Differential Geometry and Relativity Theory: An Introduction

265 Pages·1983·6.386 MB·English

by  Richard L. Faber

Differential Geometry
and Relativity Theory
An Introduction
Richard L. Faber
Department of Mathematics
Boston College
Chestnut Hill, Massachusetts
MARCEL DEKKER, INC. NewY0rk and Basel
Library ofCongress Catalogingin Publication Data
Faber, Richard L.
Differential geometryand relativity theory.
(Monographsand textbooksin prueand applied
mathematics : v. 75)
Bibliography: p.
Includesindex.
1. Relativity(Physics) 2. Geometry, Differential.
3. Geometry, Riemannian. I. Title. II. Series.
QC173.55.F33 1983 530.1’l’015l636 82-19938
ISBN0-8247-1749-X
COPYRIGHT © 1983 by MARCEL DEKKER, INC. ALL RIGHTS RESERVED
Neither this book nor any part may be reproduced or transmitted in any
form or by any means, electronic or mechanical, including photocopying,
microfilming, and recording, or by any information storage and retrieval
system, without permission in writing from the publisher.
MARCEL DEKKER, INC.
270 Madison Avenue, New York, New York 10016
Current printing (last digit):
10 9 8 7 6 5 4 3 2 l
PRINTED IN THE UNITED STATES OF AMERICA
To Susan and Lynn

Preface
This text is an introduction to differential and Riemannian geometry
and to the rudiments of special and general relativity. It is the
author’s hope that, for perhaps the first time, the general theory
ofrelativity is herein made accessible to mathematics majors whose
background includes only multivariable calculus and linear algebra.
Chapter I is in itself a mini—course in differential geometry.
Topics include curves in the plane and 3—space, the first and second
fundamental forms ofsurfaces in 3—space, curvature and geodesics,
the Theorema Egregium of Gauss, and abstract surfaces and
manifolds.
Chapter II presents the physical foundations ofspecial relativity
and its geometric interpretation as the geometry offlat spacetime.
Here the emphasis is on coordinates, the Lorentz metric, and space-
time diagrams.
'
Chapter III begins with an intuitive and non-technical account
ofthe key ideas ofEinstein’s general theory ofrelativity. Following
this is a more mathematical treatment using the tools developed in
Chapter I. The requirement that general relativity should give close
agreement with classical Newtonian physics is used to explain how
Einstein arrived at his field equations. The Schwarzschild solution
is derived, and the orbital equations are deduced, both for classical
physics and general relativity. Additional consequences of general
relativity, such as the “bending” oflight, the gravitational redshift
and perihelial precession, are also discussed.
vi PREFACE
Although all of the details are worked out, the reader is
required only to have mastered the mathematical machinery set
out in the first chapter for two-dimensional surfaces. Indeed the
calculation of relativistic orbits is seen as a higher dimensional
version ofthe calculation ofgeodesics in the plane in terms ofpolar
coordinates.
The point ofview ofthe treatment is that relativity is a geo-
metric theory ofspace and time, in which gravity is not a force, as
Newton conceived it, but rather a manifestation of the curvature
of (4—dimensional) spacetime. It is the distribution of matter
throughout the universe that determines this curvature, which in
turn defines the geodesics along which both matter and light are
constrained to move.
Abundant exercises appear at the end of nearly all sections.
These are indispensable for the student’s progress through the book.
Most ofthe exercises are designed to give the student experience in
applying the concepts presented within the text; others call for
verification of basic facts used in the text; some give additional
details or alternative explanations of the more technical matters;
and some introduce optional new material.
For students of average preparation and ability, there is
probably enough material here to fill three quarters or even an entire
year, especially ifa majority ofthe exercises are to be assigned and
discussed in class. On the otherhand, with some trimming, I have
been able to teach most ofthe text to better prepared classes in one
semester. I have also found Chapter I useful for independent reading
projects.
Acknowledgments
First and foremost, I would like to thank my beautiful wife Susan,
who typed much ofthe manuscript, for abundant patience, under-
standing and love. I wish also to thank the following mathematicians,
physicists, and historians ofscience for their helpful comments and
suggestions: Jay P. Fillmore, Thomas Hawkins, Meyer Jordan, John
Kenelly, Bennett Kivel, Edwin Taylor, Thomas Tucker, and R. L.
Wilder.
I am indebted also to my students for many text improvements
which resulted from their questions and criticisms.
vii

Contents
PREFACE
ACKNOWLEDGMENTS vii
I SURFACES AND THE CONCEPT OF CURVATURE
“ Curves
!
!" Gauss Curvature (Informal Treatment) 14
’
*
S Surfaces in E3 22
“ The First Fundamental Form 30
:
"
$ The Second Fundamental Form 41
"
F The Gauss Curvature in Detail 49
.\' Geodesics 57
°
9 The Curvature Tensor and the Theorema
Egregium 74
9. Manifolds 81
II SPECIAL RELATIVITY (THE GEOMETRY OF
FLAT SPACETIME) 95
“ Inertial Frames of Reference 96
!
°
!‘ The Michelson-Morley Experiment 98
5*’ The Postulates of Relativity 109
:‘*‘ Relativity of Simultaneity 110
"
S Coordinates 116
F" Invariance ofthe Interval 121
x CONTENTS
7. The Lorentz Transformation 128
8. Spacetime Diagrams 138
9. Lorentz Geometry 145
10. The Twin Paradox 149
11. Temporal Order and Causality 159
III GENERAL RELATIVITY (THE GEOMETRY
OF CURVED SPACETIME) 165
1. The Principle ofEquivalence 166
2. Gravity as Spacetime Curvature 172
3. The Consequence’s ofEinstein’s Theory 182
4. The Universal Law ofGravitation 185
5. Orbit’s in Newton’s Theory 190
6. Geodesics 197
7. The Field Equations 200
8. The Schwarzschild Solution 21
1
9. Orbits in General Relativity 223
10. The Bending of Light 230
APPENDIX A: Vector Geometry and Analysis 235
APPENDIX B: Hyperbolic Functions 243
BIBLIOGRAPHY 247
INDEX 25
1