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Book review

Author(s) Naber, Gregory L.
Title The Geometry of Minkowski Spacetime
Publisher Springer
Year of publication 1992
Reviewed by Paul Blaga

Since two or three decades, we assist at a process of geometrization of physics. The geometry is more and more involved in most of the hot problems of contemporary theoretical physics. The book under review represents another step in this process. The author's confessed aim is to provide an introduction to special relativity that is mathematically rigorous and, on the other hand, to emphasize the physical significance of the mathematics involved.

The first chapter of the book presents the basic informations about the geometrical and the causal structure of the Minkowski spacetime. I have to quote, between other important results of this chapter the Zeeman characterization of the causal automorphisms of Minkowski spacetime and the Penrose theorem on the apparent shape of a relativistically moving sphere. The second chapter is devoted to the construction of a geometric theory of the electromagnetic field, represented as a skew-symmetric linear transformation. It is proved that the energy-momentum transformation associated to such a skew-symmetric linear map verifies the Dominant Energy Condition. There are written, also, the Maxwell equations by using skew-symmetric bilinear forms. The last chapter of the book is an introduction to the theory of spinors in Minkoviski spacetime. Several applications of spinor formalism are given here. Among others, there is given a classification of electromagnetic fields, similar to the Petrov classification of spacetime manifolds. The book contains, also, two appendices. The first introduces the so-called Zeeman topology for Minkowski spacetime, a topology that is not equivalent to Euclidian topology, but has more physical significance.The second appendix is concerned with Dirac's "Scissors Problem" and its relation with the representations of the Lorentz group. The preresquisites include only a knowledge of linear algebra and some point-set topology (for the two appendices).

This monograph represents, without any doubt, a valuable addition to the literature. It can be of a real help for anyone interested in special relativity, especially for mathematicians searching for rigor. In particular, the careful discussion of the geometry and causal structure of Minkowski spacetime could contribute to a better understanding of the structure of a general spacetime manifold. On the other hand, I should emphasize that this book contains more informations than the classical books on this topic. Beside the traditional menu (treated in a modern, geometrical language), it is presented much material hard to be find in other books, or even published for the first time in monograph literature. The reading of the book not presuming too many knowledges, the range of readers is very large, from undergraduate students to experts. I don't know if the reader is an expert active in special relativity, but 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. Gregory Naber is, also, the author of "Spacetime and singularities", Cambridge University Press, 1990.