The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity

by Gregory L. Naber

Overview

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 is accustomed to finding in introductions to special relativity, a wide variety of results of more contemporary origin. These include Zeeman’s characterization of the causal automorphisms of Minkowski spacetime, the Penrose theorem on the apparent shape of a relativistically moving sphere, a detailed introduction to the theory of spinors, a Petrov-type classification of electromagnetic fields in both tensor and spinor form, a topology for Minkowski spacetime whose homeomorphism group is essentially the Lorentz group, and a careful discussion of Dirac’s famous Scissors Problem and its relation to the notion of a two-valued representation of the Lorentz group. This second edition includes a new chapter on the de Sitter universe which is intended to serve two purposes. The first is to provide a gentle prologue to the steps one must take to move beyond special relativity and adapt to the presence of gravitational fields that cannot be considered negligible. The second is to understand some of the basic features of a model of the empty universe that differs markedly from Minkowski spacetime, but may be recommended by recent astronomical observations suggesting that the expansion of our own universe is accelerating rather than slowing down. The treatment presumes only a knowledge of linear algebra in the first three chapters, a bit of real analysis in the fourth and, in two appendices, some elementary point-set topology.

The first edition of the book received the 1993 CHOICE award for Outstanding Academic Title.

Reviews of first edition:

“… a valuable contribution to the pedagogical literature which will be enjoyed by all who delight in precise mathematics and physics.” (American Mathematical Society, 1993)

“Where many physics texts explain physical phenomena by means of mathematical models, here a rigorous and detailed mathematical development is accompanied by precise physical interpretations.” (CHOICE, 1993)

“… his talent in choosing the most significant results and ordering them within the book can’t be denied. The reading of the book is, really, a pleasure.” (Dutch Mathematical Society, 1993)

Product Details

ISBN-13:	9781493902415
Publisher:	Springer New York
Publication date:	03/03/2014
Series:	Applied Mathematical Sciences , #92
Edition description:	2nd ed. 2012
Pages:	324
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

About the Author

Gregory L. Naber is a Professor in the Department of Mathematics at Drexel University in Philadelphia, PA.

Table of Contents

	Preface	v
	Acknowledgments	xi
	Introduction	1
Chapter 1	Geometrical Structure of M	7
1.1	Preliminaries	7
1.2	Minkowski Spacetime	9
1.3	The Lorentz Group	15
1.4	Timelike Vectors and Curves	46
1.5	Spacelike Vectors	61
1.6	Causality Relations	64
1.7	Spin Transformations and the Lorentz Group	74
1.8	Particles and Interactions	87
Chapter 2	Skew-Symmetric Linear Transformations and Electromagnetic Fields	100
2.1	Motivation via the Lorentz Law	100
2.2	Elementary Properties	101
2.3	Invariant Subspaces	107
2.4	Canonical Forms	113
2.5	The Energy-Momentum Transformation	117
2.6	Motion in Constant Fields	121
2.7	Variable Electromagnetic Fields	126
Chapter 3	The Theory of Spinors	143
3.1	Representations of the Lorentz Group	143
3.2	Spin Space	161
3.3	Spinor Algebra	170
3.4	Spinors and World Vectors	178
3.5	Bivectors and Null Flags	188
3.6	The Electromagnetic Field (Revisited)	196
Appendix A	Topologies For M	210
A.1	The Euclidean Topology	210
A.2	E-Continuous Timelike Curves	211
A.3	The Path Topology	215
Appendix B	Spinorial Objects	226
B.1	Introduction	226
B.2	The Spinning Electron and Dirac's Demonstration	227
B.3	Homotopy in the Rotation and Lorentz Groups	229
	References	240
	Symbols	243
	Index	251

Editorial Reviews

From the reviews of the second edition:

“This text brings sophisticated mathematical structures and tools to play, yet much of the work would be accessible to a motivated undergraduate. … The author lays out his goal very clearly: ‘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.’ He then proceeds to accomplish this admirably. … the underlying mathematics is wonderful, worth studying for its own sake.” (William J. Satzer, The Mathematical Association of America, May, 2012)

From the Publisher

Incorporates a broader scope of topics and more recent findings than most introductions to special relativity, and distinguishes rigorously between the physical world and the mathematical model of it. Assumes a solid knowledge of linear algebra in the context of general mathematical maturity. (The appendices assume point-set topology as well.) Annotation c. Book News, Inc., Portland, OR (booknews.com)

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