BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 47, Number 4, October 2010, Pages 567–638 S 0273-0979(2010)01304-5 Article electronically published on July 30, 2010
MATHEMATICAL GENERAL RELATIVITY: A SAMPLER
PIOTR T. CHRUSCIEL,´ GREGORY J. GALLOWAY, AND DANIEL POLLACK
Abstract. We provide an introduction to selected recent advances in the mathematical understanding of Einstein’s theory of gravitation.
Contents 1. Introduction 568 2. Elements of Lorentzian geometry and causal theory 569 2.1. Lorentzian manifolds 569 2.2. Einstein equations 571 2.3. Elements of causal theory 571 2.3.1. Pasts and futures 572 2.3.2. Causality conditions 573 2.3.3. Domains of dependence 575 2.4. Submanifolds 576 3. Stationary black holes 577 3.1. The Schwarzschild metric 577 3.2. Rotating black holes 579 3.3. Killing horizons 580 3.3.1. Surface gravity 581 3.3.2. Average surface gravity 582 3.4. Asymptotically flat metrics 582 3.5. Asymptotically flat stationary metrics 583 3.6. Domains of outer communications, event horizons 583 3.7. Uniqueness theorems 584 4. The Cauchy problem 586 4.1. The local evolution problem 587 4.1.1. Wave-coordinates 587 4.2. Cauchy data 589 4.3. Solutions global in space 590 5. Initial data sets 592 5.1. The constraint equations 592 5.2. Mass inequalities 595 5.2.1. The Positive Mass Theorem 595
Received by the editors September 11, 2008 and, in revised form, November 6, 2008. 2010 Mathematics Subject Classification. Primary 83-02. Support by the Banff International Research Station (Banff, Canada), and by Institut Mittag- Leffler (Djursholm, Sweden) is gratefully acknowledged. The research of the second author has been supported in part by an NSF grant DMS 0708048.