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The General Theory of Relativity: A Mathematical Exposition

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“General relativity is to date the most successful theory of gravity. In this theory, the gravitational field is not a conventional force but instead is due to the geometric properties of a manifold commonly known as space-time.” This evocative statement, with which the book under review opens, is both pithy and suggestive of a dramatic tension that exists in physics: the radical departure from the standard idea of force engendered by Einstein’s theory of ca. 1917 and everything that it sired, and the subsequent problem of finding a common basis for all known forces in the universe, in particular in the sense of reconciling gravitation and quantum mechanics (in its broad sense, heading toward quantum field theory, or QFT). The authors, Das and DeBenedictis, note the dramatic success of GR, meaning of course the agreement between its numbers and those coming from experiments and observations, which introduces another angle on the question of how to make GR jive with QFT: is either theory in need of refinement or even replacement, or is there an overarching theory that covers both domains of discourse? In a way this is the ultimate question in physics, of course, and something of a Holy Grail.

But before one starts on the metaphysics, there is the mathematics, and along these lines the authors point out that “this book is well suited for the advanced mathematics or physics student, as well as researchers, … [with] a balance of rigorous mathematics and physical insights and applications …” They proceed to note that “[t]he mathematics of the theory of general relativity is mostly derived from tensor algebra and tensor analysis …” and they devote Chapter 1 to this material, focusing on Riemannian and pseudo-Riemannian manifolds (in an arbitrary dimension). Special relativity appears in Chapter 2, quickly followed by the transition from flat 4-dimensional space-time to curved 4-dimensional space-time and Einstein’s famous field equations “which govern gravitational phenomena’: the birth of GR along historical lines.

The rest of this beefy book is devoted to some of what Feynman was known to call “the good stuff,” e.g. spherically symmetric solutions to Einstein’s field equations, black holes, cosmological models (in connection with which Das and DeBenedictis say that “the impact of Einstein’s theory is very deep and revolutionary indeed!”), and Einstein-Maxwell-Klein-Gordon field equations.

Taking a somewhat perverse stance in regard to this book, it is possible to split it into two classes, so to speak, namely, geometry per se and physics. The geometry sections are comprised by Chapters 1 and 7, the latter titled, “algebraic classification of field equations,” with the focus placed on Petrov’s classification of the curvature tensor. The first chapter is de rigeur, covering in little more than 100 pages the topics of differentiable manifolds, tensor fields, Riemannian and pseudo-Riemannian structures, and extrinsic curvature. The rest of the book, as already indicated (and delineated above, if in brief), is physics — but very, very beautiful physics.

Yes, The General Theory of Relativity: A Mathematical Exposition is beefy, as already mentioned; the book runs to well over 600 pages, but it is also written in a very clear style and the mathematics is done carefully and in detail. There are also a lot of “examples, worked-out problems, and exercises (with hints and solutions),” as the back-cover advertises, so it is certainly a pedagogically sound enterprise well worth the price of admission. I am happy to be able to recommend it.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.

Date Received: 
Friday, July 27, 2012
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Anadijiban Das and Andrew DeBenedictis
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Michael Berg
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Friday, August 24, 2012