The Geometry of Minkowski Spacetime An Introduction to the Mathemati
This book offers a presentation of the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. It treats, in addition to the usual menu of topics one
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Gregory L. Naber
The Geometry of Minkowski Spacetime An Introduction to the Mathematics of the Special Theory of Relativity Second Edition
With 66 Illustrations
Gregory L. Naber Department of Mathematics Drexel University Korman Center 3141 Chestnut Street Philadelphia, Pennsylvania 19104-2875 USA [email protected]
ISBN 978-1-4419-7837-0 e-ISBN 978-1-4419-7838-7 DOI 10.1007/978-1-4419-7838-7 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011942915 Mathematics Subject Classification (2010): 83A05, 83-01 © Springer Science+Business Media, LLC 2012 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)
For Debora
Preface
It is the intention of this monograph to provide an introduction to the special theory of relativity that is mathematically rigorous and yet spells out in considerable detail the physical significance of the mathematics. Particular care has been exercised in keeping clear the distinction between a physical phenomenon and the mathematical model which purports to describe that phenomenon so that, at any given point, it should be clear whether we are doing mathematics or appealing to physical arguments to interpret the mathematics. The Introduction is an attempt to motivate, by way of a beautiful theorem of Zeeman [Z1 ], our underlying model of the “event world.” This model consists of a 4-dimensional real vector space on which is defined a nondegenerate, symmetric, bilinear form of index one (Minkowski spacetime) and its associated group of orthogonal transformations (the Lorentz group). The first five sections of Chapter 1 contain the basic geometrical information about this model including preliminary material on indefinite inner product spaces in general, elementary properties of spacelike, timelike and null vectors, time orientation, proper time parametrization of timelike curves, the Reversed Schwartz and Triangle Inequalities, Robb’s Theorem on measuring proper spatial separation with clocks and the decomposition of a general Lorentz transformation into a product of two rotations and a special Lorentz transformation. In these sections one will also find the usual kinematic discussions of time dilation, the relativity of simultaneity, length contrac
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