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P.T. Chrusciel, Lectures on Energy in General Relativity Lectures on Energy in General Relativity Krak´ow, March-April 2010 revisions and additions, Vienna, May 2012 and February 2013 comments on mistakes and misprints, and suggestions for improvements welcome Piotr T. Chru´sciel University of Vienna [email protected] http://homepage.univie.ac.at/piotr.chrusciel February 22, 2013 ii Contents Contents iii I Energy in general relativity 1 1 Mass and Energy-momentum 3 1.1 The mass of asymptotically Euclidean manifolds . 4 1.1.1 Mass in two dimensions . 13 1.1.2 Coordinateindependence . 18 1.1.3 Moving hypersurfaces in space-time . 27 1.1.4 Alternative expressions . 28 1.1.5 Energy in stationary space-times, Komar mass . 30 1.2 A space-time formulation . 30 1.2.1 Hamiltonian dynamics . 30 1.2.2 Hamiltonian general relativity . 32 1.2.3 Poincar´echarges, Lorentz invariance . 37 1.3 Mass of asymptotically anti-de Sitter space-times . ..... 42 1.4 The mass of asymptotically hyperboloidal Riemannian manifolds 45 1.5 Quasi-localmass ........................... 45 1.5.1 Hawking’s quasi-local mass . 46 1.5.2 Kijowski’s quasi-local mass . 46 1.5.3 Bartnik’s quasi-local mass . 47 2 Non-spinorial positive energy theorems 49 2.1 Spherically symmetric positive energy theorem . 49 2.2 Axi-symmetry............................. 54 2.2.1 Axi-symmetric metrics on simply connected asymptoti- cally flat three dimensional manifolds . 54 2.2.2 Global considerations . 55 2.2.3 Regularity at the axis . 56 2.2.4 Axisymmetry and asymptotic flatness . 58 2.2.5 Isothermal coordinates . 64 2.2.6 ADMmass .......................... 71 2.2.7 Several asymptotically flat ends . 77 2.2.8 Nondegenerate instantaneous horizons . 77 2.2.9 Conical singularities . 80 iii iv CONTENTS 2.2.10 A lower bound on the mass of black-holes . 80 2.3 The Bartnik-Witten rigidity theorem . 81 2.4 Small data positive energy theorem . 83 2.4.1 Negative mass metrics . 86 2.5 Schoen and Yau’s positive energy theorem . 87 2.6 ALorentzianproof .......................... 88 2.6.1 Galloway’s timelike splitting theorem . 88 2.6.2 Theproofofpositivity . 89 2.7 TheRiemannianPenroseInequality . 93 3 Spinors, and Witten’s positive energy theorem 95 3.1 Spinors:aworkingapproach . 95 3.1.1 Introductoryremarks . 95 3.1.2 The spinorial connection . 97 3.1.3 Transformationlaw. 98 3.1.4 Spinorcurvature . .105 3.1.5 The origin of the spinorial connection . 106 3.2 Witten’spositivityproof. .112 3.3 Generalised Schr¨odinger-Lichnerowicz identities . ........121 3.3.1 A connection involving extrinsic curvature . 123 3.3.2 Maxwellfields. .125 3.3.3 Cosmological constant . 131 4 The Trautman-Bondi mass 133 4.1 Introduction..............................133 4.2 Bondimass ..............................138 4.3 Global charges of initial data sets . 140 4.3.1 The charge integrals . 140 4.4 Initial data sets with rapid decay . 142 4.4.1 Thereferencemetrics . .142 4.4.2 Thecharges..........................143 4.5 Problems with the extrinsic curvature of the conformal boundary 147 4.6 Convergence, uniqueness, and positivity of the Trautman-Bondimass . .148 4.6.1 The Trautman-Bondi four-momentum of asymptotically hyperboloidal initial data sets – the four-dimensional definition ...........................149 4.6.2 Geometric invariance . 150 4.6.3 The Trautman-Bondi four-momentum of asymptotically CMC hyperboloidal initial data sets – a three-dimensional definition ...........................152 4.6.4 Positivity ...........................154 4.7 The mass of approximate Bondi foliations . 156 4.7.1 Polyhomogeneous metrics . 158 4.7.2 The Hawking mass of approximate Bondi spheres . 160 4.8 Proofof(4.3.11) ...........................161 4.9 3 + 1 charge integrals vs the Freud integrals . 162 CONTENTS v 4.10 ProofofLemma4.6.8 . .166 4.11 ProofofLemma4.6.9 . .174 4.12 Asymptotic expansions of objects on S ..............177 4.12.1 Smoothcase .........................177 4.12.2 The polyhomogenous case . 182 4.13 Decomposition of Poincar´egroup vectors into tangential and nor- malparts ...............................182 II Background Material 185 A Introduction to pseudo-Riemannian geometry 187 A.1 Manifolds ...............................187 A.2 Scalarfunctions. .188 A.3 Vectorfields..............................188 A.3.1 Liebracket ..........................191 A.4 Covectors ...............................191 A.5 Bilinear maps, two-covariant tensors . 193 A.6 Tensorproducts.. .. .. .. .. .. .. .. .194 A.6.1 Contractions . 196 A.7 Raising and lowering of indices . 196 A.8 TheLiederivative . .. .. .. .. .. .. .. .198 A.8.1 A pedestrian approach . 198 A.8.2 The geometric approach . 201 A.9 Covariant derivatives . 206 A.9.1 Functions ...........................207 A.9.2 Vectors ............................208 A.9.3 Transformation law . 209 A.9.4 Torsion ............................210 A.9.5 Covectors ...........................210 A.9.6 Higher order tensors . 212 A.9.7 Geodesics and Christoffel symbols . 212 A.10 The Levi-Civita connection . 213 A.11 “Local inertial coordinates” . 215 A.12Curvature ...............................217 A.12.1 Bianchi identities . 220 A.12.2 Pair interchange symmetry . 223 A.13 Geodesic deviation (Jacobi equation) . 224 A.14Nullhyperplanesandhypersurfaces. 226 A.15Isometries ...............................229 A.16Killingvectors .. .. .. .. .. .. .. .. .229 A.17Movingframes ............................235 A.18Cliffordalgebras . .241 A.18.1 Eigenvalues of γ-matrices .. .. .. .. .. .246 A.19 Elements of causality theory . 247 A.19.1Geodesics . .. .. .. .. .. .. .. .248 vi CONTENTS B Weighted Poincar´einequalities 251 Bibliography 255 Part I Energy in general relativity 1 Chapter 1 Mass and Energy-momentum There exist various reasons why one might be interested in the notion of energy in general relativity: 1. First, one expects that generic gravitating systems will emit gravitational waves. Detecting such waves requires a transfer of energy between the field and the detector, and to quantify such effects it is clearly useful to have a device that measures the energy carried away by the gravitational field. A A 2. Next, in Lagrangean field theories with Lagrange function L (φ , ∂µφ ) there is a well defined notion of energy-density, discussed shortly in Sec- tion 1.2. There is no known geometrically defined notion of energy in general relativity, which makes the question more interesting. 3. Now, it turns out that there is a well defined notion of global energy and momentum for isolated gravitating systems. The positivity of this total energy has proved surprisingly difficult to establish, a problem eventually settled by Schoen and Yau [163, 164] in space-dimension lower than or equal to seven [162], and by completely different methods by Witten [187] under the hypothesis that the manifold is spin; the general case remaining open. 4. The positive energy theorem has a surprising application in the proof by Schoen [161] in 1984 of the Yamabe conjecture: for every metric g on a compact manifold there exists a strictly positive function φ such that the metric φg has constant scalar curvature. See [131]. (A solution of the Yamabe problem has also be given by Bahri [9] by completely different methods in 1993.) 5. Yet another surprising application of the positive of energy is the proof, by Bunting and Masood-ul-Alam [43], that event horizons in regular, static, vacuum black holes are connected. 6. The hunting season for an optimal definition of “quasi-local” energy is still open! 3 4 CHAPTER 1. MASS AND ENERGY-MOMENTUM 1.1 The mass of asymptotically Euclidean manifolds There exist various approaches to the definition of mass in general relativity, the first one being due to Einstein [83] himself. In Section 1.2 below we will outline two geometric Hamiltonian approaches to that question. However, those approaches require some background knowledge in symplectic field theory, and it appears useful to present an elementary approach which quickly leads to the correct definition for asymptotically flat Riemannian manifolds without any prerequisites. In the remainder in this chapter we will restrict ourselves to dimensions greater than or equal to three, as the situation turns out to be completely different in di- mension two: Indeed, it should be clear from the considerations below that the mass is an object which is related to the integral of the scalar curvature over the man- ifold. Now, in dimension two, that integral is a topological invariant for compact manifolds, while it is related to a “deficit angle” in the non-compact case. This an- gle, to which we return in Remark 1.1.3, appears to be the natural two-dimensional equivalent of the notion of mass. The Newtonian approximation provides the simplest situation in which it is natural to assign a mass to a Riemannian metric: recall that in this case the space-part of the metric takes the form gij = (1 + 2φ)δij , (1.1.1) where φ is the Newtonian potential, ∆ φ = 4πµ , (1.1.2) δ − with µ – the energy density. (In the Newtonian approximation we also have g = 0, g = 1 + 2φ, but this is irrelevant for what follows.) When µ has 0i 00 − compact support supp µ B(0,R) we have, at large distances, ⊂ M 2 φ = + O(r− ) , (1.1.3) r where M is the total Newtonian mass of the sources: M = µ d3x R3 Z 1 = ∆δφ −4π R3 Z 1 = ∆ φ −4π δ ZB(0,R) 1 = iφ dS −4π ∇ i ZS(0,R) 1 i = lim φ dSi . (1.1.4) − R 4π ∇ →∞ ZS(0,R) Here dSi denotes the usual coordinate surface element, dS = ∂ dx dy dz , (1.1.5) i i⌋ ∧ ∧ 1.1. THE MASS OF ASYMPTOTICALLY EUCLIDEAN MANIFOLDS 5 with denoting contraction. Then the number M appearing in (1.1.3) or, ⌋ equivalently, given by (1.1.4), will be called the mass of the metric (1.1.1). In Newtonian theory it is natural to suppose that µ 0. We then obtain ≥ the simplest possible version of the positive mass theorem: Theorem 1.1.1 (Conformally flat positive mass theorem) Consider a C2 met- ric on R3 of the form (1.1.1) with a strictly positive function 1 + 2φ satisfying 4πµ := ∆δφ 0 , φ r 0 .
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