In December, 2003, I had the pleasure of reviewing the admirable book, Special Relativity, by N. M. J. Woodhouse, and now I have the opportunity to comment on *General Relativity* by the same author. I am happy to recommend not just this sequel, but the indicated pair, for an advanced undergraduate course on relativity or for self-study.

Indeed, Woodhouse presents *General Relativity* as “a sequel, with some overlap in the treatment of tensors, to my *Special Relativity* in this same [Springer Undergraduate Mathematics S]eries,” and states in the Preface that his present objective is to meet the “challenging but rewarding task to teach general relativity to undergraduates,” aiming his discourse at Oxford’s final-year mathematics students. Thus, the reader, if he is indeed an undergraduate, had better be prepared for a pretty serious treatment of the subject and for a decent amount of hard work (and, *qua* preparation, solid knowledge of multivariable calculus and some background in PDEs are certainly called for).

This is not to say that the book, or rather the set of books, is not accessible: quite the contrary. But serious mathematics (or physics) requires a serious reader or pupil, and what I said three years ago about *Special Relativity* is equally relevant to *General Relativity*: “A lot of very nice material is touched on in its pages, presented in a natural sequence consonant with history, and is not improperly belabored. It's also rather informal in style. One gets the sense of breezing along pretty fast while, in actuality, a lot of material is being dealt with. So, an autodidactic reader had better be prepared to cover the book with marginal notes, to fill a thickish note-book with more carefully worked-out material, and to do most if not all of the exercises.” Fair enough.

One particularly noteworthy feature of *General Relativity* is that Woodhouse seeks to present the subject neither as a branch of differential geometry nor as the kind of physics mathematicians like me find unapproachable (and I’m afraid this doesn’t particularly narrow the field). When just a rookie I dabbled in relativity largely from popularizations and biographical writings, and when I tried to learn some real general relativity in graduate school — for cultural reasons, I guess — it simply didn’t take. But my interest in the subject, both specially and generally, has never flagged and Woodhouse’s books are tailor-made for even my lingering ambitions. In other words, for any slacker who feels he should have learned this beautiful material in his mathematical youth, but didn’t, and is now secretly (or not so secretly) desirous of doing it right, this is the book, or, more correctly, these are the books to read. Furthermore, as I already hinted, as far as teaching courses on these important subjects is concerned, obviously these books fit that bill very well, too, given Woodhouse’s specific pedagogical intent.

When it comes to the specific style and presentation of general relativity chosen by Woodhouse, marvelous faithfulness to historical developments, in particular Einstein’s own writings, characterizes the entire treatment. On p.7, already, the weak and strong equivalence principles are presented and analyzed in a succinct and historically rooted fashion. The former, going back to Galileo’s pendulums (Woodhouse correctly says “pendula,” of course) and famously connected with Eötvös’ experiment, entails that inertial mass and gravitational mass are the same; and the latter says that there are no observable differences between the local effects of gravity and acceleration. Woodhouse’s brief discussion of these incomparable axioms underlying Einstein’s revolution is a gem of exposition, covering the historical sweep of the attendant experiments (he even mentions a planned space experiment, “STEP,” which will test the latter principle to within one part in 10^{18} ) and conveying what is to come as a result of these stipulations.

In fact, the whole book is distinguished by this high quality of exposition. For instance, on p. 145, Woodhouse sets out to discuss the subject of gravitational waves with the following phrases: “Maxwell’s equations imply that moving charges generate electromagnetic waves. Einstein’s equations imply that moving masses generate gravitational waves… [W]e explore how this works, for the most part in the linearized theory. Gravitational waves have yet to be detected directly, although the predicted loss of energy through gravitational radiation in the binary pulsar PSR 1913+16 has been verified.” I cannot imagine a clearer and more successful teaser juxtaposing a promise of real mathematics in the service of serious physics with unabashed experimental science. It almost makes me wish that I hadn’t run away from science to philosophy as far as my undergraduate minor was concerned…

Finally, I want to draw special attention to pp. 23–27, where Woodhouse does a phenomenally good job of explicating the subject of tensors in Minkowski space, a subject which has always been a bit unsettling to me who was raised to visit tensor products in their homological algebraic home, and I cannot resist mentioning Problem 1.5 on p. 13, dealing with “Einstein’s birthday present.”

It’s a fine book, beautifully written and clear, and I highly recommend it.

Michael Berg is Professor of Mathematics at Loyola Marymount University in California.