The Classical Theory of Fields

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The last three chapters are devoted to the presentation of the theory of gravitational ..... signal at point x2 y2 z2 at the moment of time t 2 . ..... The inverse formulas, expressing x', y', z\ t' in terms of x, y, z, t, are most ... For velocities V small compared with thevelocity of light, we can use in place of (4.3) ...... Mx)] = Vu^.6(x-at),.
Landau Lifshitz

The Classical Theory of Fields Third Revised English Edition

Course of Theoretical Physics Volume 2

L.

D. Landau (Deceased) and E.

Institute of Physical

USSR Academy

Problems

of Sciences

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Pergamon Press

ML

Lifshitz

Course of Theoretical Physics

Volume 2

THE CLASSICAL THEORY OF FIELDS Third Revised English Edition

LANDAU

L,

D.

E,

M. LIFSHITZ

(Deceased) and

Institute of Physical

Problems,

USSR Academy

of Sciences

This third English edition of the book has been translated from the fifth revised and extended Russian edition

1967. Although much been added, the subject matter is basically that of the second English translation, being a systematic presentation of electromagnetic and gravitational fields for postgraduate courses. The largest published

new

in

material has

additions are four new sections entitled "Gravitational Collapse", "Homogeneous Spaces", "Oscillating Regime of Approach to a Singular Point", and "Character of the Singularity in the General Cosmological Solution of the Gravitational Equations" These additions cover some of the main areas of research in general relativity.

Mxcvn

COURSE OF THEORETICAL PHYSICS Volume 2

THE CLASSICAL THEORY OF FIELDS

OTHER TITLES IN THE SERIES Vol.

1.

Vol.

3.

Mechanics Quantum Mechanics

—Non

Vol. 4. Relativistic Vol.

5.

Statistical Physics

Vol.

6.

Fluid Mechanics

Vol. 7. Vol.

8.

Relativistic

Theory

Quantum Theory

Theory of Elasticity Electrodynamics of Continuous Media

Vol. 9. Physical Kinetics

THE CLASSICAL THEORY OF FIELDS Third Revised English Edition L.

D.

LANDAU AND

Institute for Physical Problems,

E.

M. LIFSHITZ

Academy of Sciences of the

Translated from the Russian

by

MORTON HAMERMESH University of Minnesota

PERGAMON PRESS OXFORD NEW YORK TORONTO SYDNEY BRAUNSCHWEIG •



'

U.S.S.R.

Pergamon Press Ltd., Headington Hill Hall, Oxford Pergamon Press Inc., Maxwell House, Fairview Park, Elmsford,

New York

10523

Pergamon of Canada Ltd., 207 Queen's Quay West, Toronto Pergamon Press (Aust.) Pty. Ltd., 19a Boundary Street, Rushcutters Bay, N.S.W. 2011, Australia Vieweg & Sohn GmbH, Burgplatz 1, Braunschweig Copyright

©

1971 Pergamon Press Ltd.

All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of Pergamon Press Ltd. First English edition 1951

Second English edition 1962 Third English edition 1971 Library of Congress Catalog Card No. 73-140427 Translated from the 5th revised edition

of Teoriya Pola, Nauka, Moscow, 1967

Printed in Great Britain by THE WHITEFRIARS PRESS LTD., LONDON AND TONBRIDGE

08 016019

1

1

CONTENTS Preface to the Second English Edition Preface to the Third English Edition

ix

x

Notation

Chapter

1.

xi

The Principle of Relativity

1

1 Velocity of propagation of interaction 2 Intervals 3 Proper time 4 The Lorentz transformation 5 Transformation of velocities 6 Four-vectors 7 Four-dimensional velocity

Chapter

2.

1

3

7 9 12 14 21

Relativistic Mechanics

24

8 The principle of least action 9 Energy and momentum 10 Transformation of distribution functions 1 Decay of particles 12 Invariant cross-section 13 Elastic collisions of particles 14 Angular momentum

Chapter 15 16 17 18 19

20 21

22 23 24 25

Charges in Electromagnetic Fields

43

Elementary particles in the theory of relativity Four-potential of a field Equations of motion of a charge in a field Gauge invariance Constant electromagnetic field Motion in a constant uniform electric field Motion in a constant uniform magnetic field Motion of a charge in constant uniform electric and magnetic fields The electromagnetic field tensor Lorentz transformation of the field Invariants of the field

Chapter 26 27 28 29 30

3.

24 25 29 30 34 36 40

4.

The Electromagnetic Field Equations

The first pair of Maxwell's equations The action function of the electromagnetic The four-dimensional current vector The equation of continuity The second pair of Maxwell equations

31 Energy density

and energy flux 32 The energy-momentum tensor 33 Energy-momentum tensor of the electromagnetic field 34 The virial theorem 35 The nergy-momentum tensor for macroscopic bodies

53 55

60 62 63 66



field

43

44 46 49 50 52

66 67 69 71 73 75 77 80 84 85

CONTENTS

VI

Chapter 36 37 38 39 40

5.

Constant Electromagnetic Fields

88

Coulomb's law

88 89

Electrostatic energy of charges

The field of a uniformly moving charge Motion in the Coulomb field The dipole moment 41 Multipole moments 42 43 44 45

System of charges in an external Constant magnetic field Magnetic moments Larmor's theorem

Chapter 46 47 48 49 50

6.

96 97 100

field

101

103 105

Electromagnetic Waves

108

The wave equation Plane waves

Monochromatic plane waves Spectral resolution Partially polarized light

The Fourier

resolution of the electrostatic 52 Characteristic vibrations of the field

51

91 93

Chapter

7.

field

The Propagation of Light

129

53 Geometrical optics

54 55 56 57 58 59 60

Intensity

The angular eikonal Narrow bundles of rays Image formation with broad bundles of rays The limits of geometrical optics Diffraction

Fresnel diffraction

61 Fraunhofer diffraction

Chapter

8.

The Field of Moving Charges

66 67 68 69 70 71

9.

Radiation of Electromagnetic Waves

The

field of a system of charges at large distances Dipole radiation Dipole radiation during collisions Radiation of low frequency in collisions Radiation in the case of Coulomb interaction Quadrupole and magnetic dipole radiation The field of the radiation at near distances Radiation from a rapidly moving charge Synchrotron radiation (magnetic bremsstrahlung) Radiation damping Radiation damping in the relativistic case

72 73 74 75 76 77 Spectral resolution of the radiation in the 78 Scattering by free charges 79 Scattering of low-frequency waves 80 Scattering of high-frequency waves

129 132 134 136 141

143 145 150 153

158

62 The retarded potentials 63 The Lienard-Wiechert potentials 64 Spectral resolution of the retarded potentials 65 The Lagrangian to terms of second order

Chapter

108 110 114 118 119 124 125

ultrarelativistic case

158 160 163 165

170 170 173 177 179 181 188 190 193 197 203 208 211

215 220 221

CONTENTS Chapter

10.

Vii

Particle in a Gravitational Field

225

81 Gravitational fields in nonrelativistic mechanics 82 The gravitational field in relativistic mechanics

83 84 85 86 87 88 89 90

Curvilinear coordinates Distances and time intervals Covariant differentiation The relation of the Christoffel symbols to the metric tensor Motion of a particle in a gravitational field The constant gravitational field Rotation The equations of electrodynamics in the presence of a gravitational

Chapter 91

11.

field

The Gravitational Field Equations

The curvature

258

tensor

92 Properties of the curvature tensor 93 The action function for the gravitational

94 95 96 97 98 99

field

The energy-momentum tensor The gravitational field equations Newton's law

The

centrally symmetric gravitational field

Motion in a centrally symmetric gravitational The synchronous reference system

field

100 Gravitational collapse 101

The energy-momentum pseudotensor

1 02

Gravitational waves 103 Exact solutions of the gravitational field equations depending on one variable 104 Gravitational fields at large distances from bodies 105 Radiation of gravitational waves 106 The equations of motion of a system of bodies in the second approximation

Chapter

12.

Cosmological Problems

107 Isotropic space 108 Space-time metric in the closed isotropic model 109 Space-time metric for the open isotropic model 110 The red shift 111 Gravitational stability of an isotropic universe 112 Homogeneous spaces 113 Oscillating regime of approach to a singular point 114 The character of the singularity in the general cosmological solution of the gravitational equations

Index

225 226 229 233 236 241 243 247 253 254

258 260 266 268 272 278 282 287 290 296 304 311

314 318 323 325

333 333 336 340 343 350 355 360

367 371

PREFACE

TO THE SECOND ENGLISH EDITION This book

is devoted to the presentation of the theory of the electromagnetic and gravitational fields. In accordance with the general plan of our "Course of Theoretical Physics", we exclude from this volume problems of the electrodynamics of continuous

media, and

restrict the exposition to "microscopic electrodynamics", the electrodynamics of the vacuum and of point charges. complete, logically connected theory of the electromagnetic field includes the special theory of relativity, so the latter has been taken as the basis of the presentation. As the starting-point of the derivation of the fundamental equations we take the variational

A

principles,

which make possible the achievement of maximum generality, unity and simplicity

of the presentation.

The

last three chapters are

devoted to the presentation of the theory of gravitational The reader is not assumed to have any previous knowledge of tensor analysis, which is presented in parallel with the development of the fields, i.e.

the general theory of relativity.

theory.

The present edition has been extensively revised from the first English edition, which appeared in 1951. We express our sincere gratitude to L. P. Gor'kov, I. E. Dzyaloshinskii and L. P. Pitaevskii for their assistance in checking formulas.

Moscow, September 1961

L.

D. Landau, E. M. Lifshitz

PREFACE

TO THE THIRD ENGLISH EDITION This third English edition of the book has been translated from the revised and extended Russian edition, published in 1967. The changes have, however, not affected the general plan or the

of presentation. change is the shift to a different four-dimensional metric, which required the introduction right from the start of both contra- and covariant presentations of the four- vectors. We thus achieve uniformity of notation in the different parts of this book and also agreement with the system that is gaining at present in universal use in the physics literature. The advantages of this notation are particularly significant for further appli-

An

cations in I

style

essential

quantum

theory.

should like here to express

valuable

many

comments about

my

the text

sincere gratitude to all

and

my

colleagues

especially to L. P. Pitaevskii, with

who have made

whom

I

discussed

problems related to the revision of the book.

For the new English edition, it was not possible to add additional material throughout the text. However, three new sections have been added at the end of the book, §§ 112-114.

April, 1970

E.

M.

Lifshitz

NOTATION Three-dimensional quantities

Three-dimensional tensor indices are denoted by Greek Element of volume, area and length: dV, di, d\ Momentum and energy of a particle: p and $

Hamiltonian function:

letters

2tf

and vector potentials of the electromagnetic Electric and magnetic field intensities: E and Charge and current density p and j Electric dipole moment: d Magnetic dipole moment: m Scalar

field:

and

A

H

:

Four-dimensional quantities

Four-dimensional tensor indices are denoted by Latin values 0,

1, 2,

letters

i,

k,

I,

.

.

.

and take on the

3

We use the metric with signature

(H

)



Rule for raising and lowering indices see p. 14 Components of four-vectors are enumerated in the form A 1 = (A Antisymmetric unit tensor of rank four is e iklm where e 0123 = ,

,

1

A) (for the definition

see

P- 17)

= (ct, r) = dx \ds Momentum four-vector: p = {Sic, Current four-vector j* = (cp, pi) Radius four-vector:

x*

Velocity four- vector: u l

l

p)

:

Four-potential of the electromagnetic

Electromagnetic

F

ik

to the

field four-tensor

four-tensor

T

A =

F = j± ik

components of E and H,

Energy-momentum

field:

ik

1

($,

A)



{ (for the relation of the components of

see p. 77)

(for the definition of its

components, see

p. 78)

CHAPTER

1

THE PRINCIPLE OF RELATIVITY § 1. Velocity of propagation of interaction

For the description of processes taking place reference.

in nature,

one must have a system of

By a system of reference we understand a system of coordinates serving to indicate

the position of a particle in space, as well as clocks fixed in this system serving to indicate the time.

is

There exist systems of reference in which a freely moving body, i.e. a moving body which not acted upon by external forces, proceeds with constant velocity. Such reference systems

are said to be inertial.

one of them is an inertial system, then clearly the other is also inertial (in this system too every free motion will be linear and uniform). In this way one can obtain arbitrarily many inertial systems of reference, moving uniformly relative to one another. Experiment shows that the so-called principle of relativity is valid. According to this If

two reference systems move uniformly

relative to

each other, and

if

principle all the laws of nature are identical in all inertial systems of reference. In other

words, the equations expressing the laws of nature are invariant with respect to transformations of coordinates and time from one inertial system to another. This means that the equation describing any law of nature, different inertial reference systems, has

The

when

written in terms of coordinates

and time

in

one and the same form.

is described in ordinary mechanics by means of a which appears as a function of the coordinates of the inter-

interaction of material particles

potential energy of interaction,

acting particles. It

is

easy to see that this

manner of

describing interactions contains the

assumption of instantaneous propagation of interactions. For the forces exerted on each of the particles by the other particles at a particular instant of time depend, according to this description, only on the positions of the particles at this one instant. A change in the position of any of the interacting particles influences the other particles immediately.

However, experiment shows that instantaneous interactions do not exist in nature. Thus a mechanics based on the assumption of instantaneous propagation of interactions contains within itself a certain inaccuracy. In actuality, if any change takes place in one of the interacting bodies, time. It

is

it

will influence the other bodies only after the lapse

only after this time interval that processes caused by the

of a certain interval of initial

change begin to

take place in the second body. Dividing the distance between the two bodies by this time interval,

We

we

obtain the velocity of propagation of the interaction. strictly speaking, be called the

note that this velocity should,

propagation of interaction.

It

maximum

velocity of

determines only that interval of time after which a change

occurring in one body begins to manifest

itself in

another. It

is

clear that the existence of

a

THE PRINCIPLE OF RELATIVITY

2

§

1

maximum

velocity of propagation of interactions implies, at the same time, that motions of bodies with greater velocity than this are in general impossible in nature. For if such a motion could occur, then by means of it one could realize an interaction with a velocity exceeding

the

maximum

possible velocity of propagation of interactions.

Interactions propagating

from the

sent out first

from one particle to another are frequently called "signals", and "informing" the second particle of changes which the

first particle

has experienced. The velocity of propagation of interaction

is

then referred to as the

signal velocity.

From

the principle of relativity

of interactions

is

the

tion of interactions

same

is

and

its

follows in particular that the velocity of propagation

Thus the

a universal constant. This constant velocity (as

also the velocity of light in letter c,

it

in all inertial systems of reference.

empty

numerical value

space.

The

velocity of light

is

velocity of propaga-

we

shall

show

later) is

usually designated by the

is

c

=

2.99793 x 10

10

cm/sec.

(1.1)

The large value of this velocity explains the fact that in practice classical mechanics appears to be sufficiently accurate in most cases. The velocities with which we have occasion compared with the velocity of light that the assumption that the does not materially affect the accuracy of the results. The combination of the principle of relativity with the finiteness of the velocity of propagation of interactions is called the principle of relativity of Einstein (it was formulated by to deal are usually so small

latter is infinite

Einstein in 1905) in contrast to the principle of relativity of Galileo, which infinite velocity

The mechanics based on

the Einsteinian principle of relativity (we shall usually refer to

simply as the principle of relativity) velocities

the effect

was based on an

of propagation of interactions.

is

called relativistic. In the limiting case

when

it

the

of the moving bodies are small compared with the velocity of light we can neglect on the motion of the finiteness of the velocity of propagation. Then relativistic

mechanics goes over into the usual mechanics, based on the assumption of instantaneous propagation of interactions; this mechanics is called Newtonian or classical. The limiting

from relativistic to classical mechanics can be produced formally by the transition to the limit c -* oo in the formulas of relativistic mechanics.

transition

In classical mechanics distance

is already relative, i.e. the spatial relations between depend on the system of reference in which they are described. The statement that two nonsimultaneous events occur at one and the same point in space or, in general, at a definite distance from each other, acquires a meaning only when we indicate the system of reference which is used. On the other hand, time is absolute in classical mechanics in other words, the properties of time are assumed to be independent of the system of reference; there is one time for all reference frames. This means that if any two phenomena occur simultaneously for any one

different events

;

observer, then they occur simultaneously also for all others. In general, the interval of time between two given events must be identical for all systems of reference. It is easy to show, however, that the idea of an absolute time is in complete contradiction to the Einstein principle of relativity.

For

this it is sufficient to recall that in classical

mechanics, based on the concept of an absolute time, a general law of combination of velocities is valid, according to

the (vector)

sum of

which the velocity of a composite motion

is

simply equal to

the velocities which constitute this motion. This law, being universal,

should also be applicable to the propagation of interactions.

From

this it

would follow

C

§

VELOCITY OF PROPAGATION OF INTERACTION

2

that the velocity of propagation

must be

3

different in different inertial systems of reference,

in contradiction to the principle of relativity. In this matter experiment completely confirms

performed by Michelson (1881) showed its direction of propagation; whereas mechanics the velocity of light should be smaller in the direction of the

the principle of relativity. Measurements

first

complete lack of dependence of the velocity of light on according to classical

motion than in the opposite direction. Thus the principle of relativity leads to the result

earth's

differently in different systems of reference.

interval has elapsed

is

not absolute. Time elapses definite time

between two given events acquires meaning only when the reference statement applies is indicated. In particular, events which are simul-

frame to which this taneous in one reference frame

To

that time

Consequently the statement that a

will

not be simultaneous in other frames.

clarify this, it is instructive to consider the following simple

example. Let us look at

two inertial reference systems K and K' with coordinate axes XYZ and X' Y'Z' respectively, where the system K' moves relative to K along the X(X') axis (Fig. 1).

B— A— -1

1

1

X'

x

Y

Y'

Fig.

Suppose

signals start out

from some point

1.

A on

Since the velocity of propagation of a signal in the

equal (for both directions) to

the

X'

axis in

K' system,

two opposite

directions.

as in all inertial systems,

B and

is

from A, at one and the same time (in the K' system). But it is easy to see that the same two events (arrival of the signal at B and C) can by no means be simultaneous for an observer in the K c,

the signals will reach points

C, equidistant

system. In fact, the velocity of a signal relative to the A" system has, according to the principle

K

of relativity, the same value c, and since the point B moves (relative to the system) toward the source of its signal, while the point C moves in the direction away from the signal (sent from A to C), in the AT system the signal will reach point B earlier than point C. Thus the principle of relativity of Einstein introduces very drastic and fundamental changes in basic physical concepts. The notions of space and time derived by us from our daily experiences are only approximations linked to the fact that in daily life we happen to deal only with velocities which are very small compared with the velocity of light. § 2. Intervals shall frequently use the concept of an event. An event is described by occurred and the time when it occurred. Thus an event occurring in a certain material particle is defined by the three coordinates of that particle and the time when the event occurs.

In what follows

the place where

we

it

It is frequently useful for

space,

on

reasons of presentation to use a fictitious four-dimensional

the axes of which are

marked

three space coordinates

and the

time. In this space

4

THE PRINCIPLE OF RELATIVITY

§

2

events are represented by points, called world points. In this fictitious four-dimensional space there corresponds to each particle a certain line, called a world line. The points of this line

determine the coordinates of the particle at

uniform

all

moments of time.

motion there corresponds a

easy to show that to a

It is

world line. We now express the principle of the invariance of the velocity of light in mathematical form. For this purpose we consider two reference systems and K' moving relative to each other with constant velocity. We choose the coordinate axes so that the axes and X' coincide, while the Y and Z axes are parallel to Y' and Z'; we designate the time in the systems and K' by t and t'. Let the first event consist of sending out a signal, propagating with light velocity, from a point having coordinates x t y ± z x in the system, at time 1 1 in this system. We observe the propagation of this signal in the system. Let the second event consist of the arrival of the signal at point x 2 y 2 z 2 at the moment of time t 2 The signal propagates with velocity c; the distance covered by it is therefore c^ — 1 2 ). On the other hand, this same distance equals [(x 2 — 1 ) 2 + (y 2 -y 1 ) 2 + (z 2 —z 1 ) 2 ] i Thus we can write the following relation between the coordinates of the two events in the K system: particle in

rectilinear

straight

K

X

K

K

K

.

.

(x 2

The same two

- Xl ) 2 + (y 2 - ytf + izi-tiY-fih-h) 2 =

events,

0-

(2-1)

the propagation of the signal, can be observed

i.e.

from the K'

system:

Let the coordinates of the

x 2 y'2 z'2 t 2 Since the

first

K' system be xi y[ z[ t\, and of the second: same in the K and K' systems, we have, similarly

event in the

velocity of light

.

is

the

to (2.1):

{A-AYHy'z-ytfHz'z-Af-c^-ttf = o. If

xx y x z t



and x 2 y 2 z 2

=

12

are the coordinates of any 2

2

2

two

(2.2)

events, then the quantity 2

2

(2-3) (^-*i) -(*2-*i) -(y2-yi) -(z2-Zi) 3* is called the interval between these two events. Thus it follows from the principle of invariance of the velocity of light that if the interval between two events is zero in one coordinate system, then it is equal to zero in all other

S12

[c

systems. If

two events are

infinitely close to

ds

The form of expressions

(2.3)

2

=

and

each other, then the interval ds between them 2

c dt

2

-dx -dy - dz 2

2

is

2

(2.4)

.

permits us to regard the interval, from the formal

(2.4)

point of view, as the distance between two points in a fictitious four-dimensional space

(whose axes are labelled by x, y, z, and the product ct). But there is a basic difference between the rule for forming this quantity and the rule in ordinary geometry: in forming the square of the interval, the squares of the coordinate differences along the different axes are

summed, not with the same

As

already shown,

if ds =

but rather with varying signs.f in any other system. in one inertial system, then ds' =

sign,

the other hand, ds and ds' are infinitesimals of the it

follows that ds

2

and

ds'

2

t

coefficient

order.

From

these

On

two conditions

must be proportional to each other: ds

where the

same

2

=

ads'

2

a can depend only on the absolute value of the

relative velocity of the

The four-dimensional geometry described by the quadratic form (2.4) was introduced by H. Minkowski,

in connection with the theory of relativity. This

euclidean geometry.

geometry

is

called pseudo-euclidean, in contrast to ordinary

§

INTERVALS

2

5

cannot depend on the coordinates or the time, since then different moments in time would not be equivalent, which would be in contradiction to the homogeneity of space and time. Similarly, it cannot depend on the direction of the relative velocity, since that would contradict the isotropy of space. Let us consider three reference systems K, X ,K2 and let V± and V2 be the velocities of

two

inertial systems. It

points in space

systems

K

x

and

and

different

K K2 relative to K. We then have ds

Similarly

we can

2

=

ds

a{Vi)ds\,

,

:

= a(V2 )ds 22

2

.

write

ds\

=

a(Vx2 )ds\,

where V12 is the absolute value of the velocity of with one another, we find that we must have -777\

=

K2 relative to K

x

.

Comparing these relations

a(V12 ).

(2.5)

V

V12 depends not only on the absolute values of the vectors x and V 2 but also on the angle between them. However, this angle does not appear on the left side of formula (2.5). It is therefore clear that this formula can be correct only if the function a(V) reduces to a But

constant, which

is

,

equal to unity according to this same formula.

Thus, ds

and from the equality of the intervals: s

2

=

ds'

2 ,

infinitesimal intervals there follows the equality of finite

= s'.

Thus we arrive

at a very important result: the interval

of reference,

inertial systems

system to any other. This invariance

is

between two events is the same in

all

invariant under transformation from one inertial

it is

i.e.

the mathematical expression of the constancy of the

velocity of light.

Again

let

x^y^Zxt^ and x 2 y 2 z 2

reference system K.

Does there same point

occur at one and the We introduce the notation

h-h = hi, Then

be the coordinates of two events in a certain system K\ in which these two events

t2

exist a coordinate

in space ?

(x 2

-x

the interval between events in the

in the

2 2 +(y 2 -y 1 ) +(z 2 -z 1 ) =

K system

i 12

_ —

r 2,2 l 12 C

~'2 s 12

_ —

_2,/2 c '12

2

and

2 1)

K' system

\\ 2 .

is

_;2

Ixi j/2 f

12'

whereupon, because of the invariance of intervals, 2 2 _;2 _ _//2 l l C f — c 2./2 Ii2

We I'12

want the two events to occur = 0. Then ^12

\2' \2 H2 same point in

at the

=

£ ^12

'l2

== C ^12

^

the

K' system,

that

is,

we

require

^*

Consequently a system of reference with the required property exists if s\ 2 > 0, that is, if is a real number. Real intervals are said to be timelike. Thus, if the interval between two events is timelike, then there exists a system of reference

the interval between the two events

in

which the two events occur

at

one and the same place. The time which elapses between

THE PRINCIPLE OF RELATIVITY the two events in this system

§2

is

S

t'i2

= Uchl 2 -li 2 = ^.

(2.6)

two events occur in one and the same body, then the interval between them is always which the body moves between the two events cannot be greater than ct 12 since the velocity of the body cannot exceed c. So we have always If

timelike, for the distance ,

l

Let us

12


y '"'•

z

=z

""•

the required transformation formula. It

is

>

— — "7t=

f

(4.3)

-

2

called the Lorentz transformation,

and is of

fundamental importance for what follows. t

Note

that to avoid confusion

inertial systems,

and v for the

we

shall

velocity of a

always use

moving

V to

particle,

signify the constant relative velocity

not necessarily constant.

of two

:

§

THE LORENTZ TRANSFORMATION

4

The

inverse formulas, expressing x', y', z\

-V (since

t'

11

in terms of x, y, z,

t,

are

most easily obtained

-V

K

relative to the K' system moves with velocity system). The same formulas can be obtained directly by solving equations (4.3) for x', y', z', t'.

by changing

V

to

the

easy to see from (4.3) that on making the transition to the limit c -» co and classical mechanics, the formula for the Lorentz transformation actually goes over into the Galileo It is

transformation.

For V > c in formula (4.3) the coordinates x, t are imaginary; this corresponds to the fact that motion with a velocity greater than the velocity of light is impossible. Moreover, one cannot use a reference system moving with the velocity of light—in that case the denominators in (4.3) would go to zero. For velocities V small compared with the velocity of light, we can use in place of (4.3) the approximate formulas

x

=

x'

+ Vf,

v

=

z

v\

=

z',

t

V

=

t'+-^x'.

(4.4)

Suppose there is a rod at rest in the K system, parallel to the X axis. Let its length, measured in this system, be Ax = x 2 -x 1 (x 2 and Xj are the coordinates of the two ends of the rod in the K system). We now determine the length of tliis rod as measured in the K' system. To do this we must find the coordinates of the two ends of the rod (x'2 and xi) in this system at one and the same time t'. From (4.3) we find: Xi

_ —

x[

+ Vt'

x2

^=«



V -?

J

1

The length of

the rod in the

K' system

is

x'2

Ax'

=

+ Vt' 1

x^-x'j

;

V

subtracting

x x from x 2 we

find

,

Ax'

Ax =

J -£ The proper length of a rod is its length in a reference system in which it is at rest. Let us it by l = Ax, and the length of the rod in any other reference frame K' by /. Then

denote

(=! Thus a rod has in a system in

its

which

0N/l-J

(4.5)

greatest length in the reference system in it

moves with

velocity

V is

which

it is

decreased by the factor

at

rest. Its

VI - V

2

/c

l

ength

2 .

This

Lorentz contraction. Since the transverse dimensions do not change because of its motion, the volume "T of a body decreases according to the similar formula result of the theory

of

relativity is called the

/

where y*

is

V2

the proper volume of the body.

we can obtain anew the results already known to us concerning the proper time (§ 3). Suppose a clock to be at rest in the K' system. We take two events occurring at one and the same point x', y', z' in space in the K' system. The time between these events in the K' system is Af' = t'2 -t\. Now we find the time At which

From

the Lorentz transformation

12

THE PRINCIPLE OF RELATIVITY

elapses between these

two events

K system.

in the

From

V

(4.3),

we have

V

'2+ -2*'

*i+-2*' C

1

c

C

=

t2

V

§ 5

V

2

1

c

2

one from the other,

or, subtracting

=

-t
oo, they go over into the formulas vx = v'x + V, v = v' vz = v'z of classical mechanics. y,

y

In the special case of motion of a particle parallel to the

Then

=

v' y

v'z

=

0, v'x

=

easy to convince oneself that the

v,

vy

=

vx

=

0.

(5.2)

V' + v'-*

sum of two

velocities

each smaller than the velocity

again not greater than the light velocity.

is

For a

=

+V

v

= 1

of light

vx

so that

i/,

v

It is

X axis,

velocity

arbitrary),

V

we have approximately,

vx

=

2

(

v'x

than the velocity of light (the velocity v can be

significantly smaller

v'

to terms of order V/c:

\

+ V yl--JL}>

vy

V

=

V'y- V 'A

^

v*

=