Mathematical Problems of General Relativity I
In this book, the mathematics of general relativity and especially questions related to asymptotic flatness, conserved quantities and stability of Minkowski space-time are treated. In the first chapter, the author introduces the Einstein equation of gravitation on four dimensional Lorentzian manifolds. One can find a concise treatment of the Cauchy problem for the Einstein equation as well as the concepts of causal past and future, regular hyperbolicity and maximal development. The chapter devoted to asymptotic flatness contains (besides other notions) a definition of asymptotically flat initial data, total energy and linear and angular momenta. A Noether-type theorem adapted to general relativity and the result of Schoen and Yau on the positivity of the total energy are also presented. In the last chapter, the problem of stability of Minkowski space-time is introduced and a sketch of a proof of this result is given. Roughly speaking, the theorem predicts that under some given conditions, one can construct a geodesically complete solution to the Einstein equation on a strongly asymptotically Euclidean Lorentzian manifold, which tends to Minkowski space-time along any geodesic. The book is based on a lecture course of the author given at ETH Zurich. It is very well-written and gives a balanced description of the subject, balancing the level of full-length proofs and citations of proofs, disturbed neither by a lack of ideas motivated by physics nor by an absence of mathematical proofs that are written concisely and carefully. The book can serve as an introduction to mathematics of general relativity for physicists as well as mathematicians who would like to familiarise themselves with general concepts of this theory and who would like to know some recent applications that give answers to up-to-date problems of gravitation theory.