Wolfgang Rindler. Southwest Ceuter for Advanced Studies, Dallas, Texas ESSENTIAL

Transcription

1 ESSENTIAL RELATIVITY

2 Wolfgang Rindler Southwest Ceuter for Advanced Studies, Dallas, Texas ESSENTIAL

3 RELATIVITY Special, General, and Cosmological Springer Science+Business Media, LLC

4 ISBN ISBN (ebook) DOI / Van Nostrand Reinhold Company Regional Offices: Cincinnati, New Y ork, Chicago, Millbrae, Dallas Van Nostrand Reinhold Company Foreign Offices: London, Toronto, Melbourne Copyright 1969, by Springer Science+Business Media New York Originally published by Reinhold Book Corporation in Softcover reprint of the hardcover 1 st edition 1969 Library of Congress Catalog Card Number All rights reserved. No part of this work covered by the copyrights hereon may be reproduced or used in any form or by any means-graphic, electronic, or mechanical, including photocopying, recording, taping, or information and retrieval systems-without written permission of the publisher. Manufactured in the United States of America. Published by Van Nostrand Reinhold Company 450 West 33rd Street, New York, N.Y. 1000\ Published simultaneously in Canada by D. Van Nostrand Company (Canada), Ltd. 15 \4 \

5 Preface This book is an attempt to bring the full range of relativity theory within reach of advanced undergraduates, while containing enough new material and simplifications of old arguments so as not to bore the expert teacher. Roughly equal coverage is given tospecial relativity, general relativity, and cosmology. With many judicious omissions it can be taught in one semester, but it would better serve as the basis of a year's work. It is my hope, anyway, that its level and style of presentation may appeal also to wider c1asses of readers unrestricted by credit considerations. General relativity, the modern theory of gravitation in which free particles move along "straightest possible" lines in curved spacetime, and cosmology, with its dynamics for the whole possibly curved universe, not only seem necessary for a scientist's balanced view of the world, but offer some of the greatest intellectual thrills of modern physics. Nevertheless, considered luxuries, they are usually squeezed out of the graduate curriculum by the pressure of specialization. Special relativity escapes this tag with a vengeance, and tends to be taught as a pure service discipline, with too little emphasis on its startling ideas. What better time, therefore, to enjoy these subjects for their own sake than as an underv

6 vi PREFACE graduate? In spite of its forbidding mathematical reputation, even general relativity is accessible at that stage. Anyone who knows the calculus up to partial differentiation, ordinary vectors to the point of differentiating them, and that most useful method of approximation, the binomial theorem, should be able to read this book. Its mathematical level rises very gradually. Fourvectors are introduced half way through special relativity, and are then used sufficiently to leave no doubt that they are tools and not just ornaments. And, of course, no serious approach to general relativity is possible without tensors. Accordingly, these are introduced at the appropriate point (in Chapter 8), but as unobtrusively as possible, so that those who habitually skip a few paragraphs will perhaps hardly notice them. Yet the more am bitious reader, who is prepared to pause and think, will find enough tensor theory outlined here to give hirn a basic understanding of all relativistic tensor arguments. The experience of seeing tensors and Riemannian geometry in action may motivate hirn to learn more about these subjects eventually. In its order of presentation the book breaks with tradition. With few exceptions, the usual procedure has been to accept Newton's definition of inertial frames for special relativity, and not to introduce the equivalence principle until just before it is needed in general relativity. It is then that the student suddenly learns that he has been working against the wrong background: that Newton's frames are not, after all, the frames relevant to special relativity. This is excusable only as long as general relativity, and, in particular, the equivalence principle, are regarded as doubtful. Once we accept them, as do most experts today, we should paint a consistent picture from the beginning. And so, in this book, not only the equivalence principle, but even Mach's principle (with due reservations) and some cosmology precede special relativity in order to set it in its proper perspective. Special relativity is then developed rigorously, with an eye on the physics at all times, but emphasizing ideas, not experimental detail. There is some unavoidable overlap of subject matter with my previous book "Special Relativity"* (referred to hereafter as RSR), but the actual duplication is minimal, and many common topics are treated quite differently. I trust that friends of RSR can read even *Oliver and Boyd, Edinburgh and London; New York: Interseienee Publishers, Inc., a Division of John Wiley & Sons, Ine., seeond edition, 1966.

7 PREFACE vii this part of the present book with profit. There follows an "easy" chapter on general relativity, which represents about the limit to which one can go without the use of tensors. The next chapter (Chapter 8) is probably the hardest in the book, but after Section 75 it can at first be omitted, except for a "light" reading of Section 79. The last chapter, on cosmology, though rigorous and detailed, should be easier again. Those whose primary interest lies in this field can read it alm ost independently of the rest of the book. At the end of the text there is a collection of over 130 exercises, put together rat her carefully. I would urge even the most casual reader not quite to ignore these. Though their full solution often requires ingenuity, even a mere perusal should provide some useful insights. In addition, students should give their curiosity full rein and develop the habit of continually inventing and answering their own problems. Finally a warning: in several sections the units are chosen so as to make the speed of light unity, which should be borne in mind when comparing formulas. These remarks would not be complete without an expression of gratitude to my friend Professor Jürgen Ehlers for reading the manuscript in detail and making innumerable suggestions for improvement, which I was only too glad to accept. Furthermore, it is a pleasure to have this opportunity of thanking Professor Carlo Cattaneo and members of the Istituto Matematico "Guido Castelnuovo" of the U niversity of Rome, where part of this book was written, for their warm hospitality and most stimulating discussions, and the authorities of the Southwest Center for Advanced Studies for granting me leave. Rome April 1969 WOLFGANG RINDLER

8 ABBREVIATIONS USED IN THIS BOOK RSR Rindler, "Special Relativity," 2nd ed SR Special relativity GR General relativity AS Absolute space GT Galilean transformation L T Lorentz transformation RP Relativity principle EP Equivalence principle LIF Local inertial frame CM Center of momentum CP Cosmological principle PCP Perfect cosmological principle R W Robertson-Walker viii

9 Contents PREFACE v CHAPTER I THE RISE AND FAll OF ABSOlUTE SPACE 1. Definition of Relativity Newton's Laws The Galilean Transformation The Set of All Inertial Frames Newtonian Relativity Newton's Absolute Space Objections to Newton's Absolute Space Maxwell's Ether Where is Maxwell's Ether? Lorentz Theory The Relativity Principle Arguments for the Relativity Principle Maxwellian Relativity Origins ofgeneral Relativity Mach'sPrinciple Consequences of Mach's Principle Cosmology Inertial and Gravitational Mass The Equivalence Principle The Semi-Strong Equivalence Principle Consequences ofthe Equivalence Principle ix

10 x CONTENTS CHAPTER 2 EINSTEINIAN KINEMATICS 22. Basic Features of Special Relativity On the Nature of Physical Laws An Archetypal Relativistic Argument The Relativity ofsimultaneity The Coordinate Lattice The Lorentz Transformation Properties of the Lorentz Transformation Alternative Form ofthe Lorentz Transformation Graphical Representation ofthe Lorentz Transformation World-Picture and World-Map Length Contraction Length Contraction Paradoxes Time Dilation The Twin Paradox Velocity Transformation Proper Acceleration Special Relativity Without the Second Postulate CHAPTER 3 EINSTEINIAN OPTICS 39. The Drag Effect The Doppler Effect Aberration and the Visual Appearance of Moving Objects CHAPTER 4 SPACETIME AND FOUR-VECTORS 42. Spacetime Three-Vectors Four-Vectors Four-Tensors The Null Co ne An Example: Wave Motion CHAPTER 5 RElATIVISTIC PARTICLE MECHANICS 48. Domain ofsufficient Validity ofnewton's Laws Why Gravity Does not Fit into Special Relativity Shortcut to Relativistic Mechanics Formal Approach to Relativistic Mechanics A Note on Galilean Four-Vectors Equivalence of Mass and Energy The Center of Momentum Frame Relativistic Billiards p. E Diagrams and Threshold Energies Three-Force and Four-Force... III 58. De Broglie Waves Photons. The Compton Effect The Matter Tensor of Dust CHAPTER 6 RELATIVITY AND ELECTRODYNAMICS 61. Transformation ofthe Field Vectors Magnetic Deflection of Charged Particles The Field of a U niformly Moving Charge The Field of an Infinite Straight Current

11 CONTENTS xi CHAPTER 7 BASIC IDEAS CF GENERAL RElATIVITY 65. Curved Surfaces Curved Spaces of Higher Dimensions Riemannian Spaces A Plan for General Relativity The Gravitationa1 Doppler Effect The Spacetime Around a Spherical Mass Static Fields, Geodesics, and Hamilton's Principle CHAPTER 8 FORMAL DEVElOPMENT OF GENERAL RElATIVITY 72. Tensors in General Relativity The Vacuum Field Equations ofgeneral Relativity The Schwarzschild Solution Rays and Orbits in Schwarzschild Space A Genera1-Re1ativistic "Proof" of E = mc The Schwarzschild Radius The Laws of Physics in Curved Spacetime The Field Equations in the Presence of Matter Modified Schwarzschild Space CHAPTER 9 COSMOlOGY 81. The Basic Facts Apparent Difficulties of Pre-Relativistic Cosmology Cosmological Relativity: The Cosmological Principle Milne's Model The Robertson-Walker Metric Rubber Models, Red Shifts, and Horizons Comparison with Observation Cosmic Dynamics According to Pseudo-Newtonian Theory Cosmic Dynamics According to General Relativity The Friedmann Models Once Again, Comparison with Observation Mach's Principle Re-examined EXERCISES 275 INDEX 313

12