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MR1316662 (95k:83006) 83C05; 35Q75, 58G16, 83C35 Christodoulou, Demetrios; Klainerman, Sergiu The global nonlinear stability of the Minkowski space. (English) Princeton Mathematical Series, 41. Princeton University Press , Princeton, NJ , 1993. x+514 pp. $69.50. ISBN 0-691-08777-6 This book presents the authors’ theorem on the stability of Minkowski space, a landmark in the development of mathematical relativity. The book is quite selfcontained but it is worth mentioning two useful sources of background information. An article by J.-P. Bourguignon [Astérisque No. 201–203 (1991), Exp. No. 740, 321–358 (1992);MR1157847 (93d:58164)] provides an introduction to various geometrical aspects of the proof while a paper of the authors [Comm. Pure Appl. Math. 43 (1990), no. 2, 137–199;MR1038141 (91a:58202)] shows some of the central analytic tools at work in a simpler setting. The main statement of the theorem is, informally, that, given any initial data set for the vacuum Einstein equations which is sufficiently close to the initial data induced on a hyperplane in Minkowski space, there exists a corresponding solution which is global in the sense of being geodesically complete, and whose asymptotic structure resembles that of Minkowski space. In the book there are three statements of versions of the main theorem which are increasingly precise (and technical). The first two are contained in Chapter 1 (the introduction) while the third is contained in Chapter 10, where the highest level steps of the proof are carried out. The book is not easy to read, due to the very technical nature of its contents, but under the circumstances the quality of the exposition is excellent. It is impossible to give a useful idea of the proof in this review but some elements of its structure will be used to organize the description of the contents of the various chapters which follows. Energy estimates are the motor which drives the machinery of the proof. They are obtained using the Bel-Robinson tensor, a fourth rank tensor quadratic in the curvature. This tensor is such that, given any timelike conformal Killing vector field in a spacetime, it provides a conservation law which represents a weighted L estimate for the curvature. Commuting with vector fields then gives estimates for derivatives of the curvature. (In the context of this proof this step requires a great deal of care.) This part of the proof is carried out in Chapters 7 and 8. Since the proof deals with arbitrary small perturbations of Minkowski space, in general the spacetimes considered possess no Killing vectors. In fact what is used is that these spacetimes do have approximate Killing vectors, which correspond to the exact Killing vectors of Minkowski space. Combining these approximate Killing vectors with the Bel-Robinson tensor leads to approximate conservation laws. The construction of approximate Killing vectors which give useful estimates is very delicate. This construction, and the estimates it gives rise to, forms the content of Chapters 9 and 11–16. As is clear from what has already been said, the primary estimates for the geometry which are obtained are estimates for the