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Mathematical General Relativity: a Sampler

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Mathematical General Relativity: a Sampler 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 2 2. ElementsofLorentziangeometryandcausaltheory 3 2.1. Lorentzian manifolds 3 2.2. Einstein equations 5 2.3. Elements of causal theory 5 2.3.1. Past and futures 6 2.3.2. Causality conditions 7 2.3.3. Domains of dependence 9 2.4. Submanifolds 11 3. Stationary black holes 11 3.1. The Schwarzschild metric 12 3.2. Rotating black holes 14 3.3. Killing horizons 15 3.3.1. Surface gravity 16 3.4. The orbit-space geometry near Killing horizons 17 3.4.1. Average surface gravity 18 3.5. Near-horizon geometry 18 3.6. Asymptotically flat metrics 19 3.7. Asymptotically flat stationary metrics 20 3.8. Domains of outer communications, event horizons 20 arXiv:1004.1016v2 [gr-qc] 3 Aug 2010 3.9. Uniqueness theorems 21 3.9.1. Static case 24 3.9.2. Multi-black hole solutions 25 3.10. Non-zero cosmological constant 25 4. The Cauchy problem 26 4.1. The local evolution problem 27 4.1.1. Wave coordinates 27 4.2. Cauchy data 30 4.3. Solutions global in space 31 4.4. Other hyperbolic reductions 32 4.5. The characteristic Cauchy problem 32 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 GG has been supported in part by an NSF grant DMS 0708048. 1 2 PIOTR T. CHRUSCIEL,´ GREGORY J. GALLOWAY, AND DANIEL POLLACK 4.6. Initial-boundary value problems 32 5. Initial data sets 32 5.1. The constraint equations 33 5.1.1. The constraint equations on asymptotically flat manifolds 36 5.2. Mass inequalities 36 5.2.1. The Positive Mass Theorem 36 5.2.2. Riemannian Penrose Inequality 37 5.2.3. Quasi-local mass 38 5.3. Applications of gluing techniques 38 5.3.1. The linearized constraint equations and KIDs 38 5.3.2. Corvino’s result 39 5.3.3. Conformal gluing 40 5.3.4. Initial data engineering 40 5.3.5. Non-zero cosmological constant 42 6. Evolution 42 6.1. Taub–NUT space-times 43 6.2. Strong cosmic censorship 44 6.2.1. Gowdy toroidal metrics 45 6.2.2. Other U(1) U(1)-symmetric models 47 6.2.3. Spherical symmetry× 47 6.3. Weak cosmic censorship 48 6.4. Stability of vacuum cosmological models 48 6.4.1. U(1) symmetry 48 6.4.2. Future stability of hyperbolic models 49 6.5. Stability of Minkowski space-time 49 6.5.1. The Christodoulou-Klainerman proof 50 6.5.2. The Lindblad-Rodnianski proof 51 6.6. Towards stability of Kerr: wave equations on black hole backgrounds 53 6.7. Bianchi A metrics 54 6.8. The mixmaster conjecture 57 7. Marginally trapped surfaces 58 7.1. Null hypersurfaces 59 7.2. Trappedandmarginallytrappedsurfaces 61 7.3. Stability of MOTSs 63 7.4. On the topology of black holes 65 7.5. Existence of MOTSs 67 Appendix A. Open problems 69 References 71 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 [178], MATHEMATICAL GENERAL RELATIVITY 3 subsequently globalized by Choquet-Bruhat and Geroch [93], the singularity the- orems of Penrose and Hawking [201, 317], the proof of the positive mass theorem by Schoen and Yau [344], and the proof of stability of Minkowski space-time by Christodoulou and Klainerman [107]. There has recently been spectacular progress in the field on many fronts, includ- ing the Cauchy problem, stability, cosmic censorship, construction of initial data, 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. 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 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. A Lorentzian manifold 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 T M of signature∈ ( + +). Hence, if e ,e , ..., e is an orthonormal p − ··· { 0 1 n} basis for TpM 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: For v = 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 4 PIOTR T. CHRUSCIEL,´ GREGORY J. GALLOWAY, AND DANIEL POLLACK index is implied.) Classically the metric in coordinates is displayed via the “line 2 α β element”, 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 . − ··· 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 Y ∇X = [X, Y ] for all X, Y X(M), and (ii) compatible with the metric, ∇X − ∇Y ∈ 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, (2.2) Y = (X(Y µ)+Γµ XαY β)∂ , ∇X αβ µ where Xα, Y α are the components of X and Y , respectively, with respect to the co- ∂ µ ordinate 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, (2.4) Y = Xα Y µ∂ , ∇X ∇α µ where Y µ (often written classically as Y µ ) is given by, ∇α ;α (2.5) Y µ = ∂ Y µ +Γµ Y β . ∇α α αβ We shall feel free to interchange between coordinate and coordinate free no- tations. 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) is the map R : X(M) X(M) X(M) X(M), (X,Y,Z) R(X, Y )Z, given by × × → → (2.6) R(X, Y )Z = Z Z Z.
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