Introduction to the THEORY OF RELATIVITy Peter Gabriel Bergmanh \f With a Foreword by Albert£lnstein£ M / 3 l Digitized by the Internet Archive 2012 in http://archive.org/details/introductiontothOOberg INTRODUCTION TO THE THEORY OF RELATIVITY by PETER GABRIEL BERGMANN PROFESSOR OF PHYSICS SYRACUSE UNIVERSITY WITH A FOREWORD BY ALBERT EINSTEIN DOVER PUBLICATIONS, INC. NEW YORK sD */f6 © Copyright 1942 by Prentice-Hall, Inc. Copyright (c) 1976 by Peter G. Bergmann. All rights reserved under Pan American and Inter- national Copyright Conventions. Published in Canada by General Publishing Com- pany, Ltd., 30 Lesmill Road, Don Mills, Toronto, Ontario. Published in the United Kingdom by Constable and Company, Ltd. This Dover edition, first published in 1976, is a corrected and enlarged republication of the work originally published by Prentice-Hall, Inc., Engle- wood Cliffs, New Jersey, in 1942. The author has written, especially for the Dover edition, a new Preface; Appendix A: "Ponderomotive Theory by Surface Integrals"; and Appendix B: "Supple- mentary Notes." International Standard Book Number: 0-486-63282-2 Library of Congress Catalog Card Number: 75-32903 Manufactured in the United States of America Dover Publications, Inc. 180 Varick Street New York, N.Y. 10014 Foreword by Albert Einstein Although a number of technical expositions of the theory of rela- tivity have been published, Dr. Bergmann's book seems to me to satisfy a definite need. It is primarily a textbook for students of physics and mathematics, which may be used either in the classroom or for individual study. The only prerequisites for reading the book are a familiarity with calculus and some knowledge of differential equations, classical mechanics, and electrodynamics. This book gives an exhaustive treatment of the main features of the theory of relativity which is not only systematic and logically com- plete, but also presents adequately its empirical basis. The student who makes a thorough study of the book will master the mathematical methods and physical aspects of the theory of relativity and will be in a position to interpret for himself its implications. He will also be able to understand, with no particular difficulty, the literature of the field. I believe that more time and effort might well be devoted to the systematic teaching of the theory of relativity than is usual at present at most universities. It is true that the theory of relativity, par- ticularly the general theory, has played a rather modest role in the correlation of empirical facts so far, and it has contributed little to atomic physics and our understanding of quantum phenomena. It is quite possible, however, that some of the results of the general theory of relativity, such as the general covariance of the laws of nature and their nonlinearity, may help to overcome the difficulties encountered at present in the theory of atomic and nuclearprocesses. Apartfrom this, the theory of relativity has a special appeal because of its inner con- sistency and the logical simplicity of its axioms. Much effort has gone into making this book logically and pedagog- ically satisfactory, and Dr. Bergmann has spent many hours with me which were devoted to this end. It is my hope that many students will enjoy the book and gain from it a better understanding of the ac- complishments and problems of modern theoretical physics. A. Einstein The Institute for Advanced Study in Preface to the Dover Edition ThisbookwasfirstpublishedbyPrentice-Hall, Inc.,in 1942. The new Dover editionreproduces and expands the original text. In the intervening three decades entirely new aspects of the theory of relativity have been opened up, and related laboratory and astro- nomicalinvestigations haveledto newdiscoveries aswell. Together with my co-workers and students I have taken part in this research. Mywish torevisethe book completely has hadto be postponed again and again. For this edition I have derived anew, in Appendix A, the laws of motion of ponderable bodies. Whereas the pioneering work of Einstein, Infeld, and Hoffmann relied on a weak-field slow-motion approximation, the new approach, which was originated by my colleague and former student J. N. Goldberg and by myself, is based on the full field equations and leads to rigorous relations between surface integrals. In Appendix B I have collected a number ofbrief notes that deal with recent progress in specific areas discussed in the original text; some include selected references to the literature. Footnotes added to the text indicate entries in Appendix B. For anyone seriously concerned with the foundations ofphysics I consider the study ofrelativity indispensable. Twice in the history of physics gravitation has been crucial. When Newton related the lawsoffreefalltothelawsofplanetarymotion, heestablishedphysics as an exact science. Three hundred years later Einstein revolution- izedourconceptsofspaceandtimewithhisnewtheoryofgravitation. These new concepts are still evolving. No doubt they will influence the development of physics for a long time to come, though their ultimate impact remains conjectural. I hope that my book will continue to be useful as an introduction to the fundamentals of relativity. July, 1975 P. G. B. Contents Part I THE SPECIAL THEORY OF RELATIVITY CHAPTER I. Frames of Reference, Coordinate Systems, and Co- ordinate Transformations 3 Coordinate transformations not involving time. Coordinate transformations involving time. II. Classical Mechanics 8 The law of inertia, inertial systems. Galilean transformations. The force law and its transformation properties. III. The Propagation of Light 16 The problem confronting classical optics. The corpuscular hy- pothesis. The transmitting medium as the frame of reference. Theabsoluteframeofreference. TheexperimentofMichelsonand Morley. The ether hypothesis. IV. The Lorentz Transformation 28 The relative character of simultaneity. The length of scales. The rate of clocks. The Lorentz transformation. The "kine- matic" effects of the Lorentz transformation. The proper time interval. The relativistic law of the addition of velocities. The proper time of a material body. Problems. V. Vector and Tensor Calculus in an n Dimensional Continuum 47 Orthogonal transformations. Transformation determinant. Im- proved notation. Vectors. Vector analysis. Tensors. Tensor analysis. Tensor densities. The tensor density of Levi-Civita. Vector product and curl. Generalization, n dimensional con- tinuum. General transformations. Vectors. Tensors. Metric tensor, Riemannian spaces. Raising and lowering of indices. Tensor densities, Levi-Civita tensor density. Tensor analysis. Geodesic lines. Minkowski world and Lorentz transformations. Paths, world lines. Problems. v CONTENTS VI. Relativistic Mechanics of Mass Points 85 Program for relativistic mechanics. The form of the conservation laws. Amodelexample. Lorentz covarianceofthenewconserva- tion laws. Relation between energy and mass. The Compton effect. Relativistic analytical mechanics. Relativistic force. Problems. VII. Relativistic Electrodynamics 106 Maxwell's field equations. Preliminary remarks on transforma- tion properties. The representation of four dimensional tensors inthreeplusonedimensions. TheLorentzinvarianceofMaxwell's field equations. The physical significance of the transformation laws. Gauge transformations. The ponderomotive equations. VIII. The Mechanics of Continuous Matter 121 Introductory remarks. Nonrelativistic treatment. A special coordinate system. Tensor form of the equations. The stress- energy tensor of electrodynamics. Problem. IX. Applications ofthe Special Theory of Relativity 133 . . Experimental verifications of the special theory of relativity. Charged particles in electromagnetic fields. The field of a rapidly moving partiole. Sommerfeld's theory of the hydrogen fine struc- ture. De Broglie waves. Problems. Part II THE GENERAL THEORY OF RELATIVITY X. The Principle of Equivalence 151 Introduction. The principle of equivalence. Preparations for a relativistic theory of gravitation. On inertial systems. Ein- stein's "elevator." The principle of general covariance. The nature of the gravitational field. XI. The Riemann-Christoffel Curvature Tensor .161 . . The characterization of Riemannian spaces. The integrability of theaffineconnection. Euclidicityandintegrability. Thecriterion of integrability. The commutation law for covariant differentia- tion, the tensor character of Rutf Properties of the curvature tensor. The covariantformofthecurvaturetensor. Contracted forms of the curvature tensor. The contractedBianchi identities. Thenumberofalgebraicallyindependentcomponentsofthecurva- ture tensor. XII. The Field Equations of the General Theory of Relativity 175 The ponderomotive equations of the gravitational field. The rep- resentation of matter in the field equations. The differential identities. The field equations. The linear approximation and the standard coordinate conditions. Solutions of the linearized field equations. The field of a mass point. Gravitational waves. The variational principle. The combination of the gravitational and electromagnetic fields. The conservation laws in thegeneral theory of relativity. CONTENTS vii XIII. Rigorous Solutions of the Field Equations of the General Theory of Relativity 198 The solution of Schwarzschild. The "Schwarzschild singularity." The field of an electrically charged mass point. The solutions with rotational symmetry. XIV. The Experimental Tests of the General Theory of Relativity 211 The advanceof the perihelion ofMercury. The deflection oflight in a Schwarzschild field. The gravitational shift of spectral lines. XV. The Equations of Motion in the General Theory of Relativity 223 Force laws in classical physics and in electrodynamics. The law of motion in the general theory of relativity. The approximation method. The first approximation and the mass conservation law. The second approximation and the equations of motion. Conclusion. Problem. Part III UNIFIED FIELD THEORIES XVI. Weyl's Gauge-Invariant Geometry 245 The geometry. Analysis in gauge invariant geometry. Physical interpretation of Weyl's geometry. Weyl's variational principle. The equations G^v = 0. XVII. Kaluza's Five Dimensional Theory and the Projec- tive Field Theories 254 Kaluza's theory. A four dimensional formalism in a five dimen- sional space. Analysis in the p-formalism. A special type of coordinate system. Covariant formulation of Kaluza's theory. Projective field theories. XVIII. A Generalization of Kaluza's Theory 271 Possible generalizations of Kaluza's theory. The geometry of the closed, five dimensional world. Introduction of the special co- ordinate system. The derivation of field equations from a varia- tional principle. Differential field equations. Appendix A 281 Ponderomotive Theory by Surface Integrals Appendix B 300 Supplementary Notes Index 303 Preface to the First Edition This book presents the theory of relativity for students of physics and mathematics who have had no previous introduction to the subject and whose mathematical training does not go beyond the fields which are necessary for studying classical theoretical physics. The specialized mathematical apparatus used in the theory of rela- tivity, tensor calculus, and Ricci calculus, is, therefore, developed in the bookitself. Themainemphasisofthebookisonthedevelopmentofthe basic ideas of the theory of relativity; it is these basic ideas rather than special applications which give the theory its importance among the various branches of theoretical physics. The material has been divided into three parts, the special theory of relativity, the general theory of relativity, and a report on unified field theories. The three parts form a unit. The author realizes that many students are interested in the theory of relativity mainly for its applications to atomic and nuclear physics. It is hoped that these readers will find in the first part, on the special theory of relativity, all the information which they require. Those readers who do not intend to go beyond the special theory of relativity may omit one section of Chapter V (p. 67) and all of Chapter VIII; these passages contain material which is needed only for the development of the general theory of relativity. The second part treats the general theory of relativity, including the work by Einstein, Infeld, and Hoffmann on the equations of motion. The third part deals with several attempts to overcome defects in the general theory of relativity. None of these theories has been com- pletely satisfactory. Nevertheless, the author believes that this report rounds out the discussion of the general theory of relativity by indi- cating possible directions of future research. However, the third part may be omitted without destroying the unity of the remainder. The author wishes to express his appreciation for the help of Pro- fessor Einstein, whoread thewholemanuscriptandmademanyvaluable suggestions. Particular thanks are due to Dr. and Mrs. Fred Fender, who read the manuscript carefully and suggested many stylistic and other improvements. The figures were drawn by Dr. Fender. Margot Bergmann read the manuscript, suggested improvements, and did almost all of the technical work connected with the preparation of the manuscript. The friendly co-operation of theEditorial Department of Prentice Hall, Inc. is gratefully acknowledged. viii P. G. B.