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Mathematical General Relativity: a Sampler 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. c 2010 American Mathematical Society Reverts to public domain 28 years from publication 567 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 568 PIOTR T. CHRUSCIEL,´ GREGORY J. GALLOWAY, AND DANIEL POLLACK 5.2.2. Riemannian Penrose Inequality 596 5.2.3. Quasi-local mass 597 5.3. Applications of gluing techniques 597 5.3.1. The linearized constraint equations and KIDs 597 5.3.2. Corvino’s result 598 5.3.3. Conformal gluing 599 5.3.4. Initial data engineering 600 6. Evolution 601 6.1. Strong cosmic censorship 601 6.1.1. Gowdy toroidal metrics 603 6.1.2. Other U(1) × U(1)-symmetric models 604 6.1.3. Spherical symmetry 604 6.2. Weak cosmic censorship 605 6.3. Stability of vacuum cosmological models 606 6.3.1. U(1) symmetry 606 6.3.2. Future stability of hyperbolic models 607 6.4. Stability of Minkowski space-time 607 6.4.1. The Christodoulou–Klainerman proof 607 6.4.2. The Lindblad–Rodnianski proof 608 6.5. Toward stability of Kerr: Wave equations on black hole backgrounds 610 6.6. Bianchi A metrics 610 6.7. The mixmaster conjecture 613 7. Marginally trapped surfaces 615 7.1. Null hypersurfaces 615 7.2. Trapped and marginally trapped surfaces 617 7.3. Stability of MOTSs 620 7.4. On the topology of black holes 621 7.5. Existence of MOTSs 623 Appendix A. Open problems 625 Acknowledgments 627 About the authors 627 References 627 1. Introduction Mathematical general relativity is, by now, a well-established vibrant branch of mathematics. It ties fundamental problems of gravitational physics with beautiful questions in mathematics. The object is the study of manifolds equipped with a Lorentzian metric satisfying the Einstein field equations. Some highlights of its his- tory include the discovery by Choquet-Bruhat of a well-posed Cauchy problem [140], subsequently globalized by Choquet-Bruhat and Geroch [75], the singularity the- orems of Penrose and Hawking [151, 238], the proof of the positive mass theorem by Schoen and Yau [257], and the proof of stability of Minkowski space-time by Christodoulou and Klainerman [88]. There has recently been spectacular progress in the field on many fronts, includ- ing the Cauchy problem, stability, cosmic censorship, construction of initial data, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use MATHEMATICAL GENERAL RELATIVITY 569 and asymptotic behaviour, many of which will be described here. Mutual bene- fits are drawn, and progress is being made, from the interaction between general relativity and geometric analysis and the theory of elliptic and hyperbolic partial differential equations. The Einstein equation shares issues of convergence, collapse, and stability with other important geometric PDEs, such as the Ricci flow and the mean curvature flow. A steadily growing overlap between the relevant scientific communities can be seen. For all these reasons it appeared timely to provide a mathematically oriented reader with an introductory survey of the field. This is the purpose of the current work. In Section 2 we survey the Lorentzian causality theory, the basic language for de- scribing the structure of space-times. In Section 3 the reader is introduced to black holes, perhaps the most fascinating prediction of Einstein’s theory of gravitation, and the source of many deep (solved or unsolved) mathematical problems. In Sec- tion 4 the Cauchy problem for the Einstein equations is considered, laying down the foundations for a systematic construction of general space-times. Section 5 exam- ines initial data sets, as needed for the Cauchy problem, and their global properties. In Section 6 we discuss the dynamics of the Einstein equations, including questions of stability and predictability; the latter question is known under the baroque name of “strong cosmic censorship”. Section 7 deals with trapped and marginally trapped surfaces, which signal the presence of black holes and have tantalizing connections with classical minimal surface theory. The paper is sprinkled with open problems, which are collected in Appendix A. 2. Elements of Lorentzian geometry and causal theory 2.1. Lorentzian manifolds. In general relativity, and related theories, the space of physical events is represented by a Lorentzian manifold.ALorentzianmanifold is a smooth (Hausdorff, paracompact) manifold M = M n+1 of dimension n +1, equipped with a Lorentzian metric g. A Lorentzian metric is a smooth assignment to each point p ∈ M of a symmetric, nondegenerate bilinear form on the tangent space TpM of signature (− + ···+). Hence, if {e0,e1,...,en} is an orthonormal M basis for Tp with respect to g, then, perhaps after reordering the basis, the matrix [g(e ,e )] equals diag (−1, +1,...,+1). A vector v = vαe then has square norm, i j α (2.1) g(v, v)=−(v0)2 + (vi)2 , which can be positive, negative or zero. This leads to the causal character of vectors, and indeed to the causal theory of Lorentzian manifolds, which we shall discuss in Section 2.3. On a coordinate neighborhood (U, xα)=(U, x0,x1,...,xn), the metric g is com- ∂ ∂ pletely determined by its metric component functions on U, gαβ := g( ∂xα , ∂xβ ), ≤ ≤ α ∂ β ∂ ∈ M ∈ α β 0 α, β n:Forv = v ∂xα ,w = w ∂xβ Tp , p U, g(v, w)=gαβv w . (Here we have used the Einstein summation convention: if, in a coordinate chart, an index appears repeated, once up and once down, then summation over that index is implied.) Classically the metric in coordinates is displayed via the line element, 2 α β ds = gαβdx dx . The prototype Lorentzian manifold is Minkowski space R1,n, the space-time of special relativity. This is Rn+1, equipped with the Minkowski metric, which, with respect to Cartesian coordinates (x0,x1,...,xn), is given by ds2 = −(dx0)2 +(dx1)2 + ···+(dxn)2 . License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 570 PIOTR T. CHRUSCIEL,´ GREGORY J. GALLOWAY, AND DANIEL POLLACK Each tangent space of a Lorentzian manifold is isometric to Minkowski space, and in this way the local accuracy of special relativity is built into general relativity. Every Lorentzian manifold (or, more generally, pseudo-Riemannian manifold) (M n+1,g) comes equipped with a Levi-Civita connection (or covariant differentia- tion operator) ∇ that enables one to compute the directional derivative of vector fields. Hence, for smooth vector fields X, Y ∈ X(M ), ∇X Y ∈ X(M ) denotes the co- variant derivative of Y in the direction X. The Levi-Civita connection is the unique connection ∇ on (M n+1,g) that is (i) symmetric (or torsion free), i.e., that satisfies ∇X Y −∇Y X =[X, Y ] for all X, Y ∈ X(M), and (ii) compatible with the metric, i.e. that obeys the metric product rule, X(g(Y,Z)) = g(∇X Y,Z)+g(Y,∇X Z), for all X, Y, Z ∈ X(M). In a coordinate chart (U, xα) one has ∇ μ μ α β (2.2) X Y =(X(Y )+ΓαβX Y )∂μ , where Xα, Y α are the components of X and Y , respectively, with respect to the ∂ μ coordinate basis ∂α = ∂xα , and where the Γαβ’s are the classical Christoffel symbols given in terms of the metric components by 1 (2.3) Γμ = gμν (∂ g + ∂ g − ∂ g ) . αβ 2 β αν α βν ν αβ Note that the coordinate expression (2.2) can also be written as α μ (2.4) ∇X Y = X ∇αY ∂μ , μ μ where ∇αY (often written classically as Y ;α)isgivenby ∇ μ μ μ β (2.5) αY = ∂αY +ΓαβY . We shall feel free to interchange between coordinate and coordinate-free notation. The Levi-Civita connection ∇ extends in a natural way to a covariant differentiation operator on all tensor fields. The Riemann curvature tensor of (M n+1,g)isthemapR : X(M) × X(M) × X(M) → X(M), (X, Y, Z) → R(X, Y )Z,givenby (2.6) R(X, Y )Z = ∇X ∇Y Z −∇Y ∇X Z −∇[X,Y ]Z.
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