Introduction to the theory of relativity

340 Pages·1976·44.205 MB·English

by  Peter Gabriel Bergmann

Introduction to the
THEORY
OF
RELATIVITy
Peter Gabriel Bergmanh
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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
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©
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.