James Hammond Smith - Introduction to Special Relativity

Introduction to

SPECIAL RELATIVITY

James H. Smith

Dover Publications, Inc.

Mineola, New York

Copyright

Copyright 1965, 1993 by James H. Smith.

All rights reserved.

Bibliographical Note

This Dover edition, first published in 1995 and reissued in 2015, is an unabridged, unaltered republication of the work first published by W. A. Benjamin, Inc., New York, 1965.

Library of Congress Cataloging-in-Publication Data

Smith, James H. (James Hammond), 1925-

Introduction to special relativity / James H. Smith. Dover ed.

p. cm.

Includes index.

Originally published: New York : W.A. Benjamin, 1965.

eISBN-13: 978-0-486-80896-3

1. Relativity (Physics) 2. Special relativity (Physics) I. Title.

QC173.55.S63 1995

Manufactured in the United States by RR Donnelley

68895X02 2015

www.doverpublications.com

Preface

T HIS BOOK is intended to fill a need for an elementary textbook on special relativity.

The original material was developed for a junior course in mechanics for physics majors. It was revised for use by seven entering freshmen in a seminar taught at the Massachusetts Institute of Technology under the auspices of the Science Teaching Center. The final form of the notes was developed for a one semester-hour course at the University of Illinois in special relativity, for which the prerequisites are one semester of freshman mechanics and concurrent registration in later semesters of elementary physics. Each chapter corresponds roughly to one lecture in that course. While conceived as an elementary text, it is hoped that this book will introduce relativity to any beginner in the field, although readers with more advanced preparation will find the mathematics rudimentary and the examples numerous.

The treatment is considerably more lengthy than standard treatments at this level. It is hoped that the book thereby becomes adequate for self-study. The only mathematics necessary is algebra and little attempt has been made to introduce higher mathematics in order to simplify the formalism. The first concepts are introduced from the gedanken experiment approach, but later in the book energy is introduced more formally as a logical consequence of conservation of momentum and the symmetry of moving frames of reference.

The historical introduction in the first chapters is made with an eye to subsequent use in the book and with no idea that it gives a balanced historical approach. For this, the reader is referred to Einsteins Theory of Relativity by Max Born (Dover Publications, New York, 1962). History has been distorted toward an overemphasis on the Michelson-Morley experiment and mechanics, and away from electrodynamics, on the grounds that elementary students have little familiarity with or faith in their ability to manipulate fields. There has been no attempt to discuss the philosophical implications of relativity nor is this book in any sense a popular treatment. For this, the reader is referred to Einsteins Relativity; the Special and the General Theory, a Popular Exposition (15th ed., Methuen, London, 1954). The emphasis is on a real working knowledge at an elementary level. To this end the treatment includes three-dimensional space rather than motion in one direction only, since most real problems seem to occur in three dimensions!

When Einstein wanted a familiar example of a rapidly moving object he chose the railroad train. Times have changed. The reader will find rocketships used here. But please be reminded that rockets still go too slowly for relativistic considerations to enter in their navigation. Therefore, when rocketships are used in exercises and examples, it is simply as a more modern version of a rapidly moving object. In order to find examples which are not too pedantic, liberal use has been made of elementary particle physics. It is elementary particle physicists who find special relativity a part of their daily lives. If you are a reader who finds a muon or K+ meson a very new thing, do not panic. You need know almost nothing about such particles. For further information than that given in this book, there are several paperbacks on the elementary particles: Tracking Down Particles by R. D. Hill (W. A. Benjamin, Inc., New York, 1963), Elementary Particles by D. H. Frisch and A. M. Thorndike (Van Nostrand, Princeton, New Jersey, 1964), and The World of Elementary Particles by K. W. Ford (Blaisdell, New York, 1963).

A word of caution is in order concerning the use in this book of the term proper time. The proper time interval between two events is here defined to be the interval between them as measured by a single clock which is present at both events; the clock has moved from one to the other; it is the time for a race as measured by the runner. Throughout the treatment it is tacitly assumed that the clock moves between the events at a constant velocity. Therefore the proper time interval between two events becomes a unique property of the events. Our use of proper time is thus a special case of the more usual definition of the term that allows accelerated motion and that is a property not only of the events, but also of the world line of the clock joining them. It is easy to generalize this restricted use to infinitesimal intervals and thence to the conventional proper time. I have found this treatment a good one for teaching, and I hope the simplification is permissible on this basis. I should like to thank C. Sherwin for introducing me to the pedagogical use of proper time and G. Ascoli for the basic idea behind the Twin Paradox treatment. Especially I commend the patience of the many students who have suffered through the development of these notes; to them, much thanks.

J AMES H. SMITH

Urbana, Illinois

June 1965

Contents

Introduction

B Y THE YEAR 1900 PHYSICISTS HAD HAD experience with Newtonian mechanics for over two centuries. Its predictions were not only satisfyingly simple, but they were borne out by extensive terrestrial experiments as well as minutely detailed astronomical observations. The other great fields of physical research were electricity, magnetism, optics, and thermodynamics. In the latter part of the nineteenth century thermal phenomena had been shown to be susceptible of a mechanical interpretation, and the genius of Maxwell had shown that light was an electromagnetic wave and therefore that optics, electricity, and magnetism were really different aspects of one unified electromagnetic theory. It is probably a pardonably small simplification, then, to say that at the beginning of the present century, all physics was encompassed in the two great theories of mechanics and electromagnetism.

But there was trouble. These two theories as they stood were basically inconsistent. Not many recognized it, but the trouble was there, and the seeds of relativity were sown. No one can say, of course, how long it would have taken for relativity to be developed without the ideas of Einstein, but there was other contemporary, independent work, notably that of Lorentz and Poincar, which contained many of the ideas which are now known to be consequences of the theory of relativity. It took the keen insight of Einstein, however, to see that the troubles were more basic than those of either mechanics or electromagnetic theory, and were concerned with our most elementary ideas of space and time. He reduced the apparent inconsistencies between mechanics and electromagnetism to two postulates. The first was so fundamental to mechanics and the second to electromagnetism, that they seemed necessary to the success of the two highly successful theories. The demand that the two postulates be consistent led to a new formulation of ideas about space and time, the special theory of relativity. The two postulates are: