Mathematics – Logic
Scientific paper
Oct 2005
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=2005cqgra..22.4388w&link_type=abstract
Classical and Quantum Gravity, Volume 22, Issue 20, pp. 4388-4389 (2005).
Mathematics
Logic
Scientific paper
I have had a love/hate relationship with this book ever since it first came out. On the one hand, this is an excellent introduction for mathematicians to the differential geometry underlying general relativity. On the other hand, this is definitely a book for mathematicians.
The book's greatest strength is its clear, precise presentation of the basic ideas in differential geometry, combined with equally clear and precise applications to theoretical physics, notably general relativity. But the book's precision is also its greatest weakness; this is not an easy book to read for non-mathematicians, who may not appreciate the notational complexity, some of which is nonstandard.
The present edition is very similar to the original, published in 1992. In addition to minor revisions and clarifications of the material, there is now a brief introduction to fibre bundles, and a (very) brief discussion of the gauge theory description of fundamental particles. The index to the symbols used is also a more complete than in the past, but without the descriptive material present in the previous edition.
The bulk of the book consists of a careful introduction to tensors and their properties. Tensors are introduced first as linear maps on vector spaces, and only later generalized to tensor fields on manifolds. The differentiation and integration of differential forms is discussed in detail, including Stokes' theorem, Lie differentiation and Hodge duality, and connections, curvature and torsion. To this point, Wasserman's text can be viewed as an expanded version of Bishop and Goldberg's classic text [1], one major difference being Wasserman's inclusion of the pseudo-Riemannian case from the beginning (in particular, when discussing Hodge duality). Whether one prefers Wasserman's approach to Bishop and Goldberg's is largely a matter of taste: Wasserman's treatment is both more complete and more precise, making it easier to check calculations in detail, but occasionally more difficult to remember what one is calculating. An instructor using this text would be well advised to think carefully about which topics to cover, rather than trying to do everything.
The remainder of the book contains applications to mechanics, relativity and gauge theory. In each case, better treatments exist elsewhere. However, each such treatment typically introduces its own notation; it is not without some truth that differential geometry is often described as the study of objects under changes of notation. Having several short treatments of these different topics in one place makes it easy for the instructor to choose those he or she wishes to emphasize, while providing a clear transition to more advanced treatments.
The presentation does have some idiosyncrasies. The key concept of a derivation is not clearly defined. Some subtleties are referred to in cryptic comments ('This space is too large') which are never explained. The occasionally nonstandard notation is not always easy to follow, although this is a common criticism of introductory texts in differential geometry, which must balance precision with understanding. And appropriate cross references are not always given in sentences of the form 'Recall that ...', which on occasion contained (correct) results which I did not find obvious, and which I could not quickly find explicitly stated in earlier sections. I also have one minor gripe about the publishing format, namely that the outside margins are too small, at only 1 cm; I found this extremely distracting. But all of these criticisms are minor.
Wasserman's book would unquestionably be an excellent introduction to tensor analysis for mathematicians, especially those who are interested in the physics applications. The readers of Classical and Quantum Gravity will want to know whether the book is equally suitable as a text for an introductory course in general relativity. At first glance, this text appears to fill a niche in mathematical sophistication between, say, the undergraduate texts by Schutz [2] or d'Inverno [3], which do not require prior background in differential geometry, and the graduate text by Sachs [4], which does. Nonetheless, the answer, unfortunately, is no.
I teach a course in general relativity primarily aimed at advanced undergraduate mathematics majors, which is, however, taken by both undergraduate and graduate students in both mathematics and physics; it is the only course in general relativity taught at my university. Each time I teach the course, I struggle to choose an appropriate text. Wasserman's book has frequently been one of the finalists, but I have never quite been willing to inflict it on my physics students, choosing instead to list it as an optional text. (I have been equally unwilling to inflict Hartle's marvellous book [5] on my mathematics students.)
My main objection to this book as a relativity text is not, however, due to concern that the mathematical sophistication is too much for undergraduate physics students. Rather, as a mathematical text in relativity, it does not go far enough. The standard applications are here, but just barely. There is a good discussion of the Schwarzschild solution, including applications to solar system tests and its interpretation as a black hole. However, the Kerr solution is barely mentioned, and the Reissner-Nordström solution is conspicuously absent. Similarly, the cosmological implications of the Friedmann-Robertson-Walker solutions are discussed only briefly. And that's it - no further cosmology, no mention of gravitational waves, etc. Fair enough for a mathematics text in tensor analysis, but not enough to permit this book to be used as a standalone relativity text. This is, ultimately, why I have never used this book as a primary text: I want to spend more time doing relativity and less time proving theorems.
For a course for mathematicians (only), however, or as a supplementary text filling in some of the missing mathematical details, this book would be an excellent choice.
References
[1] Bishop R L and Goldberg S I 1980 Tensor Analysis on Manifolds (New York: Dover)
[2] Schutz B F 1985 A First Course in General Relativity (Cambridge: Cambridge University Press)
[3] d'Inverno R 1991 Introducing Einstein's Relativity (Oxford: Oxford University Press)
[4] Sachs R K 1977 General Relativity for Mathematicians (New York: Springer)
[5] Hartle J B 2003 Gravity: An Introduction to Einstein's General Relativity (San Francisco: Addison Wesley)
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