This is a very impressive piece of scholarship. The acknowledgements the author, Yvonne Choquet-Bruhat, provides at the outset include a rather poignant observation to the effect that due to her advanced age she was unable to wait for her long-time collaborator, James H. York, to be free to assist her in writing this tome. Accordingly, General Relativity and the Einstein Equations is presented to us as an impressive solo performance by a prominent scholar who has spent sixty years in the field. She has given us a beautifully written book, encyclopaedic in scope even as it is expressly pedagogical, and significant both for its mathematical and physical content. Indeed, the first paragraph of the Foreword tells us that Choquet-Bruhat has throughout her long career taken pains to stay keep in close touch with the discoveries in (e.g.) astrophysics afforded by the advances in technology, so her that mathematics might always be complemented, as it were, by hard-core physics.
But the nigh-on 800 pages of General Relativity and the Einstein Equations abound with mathematics proper, from differential geometry and PDE to singularity theory and global analysis. After all, it is arguably the case that of all branches of physics general relativity is the farthest removed from, for lack of a better word, laboratory physics (although string theory might seek to vie for that title, CERN’s search for the Higgs boson notwithstanding). On the other hand, with everything from solar eclipses to NASA projects in the game, general relativity has had the universe itself as a laboratory from the beginning. But, tellingly, this was not Einstein’s way: recall the famous episode in which the originator of relativity offered his fountain pen as the answer to an admirer’s query about his “laboratory.” In any case, there is no question today about the titanic importance of general relativity, and the book under review is a serious contribution to the subject qua mathematics.
Choquet-Bruhat describes the first five chapters of her book as an introductory course in general relativity requiring little more than calculus and sufficiently advanced standing as a student: de facto advanced undergraduates and beginning graduate students are targeted, at least at first. The material in question ranges from Lorentz geometry and special relativity space-time and cosmology. Her treatment of differential geometry à la Lorentz (and, presumably, Minkowski, Ricci, Levi-Civita, and others) is clear and neat, even though it is merely meant “to save the reader’s time and fix notation”: the reader should go elsewhere for the nuts and bolts of differential geometry, but Choquet-Bruhat’s introduction is very effective for her purposes. And this is undoubtedly par for the course: I recall a very similar state of affairs in a class in general relativity I attended in graduate school at UCSD with Ted Frankel — one is immediately struck by the idiosyncratic quality of the geometry of the universe.
So it is that the first chapter of General Relativity and the Einstein Equations is full of indispensable material presented at a clip pace: Riemann-Christoffel on p. 11, Ricci curvature on p. 13, geodesics on p. 15, and on to special relativity on p. 19. This in fact sets the tone for everything that follows, inasmuch as Choquet-Bruhat has a lot of physics to cover and doesn’t waste any time doing so. Make no mistake, however: as the book unfolds it exhibits the characteristic theorem-proof cadences of our discipline and comfortably takes its place as a book in the mainstream of mathematical exposition. Indeed, it is at the forefront: General Relativity and the Einstein Equations is exceptionally well-written and will richly reward the diligent student who navigates through these (many) pages.
As already indicated, one encounters PDE, singularity theory, and global analysis in the present physical context, and this is exemplified by the book’s middle chapters. However, it is certainly the case that the required material from these fields, and others besides, is specialized, which is to say, tailored to the specific objective of understanding the workings of the universe we live in. (Said Einstein, famously: “I want to know God’s thoughts; all the rest is details…”) Thus, while the attendant mathematics is deep and serious, one should not come to general relativity for a comprehensive course in differential geometry of differential equations — not even in 800 pages. Still, General Relativity and the Einstein Equations contains over 100 pages of appendices developing such themes as “Sobolev spaces on Riemannian manifolds” and “Conformal methods.” It is all there (even a quartette of related reprints of papers covering such things as causality and supergravity and the interaction between gravitational and fluid waves: all fascinating science).
Again, Choquet-Bruhat’s General Relativity and the Einstein Equations is an impressive book by an expert in the field whose sixty years of serious scholarship shines forth from every page.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.