Einstein’s theory of unified fields 9781306972789, 1306972787, 9781315779218, 1315779218, 9781317698784, 1317698789, 9781317698791, 1317698797, 9781317698807, 1317698800

Table of contents :
Introduction : purpose and methods of unified theories --
1. Mathematical introduction --
2. Field equations, variational principles, conservation equations --
3. The first group of Einstein's equations : expression of the affine connection as a function of the fields --
4. The second group of Einstein's equations --
5. Spherically symmetric solution --
6. The field and the sources --
7. Some problems raised by the unified field theory.

Citation preview

ROUTLEDGE LIBRARY EDITIONS: 20TH CENTURY SCIENCE

Volume 19

EINSTEIN'S THEORY OF UNIFIED FIELDS

This page intentionally lefi blank

EINSTEIN'S THEORY OF UNIFIED FIELDS

M. A. TONNELAT Translated from the French by RICHARD AKERIB

First published in English in 1966 Second edition published in 1982 Published as La Théorie du champ unifié d’Einstein et quelques-uns de ses développements in French. This edition published in 2014 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN and by Routledge 711 Third Avenue, New York, NY 10017 Routledge is an imprint of the Taylor & Francis Group, an informa business © 1966 Gordon and Breach, Science Publishers, Inc. © 1955 Gauthier-Villars, French edition All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-415-73519-3 (Set) eISBN: 978-1-315-77941-6 (Set) ISBN: 978-1-138-01362-9 (Volume 19) eISBN: 978-1-315-77921-8 (Volume 19) Publisher’s Note The publisher has gone to great lengths to ensure the quality of this book but points out that some imperfections from the original may be apparent. Disclaimer The publisher has made every effort to trace copyright holders and would welcome correspondence from those they have been unable to trace.

EINSTEIN'S THEORY OF UNIFIED FIELDS M. A. Tonne/at Mattre de Conferences

ala Sorbonne

With a Preface by Andre L ichnerowicz

Translatedfrom the French by Richard Akerib

GORDON AND BREACH

New York

Science Publishers

London

Paris

Copyright

©

1966 by Gordon and Breach, Science Publishers, Inc.

Gordon and Breach, Science Publishers, Inc. One Park A venue New York, NY 10016

Gordon and Breach Science Publishers Ltd. 42 William IV Street London WC2N 4DE

Gordon & Breach 58, rue Lhomond 75005 Paris

First Published September 1966 Second Printing February 1982

French edition originally published as La Theorie du champ unifie d' Einstein et quelques-uns de ses developpements. Copyright © Gauthier-Villars, 1955.

Library of Congress Catalog Card Number: 65-16831. ISBN0677008104. All rights reserved. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage or retrieval system, without permission in writing from the publishers. Printed in the United States of America.

Preface to the French Edition

Some time ago, in a small office at the Henri Poincare Institute, I met Mrs. Tonnelat. We used to discuss our views on fields and sources within the frame of classical relativistic theory. At the time, it did not enter my mind that I would some day have the privilege of presenting the following book. Its author was already exhibiting a highly synthesizing mind and an intellectual honesty which make this book. As a legacy, Einstein has left an enigmatic theory that scientists view with suspicion and hope. A large amount of work is necessary to either prove or disprove the theory. Even if, as is probable, this theory does not yield the final understanding of physical fields, the work will still be fruitful in that it will lead to a better understanding of the unsuccessful geometrization of the fields and of the ambition itself. This book does not pretend to be a complete treatise on the theory. We are, by no means, at the stage where such a treatise is necessary. However, the book may prove to be a good research tool to further work on the theory. Cutting across the multitude of original papers, v

vi

PREFACE

Mrs. Tonnelat has succeeded in synthesizing and criticizing the various points of view. She starts with an exposition of the principles of the theory and analyzes the nature of the geometrical synthesis of an affine connection and a fundamental tensor of rank two which satisfies the field equations derived from a variational principle. These equations exhibit the necessary geometrical and physical invariance. The first sides of these equations satisfy also conservation identities whose importance has been exhibited in general relativity. The difficulties begin when one has to interpret the geometry to find, within the theory, gravitation and electromagnetism and to compare these equations with those of general relativity. Two approaches are available: to find rigorous and as simple as possible solutions of the field equations or to construct approximate solutions. With respect to the first method, the spherically symmetric solutions are the simplest and the most amenable to interpretation and discussion. With respect to the second approach, the manner of approximating is equivocal and implies an a priori interpretation. This labor was started allover the world and it was quite difficult to obtain an idea of the total work carried out with various methods, ideas and implications. Mrs. Tonnelat's book put in evidence the considerable difficulties involved. Appendices detail certain delicate calculations and discuss the isothermic coordinates which seem, to Mrs. Tonnelat and myself, most important for the future developments of the theory. It was up to the scientist who, in addition to her other achievements, was the first to explicitly solve the equations tying the affine connection to the fundamental tensor to give us this book. Andre Lichnerowicz

Acknowledgments

"God is sophisticated but not malicious" -Albert Einstein I wanted to assemble here the principles and some of the developments of the Einstein-Schr5dinger Unified Field Theory. This is not a complete exposition of the theory. Despite their intrinsic interest, I have systematically left out certain original work of which the point of view would have detracted form the homogeneity of this book. Notably among these researches are those referring to Cauchy's problem treated in previous texts. In a large measure, I have unified the notations used by various authors. However, in some particular cases, I have adhered to the original notation so as to facilitate reference to the papers. Most of the chapters arrive at conclusions which raise more or less important difficulties. This is the common lot of all the theories. The unified field theory continues the simplicity of its principles with a profusion of calculations and a wealth of formalism. It is thus difficult to surmount the mathematical complications and to decide between vii

viii

ACKNOWLEDGMENTS

the various physical interpretations to enable one to compare the hopes and realizations of the theory. One must not seek here a didactic exposition of the acquired results but the more or less happy and complete development of a theory in the process of formation. This book is just a collection of work whose unique goal is to facilitate research on the subject. The fragmentary conclusions which are reached can be only headings of chapters for further work. M. A. Tonnelat

Contents Preface . . . . . .

v

Acknowledgments.

vii

INTRODUCTION--Purpose and Methods of Unified Theories Field and Charges in the General Theory of Relativity. . . . . . . . . . . . . . . . . Role and Possibilities of Unified Theories. Unified Theories with more than Four Dimensions . . . . . . . . . . . . . . . . . Four-Dimensional Unified Theories. Spaces with General Affine Connections . Einstein's Theory . . . . . . . . .

1 5 7

9 11

1. MATHEMATICAL INTRODUCTION Relations between the Symmetrical and AntiSymmetrical Parts of the Tensors g and Jll)

g

•

Jll) ....................

Absolute Differential calculus in a Space With a General Affine Connection. . . . . . . . . The Choice of Ricci's Tensor in a Space With a General Affine Connection. . . . . . . . . 2. FIELD EQUATIONS, VARIATIONAL PRINCIPLES, CONSERVATION EQUATIONS Method of Application of a Variational Principle. . . . . . . . . . . . . . . . . . . . . . .

ix

15 21 25

31

x

CONTENTS Application of the Variational Principle to Ricci's Tensor RJlv (r) . Conservative Equations. . . . . . . . . . .

35 41

3. THE FIRST GROUP OF EINSTEIN'S EQUATIONS- - Expression of the Affine Connection as a Function of the Fields. . . . . . . . . .

47

4. THE SECOND GROUP OF EINSTEIN'S EQUATIONS The Rigorous Field Equations Approximate Equations .

65 69

5. SPHERICALLY SYMMETRIC SOLUTION Differential Equations of the Spherically Symmetric Case . . . . . . . . . . . . . . . . The Various Forms of the Spherically Symmetric Solutions . . . . . . . . . . . . . . 6. THE FIELD AND THE SOURCES Choice of the Metric and Fields in the Unified Theory. . . . . . . . . . . . . . . . . . Principles of the Born-Infeld Theory . . . The Non-linear Relations Between the Displacement and the Electric Fields in the Unified Theory. . . . . . . Definition of Conjugate Fields Current and Charge. . . . . .

81 95

111 112

116 122 129

7. SOME PROBLEMS RAISED BY THE UNIFIED FIELD THEORY Energy-Momentum Tensor. . . . . Geodesics and Equations of Motion.

135 145

Appendix I. RELATIONS BETWEEN DETERMINANTS . . . . . . . . . . . . . . .

155

Appendix

n.

APPLICATION OF THE VARIATIONAL PRINCIPLE TO A DENSITY CONsTRucTED WITH RICCI'S TENSOR (J RJlv(L) (55) (I..p=Lp(J=O). . . • . . . v

159

CONTENTS

xi

Appendix III. PROOF OF THE RELATION = cp 1 A== --A --(g-y- cp) A=.

165

Appendix IV. CALCULATION OF THE AFFINE CONNECTION IN THE STATIC SPHERICALL Y SYMMETRIC CASE.

167

p

Appendix V.

ypy

p

ISOTROPIC COORDINATE SYSTEM IN THE UNIFIED FIELD THEORY .

175

References .

179

Index . . . .

185

This page intentionally lefi blank

Introduction

PURPOSE AND METHODS OF UNIFIED THEORIES 1.

FIELD AND CHARGES IN THE GENERAL THEORY OF RELATIVITY.

In developing the General Theory of Relativity, Einstein succeeded in giving a purely geometrical interpretation of gravitation. Newton's law of gravitation was replaced by six independent conditions imposed on the structure of the universe and the trajectories of material particles became geodesics in a four dimensional Riemann space. This Riemannian space which serves as a vehicle for the general theory is completely determined by the line element: * d s 2 = g/.l v dxJ.l dxv.

( 1)

The affine connection between Riemannian spaces is well defined in terms of the metric tensor g/.l v. It is equal to the Christoffel symbols:

* We use here the usual summation rule that any repeated index indicates a sum over the index. 1

2

EINSTEIN1 S UNIFIED FIELD THEORY

°

{tv} = gPa{ofl gav + °v gfla - °a gflV}' 0iJ. = oxiJ.

( 2) The structure of a Riemannian space is completely determined , It differs from the euclidean ~tructure by the existence of a curvature which is given by the RjemannChristoffel tensor.

G~va= °a{/v}

- °v{:a} +

{:J

{:a}

-{iJ.~)J{:v}

(3)

and the curvature is thus defined in terms of the giJ. V· In empty space (exterior case), one can obtain the field equations by imposing the following conditions on the space: 10.

( 4)

S fl v -- 0,

where Sfl v is a tensor of rank two subject to being a function of the gfl v and its first two derivatives. The ten equations given by (4) are not all independent since theoretically at least, they would then determine completely the metric tensor and would restrict the choice of the reference system. There can therefore be no more than six independent conditions between the ten components of 8fl v to enable us to choose freely the coordinate system in a four dimensional space. We will therefore assume that the Sfl v must satisfy the four conservation equations V pSflP==

o.

(5)

V is the covariant gradient formed with the aid of the symbols. We thus have four arbitrary conditions in the determination of the metric tensor. E. Cartan (2) has shown that the only tensor Sfl v satisthe above conditions must have the form 1

C~ristoffel

S fl v

=h

{G flVl -

2" gfl v (G

- 2A)},

( 6)

h and A being constants and Gflv being the contracted Riemann tensor known as the Ricci tensor: G fl v == G~ vp ==

ap{ tv} A

P

-

- {J-Lp} {A)

a v{~} + { ~) {~} - { p } (7)

3

INTRODUCTION If, from (4) and (6), we form the invariant

S = gJ.1. v S

J.1.V

= 0

we obtain G=g

J..Lv

G

J.1.V

= 4A.

( 8)

Thus, in vacuum, the condition (4) is equivalent to ( 9)

G/-Lv = AgJ.1.v·

20. In matter or in the presence of an electromagnetic field (interior case) one must balance the effects of the conservative tensor SJ.1.v by those of another conservative tensor: this tensor will represent the energy-momentum of the field or the matter. We will then have S J.1. v

= XT J..L v with V P T J.1.p = 0,

X being a constant. GJ.1.V -

For h

1

=

2 g J.1.V(G -

( 10)

1, we have 2A) = X T J.1.v

( 11)

The above relation has a particular character, for while the interpretation of SJ.1. v is a purely geometrical one, that of TJ..L v is not. Further TJ.1. v contains: a. The energy momentum contribution of the electromagnetic field which in Maxwell's theory is represented by T J.1.

v

=

~ gJ.1. v

Fpa FPa - F J.1.p Fe •

(12)

b. The contribution of a distribution of matter of density J.1.. Mpa = J.1. up Ua, up being the four-vector velocity of space: dxJ.1. II uJ.1. = U,.... U II = 1. ds ' ,. .

( 13)

( 14)

If we assume the simultaneous presence of charged

material particles and an electromagnetic field, the condition

EINSTEIN'S UNIFIED FIELD THEORY

4

VpT/-/=O

p

or

VP Mf-l

p = -

Vp T /1

( 15)

becomes V A(f-l u CT u A) = - FCTA JA if

T /1

( 16)

v is the Maxwell tensor as expressed in (12) and JA = V P ~p.

( 17)

Equation (16) is identical to the fir st group of Maxwell's equations if we assume that JA is the four vector current. If this is true and if we accept Lorentz's hypothesis that all currents are convection currents, then we always have: JA = P u A

(18)

p being the charge density.

If we multiply (16) by UCT and sum over CT taking into account (14), we have V A( /1 u A) = O.

( 19)

This is just the continuity equation; (16) can then be written as A P A u 'V'AuCT= -"'j:iFCTA u.

( 20)

10. The trajectory of an uncharged material particle (p = will be u A V A u CT = 0

~

(21)

which can be rewritten in view of (14), in the form: d2 x P p dxf-l dx v d S2 + {f-l v} ds ds = 0 and can also be obtained from the variational principle

o Jds

=

O.

This is the equation for a geodesic in a Riemannian space defined by the line element ds 2 given in ( 1): thus the trajectories of material particles coincide with the geodesics of a Riemannian space.

INTRODUCTION

5

2°. In the case of a charged particle (p ~ 0), Eq. (20) shows that the trajectories differ from the geodesics of a Riemannian space. They could be interpreted as geodesics of a Finsler space in which the metric tensor is determined from the element 1 e ds' = (gil v dx ll dx V) 2" + m cP /l dx/l, by putting elm = pill and assuming that cP /l is the four vector electromagnetic potential such that F 11 v = all cP v - a vCP 11' The ratio elm varies from one particle to another. We must therefore associate with each particle a particular Finsler space whose geodesics will be the trajectories. 2.

ROLE AND POSSIBILITIES OF UNIFIED THEORIES.

General Relativity separates quite radically the gravitational field from a. the electromagnetic field which has not had any geometrical interpretation and forms, with the gravitational field, a heterogenous ensemble. b. the sources of field which conserve a phenomenological interpretation even when one talks about uncharged particles. The so-called unified theories attempt to remove the heterogenous features of the combined gravitation and electromagnetic fields. Other theories, neglecting this problem, have attacked the duality of fields and particles in the purely electromagnetic domain. Such are, for example, the theories of Mie and Born-Infeld (122). Their theories have not attempted a geometrical interpretation of the fields. Despite this, they have often been labeled unified theories. To remove any confusion in the use of this term, we shall refer to them as non-dual theories. If we were to take into account all the new problems which would arise in the integration of material tensors in a purely geometrical scheme, it would seem that the

6

EINSTEIN'S UNIFIED FIELD THEORY

difficulties, which would be encountered by a non-dual unified theory, would be insurmountable. It was Einstein's ambition to resolve these difficulties. This, perhaps, is not impossible, since, quite often, a question properly asked would remove in one step all the difficulties which, separately, are irreducible. We shall say that a unified theory is one which will unite the electromagnetic and the gravitational fields into a single hyperfield whose equations are the conditions imposed on the geometrical structure of the universe. 10. The equations which will describe the behavior of the gravitational and the electromagnetic field in empty space are, by definition, relative to a unified" exterior" case and are always written as

S

/.LV

=

0

or

R

/.LV

=

0

R/.L V being the Ricci tensor formed with some given affine connection. These equations correspond to the" interior" case of gravitation and electromagnetism proposed by the general theory. 2 o. In the presence of matter, the equations of a unified theory will include an additional term since we are not, in general, attempting to integrate the field with its sources. To impose on the structure of the universe the additional conditions that will lead to the electromagnetic field, we cannot stay in purely Riemannian space. The description of such a space is complete when the metric tensor is given. The conditions imposed on the curvature tensor in such a space are sufficient to lead to the interpretation of the gravitational field. The electromagnetic field cannot be described in it. Additional conditions on the curvature can be obtained by going to a more complex space than the one used by Einstein for General Relativity. This can be done in one of two ways: a. by increasing the number of dimensions of a Riemannian space.

INTRODUCTION

7

b. by going into a more general variety of spaces with any affine connection which will give us more latitude in the definition of the parallel displacement of a vector along an infinitely small closed contour. 3.

UNIF1ED THEORIES WITH MORE THAN FOUR DIMENSIONS.

The theories with (in spaces of) more than four dimensions occupy an important place in the development of unitary theories. The first of these was advanced by Kaluza in 1921 (1), (10), (12). This was further developed by Einstein and Mayer (1) , (12), in 1932. A recent modification of the basic postulates has led to theories with 15 field variables by Jordan (1947) and Thiry (1948) (10), (12). All these theories are developed in a five-dimensional space whose geometry leads to a formalism which is convenient for the interpretation of the laws of the generalized fields. For this reason, these theories are all subject to a cylindrical condition which displays clearly the peculiarity of the fifth dimension. Only those coordinate transformations, which satisfy the cylindrical condition, will lead to covariant field equations. A further development of the five-dimensional formalism has been carried out by the" projective" theories. The cylindrical condition imposed on these theories can be interpreted in a natural way as a projective condition and shows more clearly the purely auxiliary role of the five-dimensional space. Along with these theories, other developments have been carried out in a different way. To circumvent the introduction of a pure formalism, one can assume that the physical space actually is five-dimensional. The difference between this postulate and the above (i. e., between a geometry and a geometrical formalism) lies in the interpretation of the cylindrical condition. The condition becomes a structural hypothesis which must be satisfied by the fivedimensional physical space. The Einstein-BergmannBargman theory (1), (12) , assumes that the five-dimensional space is closed by the coordinate x 5 • The six-dimenSional theory of Podolanski (12) gives the space a sheet

8

EINSTEIN'S UNIFIED FIELD THEORY

structure. These assumptions, although gratuitous, are nearer to the spirit of General Relativity which lead to a geometrical character of gravitation. The great advantage of five-dimensional theories, or at least of some of them, is the following: in a four-dimensional Riemann space, the trajectories of charged particles are not geodesics; it is possible to interpret these trajectories as geodesics in a Finsler space under the condition that for each type of particle characterized by the ratio e 1m, a different Finsler space is used. It is possible to show however that one can obtain a parametric representation in a five-dimensional space corresponding to a family of trajectories relative to a given elm: these trajectories then coincide with the geodesics of a five-dimensional Riemann space. This remarkable result would push us to develop a unified theory in a five-dimensional formalism. Einstein's and Maxwell's equations can be brought into a satisfactory unified formalism. Aside from criticisms particular to each theory, a general criticism that can be applied to fivedimensional theories is the introduction of an additional condition, namely, the necessity of using the cylindrical condition. In particular the criticism that has been leveled against them is that they lead to a simple codifying in a fivedimensional formalism of the equations of Einstein and Maxwell. No matter how interesting such a synthesis is, it should lead, as General Relativity did, to predictions that will confirm or reject the theory. It is in answer to these objections that the Jordan-Thiry theory with 15 field variables was developed. Essentially, the consequences of these theories are: 1. The gravitational constant X becomes a weakly varying factor, the variations being a function of the variations of elm. 2. The laws of Einstein and Mawell introduce additional terms due to the variation of X. If X is constant, one reverts to the classical laws.

9

INTRODUCTION

3. There is a fifteenth field equation relative to the variations of X which implies that in the absence of any charge (p = 0), the presence of matter (!l -I 0) leads to the creation of a magnetic field. In this way we arrive at the prediction of the existence of a field due to matter in motion and particularly for a rotating body (Blackett effect) . 4.

FOUR-DIMENSIONAL UNIFIED THEORIES. WITH GENERAL AFFINE CONNECTIONS.

SPACES

Along with five-dimensional theories, other theories, since 1918, have attempted the synthesis of gravitation and electromagnetism in the context of a four-dimensional spacetime. They must therefore start from more general multipliCities than the Riemannian variety in order to enable one to impose supplementary structural conditions that will coincide with or better modify, the classical electromagnetic equations. The Riemannian space, which is described by a single type of curvature, will be replaced by a more general continuum formed by considering a space with a more general affine connection. Cartan (3) has shown that the structure of such a space defined by the coefficients r ! l ! will, in general, have a. A rotational curvature defined by the tensor n~ which generalizes the Riemann-Christoffel tensor* P n !lP --!R 2!lw =

1{

-"2

P

[dxlJo(J]

o(Jr!l1J

x

PAP

- 0 IJr !l(J + r!l1J

r A(J

A - r!l(J

[dx V ox(J] .

*

The rdxl1

oxlJl denote the differential rdxl1 oxvl = d~ ox v - oxl1 dxv.

increments and are:

p}

r AV

10

EINSTEIN'S UNIFIED FIELD THEORY

b. A (homothetic) curvature which suffices to characterize the invariant n = n ~ and which is zero in a Riemannian space. c. A torsion which is zero in a Riemann space and is given by nP = - r ~ v rdx/.l ox v ]. Most authors did not immediately perceive the nature of the elements a, b and c at their disposal. However, any affine four-dimensional unified theory assumes the existence of one or several of these elements. All these theories, although apparently different, can be classified in three distinct categories depending on whether they assume the elements a-b, c, a-b-c, which are used to characterize a Riemannian space and develop the general theory of relativ-

ity. The first theory of this type was developed by Weyl ( 13). He assumed that a parallel displacement does not only modify the direction of a vector but also its magnitude. This assumption is equivalent to postulating the non-vanishing of the homothetic curvature and the conservation of elements a and b. Analytically, this assumption introduces gauge transformation in addition to coordinate transformations. The affine connection is then expressed in terrns of the metric tensor and a four vector cp /.l tied to the gauge transformations. Thus, to determine the structure of the universe and the field equations, it is necessary to use the ensemble ( g/.l v' cP J1) and not simply ds 2 • Weyl's theory does not modify the symmetrical character of the space: this is equivalent to assuming a non-torsional space. The torsion will lead to a non-zero value for a closed infinitely small contour. The torsional spaces introduced by Cartan {3) were used in unitary theories by Eyrand (1926) (12), by Infeld (1928) (12), and finally by Einstein (5), (12), (125). A theory developed by Einstein (12) has even been based on the existence of a st.rictly torsional space which permits the definition of absolute parallelism. In many of their features, the theories of Eyrand

INTRODUCTION

11

(a, b, c) and Infeld (a, c) and Einstein's theory (1923) (5), (125) are forerunners of Einstein's unified theory which we shall describe. Unfortunately, they have unnecessarily limited the formalism at their disposal and did not succeed in exploiting the richness of the structure of a general affine connection to derive new laws. 5.

EINSTEIN'S THEORY.

Einstein's theory starts from a generalformalism which, from the beginning, conserves without restricting properties a, band c. If a four-dimensional unitary theory will give satisfactory results starting from a variational prinCiple, it will be within the context of this theory. As we shall see the price that has been paid in return for the completeness of the theory lies in the ambiguities at its foundations. A serious difficulty arises in attempting to resolve these ambiguities and to choose, in a consistent way, the contracted curvature tensor, the metric tensor and the fields. The theory has the advantage of not restricting the possibilities offered by any affine theory. We stated earlier that the often justified criticism leveled at unified theories was that they restricted themselves to synthesizing existing and well-established laws. However, although the theoretical interest in a geometrical synthesis of gravitation and electromagnetism is undeniable it is nevertheless true that it is rather useless to recover Einstein's and Maxwell's laws in theories that introduce a complicated formalism. One would wish to go further. Einstein's theory opens new perspectives for us. These are: 1. The laws of Einstein and Maxwell are modified. They contain additional terms which represent the interaction of the gravitational and the electromagnetic fields. It seems thus that one can justify the creation of a magnetic field by a purely material distribution.

2. In the general case, this type of effect is difficult to bring out due to the complicated nature of the unified

12

EINSTEIN'S UNIFIED FIELD THEORY

field equations. In the particular case of spherical or cylindrical symmetry, one can obtain rigorous laws for this particular case. One can thus determine precisely the laws obeyed by the field of a pulsating or rotating body. This would permit a justification or a modification of Blackett's empirical formula. 3. The laws of electromagnetism which result from Einstein's theory are not linear. They seem to predict, theoretically at least, some new effects, such as the scattering of light by light and the possibility of defining a field in the center of a particle. 4. The relations between inductions and fields in Einstein's theory (or at least between certain quantities tied to inductions and fields) have the same form as in Born's theory. In the case of a static spherically symmetric solution, one can show the existence of a field which is finite in the center of the particle. One would think that Einstein's hope to integrate with the field its singularities and the sources of the field in a geometrical synthesis can be realized. It is difficult to be so positive about this hope. Assuming that even this first objective has been attained, it is difficult to see the next step which will lead the theory to a reinterpretation of quantum theories, a step which Born's theory was unable to bridge in a satisfactory way. Perhaps as we stated earlier, the problem considered as a whole, namely unifying the fields and unifying fields and sources, will be more easily solved if stated more judiciously. In an immediate way, the other difficulties of the theory result on the one hand from the richness of the formalism and on the other hand from the complexity of the solutions. On the one hand, one encounters ambiguities which must be resolved, interpretations which are not always clear: the choice of the tensor RJ1 j) used in the variational principle, the "true" metric, the determination of the energy-momenttum tensor and the derivation of the geodesics. All these points raise problems, a few of which are on the way to

INTRODUCTION

13

being solved. On the other hand, one must handle complicated solutions which, except in particular cases and sometimes even in these, hide the physical contents that one could find. Despite this, a number of satisfactory results have already been obtained. Others need to be studied further. Perhaps in this fashion, we will be able to proceed toward a clearer conception of the nature of the sources of a field and from this to the scope of the theory. One can detect in both Einstein and Schrodinger a mixture of discouragement and great hopes in reference to this point. Be that as it may, Einstein's theory unites the realization of a satisfactory synthesis, obtained from a general principle, to the possibility of new predictions in the classical domain. Such are almost always the signs which characterize a fruitful and important physical theory.

This page intentionally lefi blank

1 Mathematical Introduction

A.

1.

RELATIONS BETWEEN THE SYMMETRICAL AND ANTISYMMETRICAL PARTS OF THE TENSORS iJ.LII and gJ.LII (54), (57).

DE FINITIONS.

Consider a tensor of rank two gJ.L II whose determinant is g and let ggJ.L II be the minor relative to each element gJ.L v. We have by definition: gJ.lp g

dg

iJ(J _

(J

- gpJ.l g

= ggJ.lv

dg J.lV

J.l _ a - op

== -

( 1. 1) ( 1. 2)

gg J.lV dgJ.lv

whence we have dg J.lV

= - gp v

g J.l a drf a, dgJ.l II

= -

pf II

gJ.l a dgpa '

( 1. 3)

Let us now write gfl.v and gJ.l II in terms of their symmetric parts y fl.V and h J.lvand antisymmetric parts

aV + 'cJ

rVPa

(}/la _ ",iJ v 'cJ

a r up

( 1. 50) by taking into ac count (1. 46) .

24

EINSTEIN'S UNIFIED FIELD THEORY

6.

FUNDAMENTAL IDENTITIES BETWEEN THE (J~ ~;P and between the g [ ~ z.:;P ] .

a.

Let us form Q:~;P -

Q~~;p. By contracting

( 1. 50), we have the identity:

1. J ()flP 2

t ># + -;P

-,P fl t '" ># +

0

-;P f

P

g flP _ :Ie flP

r

( 1. 51)

P

by using (1. 5) and the corresponding notations for tensorial densities. Qfl v = :Ie fl v +

g fl v

( :Ie fl v =

r:g hfl v;

g fl v =r:g ffl V). ( 1. 52)

b. Let us form the difference between the circular permutation of gfl v;p and gVfl ;p' This leads to: ++1

"2{(gflv;P + gPfl;v + gVP;fl) - (gvfl;P + gpv;fl + gflP;V)}

+-

+-

+-

+-

'" {ofl 'Pvp + 0p 'PflV + °v 'PPfl} +

+-

2{r flv, v

+

r

+-

p +rpfl,v v

( 1. 53) vp, fl}

v where we have used ( 1. 4) and defined* rflV,p= Ypargv

v

( 1. 54)

v

The two identities (1. 51) and (1. 53) are completely analogous. One refers to the vector ffl and the torsion r P: ffl = 1

f-g

aP g

flP,

r P = r pa

( 1. 55)

v

* We note here that the comma in r iJ. p simply separates the symbols and is not a sign of differentiation. lJ,

25

MATHEMATICAL INTRODUCTION

the other to the pseudo-vector Ifl and the pseudo-torsion

r'*p

=

a* r (P~l :

Ifl =

~

flVpa(ov CPpa + Ga CPvp + Gp cpav)

E

r*p -- r [ pa a* 1 -- ..f2-y Epa fl v v

_1_ {

21-y

Y

Y

Pfl

1

BY -y

~

ypl1

E fl

y

flA

f'

VT

(r

VA T ya 7T r

+r

AT, v

v

5~

a r AT

AJ,7T

av E flVAT

(1.

VA, T

} +r

v

(1. 57)

TV, A

).

v

C. THE CHOICE OF RICCI'S TENSOR IN A SPACE WITH A GENERAL AFFINE CONNECTION. 7.

POSSIBLE CHOICES FOR THE CONTRACTED CURVATURE TENSOR.

In general Relativity, the contracted curvature tensor of order two can be defined without ambiguity. This is Ricci's tensor as given in (7) . In the case of a general affine connection, we must take into account the two possible forms of the parallel displacement of a vector:

oA+P oA -P = - r Vfl p

Afl

r 11V p oX v =

Afl

0XV

- -; p

fl v

( 1. 36) Afl oX v

( 1. 37)

.

These two forms permit the definition of two tensors susceptible of generalizing the Riemann-Christoffel tensor (one being derived from and the other from v:

rev

_0

KflW(r) = 0a

p

r flv

ore

pAP

- 0v r lJ.a +

r IJ.v r Aa

-

A

P

r lJ.a r AV

(1 58) •

26

EINSTEIN'S UNIFIED FIELD THEORY

°

R P (r) = r P fJ.W (J Ill'

°II r IlP(J + 'fAIlll rA(JP - r IlA(J rPAII

P pAP A P = o(Jr l'1l - °llr (J1l + r IIIl r (JA - r (JfJ.r IIA

( 1. 59) By contracting P and (J and P and Il, we obtain from (1. 58) two contracted tensors known as tensors of the first and second kind. A similar process applied to (1. 59) leads to two other tensors. R

Ill'

=

IfIlIlP (r)

=

°Pr Ill'P - °II r IlP P +r A r P Illl

Ap

_ rA ~ IlP

P

Illl

= RPPIlII (r) =

,....

P"'"

(1. 60)

All'

° r PPIl - °fJ. r PPII

( 1. 61)

II

°

P

PAP

R Illl = R IlIlP (r) = pr IIIl - 0l'rPIl + r IIIl r PA r P

_ rA P Il -

p"'"

P Illl - Rp fJ.1I

( 1. 62)

IIA p

(r) = 0 II r IlP

- 0 Il r

~p

(1. 63)

If the connection is symmetrical, it is clear that (1. 60) and (1. 62) coincide and that (1. 61) and (1. 63) are zero. For a general connection, this is not the case and, a priori, any combination of the above four tensors and the tensor r llll = rllr ll

(1. 64)

can be considered. 8. CHOICE OF R IlIl• a. Hermitian principle. To limit the arbitrariness in the choice of the tensor generalizing Ricci's tensor, Einstein first proposed the

27

MATHEMATICAL INTRODUCTION adoption of a hermitian principle (14), fines the transposed quantities

To this end, he de-

,...., g/lv = gV/l ~p

1

-

/lV -

(1. 65)

rP

(1. 66)

V/l

According to Einstein, a tensor A/lV(r, g) is said to be hermitian with respect to the indices /l and v, if, upon interchanging /l and v and changing g and r into g and respectively, the new tensor is equal to the old one

r

A/l V( r, g) = AV/l Cf',

g),

(1. 67)

The tensor is said to be antihermitian if A/l vcr, g) = - AV/l

(r, g).

(1. 68)

According to this scheme, we can then reduce the basic tensors by considering only those linear combinations of ( 1. 60), (1. 61), (1. 62) and (1. 63) which are hermitian. We therefore would consider only the following tensors: U/l v

="21{ R/l v

"'} -"41{P/l + PV/l} ,. . , V

+ RV/l

1

(1. 69)

H/l v =

"2 {p /lV + P' V/l}

( 1.

r

r /l r

( 1. 64)

/l v =

v'

70)

ExpliCitly: U

/lV

=

a r P - ~{ a r P + a r P } P /lV 2 /l vp v JJP +rA /lV H/l V =

rP

a /l r

AP

v -

_rA /lP

( 1. 71)

rP

a v r /l'

AV

( 1. 72)

It seems somewhat advantageous to consider instead of U/l v the combination

28

EINSTEIN'S UNIFIED FIELD THEORY

1

U Jlv + 3" HJlv -

1

(1. 73)

3" r JlV·

and we can see that this combination wlJ.i~h, a priori, is really artificial, is exactly the tensor (3) RJl v given in Appendix II. (3) R

=

!{ (2) R 2

Jl v

+ (2) R

Jl v

}

VJl '

( 1. 74)

(2) RJl v being the Ricci tensor formed with the affine con-

nection

( 2) L P

JlV

=

r P

JlV

+.!.3 {aJlP

r

v

- aPr } v

Jl

( 1. 7 5)

whose torsion-vector is zero. ( 2) L

P

= (2) L 0" = 0

(1. 76)

pO" V

b. Connections with null torsions. These results lead us to consider more or less complicated linear combinations of the Ricci tensor formed with the help of special affine connections, namely those with zero torsion vector. In fact, starting with the general connection

rev ,

one can form the two connections

(l}L P

JlV

(2) L P -

Jlv-

-

r

P + ~ aP 3 Jl

JlV

r JlV P +!{ aP r 3 Jl

v

r

v

(1. 77)

- at r Jl }. ( 1. 78)

The anti-symmetrical parts of these connections are the same and in particular the torsion vector is null: Lp = (1) L gO" = ( 2) v

L~

= O.

( 1. 79)

v We can then define, from the affine connection L~ v , three tensors of the second order (54) ( 1)

P

1

RJlV= R Jlvp ( L}

(1.80)

29

MATHEMATICAL INTRODUCTION

(2) R ( 3)

Jl v

RJl V

= R PeL) Jl vp

2) ="21 {p RJl vp ( L

( 1. 81)

P 2~ } ) + R vJlP (L) ( 1. 82

of which only the third (1. 82) is hermitian and is the tensor suggested by Einstein in the third edition (1950) of the Meaning of Relativity. In short, if we do not retain the hermitian principle, it is possible to adopt a basic tensor which is some linear combination of eqs. (1. 60) through (1. 64) and, in particular one of the four tensors RJlv' (1) RJlv' (2) R/l V ' (3) R/l v • It is to the basic tensor that we will apply a variational principle: In what follows, we shall use the tensor p

P

R Jlvp = 0P rJlv -

PAP

0vrJlP + rJlv

A P rAp - rJlp rAve

( 1. 60) which we will call the Ricci tensor and to which we will apply the variational principle. However, we shall examine in Appendix II the results of a variational prinCiple applied to the tensors (1) RJl v, (2) RJ.}.v and (3) RJl v which are analogous to Ricci's tensor but which are formed with an affine connection whose torsion is null.

This page intentionally lefi blank

2 Field Equations Variational Principles Conservation Equations A. METHOD OF APPLICATION OF A VARIATIONAL PRINCIPLE.

rev are

1. MIXED THEORY (EINSTEIN). VARIATION OF A FUNCTION U) (gfl v, ,a v ). The mixed theory assumes that the variations of the affine connection and the metric of the space are independent of each other. We start from a variational principle applied to the scalar density ,\"'1:

o Jg)

dT = 0

rev ,

(2. 1)

being a function of the affine connection and its first derivatives and the generalized field density gfl v. We assume that v and gfl v are independent variables and that their variations v and ogfl v vanish at the limits of integration. With these assumptions, we obtain ,I)

rC

J[ orP a ,I)

fJ. v

orfl

0r P + flv

+

0(0

a ,I)

rP) (J fl v

o ,I)

oaMv

o(j

31

fl v]

0 0 (a

rP

flv)

(2. 2)

dT = 0

32

EINSTEIN'S UNIFIED FIELD THEORY If we integrate the first two terms by parts, we have:

J [ a rai v I)

(a

P ,\~) or /-Lv + a a a (a a rtv)

P ) 0 r /-LV

(2. 3) -

( aa

a ,I)

r P ) a (ao- /-LV

)

P

orJ.1v

]

dr.

The second term of (2. 3) will not contribute to the integral since 0 r Jl v P is zero at both limits. Thus (2. 1) can be written as

J{Gj;V

0 rev + J J.1 v ogJ.1l)- dr = 0 (2. 4)

where

a ( a,I~) ) -axo- a(ao-rCv)

GJ.1V_ aH p - ar P J.1V

J

_ a ,I) J.1V - ag"ll •

(2. 5)

(2.6)

The basic equations of the theory will therefore be:

J" a.

1b.

J.1V Gp

=

JJ.1V

= o.

0

(2. 7)

(2. 7a) are Euler's equations relative to the variation of the r P • J.1v

REMARK I. If we had imposed some conditions on the gJ.l v, for example aP g J.lP = 0, the variation O(]J.l v will not be independent and this would have to be taken into account in the variational principle. To this end, one can use the method of the Lagrange multipliers. If we impose the above condition to the g J.l v, we would have to add to H the scalar density -20- J.l aP fJ JlP, a J.l being an arbitrary four vector. The variational prinCiple leads to Jlv P J.lV! Gp 0 r J.l v + (JJl v + a v 0- J.l - a J.l 0- v) 0 r; J dr = 0

J{

(2. 8)

which leads to the field equations

33

VARIA TIONAL PRINCIPLES GPJ.lV

\ a.

:=

0 (2. 9)

b. opJJ.lv;;;:o

~ c.

JJ.lV +oVcrll -oJ.lcr v ;;;: O.

REMARK IT:

a. If we vary the scalar density -

,\.1

;;;:

0 J.lv R J.l v

(2. 10)

r.es/o

RJ.l v being some tensor which is a function of the their derivatives, we would obtain, since RJ.l v;;;: 0 = J J.l v; for the field equations (2. 7)

a Gllv = 0 • p

1,b.

v

and gil v

(2. 11)

R llv ;;;: 0

b. In varying the scalar density H' ;;;: H - 2 A r:g,

( 2. 12)

the variation of the Oil v gives the additional term -g 0r) Il v --A(lllv0r) Il v, 2 0 f .... -A

since

or-g oOIlV;;;:

(2 • 13)

1

"2 gllv'

The field equations will thus be: ~ a.
J a V '>J Y a{J Ci Y Ilv

+~f3Yr'Y +~IlYr~ av

Y

IlCi

'>J

=0

(2. 25)

contracting a and I' and then Ci and (3 leads to: - 3 8 Ci (J{3 Ci - {3 g{3a

r~ -

rev = 0

( 2. 26)

«(jCi(3 = ;rea{3 + ga(3) •

(2. 27)

3(JIlV

v 8 a gay =

0,

We solve (2. 26) for (}Il V r Ilv{3 and substitute its value in (2. 25), obtaining: 8

~{3y -~{3YrP + ~ o'y (j{3CJr +~{3CJrY

a Y

a{J

Y

3

a

+ (jay

whence

~tr.

'>J,

Ci

+~3

o'y a

r tCi

a

(1(J

( 2. 28)

= 0

~(30" r a'>J - ~{3y r

Y

Y

Ci

= 0

•

(2. 29)

The field equations in a space with a general affine connection will then be IlV 2 a ~+-'P= __ ov ~Ilar +~IlVr 8 gIlP=O • '>J, 3 'P Y a Y p' Il (I) b. RJlv = O.

37

VARIATIONAL PRINCIPLES

Equations (1) are deduced from a variational principle and are necessarily compatible. Einstein suggested that one should add to I, the condition

r P = O.

J..LV

This leads to (r -;P = 0 and as a consequence of this condition (1. 51) leads to a J..L qJ..LP = O. Thus we have the following system: J..LV

,0+-

lJ

rP

;P = 0

=

a(l

0 (--l

'] (lP = 0)

( F) RJ..LV

=

0

These are the equations of a strong system and are not deducible from a variational principle. 4.

CHANGE OF AFFINE CONNECTION. The weak system. We shall define a new affine connection Lev such that We

Lp :::: Lo/ :::: 0 and refer all the field equations to it.

shall adopt the following notations: P L(lv

Affine connection

P

P {/-LV} rJ-lv

P

~(lv

(Lp = 0) (g gv' p = 0)

+-'

Covariant differenti- Vp ation

Dp

;P

(2.30)

Ricci's tensor

GJ..Lv

RJ..LV

WJ-lV

Einstein's tensor

8J-lv

HJ..Lv

KJ-lV

Let LP = (lV

P r J..LV

2 (jP 3 J..L

+ -

r v,

(2. 31)

such that L

J..L

= Lep =

v

O.

( 2. 32)

38

EINSTEIN'S UNIFIED FIELD THEORY

Equations (2. 28) can then be written as (Jf3Y - (Jf3Y L P + (Jf3CJ L Y + (JCJY ~ a a aP aCJ

o or

D R IlV

f3y

~+- -

alJ

-

0

(2. 33)

(2.34)

as given in (1. 60) takes the form: 2 R llv =W llv - 3{oll

=0

r

v - 0v

r ll }

= 0 (2.35)

where WIlV is the Ricci tensor formed with L ~v : P

WJlv = 0 P LJlV - 0 v

PAP LJ;..p + LJ1v LAp

-

A LJ;.p

P L Av ' (2.36)

We can, from (2.34), determine all the Ltv' But can be defined up to some arbitrary ( 2. 31) shows that r P; thus the antisymmetrical part of (2.35) must satisfy the relations

rllv

Op W~ + 0v

rpt

+ oil WIf

= O.

( 2. 37)

Finally, the field equations can be obtained from I * with the un~erstanding that the affine connection in L{}. v (Lp = Lp-CJ = 0):

v

a. Dp

alii!

0p gllP = 0

= 0

(II) b. WIlV = 0

op WJlV v

+ 0 v WP JJ. + 0 Il WvP = 0 v V

* The equations equivalent to (I) but referred to L~ II are (ITa) and (2.35) Dpglill..=O (I' )

WJ.l1I

0pgJ.lP=o

2

= 3" (oJ.lr II

- all

rJ.l)

They identify the arbitrary vector for which W is a rotational tensor with the torsion (10) page 267, ( 43) . J.l1I

39

VARIATIONAL PRINCIPLES

There are the equations for a 'weak' system. They are completely derivable from a variational principle. * If we add a cosmological term to the action function or if, in a purely affine theory, we choose the denSity (2.18), we would obtain equations of type n on the condition of replacing WJ..LV by the tensor W J..Lv = WJ..LV - A g J..LV'

We would then have J..LV

\ a. Dp(J+{ b.

0pJJ..LP=O

= 0

WJ..Lv = A YJ..Lv

( 2. 38)

(II' )

opWgv + 0vWpJ..L + oJ..L WV() = AC{JJ..1VP

v v v' This is the system of equations adopted by Schr(jdinger. 5.

REMARKS ON THE SYSTEMS OF EQUATIONS DEDUCIBLE FROM THE VARIATION OF $") = (]J..Lv (a) RJ..Lv'

In Appendix n, we will apply the variational principle to a Ricci tensor (a) RJ..Lv( a = 1, 2, 3) formed with the affine connection L whose torsion vector is zero (54). We obtain then a system of equations which are weaker than (II). In fact, a transformation of the affine connection permits us to express them in terms of a connection l:l. (See (n - 7) App. IT) and to give them the form J..LV

\ a. Q+-;P = 0 (m) __--'-tb:.,;.._RJ..Lv( (a) l:l.) + (a)KJ..Lv(f P) = 0 * Mrs. J. Winogradski has shown that all hamiltonians of the type (}/l v R/l v are equivalent for the formulation of the field equations. In fact the densities g/l v R/l v' r;:/l v3v/l' (j/l v R/l v and g/l v R"'/l can be deduced from each other by a A or A transformation, i. e., in substituting for rev the connection rfl", + AI' (or + De rj.L)' The field equations obtained by varying the metric under the condition that (J P .] /lP = 0 and the connection are invariant under these A-transfurmations. The transformation of the affine connection (2. 31) which leads to the field equations (II) is a particular case of the A transformations (cf. (43) and Einstein (19)).

oC

rev

40

EINSTEIN'S UNIFIED FIELD THEORY

We can make two remarks with respect to these equations.

r

tv

I - From the variational principle, it is clear that can be determined up to an arbitrary vector r p.*

II - If we define a connection A~v such that the covariant derivative of (JJ1.v is zero with respect to this connection, we have the following results: a. From the identities (1. 51), we have:

aP J

JlP - J{',JlP IIp = 0

If we multiply by

II 0"

h

(2. 39)

1/f:g hJlO" and sum, we obtain

= _cg jJ(J

op (./-i;g f Jlp)

= h JlO"

fJl .

(2. 40)

We can substitute for hJlO" its value as given in (1. 20s) and we are led to llO" = (y jJ(J + CfJ JlA CfJo"T Y AT) f Jl = fer

- fa

(2. 41)

where f (j- -- % T Y TA (j) AJlfJl b. From the identities (1. 53), we have 1 II JlVP = - 2: CfJ JlVP

(2. 42)

with llJlVP = llJlv, P + llpJl, v + A vp , Jl v v v CfJ Jlvp

= 0 P CfJ JlV +

0v CfJp Jl + a Jl CfJ vp'

(2. 43)

(2. 44)

It is clear that the 64 coefficients of the affine connection

are determined unequivocally from (IlIa).

* ()~ r: ;p

This is due to the fact that

r

e lJ

never satisfies the equations

= 0, which would then determine it completely, unless additional conditions are imposed on it.

41

VARlATIONAL PRINCIPLES

SPECIAL CASE: In the special case where we start from the Ricci tensor formed with a generalized affine connection, the connection D. such that Dp O'~ = 0 is the same as the connection (1) = + 2/3 r whose torsion is zero. In this case, it turns out that the four vector J /l is zero. ( 1. 51) written in terms of Lev is automatically satisfied by the simultaneous vanishing of Lp and gp. The connection Lp which satisfies equations (II) is also determined unequivocally. The vanishing of its torsion vector results directly from these equations. In all our further developments, we shall use Ricci I s tensor as the basic tensor. The condition g P = 0 results directly from the variational principle and the connection coincides with the connection LEv whose torsion vector is zero. The field equations which result from a variational principle and which we shall use are given by (II) :

LEv rev

08 v

D.£v

,a.

Dp

/lV

0'+-

=

0

Lp = Lpa = 0 '" J P = 0 v

/

I b.

WflV = 0

( II)

0P W flv + 0vWP/l + oflW VP = O.

v

v

v

It will be in this context that we shall treat most of the ap-

plications and that we shall discuss the possible forms of the spherically symmetric solutions. * C. Conservation equations (41), (115), (116). In a Riemannian space, the tensor (which is a function of the g/lv' and its first and second derivatives) satisfies the identity

st

p vp S/l = 0

( 2. 45)

* Nevertheless, we must not forget the possib\lities resulting from the use of a different basic tensor (such as (1) R) than the Ricci tensor. The field equations will then be given by eq. (rna) with.p = fp - f~. The disappear ance of the conditions fp = 0 or ~p = 0 might entail useful modifications for the determination of solution.

42

EINSTEIN'S UNIFIED FIELD THEORY

and will necessarily have the form p p 1 P SJJ. = GJJ. -2 0JJ. (G - 2A).

(2. 46)

If A = 0, (2. 45) takes the form 1,n {1J oP ~~P JJ. + 2 'J pa 0JJ. (j

(2. 47)

=

0

These identities follow uniquely from the definition of Ricci's tensor as a function of the gJJ. v and their derivatives, i. e. , from the definition of the Christoffel symbols equivalent to ..6. p (jllV = O. In unified theory, the definition of the affine connection in terms of the metric is equivalent to equations Dp(jIiI!=o

0pgJJ.P=O.

(ITa)

The question arises as to whether Eq. ITa is sufficient to lead to identities between the components of Ricci's tensor corresponding to the connection r We will give here the proof of Lichnerowicz (41). Consider the integral

tv .

J

(2. 48) Yl = ~v (jJJ.v dT. Generalizing Weyl's method, we determine the variation of aJJ.v resulting from an infinitesimal change of coordinates x'P = ,f + ~p determined by the vector field ~p. We will have: (1) ogJJ.v = -(a pgJJ.ll) ~p + gJJ.p 0 p ~ v + gP vOp~ JJ. . (2. 49)

These equations determine the corresponding variations: .. r-:

-av-g

f -g = -2-

gJJ.v og

JJ.V

-

= (opf-g

HP+

= op(r-g~ p),

~

Y

-g op~

P

(2. 50)

43

VARIATIONAL PRINCIPLES -O(R{lv(lJ-LVdT) = =

-{I=g oR + RoI-g}dT {{:g( apR) ~p +

R8p ( I=g ~p) } dT

(2. 51)

= ap(R!=g~P) dT.

The variation 0 8l is expressed in terms of the integral of a divergence. From Stokes' formula, this integral is zero unless the increments ~p are zero at the limits of integration. We assume that the latter holds and obtain

081=

J{(OR{lv) r;J-Lv+ R{lv OQ{lV}dT = O.

(2. 52)

However, Eqs. (IIa) which we assume to hold are precisely the necessary and sufficient condition for

J( {j RJ-LJ r;{lv dT = 0

(2. 53)

for an increment Or~T which vanish at the limits of integration. This will be the case if we assume that the ~Pand their first two derivatives also vanish at the limits. We will then have:

J R{lVo

r; {lV

(2. 54)

dT = 0

* ogll v is then determined by the difference between the gil v computed at the two points which have the coordinates x P in the two reference sy.stems. ogllv= g'IlV(x} - gllV(x}

= rg'IlV(x'}

JJa = [ -ax'll - -ax'v a g {x> P ax

ox

- gil V(x)]

_gIlV(x)]-[g'IlV{x')_g'IlV(x)]

ag llV - ~ ~P

ox P

=[ (opll+ a:~) (oa V+ :;~) gPa _gIlV(x)] _ :!~V ~ p avo~1l agllVtp _ /J.aa~v -+g ---;y-" a ax axa ax For a scalar R: -g

aR p 6R=R'(x) -R(x) = -[R'(x') -R'(x)] = --p- ~ .

ax

(cf. Von Laue (9), p. 208).

44

EINSTEIN'S UNIFIED FIELD THEORY

But, from (2. 49) : -R/.w o(jJ.lv = RJ.lv(Eip(JJ.lVH P - (RJ.lv()PV +

Rz".J;VP)Eip~J.l

+ RJ.lv (jJ.lv Ei P ~P = Eip[RJ.lvC'J.lV~P - (RJ.lv()PV + RVJ.lC'vPHJ.l]

+[Eip (RJ.lv(JPV + RVJ.l(JvP) -O:(Jaf3EipRaf3]~J.l.

The integral of this expression must be zero independently of ~ J.l. The first term being a divergence will vanish. We will thus have: Eip(RJ.lv()PV + RVJ.l(JvP) -

oe (Jaf3EipRaf3

=

O. (2. 55)

Let p

JCJ.l

P = f-g HJ.l '

HP =!(R gPv+R gvp)_loP R J.l 2 J.lV v J.l 2" J.l • (2. 56)

We then have: * +

EipJC:

i

R af3 BJ.l(J af3 = O.

(2. 57)

This is an identity of the type (2. 47). It refers to the tensor RJ.lv = RJ.lv( r) . To write it in terms of the tensor WJ.lV = RJ.lv( L), we make the substitution RJ.lv = W J.lV -

2

"3

(a J.l r v - Ei vr J.l) •

(2. 35)

We will then have: JCP J.l

=

9

a.

--S 0. 0. ~

t-

LD

--

--

(l) (l) f/)

---N N

M "-

8

a

M

..c: .......

(l)

..c a s::

eli .......

~:> .....s::

"