20) 

is a tensor. A J being arbitrary, it follows that 

Bj kl = n k r h -rur jk+ d ^- 8 -^ (37.21) 

is a tensor. It is the Riemann-Christoffel Curvature Tensor. 
Equation (37.19) can now be written 

AA 1 = W jkl AU kl . (37.22) 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 113 

If Bjki is contracted with respect to the indices i and /, the resulting 
tensor is called the Ricci Tensor and is denoted by R Jk . Thus 



R-ik — Bji 



^jk 



*jki' 



(37.23) 



This tensor has an important role to play in Einstein's theory of gravi- 
tation. Since BJ kt is skew-symmetric with respect to the indices k and 
/, its contraction with respect to / and k yields only the Ricci tensor 
again in the form —Rji. However, contraction with respect to the 
indices / and 7* yields another second rank tensor, viz. 



Su - Baa - *?-!? 



(37.24) 



38. Geodesic coordinates. The Bianchi identities 

If the affinity P Jk is symmetric in its subscripts, it is always possible to 
find a coordinate frame in which all components of the affinity vanish 
at any chosen point. 

We shall first choose a coordinate frame in which the chosen point 
P is the origin and hence has coordinates x l = 0. We then transform to 
new coordinates x* by the equations 



x' = x % +\a) k x J x k t 



(38.1) 



where the aj k are constants whose values will be chosen later and we 
shall assume, without loss of generality, that aj k is symmetric with 
respect toj and k. Differentiating the equations (38.1), we find that 



— • - bj+a Jk x , 



3V 



(38.2) 



dx J dx 
AtPx l = and these reduce to 



k = a ik- 



8x J J ' dx J 8x k 



Bjk- 



(38.3) 



Sj^ = *i, (38.5) 



or — k = *k. (38.6) 



114 TENSOR CALCULUS AND RELATIVITY 

8x' 8x J 
Now — jTlc = S' k (38.4) 

8x J 8x K 

and hence, at the chosen point, 

Jj 8x k 
8? 
8x k 

Substituting appropriately in equation (34.8), we calculate that the 
components of the affinity at the point P in the x'-frame are given by 

= 4+4-. (38.7) 

Since the affinity is symmetric, it is now possible to choose the a'y 
to satisfy 

4 =-4- < 38 - 8 > 

The transformation (38.1) is now determined completely and, by 
equation (38.7), 

F\j = (38.9) 

as required. 

The coordinates x' are said to be geodesic at the point P. If such 
coordinates are employed, it is clear that covariant and partial de- 
rivatives are identical atP. This enables us to simplify many arguments 
leading to tensor equations. However, if such equations can, by this 
means, be proved valid in the geodesic frame, they are necessarily 
valid in all frames. As an example, we will derive the Bianchi Identity. 

Thus, assuming the affinity is symmetric and employing geodesic 
coordinates at the point being considered, the covariant derivative of 
equation (37.21) is simply 

B Jkl;m - -^y. rkiji-i nhk+ dxk dxi y 

8>n #i%_ (38JO) 



8x m 8x k dx m 8x'' 
since the I) k (but not their derivatives necessarily) all vanish at this 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 115 

point. Cyclically permuting the indices k, /, m in equation (38.10), we 
obtain 

t 8 2 rj m d 2 rji 

Bjlm > k = 'dx^~'dx T d X ^' (38J1) 

Bjmk-.i - ^r^-^r^k (38-12) 

Addition of equations (38.10), (38.11), (38.12), yields the following 
identity 

Bjkl;m + Bjlm t k + Bjmk;l ~ 0« (38.13) 

But this is a tensor equation and, having been proved true in the 
geodesic frame, must be true in all frames. Also, since the chosen point 
can be any point of £? Nt it is valid at all points of the space. It is the 
Bianchi Identity. 

39. Metrical connection. Raising and lowering indices 

In this section we shall further particularize our space S? N by suppos- 
ing it to be Riemannian. That is, we shall suppose that a ' distance ' or 
interval ds between two neighbouring points x*, x'+dx* is denned by 
the equation 

ds 2 = gydx l dx } , (39.1) 

where the N 2 coefficients gy are specified in some coordinate frame at 
every point of SP N . It will be assumed, without loss of generality, that 
the gy are symmetric. Such a relationship between all pairs of adjacent 
points is called a metrical connection and the expression (39.1) fords 2 
is termed the metric. 

For any two neighbouring points, ds will be regarded as an in- 
variant associated with them and the gy must accordingly transform 
so that this shall be so. Since dx i dx J is an arbitrary symmetric tensor, 
gy is symmetric and ds 2 is an invariant, it follows by a modified quo- 
tient theorem similar to the one proved in section 37, that gy is a 
tensor. It is called the fundamental covariant tensor. The contravariant 
tensor which is conjugate to gy (see section 31), viz. g 1 ^ is termed the 
fundamental contravariant tensor. This exists only if g= \gy\ ^0, which 
we accordingly assume to be the case. 



116 TENSOR CALCULUS AND RELATIVITY 

Let A' be any contravariant vector defined at a point of 8t N . Then 
gijA J is a covariant vector at the same point and this will be denoted 
by A h Thus 

A t = gij A } . (39.2) 

We shall regard the A 1 and A t as the contravariant and covariant 
components respectively, relative to the coordinate frame in use, of 
the same vector. The process defined by equation (39.2), of converting 
the contravariant expression for a vector into its covariant expression 
is termed lowering the index. 

If B t is a covariant vector, its contravariant expression is determined 
by raising the index with the aid of the fundamental contravariant 
vector. Thus 

B* = g ij Bj. (39.3) 

For the notation to be consistent, it is necessary that if an index is 
first lowered and then raised, the original vector should again be 
obtained. This is seen to be the case for, if A t is formed from A 1 (equa- 
tion (39.2)), the result of raising its subscript is (equation (39.3)) 

g v Aj = g iJ g Jk A k = 8 l k A k = A 1 , (39.4) 

where equations (31.9) have been used in the reduction. Similarly, if 
an index is first raised and then lowered, the original covariant vector 
is reproduced. 

Any index of a tensor can now be raised or lowered in the obvious 
way. Thus, if A ij k is a tensor, we define 

A% = g Jr A ir k . (39.5) 

To allow for the possibility that indices may be raised or lowered 
during a calculation, it will be convenient to displace the subscripts to 
the right of the superscripts. It is also often helpful to keep a record 
of these operations by. placing a dot in the gap resulting from the 
raising or lowering of an index. These conventions are illustrated in 
equation (39.5). 

Suppose an index of the fundamental tensor gy is raised. The result 

g k j = **'*„ = 8 k j, (39.6) 

i.e. the mixed fundamental tensor. The same tensor results when an 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 117 

index of g ij is lowered. If both subscripts of gy are raised, the result is 
g ri g SJ gij = *"«'/ = g rs - (39.7) 

Our notation is entirely consistent, therefore, and gy, g u , S' 7 - are taken 
to be the covariant, contravariant and mixed components respectively 
of a single fundamental tensor. 
Consider the inner product of two vectors A 1 , B h We have 

A'Bi = g i} A jgik B k , 

= gVg ik AjB\ 

= 8 J k AjB\ 

= A k B k , 

= A t B*. (39.8) 

It is clear that the dummy index occurring in the expression for an 
inner product can be raised in one factor and lowered in the other 
without affecting the result. This is obviously valid for the inner pro- 
duct of any pair of tensors. 
In 6 Nt if we confine ourselves to rectangular Cartesian frames, 

ds 2 = d^dx 1 (39.9) 

and hence gy =1, i = / 



n • ^ • • (39 - 10) 

= 0, / 5* J. 



It will now be found that g = \gy\ = 1 and that 



= 0, i*j.j 



8 *' * ■" ' (39.11) 



Hence, if A t is a vector in the space, according to our definition its 
contravariant components will be 

A 1 = g»Aj = A h (39.12) 

confirming that there is no distinction between covariant and contra- 
variant vectors in this special case of S N . 

40. Scalar products. Magnitudes of vectors 

In 6 N , the magnitude of the displacement vector dx l is taken to be ds 
as given by equation (39.9). In @ Ni the magnitude of this vector is 



118 TENSOR CALCULUS AND RELATIVITY 

taken to be ds as given by equation (39.1). If A' is any other contra- 
variant vector, it may be represented as a displacement vector and 
then its magnitude is the invariant A, where 

A 2 = g A'A J . (40.1) 

This equation is accordingly taken to define the magnitude of A 1 . 

Raising and lowering the dummy indices in equation (40.1), we 
obtain the equivalent result 

A 2 = g u A;Aj. (40.2) 

It is natural to assume that the associated vectors A h A 1 have equal 
magnitudes and hence A is also taken to be the magnitude of A\. 
Equation (40.2) indicates how this can be calculated directly from A { . 
Since gyA J ' = A t and g iJ A J = A i , equations (40.1), (40.2) are also 
both seen to be equivalent to the equation 

A 2 = A t A l . (40.3) 

The scalar product of two vectors A, B is defined to be the invariant 

A-B = A,B? = A l B t = gyA'Bl = g v ' A t Bj. (40.4) 

It will be noted that A 2 = A- A. (40.5) 

By analogy with <? 3 , we now define the angle 6 between two vectors 
A, B to be such that 

ABcosd = A-T&. (40.6) 



If 6 = in, the vectors are said to be orthogonal and 



(40.8) 



41. The Christoffel symbols. Metric affinity 

At any chosen point P of @ N , as is proved in algebra texts, the homo- 
geneous quadratic form defining the metric can, by a regular linear 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 119 

transformation from the coordinates x' to coordinates y', be reduced 
to diagonal form thus 

gu dx ! dx J = (dy l ) 2 +(dy 2 ) 2 +... + (dy N ) 2 . (41.1) 

The transformation may involve complex coefficients. The metric 
may be assumed to take this simplified form over a small neighbour- 
hood of P in which the y' will behave like rectangular Cartesian co- 
ordinates in S N . In such a neighbourhood, therefore, the components 
of the affinity P ik will be representable in the form given by equation 
(33.8) and hence must be symmetric in their subscripts at the point P. 
This will be true at every point of 0t N . We have established, therefore, 
that an affinity imposed upon a Riemannian space must be symmetric 
if parallel displacement in any small Euclidean region is to be in 
accordance with the usual definition for such a region. 

Apart from the fact that it is symmetric, the affinity is otherwise 
arbitrary. However, consider the covariant derivative A i; k . Formally, 
this can be obtained from A* in either of two ways, (i) by lowering the 
index / and then differentiating or (ii) by differentiating and then 
lowering the index. Unless these two operations commute, so that 
either process leads to the same result, confusion will clearly arise. It 
is convenient, therefore, to define the affinity, if possible, in such a 
way that these operations are commutative. Thus, we shall require 
that 

SaA 1 , k = {g ij A i ), k . (41.2) 

Expanding the right-hand member by differentiation of the factors of 
the product, the condition becomes 

gij^.k = gu AJ ;k+gu,k^ J » 

or g(/ik A J = 0. (41.3) 

This is to be true for arbitrary A J and thus 

gy.k = 0. (41.4) 

I.e. the affinity should be chosen so that the covariant derivative of 
the metric tensor gy is identically zero. This can be done as follows: 



120 



TENSOR CALCULUS AND RELATIVITY 



Cyclically permuting the indices /, j and k in equation (41.4), we 
derive the equations 



-^-rjigrk-rhgjr = o, 



8 Ski 

d x J ' 



■r r kJ g ri -r?jg kr = o. 



(41.5) 



Adding the last pair of this set of equations and subtracting the first, 
and remembering that gy, r' jk are symmetric, it will be found that 



where 



gkr^ij = W,k], 



)■ 



(41.6) 
(41.7) 



[ij,k] is termed the Christoffel Symbol of the First Kind. It is not a 
tensor, but its indices invariably behave like subscripts in any formula 
in which it occurs. It is symmetric in the indices i, j. 

Multiplying both members of equation (41.6) by g sk and summing 
with respect to k, it follows that 



where 



r?j = {,% (41.8) 

W-t-Btfl.^+^-g). (41.9) 

{,*/} is called ChristoffeVs Symbol of the Second Kind. It, also, is not 
a tensor and is symmetric with respect to the indices i, j. 

It is now easy to verify that, if the affinity is determined by equation 
(41.8), the condition (41.4) is satisfied. Further, since 

g V g kJ = S'*, (41.10) 

by taking the covariant derivative of both members of this equation, 
we obtain 

g ij g k = 0. (41.11) 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 121 

Multiplying by g kr and summing with respect to k, we then find that 

g ir ; i = 0. (41.12) 

Thus, the covariant derivative of the fundamental contravariant 
tensor is also identically zero. It now follows that 

(g V Aj) ;k = g iJ Aj, k , (41.13) 

i.e. a subscript may be raised before or after a covariant differentiation 
without affecting the result. This is clearly true for tensors of any 
rank. 

The affinity determined by equation (41.8) will be called the metric 
affinity. With this choice of affinity, all forms of the fundamental 
tensor may be treated as constants in covariant differentiation. 

In & N , if a rectangular Cartesian coordinate frame is employed, the 
gy are constants (equations (39.10)) and the Christoffel symbols are 
ail identically zero. It follows, therefore, that covariant derivatives 
then reduce to ordinary partial derivatives. Such derivatives were 
proved to be tensors relative to these frames in section 11. 

42. The coyariant curvature tensor 

The components of B) k i are not all independent since the tensor is 
skew-symmetric in the indices k, I. In addition, however, if the 
affinity is symmetric, it is easily verified from equation (37.21) that 

B) kl +B i klj +B i lJk = o. ( 42 -D 

If the affinity is metrical, by lowering the contravariant index of the 
Riemann-Christoffel tensor, a completely covariant curvature tensor 
Byki is derived. This has a number of symmetry properties, one of 
which is obtained from our last equation immediately by lowering 
the index i throughout to give 

By k i+B ikIJ +B m = 0. (42.2) 

Further such properties can be established by first calculating a par- 
ticular expression for this covariant tensor. 
From equation (37.21), it follows that 



'ijkl 



= ^[{A}{//}-(r S /}{A}+^{//}-£/{A}]- (42.3) 



k =[ik,s\+[sk,il (42.6) 



122 TENSOR CALCULUS AND RELATIVITY 

Multiplying equation (41 .9) through by g rs and summing with respect 
to s, we obtain the result 

gr.U'j) = KW,k\ = [& r]. (42.4) 

I.e. lowering an index of the symbol of the second kind, yields the 
symbol of the first kind. Equation (42.3) is therefore equivalent to 

B uk i = [^,/]{//}-[/-/,/]{A} + ^[g f -,{/,}]- 

-^^Wl-^W+^IA). (42.5) 
Now, from equation (41.7), it follows that 

dx 
and hence 

B m = [rkM/d-irlM/^+^Vl^—^Kil- 
- i/im. si + [sfe, i]) + {/*}([//, s] + [si, /]), 

= U d2 8li *gjk *gjl *8ki \ 
2\dx J 8x k Bx'dx 1 dx t dx k Bx J dx l J 

+gsr{nWk}-gsr{i r k){fi}. (42.7) 

It is now clear that B iJk i = - B Jik n (42.8) 

*M = -Byik, (42.9) 

B tjM = Bk W . (42.10) 

Equations (42.8), (42.9) indicate that B y ki is skew-symmetric in its 
first and final pair of indices. 

Also, lowering the superscript i throughout the Bianchi identity 
(38.13), we obtain 

By kl; m + Byim . k + By mk ,1=0. (42. 1 1) 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 123 

43. Divergence. The Laplacian. Einstein's tensor 

If the covariant derivative of a tensor field is found and then con- 
tracted with respect to the index of differentiation and any superscript, 
the result is called a divergence of the tensor. With respect to ortho- 
gonal coordinate transformations in & N , the partial and covariant 
derivatives are identical and then this definition of divergence agrees 
with that given in section 12. 
From the tensor A ij k , two divergences can be formed, viz. 

diViA ij k = A u k;i and diVjA iJ k = A iJ k;J . (43.1) 

A contravariant vector possesses one divergence only, which is an 
invariant. If the affinity is the metrical one, such a divergence is simply 
expressed in terms of ordinary partial derivatives thus: Since a 
derivative of a determinant can be found by differentiating each row 
separately and summing the results, we deduce that 

* G*^J = «*%- (43.2) 

8x J 8x J ** 8x 3 

Substituting for 8g ik /8x J from equation (42.6), this reduces to 

jjp = «*<W.*] + R/.A> = 2g{/j}. (43.3) 

Hence {/,} = ~jWg)- (43.4) 

Now let A i be a vector field. Its divergence is 

y/g8x l 
which is the expression required. 



124 TENSOR CALCULUS AND RELATIVITY 

In particular, if the vector field is obtained from an invariant Fby 
taking its gradient, we have 

8V 
A, = -. (43.6) 

..8V 
and hence A 1 = g lJ — -,• (43.7) 

8x J 

From equation (43.5), it now follows that the divergence of this vector 

is 

1 8 I ..8V\ 
divgrad V = W = Vg ^,[V gg " jp) • (43.8) 

The right-hand member of this equation represents the form taken by 
the Laplacian of Fin a general Riemannian space. In £ N , employing 
rectangular axes, the equations (39.1 1) are valid and 

v2y =£h- (439 > 

which is its familiar form. 

We shall now calculate the divergence of the Ricci tensor R Jk 
(equation (37.23)). 

If the metric affinity is being employed, this tensor is symmetric 
for 

&kj = B kji = S" B rkji = g" Bji r k = g ir B ijkr 

= B r j kr = R Jk , (43.10) 

having employed equations (42.8)-(42.10). Raising either index ac- 
cordingly yields the same mixed tensor Rfc. If this is contracted, an 
invariant 

R = Rj (43.11) 

is obtained. R is called the curvature scalar of &t N . 
The divergence of the Ricci tensor is 

= *"V***fif ; «. (43.12) 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 125 

In view of equation (42.10), the Bianchi identity (42. 1 1) can be written 
in the form 

Bklij;m + B lmij;k + B mkij;l ~ (43.13) 

and it follows that 

R?\m ~ —g m 'g J (Blmij;k + Bmkij;d> 
= —g™'^ (Bmlji;k — B m kji;i), 

= —g J (B [ji . k — B'kji ; /), 

= -^(Rij.^-Rkjj), 

= -R k . k +RJ ;l . (43.14) 

Thus */% = iR;' : / = ^ (43.15) 

2 ox 

is the divergence of the Ricci tensor. 
Consider now the mixed tensor 

$-i8}R. (43.16) 

dR 
Its divergence is R) ; ,- - \h) — . , 

= 0. (43.17) 

This is Einstein's Tensor. Its covariant and contravariant components 
are 

Ry-teoR, R ij -W j R, (43.18) 

respectively. Upon contraction, it yields the invariant 

R-$NR= -KN-2)R. (43.19) 

44. Geodesies 

Let C be any curve constructed in a space & N having metric (39. 1) and 
let s be a parameter defined on C such that, if s, s + ds are its values at 



126 TENSOR CALCULUS AND RELATIVITY 

the respective neighbouring points P, P' on C, then ds is the interval 
between these two points. If x l are the coordinates of any point P on 
C, then the curve will be defined by parametric equations 

x l = x*(s). (44.1) 

Since dx { are the components of a vector and ds is an invariant, 

dx'lds is a contravariant vector at P. Its magnitude is, by equation 

(40.1), 

dx'dxtV 12 

(44.2) 



/ aW<W 
\ gij ds ds 



ds) 

and this is unity by equation (39.1). dx'/ds is termed the unit tangent 
to the curve at P, its direction being that of the displacement dx l along 
the curve from P. 

Suppose C possesses the property that the tangents at all its points 
are parallel, i.e. the curve's direction is constant over its whole length. 
This property is clearly quite independent of the coordinate frame 
being employed. In ^ 3 , such a curve would, of course, be a straight 
line. In 9t Ni the curve will be called a geodesic. A geodesic is accord- 
ingly the counterpart of the Euclidean straight line in a Riemannian 
space. Suppose P, P' are neighbouring points on a geodesic having 
coordinates jc', x l + dx l respectively. If the unit tangent at P is parallel 
displaced to P ', it will then be identical with the actual unit tangent at 
this point. Now, by equation (35.4), after parallel displacement from 
P to P\ the unit tangent has components 

£♦■©-£-*£**• <-> 

But the actual unit tangent for the point P' has components 

tbt_ 
K+ds ds^ ds 2 
The vectors (44.3), (44.4) are identical provided 

dx^dx^ 

~ds T ' r ^ Jk ds ds 



fdx \ ax ax , ,, M JV 

*)- -" + -^ A (44 ' 4) 



+ rj k — — = 0. (44.5) 



If these equations are satisfied at every point of the curve (44.1), it is 
a geodesic. We shall assume, in future, that the affinity is metrical. 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 127 

The N equations (44.5) are second order differential equations for 
the functions x { (s) and their solution will involve 2N arbitrary con- 
stants. If A, B are two given points having coordinates x l = a', x' = b l 
respectively, the 2N conditions that the geodesic must contain these 
points will, in general, determine the arbitrary constants. Hence there 
is, in general, a unique geodesic connecting every pair of points. 
However, in some cases, this will not be so. For example, the geodesies 
on the surface of a sphere (^2) are great circles and, in general, there 
are two great circle arcs joining two given points, a major arc and a 
minor arc. Also, if these points are diametrically opposed to one 
another, there is an infinity of great circle arcs connecting them. 

Since dx'/ds is everywhere a unit vector, on a geodesic 

dx i dx i 

This must, accordingly, be a first integral of the equations (44.5). To 
show that this is the case, multiply equations (44.5) through by 
2g ir dx r /ds and sum with respect to / to obtain 

„ dx r d 1 x i „ rt dx J dx k dx r n , AArn 

2g »*^ +2g » r *-d;-d7* =0 - (44 - 7) 

„ dx r d 2 X l dl dx l dx r \ dgiMdXr , iAo ^ 

Now 2gi 'Ts^ - *(** *)"* ** ' (44 - 8) 

„ -mi dx 1 dx h dx r /%r , ,dx J dx k dx r 

*■" 2g - p ' k * ■* * = m - r] * * * • 

dx J dx k dx r 

_ 8g Jr dx k dx J dx r 
~ dx k ds ds ds' 

= dgjrdxJdS 

ds ds ds 



128 TENSOR CALCULUS AND RELATIVITY 

By addition of equations (44.8) and (44.9), it will be seen that 
equation (44.7) can be expressed in the form 



/ dx'dx J \ n 



^Ulf^^l-O. (44.10) 



Upon integration, there results the first integral 

dx i dx i 
git —r—r = constant. (44. 1 1) 

as as 

The constant of integration must, of course, be taken to be unity. 

The definition of a geodesic which has been given at the beginning 
of this section cannot be applied to the class of curves for which the 
interval ds between adjacent points vanishes. For such a curve, the 
parametric representation (44. 1) is not appropriate and a unit tangent 
cannot be defined. Instead, suppose that a (1-1) correspondence is set 
up between the points of the curve and the values of an invariant A in 
some interval Aq < A < A ls so that parametric equations for the curve 
can be written 

x l = x l (X). (44.12) 

It will be assumed that the derivatives dx'/dX all exist at each point of 
the curve. These derivatives constitute a contravariant vector and this 
has zero magnitude for, since ds = along the curve, 

8 ^Tx =0 - (44 ' 13) 

This vector will be in the direction of the displacement vector along 
the curve dx' and will be called a zero tangent to the curve. The curve 
will be termed a null geodesic if the zero tangents at all points of the 
curve are parallel. This implies that, when the zero tangent at P is 
parallel displaced to the adjacent point P', it must be parallel to the 
zero tangent at this latter point, and since the magnitudes of these 
two vectors atP' are the same, they will be taken to be identical. The 
condition for this to be so is found, as before, to be 

d 2 x* . dx J dx k 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 129 

These are, therefore, the equations of the null geodesies. It may now 
be shown, by an argument similar to that culminating in equation 
(44.1 1), that a first integral of these equations is 
dx dx 1 
8v ~dX ~dX = constant - (44.15) 

In this case the constant must be zero. 

Equation (44.5) may be put in an alternative form which is more 
convenient for particular calculations, as follows: Multiply through 
by 2g ri and sum with respect to /; the resulting equation is equivalent 
to 



Now 



and 



dl dx l \ ^dg ri dx' „ rii dx J dx k A ,„„ <^ 

^g^dS = 2 8g L idx^dx^ 
ds ds dx k ds ds 

= (?**+?**)<**? (4417) 

\dx k+ 8x J )ds ds Km } 

2s ,n = \ik A = 8 lH+ d l*- d -*L k . (44.18) 

lgrt*Jk-UK,n 8x k+ 8x j 8x r 



Equation (44.16) accordingly reduces to 

d 
ds 



(*-£)-££?-* ^ 



Exercises 5 

1 . Ay is a covariant tensor. If By = A Jh prove that By is a covariant 
tensor. Deduce that, if Ay is symmetric (or skew-symmetric) in one 
frame, it is symmetric (or skew-symmetric) in all. [Hint : The equations 
Ay = Aj h Ay = — Aj t are tensor equations.] 

2. (x, y, z) are rectangular Cartesian coordinates of a point P in 
^ 3 and (r, 0, <f>) are the corresponding spherical polars related to the 
Cartesians by equations (29.3). A is a contravariant vector defined at 
P having components (A x , A y , A 2 ) in the Cartesian frame and com- 
ponents (A r , A e , A^) in the spherical polar frame. Express the polar 



130 TENSOR CALCULUS AND RELATIVITY 

components in terms of the Cartesian components. Ol, 02, 03 are 
rectangular Cartesian axes such that P lies on 01 and 03 lies in the 
plane Oxy. If (A 1 , A 2 , A 3 ) are the components of A in this Cartesian 
frame, show that 

A 1 = A r , A 2 = rA e , A 3 = rsmdA^. 

[Note: Assume the Cartesian axes are right-handed.] 

3. If A t is a covariant vector, verify that By = A j-A jt t transforms 
like a covariant tensor. (This is curl A.) If A is the gradient of a scalar, 
verify that its curl vanishes. 

4. If Ay is a skew-symmetric covariant tensor, verify that 

B Vk = A ij,k+ A jk,i+ A ki,j 
transforms as a tensor. 

5. gu is the metric tensor of an ^ 3 . If g = \g v \, show that e IJk , e uk , 
where 

e Uk =t ijk fVgf em = V(g) e m , 

are tensors in the ^ 3 . Ay is a skew-symmetric tensor in the space. 
Deduce that \e iik A jk is a contravariant vector whose components are 
A 2s/Vg, A u/Vg, A n/Vg- In particular, taking A v = BiCj-BjQ, 
show that the contravariant vector is the vector product of the co- 
variant vectors B h Q, if the space is Euclidean. (If the space is not 
Euclidean, this is taken to be the definition of the vector product of 
covariant vectors.) 

6. A iJ is a skew-symmetric tensor in M 2 . Show that %e ijk A Jk is a 
covariant vector whose components are A 23 y/g, A 3l Vg, A X2 y/g. 
Hence, define the vector product of two contravariant vectors. 

7. Show that 

A i;J~ A J;i - A u- A j,i 

provided the affinity is symmetric. 

8. Show that 

A i,jk- A i;kj — Bijk-Ar + iTkj— Tj k ) A i;r 

and deduce that B uk is a tensor and that covariant differentiations are 
commutative in a space for which B r ijk = and the affinity is sym- 
metric. Obtain the corresponding result for a contravariant vector A . 

9. A is defined at the point x* and is parallel displaced around a 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 131 

small contour enclosing the point. Prove that the increment in A 
resulting from one circuit is given by 

AA t =* -\B l tjk A ia . Jk t 

where a Jk is defined by equation (37.10). 

10. The parametric equations of a curve in S? N are 

x* = x l (t); 

t is an invariant parameter. A tensor A) is denned over a region con- 
taining the curve. P, P' are neighbouring points t, t+ At on the curve 
and AAJ is denned to be the difference between the actual value of the 
tensor at P ' and the value of the tensor at P after it has been parallel 
displaced to P'. Prove that 

DA, ,. AAj Ai dx k 
— J = hm —r 1 = Aj. k -—~- 
Dt At-*o At at 

(DAJ/Dt is called the intrinsic derivative of the tensor along the curve.) 

1 1 . Verify that {/*}, as defined by equation (41 .9), transforms as an 
affinity. 

12. If Ay is symmetric, prove that A U; k is symmetric in i and./. 

13. Show that the number of the components of B) kl which may be 
assigned values arbitrarily is, in general, $N 3 (N- 1). If the affinity is 
symmetric, show that this number is $N 2 (N 2 -1). [Hint: Use 
equation (42.1).] 

14. Show that the number of the components of B m which may 
be assigned values arbitrarily is N 2 (N 2 - 1)/12. [Hint: Use equations 
(42.2), (42.8), (42.9), (42.10).] 

15. By differentiating the equation 

g u g Jk = si 
with respect to x 1 , show that 

til - -**#**& 
dx l ~ S g dx l 



and hence that 






132 TENSOR CALCULUS AND RELATIVITY 

Deduce that g iJ . k = 0. 

16. If the affinity is the metric one, prove that 

*,* - Bj ki = - A. {AH _f_ log Vg+ 
+ { r i k }{j r i}-{j r k}-^MVg, 

Ski = Biki — 0. 

[Hint: Employ equation (43.4).] Deduce that R jk is symmetric. 

17. If 0, <f> are co-latitude and longitude respectively on the surface 
of a sphere of unit radius, obtain the metric 

ds 2 = d0 2 +sin 2 0d<f> 2 

for the surface. Show that the only non-vanishing three index symbols 
for this St 2 are 

{2*2} = -sin^ cos 0, U 2 2 } = { 2 2 i> = cot^. 
Show also that the only non-vanishing components of B ijkl are 

^1212 = —-^1221 — -52121 = —#2112 = sin 
and that the components of the Ricci tensor are given by 
R n = R 2l = 0, R n = -1, R22 = -sin 2 0. 

Prove that the curvature scalar is given by R = — 2. 

18. Employing equation (43.8), obtain expressions for V 2 V in 
cylindrical and spherical polars. 

19. In a certain coordinate system 

where <f>, tft are functions of position. Prove that Bj k i is a function of 
tp only. If ^ = — logfa,* 1 ) prove that 

R jk = Bjki ~ 0- 



20. In the 0t 2 whose metric is 



(L.U.) 



,2 dr 2 +t*d8 2 r 2 dr 2 
ds = S-a 2 -(?=??- (r>a) > 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 133 

prove that the differential equation of the geodesies may be written 

where k 2 is a constant such that k 2 = 1 if, and only if, the geodesic is 
null. By putting rdd/dr - tan<£, show that if the space is mapped on a 
Euclidean plane in which r, 6 are taken as polar coordinates, the 
geodesies are mapped as straight lines, the null-geodesies being 
tangents to the circle r = a. 

(L.U.) 

21. A 2-space has metric 

ds 1 = gn(dx l ) 2 +g 2 2(dx 2 ) 2 . 
Prove that 

! » _ * / 8 I 1 d Sn\M±(A-?*ll\\ 

g Bl212 -~^W[vg^rdx 2 Wgex 2 )j' 

(L.U.) 

22. Prove that 

(i) A»., = -^^(VgA^ + A'HA) 

Vg 8 x 

(ii) ^ ,y ;(,= 0, 

provided X iJ is skew-symmetric. Hence prove that, for any tensor A i} 

(L.U.) 

23. A curve C has parametric equations 

x l = x\t) 
and joins two points A and B. The length of the curve is defined to be 

2? S 

L -J*-jy("!r?)* 

A A 

Write down the Euler conditions that L should be stationary with 
respect to all small variations from C and by changing the indepen- 
dent variable in these conditions from t to s, show that they are 



134 TENSOR CALCULUS AND RELATIVITY 

identical with equations (44.5). (This provides an alternative definition 
for a geodesic.) 

24. If r' jk is a symmetric affinity, show that 

I% = r) k + VjA k +h k Aj 

is also a symmetric affinity. 

If Bj k i, &* kl are the Riemann-Christoffel curvature tensors relative 
to the affinities T) k , Tf k respectively, prove that 

tyki - b m + &k a ji - 8/ A Jk + 8j(A kl -A lk \ 

where Ay = A t Aj—Af.j. 
Hence show that if A t is the gradient of a scalar, then 

(M.T.) 

25. Prove that the affinity transformations form a group. 

26. If D = | ax'/a^l , show that 

i£P _ 8 * J 82 ** 

27. Show that the transformation law for the quantities .£,• 
(equation (36.5)) is 

_ _ 8^ ajc y " 3 2 x* 

and deduce that r r ri —Ki is a tensor. 

28. Oblique cartesian axes are taken in a plane. Show that the 
contravariant components of a vector A can be obtained by project- 
ing a certain displacement vector on to the axes by parallels to the 
axes and the covariant components by projecting by perpendiculars 
to the axes. 

29. Define coordinates (r, 0) on a right circular cone having semi- 
vertical angle a so that the metric for the surface is 

ds 2 = dr 2 +r 2 sm 2 ccd<t> 2 . 

Show that the family of geodesies is given by 

r = asec(<f>sina—p), 



GENERAL TENSOR CALCULUS. RIEMANNIAN SPACE 135 

where a,/? are arbitrary constants. Explain this result by developing 
the cone into a plane. 

30. An 9t N has metric 

ds 2 = e x dx i dx i > 

where A is a function of the x*. Show that the only non-vanishing 
Christoffel symbols of the second kind are 

where A r = dX\dx r . Deduce that 

{j>}{;J = H"+2)4-*A r A r 

and that the scalar curvature of this space is given by 
R = (N-l)e- x [\ rr +W-?)KKl 

where X rr = 8 2 X/dx r dx r . 

31. is the colatitude and <j> is the longitude on a unit sphere, so 
that the metric for the surface is 

ds 2 = dd 2 +sm 2 9d<f> 2 . 

The covariant vector A t is taken with initial components (X, Y) and is 
carried, by parallel displacement, along an arc of length ^sin a of the 
circle 6 = a. Show that the components of A t attain the final values 

Ax = Xcos((f>cosa)+ ycosec a sin Oleosa), 
A 2 = — Xsin a sin Oleosa) + ycos Oleosa). 

Verify that the magnitude of the vector A t is unaltered by the displace- 
ment. 

32. An ^ 3 has metric 

ds 2 = Xdr 2 + r 2 (d6 2 + sin 2 ddcf> 2 ), 

where A is a function of r alone. Show that, along the geodesic for 
which 6 = irr, dd/ds = at s = 0, 

<f> = f X^ 2 diff, 

where r=bsecift. Interpret this result geometrically when A = 1. 



136 TENSOR CALCULUS AND RELATIVITY 

33. / = 1,2,3,4) are rectangular cartesian coordinates in <f 4 . 
Show that 

y 1 = i?cos0, 
y 2 = Rsin9cos<j>, 
y 3 = Rsin 9 sin <f> cos ifi, 
y 4 = Rsindsintfrsimf/, 

are parametric equations of a hypersphere of radius R. If (0, <f>, 0) are 
taken as coordinates on the hypersphere, show that the metric for 
this & 3 is 

ds 2 = R 2 [dd 2 + sin 2 6(d<f> 2 + sin 2 <f>dift 2 )]. 

Deduce that in this ^ 3 , 

B 12l2 = R 2 sin 2 e, 52323 = R 2 sm 4 6sm 2 <f, t 
•^31 31 = R 2 sin 2 0sm 2 <f>, 
all other distinct components being zero. Hence show that 

B mi = K(Sikgji-guZjk) 

where K = l/R 2 . (This is the condition for the space to be of constant 
Riemannian curvature K.) 

34. An ^ 2 has metric 

ds 2 = sech 2 y(dx 2 + dy 2 ). 

Find the equation of the family of geodesies. 

9, <f> are colatitude and longitude respectively on the surface of a 
sphere of unit radius. Mercator's projection is obtained by plotting 
x,y as rectangular cartesian coordinates in a plane, taking 

x = <f>, y = log cot i9. 

Calculate the metric for the spherical surface in terms of x and y 
and deduce that the great circles are represented by the curves 

sinh^ = asin(*+)8), 

where a, /? are parameters, in Mercator's projection. 



CHAPTER 6 

General Theory of Relativity 

45. Principle of equivalence 

The special theory of relativity rejects the Newtonian concept of a 
privileged observer, at rest in absolute space, and for whom physical 
laws assume their simplest form and assumes, instead, that these laws 
will be identical for all members of a class of inertial observers in 
uniform translatory motion relative to one another. Thus, although 
the existence of a single privileged observer is denied, the existence of 
a class of such observers is accepted. This seems to imply that, if 
all matter in the universe were annihilated except for a single experi- 
menter and his laboratory, this observer would, nonetheless, be able 
to distinguish inertial frames from non-inertial frames by the special 
simplicity which the descriptions of physical phenomena take with 
respect to the former. The further implication is, therefore, that 
physical space is not simply a mathematical abstraction which it is 
convenient to employ when considering distance relationships be- 
tween material bodies, but exists in its own right as a separate entity 
with sufficient internal structure to permit the definition of inertial 
frames. However, all the available evidence suggests that physical 
space cannot be defined except in terms of distance measurements 
between physical bodies. For example, such a space can be constructed 
by setting up a rectangular Cartesian coordinate frame comprising 
three mutually perpendicular rigid rods and then defining the co- 
ordinates of the point occupied by a material particle by distance 
measurements from these rods in the usual way. Physical space is, 
then, nothing more than the aggregate of all possible coordinate 
frames. A claim that physical space exists independently of distance 
measurements between material bodies, can only be substantiated if 
a precise statement is given of the manner in which its existence can 
be detected without carrying out such measurements. This has never 
been done and we shall assume, therefore, that the special properties 
possessed by inertial frames must be related in some way to the 

137 



138 TENSOR CALCULUS AND RELATIVITY 

distribution of matter within the universe and that they are not an 
indication of an inherent structure possessed by physical space when 
it is considered apart from the matter it contains. This line of argument 
encourages us to expect, therefore, that, ultimately, all physical laws 
will be expressible in forms which are quite independent of any co- 
ordinate frame by which physical space is defined, i.e. that physical 
laws are identical for all observers. This is the general principle of 
relativity. This does not mean that, when account is taken of the 
actual distribution of matter within the universe, certain frames will 
not prove to be more convenient than others. When calculating the 
field due to a distribution of electric charge, it simplifies the calcula- 
tions enormously if a reference frame can be employed relative to 
which the charge is wholly at rest. However, this does not mean that 
the laws of electromagnetism are expressible more simply in this 
frame, but only that this particular charge distribution is then des- 
cribed more simply. Similarly, we shall attribute the simpler forms 
taken by some calculations when carried out in inertial frames, to the 
special relationship these frames bear to the matter present in the 
universe. Fundamentally, therefore, all observers will be regarded as 
equivalent and, by employing the same physical laws, will arrive at 
identical conclusions concerning the development of any physical 
system. 

The main difficulty which arises when we try to express physical 
laws so that they are valid for all observers is that, if test particles are 
released and their motions studied from a frame which is being 
accelerated with respect to an inertial frame, these motions will not 
be uniform and this fact appears to set such frames apart from in- 
ertial frames as a special class for which the ordinary laws of motion 
do not apply. However, by a well-known device of Newtonian 
mechanics, viz. the introduction of inertial forces, accelerated frames 
can be treated as though they were inertial and this suggests a way out 
of our difficulty. Thus suppose a space rocket, moving in vacuo, is 
being accelerated uniformly by the action of its motors. An observer 
inside the rocket will note that unsupported particles experience an 
acceleration parallel to the axis of the rocket. Knowing that the 
motors are operating, he will attribute this acceleration to the fact 
that his natural reference frame is being accelerated relative to an 



GENERAL THEORY OF RELATIVITY 139 

inertial frame. However he may, if he prefers, treat his reference frame 
as inertial and suppose that all bodies within the rocket are being sub- 
jected to inertial forces acting parallel to the rocket's axis. If f is the 
acceleration of the rocket, the appropriate inertial force to be applied 
to a particle of mass m is - wf. Similarly, if the rocket's motors are 
shut down but the rocket is spinning about its axis, an observer within 
the rocket will again note that free particles do not move uniformly 
relative to his surroundings and he may again avoid attributing this 
phenomenon to the fact that his frame is not inertial, by supposing 
certain inertial forces (viz. centrifugal and Coriolis forces) to act upon 
the particles. Now it is an obvious property of each such inertial 
force that it must cause an acceleration which is independent of the 
mass of the body upon which it acts, for the force is always obtained 
by multiplying the body's mass by an acceleration independent of the 
mass. This property it shares with a gravitational force, for this also 
is proportional to the mass of the particle being attracted and hence 
induces an acceleration which is independent of this mass. This inde- 
pendence of the gravitational acceleration of a particle and its mass 
has been checked experimentally with great accuracy by Eotvos. If, 
therefore, we regard the equivalence of inertial and gravitational 
forces as having been established, inertial forces can be thought of as 
arising from the presence of gravitational fields. This is the Principle 
of Equivalence. By this principle, in the case of the uniformly acceler- 
ated rocket, the observer is entitled to neglect his acceleration relative 
to an inertial frame, provided he accepts the existence of a uniform 
gravitational field of intensity -f parallel to the axis of the rocket. 
Similarly, the observer in the rotating rocket may disregard his motion 
and accept, instead, the existence of a gravitational field having 
such a nature as to account for the centrifugal and Coriolis forces. 
By appeal to the principle of equivalence, therefore, an observer 
employing a reference frame in arbitrary motion with respect to an 
inertial frame, may disregard this motion and assume, instead, the 
existence of a gravitational field. The intensity of this field at any point 
within the frame will be equal to the inertial force per unit mass at the 
point. By this device, every observer becomes entitled to treat bis 
reference frame as being at rest and all observers accordingly become 
equivalent. However, the reader is probably still not convinced that 



140 TENSOR CALCULUS AND RELATIVITY 

the distinction between accelerated and inertial frames has been effec- 
tively eliminated, but only that it has been concealed by means of a 
mathematical device having no physical significance. Thus, he may 
point out that the gravitational fields which have been introduced to 
account for the inertial forces are ' fictitious ' fields, which may be com- 
pletely removed by choosing an inertial frame for reference purposes, 
whereas ' real ' fields, such as those due to the Earth and Sun, cannot be 
so removed. He may further object that no physical agency can be held 
responsible for the presence of a * fictitious ' field, whereas a * real ' field 
is caused by the presence of a massive body. These objections may be 
met by attributing such ' fictitious ' fields to the motions of distant mas- 
ses within the universe. Thus, if an observer within the uniformly ac- 
celerated rocket takes himself to be at rest, he must accept as an observ- 
able fact that all bodies within the universe, including the galaxies, 
possess an additional acceleration of — f relative to him and to this 
motion he will be able to attribute the presence of the uniform gravita- 
tional field which is affecting his test particles. Again, the whole uni- 
verse will be in rotation about the observer who regards himself and 
his space-ship as stationary when it is in rotation relative to an inertial 
frame. It is this rotation of the masses of the universe which we shall 
hold responsible for the Coriolis and centrifugal gravitational fields 
within the rocket. But, in addition, these ' inertial' gravitational fields 
will account for the motions of the galaxies as observed from the non- 
inertial frame. Thus, for the observer within the uniformly accelerated 
rocket a uniform gravitational field of intensity — f extends over the 
whole of space and is the cause of the acceleration of the galaxies ; for 
the observer within the rotating rocket, the resultant of the centrifugal 
and Coriolis fields acting upon the galaxies is just sufficient to account 
for their accelerations in their circular orbits about himself as centre 
(the reader should verify this, employing the results of Exercise 1, 
Chapter 1). On this view, therefore, inertial frames possess particu- 
larly simple properties only because of their special relationship to 
the distribution of mass within the universe. In much the same way, 
the electromagnetic field due to a distribution of electric charge takes 
an especially simple form when described relative to a frame in which 
all the charges are at rest (assuming such exists). If any other frame is 
employed, the field will be complicated by the presence of a magnetic 



GENERAL THEORY OF RELATIVITY 141 

component arising from the motions of the charges. However, this 
magnetic field is not considered imaginary because a frame can be 
found in which it vanishes, whereas for certain magnetic fields such a 
frame cannot be found. The laws of electromagnetism are taken to be 
valid in all frames, though it is conceded that, for solving particular 
problems, a certain frame may prove to be pre-eminently more con- 
venient than any other. Neither, therefore, should the centrifugal and 
Coriolis fields be dismissed as imaginary solely because they can be 
removed by proper choice of a reference frame, although it may be 
convenient to make such a choice of frame when carrying out par- 
ticular computations. In short, the general principle of relativity can 
be accepted as valid and, at the same time, the existence of the inertial 
frames accounted for by the simplicity of the motions of the galactic 
masses with respect to these particular frames. 

If it is accepted that the existence of inertial frames is bound up with 
the large-scale distribution of matter within the universe, then it fol- 
lows that the inertia possessed by a body, which causes it to move 
uniformly in such a frame, is also a consequence of this matter 
distribution. This is Mack's Principle, viz. that mass is induced in a 
body by the presence of distant matter in the universe. 

The previously unexplained identity of inertial and gravitational 
masses is easily deduced as a consequence of the principle of equiv- 
alence. For, consider a particle of mass m which is being observed 
from a non-inertial frame. A gravitational force equal to the inertial 
force will be observed to act upon this body. This force is directly pro- 
portional to the inertial mass m. But, by the principle of equivalence, 
all gravitational forces are of the same nature as this particular force 
and will, accordingly, be directly proportional to the inertial masses of 
the bodies upon which they act. Thus the gravitational 'charge' of a 
particle, measuring its susceptibility to the influences of gravitational 
fields, is identical with its inertial mass and the identity of inertial and 
gravitational masses has been explained in a straightforward and 
convincing manner. 

46. Metric in a gravitational field 

Suppose that an inertial frame has been established in a region of 
space where there are no local gravitational influences and that a 



142 TENSOR CALCULUS AND RELATIVITY 

material plane is set rotating with angular velocity <a relative to the 
frame about an axis perpendicular to it and fixed in the inertial frame. 
An observer O, moving with the plane, is entitled to consider himself as 
at rest in the presence of a gravitational field which will account for the 
centrifugal and Coriolis forces. Suppose O identifies a large number of 
points on the plane which he measures with a standard rod to be at the 
same distance r from the axis of rotation and, by this means, constructs 
a circle. If, now, O lays his measuring rod repeatedly along a radius of 
this circle to measure its length r and this operation is watched by an 
observer O' who is stationary in the inertial frame, O' will agree with 
the length found by O, for the rod will only move laterally during the 
process of measurement and hence will suffer no Fitzgerald contrac- 
tion of its length. O' will therefore agree with O that all the radii are 
of equal length and hence that the figure constructed is a circle. How- 
ever, if O now lays his rod along the circumference of the circle (we 
assume the dimensions of the rod are small by comparison with those 
of the circle) and measures its length to be /, O' will disagree with this 
measurement for the reason that, for him, throughout the measuring 
process the rod will have a speed tar along its length and hence will 
have contracted by the factor (1 — a^r^/c 2 ) 112 . Allowing for this con- 
traction, O' will assert the length of the circumference to be 

/(l-o> 2 /-V) I/2 . (46.1) 

But O' is employing an inertial frame in which the geometry is known 
to be Euclidean. It follows that 

1(1 - o>V/c 2 ) 1/2 = 2w (46.2) 

2rrr 

and hence that / = - - ,, ?v m . (46.3) 

(1— to z r/c ) ' 

This last equation implies that for O the ratio l/2r is not it, but a 
number greater than it and the larger the radius of his constructed 
circle, the larger this ratio will become. By direct measurement, there- 
fore, O will be able to ascertain that the geometry of figures drawn on 
the plane is not Euclidean. We conclude that, in the presence of a 
gravitational field of the centrifugal-Coriolis type, the geometry of 
space is not Euclidean. 



GENERAL THEORY OF RELATIVITY 143 

By the principle of equivalence, the conclusion which has just been 
reached concerning the non-Euclidean nature of space in which there 
is present a gravitational field of the centrifugal-Coriolis type, must 
be extended to all gravitational fields. However, in the case of a field 
such as that which surrounds the Earth, it will not be possible (as it is 
for the centrifugal-Coriolis field) to find an inertial frame of reference 
relative to which the field vanishes and for which the spatial geometry 
is Euclidean. Such a field will be termed irreducible. Even in an irre- 
ducible field, however, a frame can always be found which is inertial 
for a sufficiently small region of space and a sufficiently small time 
duration. Thus, within a space-ship which is not rotating relative to 
the extra-galactic nebulae and which is falling freely in the Earth's 
gravitational field, free particles will follow straight line paths at con- 
stant speed for considerable periods of time and the conditions will be 
inertial. A coordinate frame fixed in the ship will accordingly simulate 
an inertial frame over a restricted region of space and time and its 
geometry will be approximately Euclidean. 

Since a rectangular Cartesian coordinate frame can be set up only 
in a space possessing a Euclidean metric, this method of specifying 
the relative positions of events must be abandoned in an irreducible 
gravitational field (except over small regions as has just been ex- 
plained). Instead, it will be assumed that each physical event is 
allotted three space coordinates (£*, £ 2 , f 3 ) according to any con- 
venient scheme. It will be necessary in any particular case only to 
describe the procedure whereby these coordinates are to be established 
for any event. They might, for example, be determined by radar tech- 
niques. Thus radar transmitters could, in principle, be set up at three 
widely separated points and the electromagnetic pulses generated by 
them reflected back to their sources by the event in question. The time 
intervals between transmission of a pulse and reception of its 'echo' 
at the three stations, would then be suitable coordinates of the type 
we are considering for the event. All that is necessary is that there 
should exist a correspondence between points in space and triads of 
real numbers (£\ £ 2 , £ 3 ) such that, to each point there corresponds a 
unique triad, and to each triad there corresponds a unique point. 
Again, clocks will be supposed distributed throughout space so that 
to each event a time f 4 can be allocated, namely, the time indicated by 



144 TENSOR CALCULUS AND RELATIVITY 

the clock situated at the event. It will be assumed that, whatever system 
is adopted for fixing the coordinates $ , a series of contiguous events, 
such as the positions occupied by a particle at successive instants of 
time, will have coordinates which will vary continuously along the 
series and which will correspond to a continuous curve in £-space. 
This implies that adjacent clocks must be synchronized and must run 
at the same not necessarily constant rate, but no special procedure 
for the synchronization of distant clocks need be laid down. It follows 
that, as in special relativity theory, two distant events may be simul- 
taneous according to one system of time reckoning, but not so 
according to another. 

We shall now further generalize the coordinates allocated to an 
event. Let x' (/* = 1 , 2, 3, 4) be any functions of the £' such that, to each 
set of values of the £' there corresponds one set of values of the x , and 
conversely. We shall write 

x* = At 1 ,? 2 ,?,?). (46.4) 

Then the x\ also, will be accepted as coordinates, with respect to a 
new frame of reference, of the event whose coordinates were previ- 
ously taken to be the £ . It should be noted that, in general, each of 
the new coordinates x l will depend upon both the time and the position 
of the event, i.e., it will not necessarily be the case that three of the 
coordinates x' are spatial in nature and one is temporal. All possible 
events will now be mapped upon a space S? 4 , so that each event is 
represented by a point of the space and the x* will be the coordinates 
of this point with respect to a coordinate frame. y 4 will be referred to 
as the space-time continuum. 

It has been remarked that, in any gravitational field, it is always 
possible to define a frame relative to which the field vanishes over a 
restricted region and which behaves as an inertial frame for events 
occurring in this region and extending over a small interval of time. 
Suppose, then, that such an inertial frame S is found for two con- 
tiguous events. Any other frame in uniform motion relative to S will 
also be inertial for these events. Observers at rest in all such frames 
will be able to construct rectangular Cartesian axes and synchronize 
clocks in the manner described in Chapter 1 and hence measure the 
proper time interval dr between the events. If, for one such observer, 



GENERAL THEORY OF RELATIVITY 145 

the events at the points having rectangular Cartesian coordinates 
(x,y,z), (x+dx,y+dy,z+dz) occur at the times t,t+dt respectively, 
then 

dr 2 = dt 2 --.(dx 2 +dy 2 +dz 2 ). (46.5) 

cr 

The interval between the events ds will be defined by 

ds 1 = -JdT 2 = dx 2 +dy 2 +dz 2 -c 2 dt\ (46.6) 

The coordinates (x,y, z, t) of an event in this quasi-inertial frame, will 
be related to the coordinates x* defined earlier, by equations 

x = x{x x ,x 2 ,x i t x\t\c. (46.7) 

dx 
and hence dx = —idx', etc. (46.8) 

ex' 

Substituting for dx, dy, dz, dt in equation (46.6), we obtain the result 

ds 2 = g v dx f dx J , (46.9) 

determining the interval ds between two events contiguous in space- 
time, relative to a general coordinate frame valid for the whole of 
space-time. The space-time continuum can accordingly be treated as a 
Riemannian space with metric given by equation (46.9). 

47. Motion of a free particle in a gravitational field 

In a region of space which is at a great distance from material bodies, 
rectangular Cartesian axes Oxyz can be found constituting an inertial 
frame. If time is measured by clocks synchronized within this frame 
and moving with it, the motion of a freely moving test particle relative 
to the frame will be uniform. Thus, if (x,y, z) is the position of such a 
particle at time /, its equations of motion can be written 

£± = €2 = €i = o (47 1) 

dt 2 dt 2 dt 2 u - K *' A) 

Let ds be the interval between the event of the particle arriving at the 
point (x,y, z) at time t and the contiguous event of the particle arriving 



146 TENSOR CALCULUS AND RELATIVITY 

at (x+dx,y + dy>z+dz) at t+dt. Then ds is given by equation (46.6) 
and, if v is the speed of the particle, it follows from this equation that 

ds = (v 2 -c 2 ) ll2 dt. (47.2) 

Since v is constant, it now follows that equations (47.1) can be ex- 
pressed in the form 

d 2 x d 2 y d 2 z 

*? = rf? = ^ = °- (47 - 3) 

Also, from equation (47.2), it may be deduced that 

d 2 t 

- 2 = 0. (47.4) 

Equations (47.3) and (47.4) determine the family of world-lines of free 
particles in space-time relative to an inertial frame. 

Now suppose that any other reference frame and procedure for 
measuring time is adopted in this region of space, e.g. a frame which 
is in uniform rotation with respect to an inertial frame might be 
employed. Let (x l ,x 2 ,x 3 ,x 4 ) be the coordinates of an event in this 
frame. The interval between two contiguous events will then be given 
by equation (46.9). If an observer using this frame releases a test 
particle and observes its motion relative to the frame, he will note that 
it is not uniform or even rectilinear and will be able to account for this 
fact by assuming the presence of a gravitational field. He will find that 
the particle's equations of motion are 

d 2 x l ri dx J dx k n , Artry 

^ +{ ;*>-^ = °- (47 - 5) 

This must be the case for, as shown in section 44, this is a tensor 
equation defining a geodesic and valid in every frame if it is valid in 
one. But, in the xyzt-frame, the g {J are all constant and the three 
index symbols vanish. Hence, in this frame, the equations (47.5) 
reduce to the equations (47.3), (47.4) and these are known to be true 
for the particle's motion. We have shown, therefore, that the effect 
of a gravitational field of the reducible variety upon the motion of a 
test particle can be allowed for when the form taken by the metric 



GENERAL THEORY OF RELATIVITY 147 

tensor gy of the space-time manifold is known relative to the frame 
being employed. This means that theg v determine, and are determined 
by, the gravitational field. 

The ideas of the previous paragraph will now be extended to regions 
of space where irreducible gravitational fields are present. It has been 
pointed out that, for any sufficiently small region of such space and 
interval of time, an inertial frame can be found and consequently the 
paths of freely moving particles will be governed in such a small region 
by equations (47. 5). It will now be assumed that these are the equations 
of motion of free particles without any restriction, i.e. that the world- 
line of a free particle is a geodesic for the space-time manifold or that 
the world-line of a free particle has constant direction. This appears 
to be the natural generalization of the Galilean law of inertia whereby, 
even in an irreducible gravitational field, a particle's trajectory 
through space-time is the straightest possible after consideration has 
been given to the intrinsic curvature of the continuum. It will then 
follow that the motions of particles falling freely in any gravitational 
field can be determined relative to any frame when the components 
g (j of the metric tensor for this frame are known. Thus the gy will 
always specify the gravitational field observed to be present in a frame 
and the only distinction between irreducible and reducible fields will 
be that, for the latter it will be possible to find a coordinate frame in 
space-time for which the metric tensor has all its components zero 
except 

£11 = S22 = £33 = 1 #44 = ~c\ (47.6) 

whereas for the former this will not be possible. 

48. Einstein's law of gravitation 

According to Newtonian ideas, the gravitational field which exists in 
any region of space is determined by the distribution of matter. This 
suggests that the metric tensor of the space-time manifold, which has 
been shown to be closely related to the observed gravitational field, 
should be calculable when the matter distribution throughout space- 
time is known. We first look, therefore, for a tensor quantity describ- 
ing this matter distribution with respect to any frame in space-time 
and then attempt to relate this to the metric tensor. The energy- 
momentum tensor Ty, defined in Chapter 4 with respect to an inertial 



148 TENSOR CALCULUS AND RELATIVITY 

frame, immediately suggests itself. Both matter and electromagnetic 
energy contribute to the components of this tensor but since, accord- 
ing to the special theory, mass and energy are basically identical, it is 
to be expected that all forms of energy, including the electromagnetic 
variety, will contribute to the gravitational field. 

Since the energy-momentum tensor has been defined in inertial 
frames only, this definition must now be extended to apply to a gen- 
eral coordinate frame in space-time. This can be carried out thus: A 
rectangular Cartesian inertial frame can be established for the neigh- 
bourhood of any point of a gravitational field and valid for a short 
time duration. Corresponding to this frame and its associated clocks, 
rectangular Cartesian axes Oy l y 2 y 3 y 4 can be constructed in the 
region surrounding the corresponding point P of space-time. The 
transformation equations relating the coordinates y' of an event to its 
coordinates x' with respect to any other coordinate frame can now be 
found. Then, if 7^ 0) are the components of the energy-momentum 
tensor in the y-framt at the point P, its components in the *-frame at 
this point can be determined from the appropriate tensor transforma- 
tion equations. Thus, the covariant energy-momentum tensor will 
have components T tJ in the x-frame given by 

Since covariant and contravariant tensors are indistinguishable with 
respect to rectangular Cartesian axes, T^P can also be taken to be the 
components of a contravariant tensor in the ,y-frame and the com- 
ponents of this tensor in the x-frame will then be given by the equation 

T " - %%**■ < 48 - 2 > 

Similarly, the components of the mixed energy-momentum tensor are 
given by 

v-pl? 7 *- (483 > 

These transformations can be carried out at every point of space-time, 



GENERAL THEORY OF RELATIVITY 149 

thus generating for the x-frame an energy-momentum tensor field 
throughout the continuum. 
Consider the tensor equation 

T v .j = 0. (48.4) 

Expressed in terms of the coordinates y l at any point of space-time, 
this simplifies to 

If] = 0, (48.5) 

which is equation (28 .18). Being valid in one frame, therefore, equation 
(48.4) is true for all frames. Thus, the divergence of the energy- 
momentum tensor vanishes. If, therefore, this tensor is to be related to 
the metric tensor g v , the relationship should be of such a form that it 
implies equation (48.4). Now 

g iJ ,k = 0, (48.6) 
by equation (41.12) and hence, a fortiori, 

g°.j = 0. (48.7) 

The law T u = Xg iJ , (48.8) 

where A is a universal constant, would accordingly be satisfactory in 
this respect. However, over a region in which matter and energy were 
absent so that T IJ = 0, this would imply that 

g iJ = 0, (48.9) 

which is clearly incorrect. Further, according to Newtonian theory, 
if jm is the density of matter, the gravitational field can be derived from 
a potential function V satisfying the equation 

V 2 K=47ryfi, (48.10) 

where y is the gravitational constant. The new law of gravitation 
which is being sought must include equation (48.10) as an approxi- 
mation. But, as appears from equation (28.12), T44 involves /j. and it 
seems reasonable, therefore, to expect that the other member of the 
equation expressing the new law of gravitation will provide terms 
which can receive an approximate interpretation as V 2 V. This implies 



150 TENSOR CALCULUS AND RELATIVITY 

that second order derivatives of the metric tensor components will 
probably be present. We therefore have a requirement for a second 
rank contravariant symmetric tensor involving second order deriva- 
tives of the g tJ and of vanishing divergence to which T^can be assumed 
proportional. Einstein's tensor (43.18) possesses these characteristics 
and consequently we shall put 

R u -lg v R= -kTV, (48.11) 

where k is a constant of proportionality which must be related to y 
and which we shall later prove to be positive. Equation (48.11) 
expresses Einstein's Law of Gravitation; by lowering the indices suc- 
cessively, it may be expressed in the two alternative forms 

Rj-tfjR** -kT}, (48.12) 

Ry-ig v R^ -kT v . (48.13) 

If equation (48.12) is contracted, it is found that 

R = kT, (48.14) 

where T= T\. It now follows that Einstein's law of gravitation can 
also be expressed in the form 

Ry = "(.teijT-Tij), (48.15) 

with two other forms obtained by raising subscripts. 

Since the divergence of g iJ vanishes, a possible alternative to the 
law (48.11) is 

R.V- Jg y R + Xg u = - kT", (48.16) 

where A is a constant. The law (48.1 1) gives results which agree with 
observation over regions of space of galactic dimensions, so that it is 
certain that, even if A is not zero, it is exceedingly small. However, the 
extra term has entered into some cosmological investigations. It will 
be disregarded in later sections of this book. 

49. Acceleration of a particle in a weak gravitational field 

In a gravitational field, such as the one due to the Earth, the geometry 
of space is not Euclidean and no truly inertial frame exists. In spite of 
this, we experience no practical difficulty in establishing rectangular 



GENERAL THEORY OF RELATIVITY 151 

Cartesian axes Oxyz at the Earth's surface relative to which for all 
practical purposes the geometry is Euclidean and the behaviour of 
electromagnetic systems is indistinguishable from their behaviour in 
an inertial frame. It must be concluded, therefore, that such a gravi- 
tational field is comparatively weak and hence that, with respect to 
such axes and their associated clocks, the space-time metric will not 
differ greatly from that given by equation (46.6). Putting 

x l = x, **-* * 3 = z, x A = ici, (49.1) 

in terms of the x l the metric will be given by 

ds 2 = dx'dx 1 (49.2) 

approximately. With respect to the x'-frame, it will accordingly be 
assumed that 

gy = 8y + hy, (49.3) 

where the 8 y are Kronecker deltas and the hy are small by comparison. 
Consider a particle moving in a weak gravitational field whose 
metric tensor is given by equation (49.3). The contravariant metric 
tensor will be given by an equation of the form 

gV = 8 y +jfc y , (49.4) 

where the k v are of the same order of smallness as the hy. Then, since 

r ««_!/!k + !*«-£W (49 5) 

lii ' k *-l\dx i + dx> dx k )' ^* 5; 

it follows that, to a first approximation, 

The equations of motion of the particle can now be written down as 
at (47.5). 
By equation (47.2) 

& = d A d 4 = ( W 2- C 2)-1/2( V , /C)> (49.7) 

ds dt as 

where v is the particle's velocity in the quasi-inertial frame. Hence, 



152 TENSOR CALCULUS AND RELATIVITY 

if the particle is stationary in the frame at the instant under con- 
sideration, 

dx l 

^- = «U). (49.8) 

and the equations of motion (47.5) reduce to the form 

d 2 x* 

-^ + {4 / 4> = 0, (49.9) 

correct to the first order in the h v . Substituting from equation (49.6), 
this is seen to be equivalent to 

d 2 x' ldhu 8h 4i 

**-2-5?"S?' (4910 > 

Differentiating equation (49.7) with respect to s and making use of 
equation (47.2), we obtain 

£g = ( „W)-'g,o)-„f (o >-cV 2 <v,fc) (49.11) 

and, when v = 0, this reduces to 

d 2 x l 1 Ids \ 

!?--?[*>*)' (49J2 > 

From equations (49.10), (49.12), we deduce that the components of 
the acceleration of the stationary particle in the directions of the 
rectangular axes are 

-«W-i?) (4913 > 

for /= 1, 2, 3. Reverting to the original coordinates (x,y,z,t), these 
components are written 

-^l2^ + cir)- ete - (49 - 14 > 

Hence, if the field does not vary with the time, the acceleration vector 
is 

-grad(ic 2 /t44). (49.15) 



GENERAL THEORY OF RELATIVITY 153 

But, if Fis the Newtonian potential function for the field, this accelera- 
tion will be -grad V. It follows that, for a weak field, a Newtonian 
scalar potential Kexists and is related to the space-time metric by the 
equation 

V =? lc 2 h 44 . (49.16) 

Alternatively, we can write 

2V 

*44=l + -r < 49 - 17 ) 

c 

50. Newton's law of gravitation 

In this section it will be shown that Newton's Law of Gravitation may 
be deduced from Einstein's Law in the normal case when the gravi- 
tational field's intensity is weak and the matter distribution is static. 
First consider the form taken by the Riemann-Christoffel tensor in 
the space-time of a weak field. In the x'-frame, the metric tensor is 
given by equation (49.3) and the Christoffel three-index symbols by 
equation (49.6). If products of the hy are to be neglected, equation 
(37.21) shows that 

^fe-^tO'/H^Oft) (50.1) 

approximately. Hence the Ricci tensor is given by 

8 i d i 

~ 2.'dx lc [dx 1 + ^x~ J ~dx 1 j 

J_±\ e K + 8 ±kiJ_hjk\ 
2dx i \dx k d X J dx'y 

U d 2 h u e 2 h Jk d 2 h u &h ki \ 

2\dx J 8x k dx'dx* 8x i 8x k 8x'dx J j ' J 

In particular, putting j = k = 4, we find that 

If 8 2 h H d 2 h 44 8 2 h i4 \ ,„.. 

Ru = 2(a^ + a7^~ 2 i^a?j (50>3) 



154 TENSOR CALCULUS AND RELATIVITY 

If the matter distribution is static in the quasi-inertial frame being 
employed, the h u will be independent of t and equation (50.3) reduces 
to 

*44 = £V 2 /U4, (50.4) 

where V 2 = d 2 /8x 2 + 8 2 /dy 2 + d 2 /8z 2 . If V is the Newtonian potential 
for the field, equation (49.16) now shows that 



R« = \V 2 V. 



(50.5) 



Assuming that no electromagnetic field is present, the overall energy- 
momentum tensor for the mass distribution will be given in the 
quasi-inertial frame by the equation 

T u = ©<,, (50.6) 

where ©y is determined approximately by equation (28.10). Since the 
distribution is static, its 4-velocity at every point is (0,/c) and hence 
all components of Ty, with the exception of 744, are zero. In this case, 

T u = -c 2 /*oo, (50.7) 

where /xqq, for zero velocity of matter, is the ordinary mass density. 
Also 

T = T\ = T u = T44 = -c 2 /*oo. (50.8) 

The 44-component of Einstein's gravitation law in the form of 
equation (48.15) can now be expressed approximately 

i 2 V 2 F=iKc 2 / xoo, 

or V 2 V = iiccVoo. (50.9) 

This is the Poisson equation (48.10) of classical Newtonian theory, 
provided we accept 

k = ^r (50.10) 

c 
This specifies k in terms of the gravitational constant. 



GENERAL THEORY OF RELATIVITY 155 

51. Metrics with spherical symmetry 

When a change is made in the space-time coordinate frame from 
coordinates x* to coordinates x , the metric tensor gy will change to 
gy by the law of transformation of a covariant tensor. In general, the 
gy will be functions of the x l and the gy will be functions of the jc', but 
it will not usually be the case that the gy are the same functions of the 
* barred' coordinates that the gy are of the 'unbarred' coordinates. 
I.e., the functions gyix*) are not form invariant under general co- 
ordinate transformations. However, in some special cases, itis possible 
for these functions to be form invariant under a whole group of 
transformations and we shall study such a case in this section. 

In a gravitational field, the geometry can only be quasi-Euclidean 
and consequently rectangular Cartesian axes do not exist. Neverthe- 
less, no difficulty is experienced in practice in defining such axes 
approximately and we shall suppose, therefore, that the coordinates 
x, y, z, t of an event in the gravitational field about to be considered 
are interpreted physically as rectangular Cartesian coordinates and 
time. We shall now search for a metric which, when expressed in those 
coordinates, is form invariant with respect to the group of coordinate 
transformations which will be interpreted physically as rotations of 
the rectangular axes Oxyz (t is to remain unaltered). Such a metric 
will be said to be spherically symmetric about O. 

Invariants for this group of coordinate transformations, which are 
of degree no higher than the second in the coordinate differentials ax, 
dy, dz, are 

x 2 +y 2 +z 2 , xdx+ydy+zdz, dx^+df+dz 2 . (51.1) 

Introducing spherical polar coordinates (r, 6, <f>), which will be defined 
by the equations (29.3), these invariants may be written 

r 2 , rdr, dr 2 +r 2 dd 2 +r 2 sm 2 6d<i> 2 . (51.2) 

It follows that r, dr, d6 2 + sm 2 6d<l> 2 (51.3) 

are invariants. The most general metric with spherical symmetry can 
now be built up in the form 

ds 2 = A(r,t)dr 2 +B(r,t)(d6 2 + sin 2 6d<f> 2 ) + 

+ C(r,t)drdt+D(r,t)dt 2 . (51.4) 



156 TENSOR CALCULUS AND RELATIVITY 

We now replace r by a new coordinate r' according to the transfor- 
mation equation 

r' 2 = B(r,t). (51.5) 

Then ds 2 = E(r',t)dr' 2 +r ,2 (dd 2 +sin 2 dd<f> 2 ) + 

+F(r',t)dr'dt+G(r',t)dt 2 . (51.6) 

In a truly inertial frame, spherical polar coordinates can be defined 
exactly and the metric will, by equation (46.6), be expressed in the 
form 

ds 2 = Jr 2 +r 2 (^ 2 +sin 2 0^ 2 )-c 2 ^ 2 . (51.7) 

Comparing equations (51.6), (51.7), it is clear that, in a region for 
which (51.6) is the metric, r' will behave physically like a true spherical 
polar coordinate r. We shall accordingly drop the primes and write 

ds 2 = E(r,O^+rW + sin 2 0# 2 )+ 

+ F(r,t)drdt+G(r,t)dt 2 . (51.8) 

If our frame is quasi-inertial, equation (51.7) must be an approxi- 
mation for equation (5 1 .8) and the following equations must therefore 
be true approximately: 

E(r,t) = 1, F(r,t) = 0, G(r,t) = -c 2 . (51.9) 

Consider now the special case when the gravitational field is static 
in the quasi-inertial frame for which (r, 6, <f>) are approximate spherical 
polar coordinates and t is the time. The functions E, F, G will then be 
independent of t. Also, space-time will be symmetric as regards past 
and future senses of the time variable and this implies that ds 2 is 
unaltered when dt is replaced by — dt. Thus F= and we have 

ds 2 = adr 2 +r 2 (dd 2 +sin 2 dd<f> 2 )-bc 2 dt 2 , (51.10) 

where a, b are functions of r both approximating unity. 
For this metric, taking 

x l = r , x 2 = 6, x 3 = <f>, x 4 = f, (51.11) 

we have 

Su = a, S22 = r 1 , #33 = /^sin 2 ^, g 44 = -be 2 , (51.12) 
all other gy being zero. Thus 

g= -abc 2 r 4 sm 2 6, (51.13) 



GENERAL THEORY OF RELATIVITY 



157 



and hence 



*" = -, g 22 = - 2 , S 33 

a r" 



-A-T-v g 44 = -A, (51.14) 
rsurv be 



all other g ij being zero. The three-index symbols can now be calculated 
from equation (41.9). Those which do not vanish are listed below: 



i \ - 



{l 2 2> = { 2 2 1> = 



1 



1 



W = Gil = y 



A\ = tJ.\ = t, 
r 

V_ 

2b 



(iV = W = 

( i 1- -' 

12 2/ — » 

a 

{ 2 3 3> = {3 3 2> = COt0, 

{ 3 1 3 >= --sin 2 0, 
a 

(3 2 3) = — sin0cos0, 



.i \ = 



c 2 6' 



(51.15) 



{44} "¥' 

primes denoting differentiations with respect to r. 

Contracting B) M as given by equation (37.21) with respect to the 
indices /, /, it follows that 

R jk = ir'klifj} — ir iHfk) + 7T* iji) ~ T~i ijk) » 



dx* 



d x ' 



= ir t k}{t r j}-{j r k}jpioeV(-g)+ 

+ 8xk? logV( - g) ~^ {jik} ' 



(51.16) 



158 



TENSOR CALCULUS AND RELATIVITY 



by equation (43.4) (g can clearly be replaced by —g in this equation 
without affecting its validity). The non-zero components of the Ricci 
tensor are now calculated to be as follows: 



b" V 2 a'V a' 
R\\ — rr — TVt — : — : 



R 



22 



2b 4b 2 
rb' ra 
lab 



— I?*" 1 ' 



Aab 
1 
a 



Since 



^33 = i?22sin 2 ^, 

2 ( b" b' 2 a'V b'\ 
R44 = C [~2^ + 4a~b + 4^-ar)' 
R\=g n R n , Rl = g 22 R 21 , etc. 



it will be found that 
R 



i b " 
R * = ab~2ab 2 



b' 2 a'V lb' la' 



2a 2 b abr 



arr 



at 2 r 2 



(51.17) 



(51.18) 



(51.19) 



52. Schwarzschild's solution 

The static, spherically-symmetrical, metric (51.10) will determine the 
gravitational field of a static distribution of matter also having 
spherical symmetry, provided it satisfies Einstein's equations (48.15). 
We shall consider the special case when the whole of space is devoid 
of matter, apart from a spherical body with its centre at the centre of 
symmetry O. Then Ty = 0, T= at all points outside the body and 
Einstein's equations reduce in this region to 

Ry = 0. (52.1) 

By equations (51.17), these are satisfied by the metric (51.10), provided 
a, b are such that 

*" ^ "'"' "' - (52.2) 



2b 4b 2 4ab or °' 

rV__rrt_ 1 

lab la 2 + a ~ ' 

V V 2 a'V V n 

+ + — ~ = 0. 

la 4ab 4a z ar 



(52.3) 



(52.4) 



GENERAL THEORY OF RELATIVITY 159 

From (52.2), (52.4), it follows that 

ab'+a'b = (52.5) 

and hence that ab = constant. (52.6) 

But, as r-»-oo, we shall assume that our metric approaches that given 
by equation (51.7) and valid in the absence of a gravitational field. 
Thus, as r-»a>, then a->l, b-+l and hence 

ab = 1. (52.7) 

Eliminating b from equation (52.3), it will be found that 

ra' = a(l-a). (52.8) 

This equation is easily integrated to yield 

1 — (2m//-) 
where m is a constant of integration. Then 

b=l-~, (52.10) 

r 

and it may be verified that each of the equations (52.2)-(52.4) is 
satisfied by these expressions for a and b. 
We have accordingly arrived at a metric 



** = 



dP- 



l-Qm/r. 



-+P(dd 2 +sm 2 6d<j> 2 )-c 1 (l~\dt 2 (52.11) 



which is sphericaPy^symmetrical and can represent the gravitational 
field outside a spherical body with its centre at the pole of spherical 
polar coordinates (r,d,<f>). This was first obtained by Schwarzschild. 
It will be proved in the next section that the constant m is proportional 
to the mass of the body. This may also be deduced from equation 
(49.17), for the potential Vat a. distance r from a spherical body of 
mass M is given by 

V= - y — (52.12) 

r 



160 TENSOR CALCULUS AND RELATIVITY 

and hence g 44 = 1 - -^ — (52. 1 3) 

c r 

Now g 44 is the coefficient of (dx 4 ) 2 = - c 2 dt 2 in the metric and hence 

b=\ — V-- (52.14) 

err 

Comparing equations (52.10), (52.14), it will be seen that 

m = ^y * (52.15) 



It is clear from equation (52.11) that the metric is singular at a 
distance 

r = 2m = -V (52.16) 

c 

from O. We conclude that the radius of the body must certainly be 
greater than this minimum value. Since, in c.g.s. units, c = 3 x 10 10 
and for the Earth yM= 3-991 x 10 20 , the minimum radius for this 
body is about 9 mm. 



53. Planetary orbits 

The attractions of the planets upon the Sun cause this body to have 
a small acceleration relative to an inertial frame. If, therefore, a co- 
ordinate frame moving with the Sun is constructed, relative 1 to this 
frame there will be a gravitational field corresponding to this accelera- 
tion in addition to that of the Sun and planets. However, for the pur- 
pose of the following analysis, this field and the fields of the planets 
will be neglected. Thus, relative to spherical polar coordinates having 
their pole at the centre of the Sun, the gravitational field will be 
assumed determined by the Schwarzschild metric (52.1 1). The planets 
will be treated as particles possessing negligible gravitational fields, 
whose world-lines are geodesies in space-time (section 47). We pro- 
ceed to calculate these geodesies. 



GENERAL THEORY OF RELATIVITY 



161 



Substituting for a, b from equations (52.9), (52.10) in equations 
(51.15), it will be found that 



m 



u is — 


r(r-2m)' 


{l 2 2> = 


1 

- » 
r 


{l 3 3> = 


1 

— » 
r 


(l\> = 


m 


r{r-2ni) 


til) = 


-(r-2m), 


{ 2 3 3> = 


cot0, 


(sS) = 


-(r- 2m) sin 2 6, 


{ 3 2 3> = 


— sin cos 0, 


W = 


mc 2 

— s- (r—2m). 
r 3 



(53.1) 



Equations (44.5) for the geodesies accordingly take the form 
d 2 r 



ds 2 






d 2 2drdd . n „. 
-ryH — - — -sin0cos0| 
ds* rds ds 



d 2 4> 2drd<f> „ n d9d<f> n 
7I + -T -j- + 2cot0— -f = 0, 
ds* rdsds dsds 



d 2 t 2m drdt 

+ ~. — — - = 0. 



ds 2 r(r — 2m)dsds 



(53.3) 
(53.4) 
(53.5) 



162 TENSOR CALCULUS AND RELATIVITY 

There is also available the first integral (44.6) of these equations 
which, by equation (52.11), is 

Equation (53.2) will be discarded in favour of this first integral. 

We now choose the spherical polar coordinates so that the planet 
is moving initially in the plane 6 = \n. Then ddjds = initially and 
hence, by equation (53.3), d 2 6/ds 2 = at this instant and the particle 
continues to move in this plane indefinitely. Thus, putting Q=\n, 
ddjds = in the remaining geodesic equations, we find these reduce to 

%^f-0. (53.7) 

dsr rdsds 

d 2 t 2m drdt 

— =- H =0, (53.8) 

ds 2 r(r-2m)dsds 

Putting w = d<f>jds, v = dtjds, equations (53.7), (53.8) may be 
written 

d ^+ 2 w = 0, (53.10) 

dr r 

do 2m . /c - 11N 

dr r(r-2m) 

respectively. These equations can now be integrated to yield 



d<f> _ oc 
Is ~? 



w = ^ = - 2 , (53.12) 



,-£--4-. (53-13) 

as r—2m 



where a, j8 are constants of integration. 



GENERAL THEORY OF RELATIVITY 163 

Substituting for d<f>/ds, dtjds from the last two equations into 
equation (53.9), it follows that 

" (r-2m) = l + c 2 j8 2 - — • (53.14) 



(dr\ 2 a 
U) + ? 



Then, eliminating ds between this equation and equation (53.12), we 
obtain the equation for the orbit, viz. 

/adr\ 2 a 2 2o2 2m 2ma 2 ,-. ,- 

With k = 1/r, this reduces to the form 

(du\ 2 2 l + c 2 j3 2 2m „ , ,,_ 

U) +K =-^-^ M+W - (53 ' 16) 

Differentiating through with respect to ^, this equation takes a form 
which is familiar in the theory of orbits, viz. 

d 2 u m 

dj> 



2+u = ~- 2 +3mu 1 . (53.17) 



The corresponding equation governing the orbit according to 
classical mechanics is 

d 2 u yM 

d? +u= i?' (5318> 

where M is the mass of the attracting body and h is the constant 
velocity moment of the pl?net about the centre of attraction, i.e., 

r ld 4- = h. (53.19) 

dt 

The general relativity counterpart of this last equation is clearly 
equation (53.12). If t is the proper time for the orbiting body, by 
equation (46.6) icdr = ds and equation (53.12) is equivalent to 

r 2 ^ = icoi. (53.20) 

dr 



164 TENSOR CALCULUS AND RELATIVITY 

We shall identify the time variable t of classical mechanics with the 
proper time t for the body whose motion is being studied and then, 
by comparison of equations (53.19), (53.20), it is seen that 

h = ica. (53.21) 

Equation (53.17) then becomes 

d 2 u mc 2 , , « v 

372 + u = -y + 3wk 2 (53.22) 

a^ ft 

which, apart from a term 3mu 2 , is identical with the classical equation 
(53.18) provided 

m = ^r • (53.23) 

c 

This confirms equation (52. 1 5). 

The ratio of the additional term 3mu 2 to the 'inverse square law' 
term mc 2 /h 2 is 

^-?^ (53.24) 

by equation (53.19). r<f> is the transverse component of the planet's 
velocity and, for the planets of the solar system, takes its largest value 
in the case of Mercury, viz. 48 km/sec. Since c = 3 x 10 5 km/sec, the 
ratio of the terms is in this case 7-7 x 10 ~ 8 , which is very small. 
However, the effect of the additional term proves to be cumulative, 
as will now be proved, and for this reason an observational check can 
be made. 
The solution of the classical equation (53.18), viz. 

u = ^ {1 + ecos (<f> - cD)}, (53.25) 

where fi = yM = mc 2 , e is the eccentricity of the orbit and c5 is the 
longitude of perihelion, will be an approximate, though highly accur- 
ate, solution of equation (53.22). Hence the error involved in taking 

3mu 2 = -^~{l+ecos(<f>-di)} 2 (53.26) 



GENERAL THEORY OF RELATIVITY 165 

will be absolutely inappreciable, since this term is very small in any 
case. Equation (53.22) can accordingly be replaced by 

d 2 u u. 3mu 2 ~ 

^2+" = j£+-jjrU+ecos(<f>-6>)} 2 . (53.27) 

This equation will possess a solution of the form (53.25) with ad- 
ditional 'particular integral' terms corresponding to the new term 
(53.26). These prove to be as follows: 

-?Jt- {1 + ie 2 - \e 2 cos 2((f> -w) + e<f> sin (<f> - d>)}. (53.28) 

The constant term cannot be observationally separated from that 
already occurring in equation (53.25). The term in cos2(<£-c5) has 
amplitude too small for detection. However, the remaining term has 
an amplitude which increases with <f> and its effect is accordingly 
cumulative. Adding this to the solution (53.25), we obtain 



u = -2< 1 +ecos (<ft- to) + 2 0sin(<p 



-<3)i 



= j^{l + ecos(<f>-d)-8d>)}, (53.29) 

where ScD = 3mfjL^>/h 2 and we have neglected terms 0(8a> 2 ). 

Equation (53.29) indicates that the longitude of perihelion should 
steadily increase according to the equation 

8<5 - iJ w* - S* - %*• (53 - 30) 

where / = h 2 /fi is the semi-latus rectum of the orbit. Taking 

P = 1-33 xlO 26 

c.g.s. units for the Sun, c = 3 x 10 10 cm/sec and / = 5-79 x 10 12 cm for 
Mercury, it will be found that the predicted angular advance of 
perihelion per century for this planet's orbit is 43". This is in agree- 
ment with the observed value. The advances predicted for the other 
planets are too small to be observable at the present time. 



166 TENSOR CALCULUS AND RELATIVITY 

54. Gravitational deflection of a light ray 

In section 7 it was shown that the proper time interval between the 
transmission of a light signal and its reception at a distant point is 
zero. It was there assumed that the signal was being propagated in an 
inertial frame and hence that no gravitational field was present. This 
result can be expressed by saying that 

ds = (54.1) 

for any two neighbouring points on the world-line of a light signal. 
Now, null-geodesies in the space-time having metric (46.6) are defined 
by equation (54.1) and the equations 

d^_<Py_d*z_dh ( , A1 . 

dX 2 ~ dX 2 ~ dX 2 ~ dX 2 ' K } 

for the three index symbols are all zero. Equations (54.2) imply that 
along a null-geodesic x, y, z are linearly dependent upon t . But this 
is certainly true for the coordinates of a light signal being propagated 
in an inertial frame. We conclude that the world-lines of light signals 
are null-geodesies in space-time. 

Since an inertial frame can always be found for a sufficiently small 
space-time region even in the presence of a gravitational field, it fol- 
lows that the world-line of a light signal in any such region is a null- 
geodesic. We shall accept the obvious generalization of this result, 
viz. that the world-lines of light signals over an unlimited region of 
space-time are null-geodesies. 

We shall now employ this principle to calculate the path of a light 
ray in the gravitational field of a spherical body. Taking the space- 
time metric in the Schwarzschild form (equation (52.11)), the three- 
index symbols are given at equation (53.1) and the equations govern- 
ing a null-geodesic (equations (44. 14) ) are identical with the equations 
(53.2)-(53.5) after s has been replaced by A. The first integral (44.13) 
takes the form 

(54.3) 



GENERAL THEORY OF RELATIVITY 167 

Without loss of generality, we shall again put 6 = \n, so that a ray 
in the equatorial plane is being considered and then proceed exactly 
as in the last section to derive the equation 

— \+u= 3mu 2 , (54.4) 

d<p 

where u = I jr. This equation determines the family of light rays in the 
equatorial plane. 

As a first approximation to the solution of equation (54.4), we shall 
neglect the right-hand member. Then 

u = -cos(<£ + a), (54.5) 

R 

where R, a are constants of integration. This is the polar equation of 
a straight line whose perpendicular distance from the centre of 
attraction is R. As might have been expected, therefore, provided the 
gravitational field is not too intense, the light rays will be straight lines. 
This deduction is, of course, confirmed by observation. Thus, as the 
Moon's motion causes its disc to approach the position of a star on the 
celestial sphere and ultimately to occult this body, no appreciable 
deflection of the position of the star on the celestial sphere can be 
detected. 

Again, without loss of generality, we shall put a = so that the 
light ray, as given by equation (54.5), is parallel to the y-axis 
(<j>= ± %n). Then, putting u — cos<f>/R in the right-hand member of 
equation (54.4), this becomes 

d 2 u 3m , , 

-772 + « = r-j cos 9- < 54 - 6 ) 

ct<f> R 

The additional 'particular integral' term is now found to be 

^(2-cos 2 ^) (54.7) 

and hence the second approximation to the polar equation of the 
light ray is 

1 m 

« = - cos ^ + -j (2 - cos 2 <f>). (54.8) 

R R 



168 TENSOR CALCULUS AND RELATIVITY 

At each end of the ray u = and hence 

^ cos 2 <f>-cos<f> — ^ = 0. (54.9) 

R R 

Assuming m/R to be small, this quadratic equation has a small root 
and a large root. The small root is approximately 

2/w 
cos«£ = — - (54.10) 

R 



andhence <f> = ± (^ + ~7r) (54.11) 



at the two ends of the ray. The angular deflection in the ray caused 
by its passage through the gravitational field is accordingly 

- (54.12) 

R 

approximately. 
For a light ray grazing the Sun's surface, 

R = Sun's radius = 6-95 x 10 10 cm and m = 1-5 x 10 5 cm. 

Thus the predicted deflection is 8-62 x 10 -6 radians, or about 1-77*. 
This prediction has been checked by observing a star close to the 
Sun's disc during a total eclipse. The experimental findings are in 
accord with the theoretical result. 

55. Gravitational displacement of spectral lines 

A standard clock will be taken to be any device which experiences a 
periodic motion, each cycle of which is indistinguishable from every 
other cycle. The passage of time between two events which occur in 
the neighbourhood of the clock is then measured by the number of 
cycles and fraction of a cycle which the device completes between 
these two instants. The clocks employed to determine the time co- 
ordinate | 4 of an event in section 46 were not, necessarily, standard 
clocks. Such coordinate clocks can have arbitrary variable rates, the 
only requirement being that, if A, B are two events in the vicinity of 



GENERAL THEORY OF RELATIVITY 169 

a coordinate clock and B occurs after A, then the coordinate-time for 
B must be greater than the coordinate-time for A. 

Consider, then, a standard clock which is moving in any manner 
within some reference frame. Let A, B be the two events representing 
the commencement and termination of one clock cycle and let C, D 
be two events separated by another clock cycle. Since we are assuming 
that the clock cycles are indistinguishable, the geometrical relation- 
ship between A and B in space-time must be identical with the relation- 
ship between C and D. It follows that the space-time intervals between 
A and B and between C and D are equal. Thus, every cycle of a stan- 
dard clock registers its advance along its world-line through a con- 
stant interval and, if ds is the interval between neighbouring points on 
this world-line, the quantity whose passage is registered by the clock 
will be 



/ 



ds, (55.1) 



integrated along the world-line. I.e. a standard clock registers the 
passage of interval along its trajectory. 

Let x l (i = 1,2,3,4) be the coordinates of an event with respect to 
some space-time reference frame, x l , x 2 , x 3 being interpreted physic- 
ally as spatial coordinates relative to a static frame and x 4 /ic as time. 
If a standard clock is at rest relative to this frame, for adjacent points 
on its world-line dx x = dx 2 = dx 3 = and hence 

ds 2 = g^idx 4 ) 2 = -c 2 g 44 dt 2 , (55.2) 

where we have put x 4 = ict. The interval s measured by the clock is 
therefore related to t, the coordinate-time at the point (x l ,x 2 ,x 3 ), by 
the equation 

s = ic j V(g«)dt. (55.3) 

s is imaginary only because we have chosen to define the interval ds 
between two events in such a way that time-like intervals are imagin- 
ary. The standard clock can be graduated to register T = s/ic and then 
standard clock time T will be related to coordinate-time by the 
equation 

T=jV(g4ddt. (55.4) 



170 TENSOR CALCULUS AND RELATIVITY 

In the special case of the coordinate frame employed in section 49 
which was stationary in a relatively weak static gravitational field, it 
was proved that g 44 is given in terms of the Newtonian scalar potential 
Ffor the field by the approximate equation (49.17). Thus 

/ 2F\ 1/2 
dT = l+-r dt (55.5) 



relates time intervals measured by a stationary standard clock and a 
coordinate-clock at a point in a gravitational field where the potential 
is V. Now, when it is emitting its characteristic spectrum, an atom is 
operating as a standard clock. If, therefore, two similar atoms are 
stationary at different positions in a static gravitational field and they 
emit their characteristic radiation, the two intervals corresponding to 
the emission of one complete cycle of a certain spectral line by each 
atom will be identical. If dTis this standard-time interval, V u V 2 are 
the gravitational potentials at the atoms, and dt\, dt 2 are the periods 
of the complete cycles as measured in coordinate-time, then by 
equation (55.5) 

/ 2F,\ 1/2 / 2F<A 1/2 

dT=ll + - 2 ±\ dt l = il+^\ dt 2 (55.6) 

/ 2F-A 1/2 / 2Fi\ 1/2 
and hence dt x : dt 2 = 1 1 + -~ I : 11 + -~\ • (55.7) 

Suppose that the radiation from the atoms is observed at some point 
P in the field. Let t a be the coordinate-time at one atom when a radi- 
ation wave crest is emitted and t b the corresponding time for the 
next crest. Let t' m t b be the respective coordinate-times of arrival of 
these crests at P. Since the field and coordinate frame are both static, 
the time delay between a crest leaving the atom and its arrival at P 
will be constant. Thus 

t'a-ta = *b-t b , (55.8) 

or t'a-ti, = t a -t b . (55.9) 

This last equation shows that the period of vibration of the atom as 
measured by the coordinate-clock at P is independent of this point's 



GENERAL THEORY OF RELATIVITY 171 

position and is equal to the period as measured by the coordinate- 
clock at the atom itself. The frequencies v u v 2 of corresponding spec- 
tral lines for the two atoms as measured at P will therefore, by 
equation (55.7), be in the ratio 

v 2 *J\l+2V 2 /c 2 J c 2 

approximately. 

In the case of an atom at the surface of the Sun and a similar atom 
at the Earth's surface, it will be found that, in c.g.s. units, 

V t = - 9-512 xlO 12 (Earth), 

V 2 = - 1-914 xlO 15 (Sun), 

and thus - = 100000212. (55.11) 

v 2 

This effect is so small, that it is very difficult to measure. However, in 
the case of the companion of Sirius, the predicted effect is 30 times 
larger and has been confirmed by observation. 



56. Maxwell's equations in a gravitational field 

In this final section, the equations (23.11) determining the electro- 
magnetic field due to the motion in vacuo of a distribution of electric 
charge, will be generalized to take account of any gravitational field 
which may be present, but we shall not elaborate upon the implications 
of the modified equations. 

Over any sufficiently small region of space and restricted interval 
of time it is possible to define a rectangular Cartesian inertial frame, 
i.e. the frame in ' free fall ' in the gravitational field. If the electric and 
magnetic components of ths electromagnetic field are measured in 
this frame, the field tensor Fy defined by equation (23.5) can be found. 
Employing the appropriate transformation equations, the com- 
ponents of this tensor relative to general coordinates x* in the gravi- 
tational field can be computed. No distinction is made between 
covariant and contravariant properties relative to the original in- 
ertial frame so that, when transforming, Fy may be treated as a 



172 TENSOR CALCULUS AND RELATIVITY 

covariant, contravariant or mixed tensor. If it is treated as a covariant 
tensor, the covariant components Fy in the general jc'-frame will be 
generated. If it is treated as a contravariant or as a mixed tensor, the 
contravariant or mixed components F iJ , F) respectively will be gener- 
ated. In this way, the field tensor is defined at every point of space- 
time. Similarly, a current density vector with covariant components 
/, and contravariant components /' is defined relative to the x'-frame. 
Consider the equations 

F iJ .j = -/', (56.1) 

c 

Fu;k + Fjk;i+F ki .j = 0. (56.2) 

These are tensor equations and hence are valid in every space-time 
frame if they are valid in any one. But, relative to the inertial coordin- 
ate frame (x,y,z,ici) which can be found for any sufficiently small 
space-time region, these equations reduce to equations (23.11) and 
hence are valid over such a region. Regarding the whole of space-time 
as an aggregate of such small elements, it follows that equations (56. 1), 
(56.2) are universally true. 
Since F ij is skew-symmetric, 

F U ;J = ~j + {r i j}F rJ +{r J j}F ir , 
8F' J 1 8 

= ^ + v(^ {V( -* )}F '' 

1 8 ~ i W(-g)F iJ }, (56.3) 



V(~g)8x J 



by equation (43.4) (# has been replaced by — g, since g is always 
negative for a real gravitational field). Equation (56.1) is accordingly 
equivalent to 

1 a .. 4-n- . 

AV(-g)F IJ } = —J'. (56.4) 



V(.-g)te J c 

Further, since g is a relative invariant of weight 2 (cf. equation (32.7)) 



GENERAL THEORY OF RELATIVITY 173 

and hence V( - s) is an invariant density, it follows that g y , 3' defined 
by the equations 

g' 7 = V(-g)F U , 3' = V(-g)J l , (56.5) 

are densities and then equation (56.4) takes the simpler form 

g^tf. (56.6, 

dx J c 

Also, in view of the skew-symmetry of the field tensor, it follows 
that equation (56.2) is equivalent to 

dx k dx l dx J 



- k +^rf + ^ri = 0. (56.7) 



Exercises 6 

1. Prove that when an index of T iJ as defined by equation (48.2) is 
lowered, T) as defined by equation (48.3) is obtained. 

2. In the space-time whose metric is given by 

ds 2 = ^(dx A ) 2 +^ e (dx l ) 2 +(.dx 2 ) 1 +{dx z f > 

where (f>, 6 are functions of x 1 only, prove that the Riemann- 
ChristofFel tensor vanishes if and only if 

<p"-d'<f>'+<f>' 2 = 

where the dashes denote differentiations with respect to x 1 . If <f> = —6, 
prove that the space is flat provided that 

<f> = \ log (a + bx\ 

where a, b axe constants. 

(L.U.) 

3. If the metric of space-time is 

ds 2 = -^{(dxif+idx^y-ix^e-Pidxif+ePidx 4 ) 2 , 

where A and p are functions of jc 1 and x 2 only, show, by calculating 
/?44, that the field equations Ry = (for a region devoid of matter) 
require that p shall satisfy 

z> 2 p , a 2 p , i gp _ ft 



(dx 1 ) 2 (dx 2 ) 2 x 2 dx 2 

(Li.U.) 



174 TENSOR CALCULUS AND RELATIVITY 

4. Find the differential equations of the paths of test particles in 
the space-time of which the metric is 

ds 1 = e 2kx [-(dx 2 +dy 2 +dz 2 ) + dt 2 ], 

where & is a constant. If 



Mir+eHf 



and if v = V when x — 0, show that 

\-v 2 = (\-V 2 )<? kx . 



(Li.U.) 



5. Use the equations 

RJ-^R = -kT) 

to find the energy-momentum tensor for the distribution of matter 
corresponding to the space-time 

ds 2 = -e g (dx 2 +dy 2 +dz 2 )+dt 2 , 



where g is a function of t only. 

6. If 

- _ <ft 2 1 dx 2 + dy 2 +dz 2 
l-kx c 2 (1-kx) 2 

where A: is a constant, and if 

,2 



ai.u.) 



^(f) + (i) + (ty 



prove that, along a geodesic, 

V 2 -v 2 = kc 2 x, 
where Vis a constant. 

(L.U.) 

7. Show that the four differential equations (53.2)-(53.5) for the 

geodesies in the Schwarzschild space-time have a solution for which 



GENERAL THEORY OF RELATIVITY 175 

6 = fa, r = a, where a is a constant greater than 3m, and that the total 
interval along this geodesic from <j> = to <f> = 2n is 

(a V> 2 

Also show that there is a geodesic along which 6 = const., <f> = const, 
and which satisfies an equation of the form 

where R is fixed. State briefly the physical interpretation of these 
results. 

(L.U.) 

8. A space-time has metric 

ds 2 = ^{(dx^+idx^ + idx^f+idx 4 ) 2 } 

where a is a function of (be 1 ,* 2 ,* 3 ,* 4 ). If t' is the unit tangent to a 
geodesic, prove that 

^+2(a,.*V = o k e- 2 °, 
ds 

da 
where a k = ^. 

For slow motions in slowly changing fields, prove that the geodesies 
are paths of particles in a gravitational field of potential — ac 2 . 

(L.U). 

9. Find the Riemann-Christoftel tensor of the space-time of the 
last exercise, and prove that the scalar curvature R vanishes if, and 
only if, 

where o pq = ^-— 



176 TENSOR CALCULUS AND RELATIVITY 

If a is a function of r = [(x l ) 2 + (x 2 ) 2 + (* 3 ) 2 ] 1/2 only, prove that this 
condition is 

a" + -a' + a' 2 = 0, 
r 

where dashes denote differentiations with respect to r. 

(L.U.) 
10. In the space-time of metric 

ds 2 = e 2a {dr 2 +r 2 dd 2 + r*sm 2 dd<f> 2 -dt 2 } 

where <x = log(l + m/r), and m is constant, prove that the scalar 
curvature is zero (see Exercise 9 for this condition). Prove that the 
geodesies satisfy the equations 

^sm 2 9 d 4 = k l e- 2a 
ds 

dt , -In 

J S - ^ • 

where k u k 2 are constants. If we choose ^ = 0, d<f>/ds = initially, 
prove that <f> is always zero, and that 



fdd . - 
r 2 — = he~ 2a , 
ds 



where h is constant. 
11. Show that 

ds 2 = e-wy 



(L-U.) 



ldt 2 -l(dx 2 +dy 2 +dz 2 )\ 



where q is an arbitrary function and 

a 2 = t 2 -\{x 2 +y 2 +z 2 ) 
c 

is invariant under a Lorentz transformation. 
If 

/ = px, f = py, ' 3 = pz, ' A = pt 



GENERAL THEORY OF RELATIVITY 177 

where p is a function of a and/ is a contravariant vector satisfying 

7^(V<-*)/) = 0, {x l = x,...,x 4 = t), 

ox 

show that 

Ae Aq 

> = -*> 
where A is a constant. 

(L.U.) 
12. By replacing the spherical polar coordinate r occurring in the 
Schwarzschild metric (52.1 1) by a new coordinate r' where 



--(■*# 



obtain this metric in 'isotropic' form, viz. 

\ 2r'J T \\+ml2r'J 

13. Employing a certain frame, an event is specified by spatial 
coordinates (x,y,z) and a time /. The corresponding space-time 
manifold has metric 

ds 2 = dx 2 + dy 2 + dz 2 +2atdxdt-(c 2 -a 2 t 2 )dt 2 . 

Show that a particle falling freely in the gravitational field observed 
in the frame has equations of motion 

x = A + Bt-^at 2 , y=C+Dt, z = E+Ft, 

where A, B, C, D, E, Fare constants. By transforming to coordinates 
(x',y,z,t), where x' = x+iat 2 , and recalculating the metric, explain 
this result. 

14. (x x i x 2 ,x y ) are spatial coordinates of an event relative to a 
frame S and x 4 is the time of the event measured by a clock in S. A 
second frame / is falling freely in the neighbourhood of P and may 
be regarded as inertial. Oy 1 y 2 y i are rectangular cartesian axes in / 
and y 4 /ic represents the time within / as measured by synchronised 



178 TENSOR CALCULUS AND RELATIVITY 

clocks attached to the frame. Show that gy, the metric tensor in S, is 
given by 

_ 8y k 8y k 

P is a point, fixed in S, having coordinates (x 1 , x 2 ,x 3 ). At the instant 
x\ I is chosen so that P is instantaneously at rest in /. Deduce that 

W g i4 



d x' V(g44> 
dl is the distance between P and a neighbouring point 

P'(x l + dx\ x 2 + dx 2 , x 3 + etc 3 ) 
as measured by a standard rod in / at the instant x 4 . Prove that 

dl 2 = dy*dy a = y^dx^dx*, 
where a, A, /* range over the values 1, 2, 3 and 

g\4gfi4 

y^ ^ g44 

(yXfi i s tne metric tensor for the &$ 3 which is 5" at the instant x*.) 

15. Oxyz is a rectangular cartesian inertial frame /. A rigid disc 
rotates in the xy-plane about its centre O with angular velocity co. 
Polar coordinates (r, 6) in a frame R rotating with the disc are defined 
by the equations 

x = rcos(d+wf), y = rsin(0+a>/), 

where t is the time measured by synchronised clocks in the inertial 
frame. If the time of an event in R is taken to be the time shown by 
an adjacent clock in J, show that the space-time metric associated 
with R is 

ds 2 = dr 2 +r 2 dd 2 +2cor 2 dddt-(c 2 -r 2 <o 2 )dt 2 . 

Deduce that the metric for geometry in R is given by 

r 2 d6 2 

dl 2 = dr 2 +^ — — -• 

1 — co 2 r 2 /c 2 



GENERAL THEORY OF RELATIVITY 179 

(Hint: employ the result of the previous exercise.) Hence show that 
the family of geodesies on the disc is determined by the equation 

6 = const. - sin" 1 1 - I - ~ 2 V(r 2 - a 7 ), 

where r x = clot and \a\ < r t . Sketch this family. What is the physical 
significance of ri ? 

16. x' (/ = 1,2,3,4) are three space coordinates and time relative 
to a reference frame S. A test particle is momentarily at rest in S at 
the point (x 1 ,* 2 ,* 3 ) at the time x 4 . If g u is the metric tensor for the 
gravitational field in S, write down the conditions that the world-line 
of the particle is a geodesic and deduce that 

d 2 x* = l/gg44 £,4^44 \ dg* t 

gi "(dx*) 2 2\ax'' g 4 4^ 4 / a* 4 ' 

where the Greek index ranges over the values 1,2,3. Hence show that 
the covariant components of the particle's acceleration in S are given 

by 

d 2 x° dU 8y„ 

y *t(dx*) 2 dx« K } ex* 

where y aj g is defined in exercise 14 and 

g 4 4 = - (c 2 + 2 U), y x = g aA l V( ~ ^44). 
(t/, y a are the gravitational scalar and vector potentials respec- 
tively.) 

Show that, in the case of the space-time metric appropriate to the 
rotating frame of exercise 15, the gravitational vector potential 
vanishes and the scalar potential is given by U = \u> 2 r 2 . Interpret 
this result in terms of the centrifugal force. 

17. De Sitter's universe has metric 

ds 2 = - A' 1 dr 2 - r 2 dd 2 -r 2 sin 2 6d<f> 2 + Ac 2 dt\ 
where A=l-r 2 /R 2 , R being constant. Obtain the differential 
equations satisfied by the null-geodesies and show that along null- 
geodesics in the plane 8 = \-n 

a %- = rir 2 -a 2 yi 2 , 
dq> 



180 TENSOR CALCULUS AND RELATIVITY 

where a is a constant. Deduce that, if r, <f> are taken to be polar 
coordinates in this plane, the paths of light rays in this universe are 
straight lines. 

18. Einstein's universe has the metric 

ds 2 = c 2 dt 2 - , — — dr 2 - r 2 dd 2 - r 2 sin 2 d d<f> 2 , 
1 — Ar 2 

where (r, 6, (f>) are spherical polar coordinates. Obtain the equations 
governing the null-geodesies and show that, in the plane 6 = %n, 
these curves satisfy the equation 

where p is a constant. Putting r 2 — l/v, integrate this equation and 
hence deduce that the paths of light rays in the plane 8 = ^n are the 
ellipses 

Ax 2 + fiy 2 = 1, 

where (x,y) are rectangular cartesian coordinates. Show, also, that 
the time taken by a photon to make one complete circuit of an 
ellipse is lir/icX 1 ' 2 ). 

1 9. If the metric of space-time is 

ds 2 = kocdt 2 -<x 2 (dx 2 +dy 2 +dz 2 ), 

where a is a function of x alone and & is a constant, obtain the 
differential equations governing the world-lines of freely falling 
particles. If x,y,z are interpreted as rectangular cartesian co- 
ordinates by an observer and t is his time variable, show that there is 
an energy equation for the particles in the form 

iv 2 — — = constant. 

2a 

20. Explain why equation (23.6) remains valid in a gravitational 
field. 

21. (r,d,<j},t) are interpreted as spherical polar coordinates and 
time. A gravitational field is caused by a point electric charge at the 



GENERAL THEORY OF RELATIVITY 181 

pole. Assuming that the space-time metric is given by equation 
(51.10) and that the 4- vector potential for the electromagnetic field 
of the charge is given by Si = (0,0, 0,x), where x = x( r )> calculate the 
covariant components of the field tensor F tj from equation (23.6) and 
deduce the contravariant components F' J . Assuming that J' = 0, 
prove that Maxwell's equations are all satisfied if 

^ = ~ 2 .cW(ab), 
dr r 2 

where e is a constant. 

Calculate the elements of the mixed energy-momentum tensor 
from the equation 



-F ik F-, - 

An Jk \(m 



T'- = — F ik F, S'" F kl F, i 



and write down Einstein's equations for the gravitational field. Show 
that these are satisfied provided 



1 , ., 2m ye 2 
- = b = l-~ + ~-' 
a r c 2 r 2 



where m is a constant. 



Appendix 

Bibliography 

1. aharoni, J., The Special Theory of Relativity, Oxford University 
Press. 

2. bergmann, p. G., Introduction to the Theory of Relativity, 
Prentice-Hall. 

3. eddington, a. s., Mathematical Theory of Relativity, Cam- 
bridge University Press. 

4. einstein, a., The Meaning of Relativity, Princeton University 
Press. 

5. fock, v., Theory of Space, Time and Gravitation, Pergamon. 

6. landau, l. d. and lifshitz, e. m., The Classical Theory of 
Fields, Pergamon. 

7. mcconnell, a. j., Applications of the Absolute Differential 
Calculus, Blackie. 

8. mccrea, w. h., Relativity Physics, Methuen. 

9. mcvittie, g. c, General Relativity and Cosmology, Chapman 
and Hall. 

10. M0LLER, c, Theory of Relativity, Oxford University Press. 

11. pauli, w., Theory of Relativity, Pergamon. 

12. rainich^ g. y., Mathematics of Relativity, Wiley. 

13. rindler, w., Special Relativity, Oliver and Boyd. 

14. schrodinger, e., Space-Time Structure, Cambridge Uni- 
versity Press. 

15. sommerfeld, a., Electrodynamics, Academic Press. 

16. spain, b., Tensor-Calculus, Oliver and Boyd. 

17. synge, J. l., Relativity -The Special Theory, and Relativity -The 
General Theory, North-Holland. 

18. tolman, r. c., Relativity, Thermodynamics and Cosmology, 
Oxford University Press. 

19. weber, J., General Relativity and Gravitational Waves, Inter- 
science. 

20. weatherburn, c. e., Riemannian Geometry and Tensor 
Calculus, Cambridge University Press. 

21. weyl, h., Space, Time, Matter, Dover. 

182 



Index 



Aberration of light, 52 
Acceleration, Lorentz transforma- 
tion of, 51 
Aether, 6 

Affine connection, 99 
Affinity, 99 

metric, 121 

symmetric, 111, 119 

transformation of, 99 
Affinities, difference of, 101 
Atomic explosion, 47 

Bianchi identity, 114, 122 
Biot-Savart law, 67 



Current density, 59 

4-, 60, 172 

Lorentz transformation of, 77 
Curvature scalar, 124 
Curvature tensor, covariant, 121 

Riemann-Christoffel, 112 

symmetry of, 122 
Curvature tensor for a weak field, 
153 



De Sitter's universe, 179 
Dilation of time, 16, 38 
Divergence, 31, 123 
Doppler effect, 78 



Cartesian tensor, 27 

covariance and contravariance of, 
89 
Charge, equation of continuity for, 
59 

field of moving, 65 
Charge density, proper, 60 
Christoffel symbols, 120 
Clock, coordinate, 168 

standard, 168 
Clock paradox, 16 
Compton effect, 55 
Conjugate tensors, 93 
Continuity, equation of, 59, 74 
Coordinate lines, 82 
Coordinates, curvilinear, 82 

geodesic, 114 

spherical polar, 81 
Coordinates of an event, 144 
Coordinate surfaces, 82 
Copernicus, 5 
Cosmical constant, 150 
Cosmic ray particles, 42 

decay of, 16 
Covariant derivative, 98 
Covariant differentiations, commut- 

ativity of, 130 
Curl, 35, 130 



Einstein's equation, 46 
Einstein's law of gravitation, 150 
Einstein's tensor, 125, 150 
Einstein's universe, 180 
Electric charge, 60 
Electric intensity, 62 

Lorentz transformation of, 65 
Electromagnetic field tensor, 63, 171 
Energy, equivalence of mass and, 46 

kinetic, 45 

particle's internal, 47 
Energy current density, 73 
Energy density in an electromagne- 
tic field, 71 
Energy-momentum tensor, 147 

electromagnetic field, 70 

kinetic, 76 
Eotvos, 139 

Equivalence, principle of, 139 
Euclidean space, 10, 81, 84, 98, 117, 

119,121,143,150 
Event, 8 

coordinates of, 144 



Fitzgerald contraction, 14, 21, 142 
Force, 2, 43 
centrifugal, 3, 21, 139, 142 



183 



184 



INDEX 



Coriolis, 3,21,139, 142 

fictitious, 3 

4-, 44 

inertial, 138 

Lorentz transformation of, 48 

rate of doing work by, 45, 49 
Force density, 68 
Fundamental tensor, 28, 89, 115 

covariant derivative of, 103 
Future, absolute, 19 

Galactic masses, field due to, 140 
Galilean law of inertia, 147 
Galilean transformation, special, 13 
General principle of relativity, 138 
Geodesic, 126, 134, 147 

null, 128, 166 
Gradient, 30, 87 
Gravitational constant, 154 
Gravitational field of point charge, 

181 
Gravitational field outside a spheri- 
cal mass, 159 
Green's theorem, 72 

Hamiltonian, 50, 79 
Hamilton's equations, 50 
Hypersphere, 136 

Index, dummy, 26 

free, 26 

raising and lowering, 116 
Inertial forces, 138 
Inertial frame, 2 

local, 143 

quasi-, 151 
Interval, 115 

proper time, 17, 144 

spacelike, 18 

timelike, 18 
Interval between events, 145 
Intrinsic derivative, 131 
Invariant, 29, 87 

covariant derivative of, 102 

relative, 95 
Invariant density, covariant deriv- 
ative of, 105 

parallel displacement of, 104 
Invariant field, 29, 87 



Irreducible gravitational field, 143 

Kronecker deltas, 25, 89 

Lagrange's equations, 50 

Laplacian, 124 

Length, 15 

Levi-Civita tensor density, 32, 94 

covariant derivative of, 106 
Light cone, 20 
Light pulse, wavefront of, 9 
Light ray, gravitational deflection 

of, 166 
Light waves, velocity of, 6 
Lorentz force, 68 
Lorentz transformation, general, 1 1 

inverse, 14 

special, 13 

Mach's principle, 141 
Magnetic intensity, 62 

Lorentz transformation of, 65 
Mass, 2 

conservation of, 42 

density of proper, 74 

equation of continuity for proper, 
74 

equivalence of energy and, 46 

inertial and gravitational, 141 

invariance of, 4 

proper, 42 

proper density of proper, 74 

rest, 42 

variable proper, 48 
Maxwell's equations, 6, 61, 64, 172 
Maxwell's stress tensor, 71 
Mercator's projection, 136 
Mercury, 164, 165 
Metric, 84, 115 

form invariance of, 155 

Schwarzschild, 159, 174, 177 

spherically symmetric, 155 
Metrical connection, 115 
Metric of a conical surface, 134 
Metric of a gravitational field, 147 
Metric of a spherical surface, 85, 

132, 136 
Michelson-Morley experiment, 6 
Minkowski, 9 
Minkowski space-time, 10, 18 



INDEX 



185 



Momentum, 3 
conservation of, 3, 41, 73, 77 
4-, 42 
Lorentz transformation of, 43 

Newtonian potential, 149, 153, 159, 

170 
Newton's first law, 1, 8 
Newton's law of gravitation, 153 
Newton's laws, covariance of, 4 
Newton's second law, 3 
Newton's third law, 3, 44 
Non-Euclidean space, 143 

Operator, substitution, 26 
Orbit, planetary, 55, 160 

equation of, 163 
Orthogonal transformation, 23, 81 

Parallel displacement of vectors, 97, 

109 
Particle, Hamiltonian for, 50 

internal energy of, 47 

Lagrangian for, 50 
Particles, collision of, 3, 41, 54, 55, 

56 
Past, absolute, 20 
Perihelion, advance of, 165 
Photon, 53, 55 
Physical space, 137 
Planck's constant, 55 
Poisson's equation, 154 
Poynting's vector, 71 
Present, conditional, 20 
Privileged observers, 137 
Product, inner, 31, 92 

outer, 91 

scalar, 31 

vector, 34, 130 

Quotient theorem, 92, 111 

Relative invariant, 95 
Relative tensor, 94 

covariant derivative of, 108 
Ricci tensor, 113 

divergence of, 124 
Ricci tensor for a weak field, 1 53 
Riemannian space, 84, 115 
Ritz, 7 



Rocket motion, 53 

Scalar, 29, 87 
Scalar potential, 61 
Simultaneity, 15 
Sirius, companion of, 171 
Space-time continuum, 144 
Special principle of relativity, 5 
Spectral lines, gravitational dis- 
placement of, 168 
Summation convention, 25 
Synchronization of clocks, 7, 144 

Tangent, unit, 126 

zero, 128 
Tensor, Cartesian, 27 

contraction of, 31, 91 

contravariant, 88 

covariant, 88 

covariant derivative of, 103 

fundamental, 28, 89 

mixed, 88 

parallel displacement of, 97 

rank of, 27 

relative, 94 

skew-symmetric, 28, 90 

symmetric, 28, 90 
Tensor density, 32, 94 

covariant derivative of, 106, 108 

Levi-Civita, 32, 94 
Tensor equation, 29, 90 
Tensor field, 30 

Tensor product, covariant deriv- 
ative of, 104 
Tensors, addition of, 27, 90 

multiplication of, 27, 90 
Tensor sum, covariant derivative of, 

103 
Time, absolute, 13 

Vector, axial, 34, 35 
Cartesian, 27 
contravariant, 86 
covariant, 87 

covariant and contravariant com- 
ponents of, 116, 134 
displacement, 26 
free, 86 

infinitesimal displacement, 86 
magnitude of, 31, 118 



186 



INDEX 



Vector field, 87 

Vector multiplication, laws of, 34 

Vector potential, 61 
4-, 61 

Vector product, 34, 130 

Vectors, angle between, 31, 118 
orthogonal, 31, 118 
scalar product of, 31, 118 

Velocities, composition of, 53 



Velocity vector, 38 
4-, 39 
Lorentz transformation of, 40 



Weak gravitational field, 151 
Wilson cloud chamber, 54 
World-line, 18 
World-lines of free particles, 146 



This text is based upon lectures given to final year Honours undergraduates. 
After a discussion of the special relativity principle and its relation to 
Newton's laws and Maxwell's equations, Cartesian tensors are introduced. 
Tensors are employed systematically to develop the Special theory in a four- 
dimensional pseudo-Euclidean space, and the tensor calculus is then gen- 
eralized to include a statement of the basic ideas of the General theory of 
relativity. Each chapter concludes with a set of exercise and there is a select 
bibliography to assist students to plan their further reading. 

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