Customer Reviews

Tensor Geometry: The Geometric Viewpoint and Its Uses (Graduate Texts in Mathematics, 130)

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The authors of this excellent text include a memorable passage in the Introduction that perfectly captures the purpose and primary strength of the book:

"The title of this book is misleading. Any possible title would mislead somebody. 'Tensor Analysis' suggests to a mathematician an ungeometric, manipulative debauch of indices, with tensors ill-defined as 'quantities that transform according to' unspeakable formulae. 'Differential Geometry' would leave many a physicist unaware that the book is about matters with which he is very much concerned. We hope that 'Tensor Geometry' will at least lure both groups to look more closely."

Dodson and Poston's text is a welcome entry in that all-too-small class of books that attempt to bridge the conceptual gulf that separates mathematicians from physicists when they write about differential geometry and general relativity. Modern mathematical treatments of both Riemannian and Lorentzian geometry are typically written primarily in concise and conceptually rich coordinate-free notation; physicists, in sharp contrast, tend to write almost exclusively in a notation that stresses the use of local coordinate systems and index manipulation. A person who is educated in one of these traditions must apply himself with diligence to become proficient in the other; however, this "bilingual" proficiency is surely necessary for the serious students of general relativity, who must study literature written in both styles.

Dodson and Poston's book provides an accessible introduction to the mathematics of general relativity, and it should be particularly useful to both mathematicians and physicists as they develop their abilities to read and write in both coordinate-free and index-based notations. The book is written at a level that should make it accessible to anyone who has studied multi-variable calculus and linear algebra. It is not a complete introduction to either modern differential geometry or general relativity, nor do the authors claim that it is. After all, Spivak devoted five volumes to Riemannian geometry alone and still failed to provide an exhaustive introduction; the subject is enormous in scope.

Mathematicians who find this book helpful in their studies of general relativity might consider looking into the following books, each of which is written in the same mathematical style: (1) Gravitational Curvature by Theodore Frankel (offers a beautiful derivation of the Raychaudhuri Equation); (2) Manifolds, Tensor Analysis and Applications by Abraham, Marsden and Ratiu (Chapters 6, 7 and 8 offer an exceptionally lucid introduction to differential forms, integration on manifolds, the Hodge star operator, the codifferential, and applications of these materials to physics); (3) The Geometry of Kerr Black Holes by Barrett O'Neill (if you want to UNDERSTAND the use of the Weyl curvature tensor in defining the Petrov Type of a spacetime, then read Chapter 5 of this wonderful book); (4) Semi-Riemannian Geometry with Applications to Relativity by Barrett O'Neill (makes an excellent companion text to Dodson and Poston as a mathematically rigorous introduction to GR); (5) General Relativity for Mathematicians by Rainer Sachs and Hung-Hsi Wu (a masterpiece, difficult to find today but worth the effort). For a more far-ranging treatment of geometry with applications beyond GR, Theodore Frankel's The Geometry of Physics is also highly recommended.

After one has used Dodson and Poston and some of these other references as a sort of "Rosetta Stone," then one can become reasonably proficient in deciphering both coordinate-free and coordinate-based literature and translating one into the other. It is sad that the educational process is necessarily so inefficient, but we must be grateful for books like Dodson and Poston's that help us in the endeavor.

I found this book much more comprehensible than the Bishop & Goldberg text (perhaps because the specifics of my particular graduate program made this approach more accessible). The only problem is that in some important areas of the text there are typographical errors; this text needs a new corrected edition--otherwise I have no complaints. The only recommendation I would make to a prospective reader is to obtain a copy of the excellent, old, out-of-print (but still readily avilable used, on the web) Vector and Tensor Analysis by Louis Brand and master the tensor chapter in it. This will prepare the reader to gain an idea of where the authors are heading with their modern, abstract functional analysis approach.
The reader will be greatly assisted by a solid understanding of linear algebra and a preparatory course in functional analysis wouldn't hurt, either. This material is challenging even for a math graduate student; any high school student who could master this book would have to be gifted, indeed.

I agree with previous reviewers, and only wish to add a few comments: 1. This book assumes very little on the part of the reader, which makes it ideal for beginners, as long as they're mature readers. 2. Like many books out there, everything in this book is real and finite dimensional, which is a bit disappointing. 3. It's not as advanced as the writers or reviewers would like to think. For instance, no differential forms, no killing vectors, and although there's a chapter on lie groups it treats only their geometrical aspects and not the algebraic ones. 4. However, it contains two (extensive) chapters on SR and GR which are pure gold, I say! Everything is done from the geometrical point of view, and only AFTER all of the math has been introduced, so the discussion is mature and elegant. In short, this is a good book to read for the geometrical intuition but don't count on it to explain everything about differential geometry. Enjoy!

I found learning GR very frustrating, as I am a mathematician by training and found the woolly "picture the vector" approach of say Graviatation (Misner et al) very loose. it drove me crazy. When I found this I was pleased, and found thye rigourous footing I wanted for the subject. Bundles! After this I could read Gravitation with some comfort.

NOT repeat NOT for anyone but mathematically trained readers.

This book is excellent, but I can't understand that this second edition contains so many "typographical" errors! So, the novice reader will have to consult reference material to make sure he doesn't overlook something, which somehow defeats the purpose of the book.

Besides providing in clear-cut fashion the mathematics essential to research in cosmology, the authors simplify many concepts the physicists make opaque.

Considering the fact that I am a high school student and I had no problem understanding the majority of this text, I would call this book wonderful! It is a book that no student of general relativity or differential geometry can do without. It develops insightful geometric premises early on so that the whole picture of tensors can be absorbed, not just those "definitive" transformation equations. The book is not dry like most math books either. It contains almost witty sections as well as enlightening mathematical ideas. I would recommend this book to anyone studying tensors on their own, or as a supplementary text book for a class.