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Geometry of Black Holes Revised July 2018 Geometry of Black Holes revised July 2018 Piotr T. Chru´sciel University of Vienna [email protected] http://homepage.univie.ac.at/piotr.chrusciel August 15, 2018 Contents Contents iii I Black holes 1 1 An introduction to black holes 3 1.1 Black holes as astrophysical objects . 3 1.2 The Schwarzschild solution and its extensions . ... 9 1.2.1 The singularity r =0..................... 14 1.2.2 Eddington-Finkelstein extension . 14 1.2.3 The Kruskal-Szekeres extension . 18 1.2.4 Other coordinate systems, higher dimensions . 23 1.2.5 Somegeodesics ........................ 29 1.2.6 The Flamm paraboloid . 31 1.2.7 Fronsdal’sembedding . 33 1.2.8 Conformal Carter-Penrose diagrams . 34 1.2.9 Weylcoordinates . 36 1.3 Somegeneralnotions. 37 1.3.1 Isometries........................... 37 1.3.2 Killinghorizons. 38 1.3.3 Surfacegravity . .. .. .. .. .. .. 40 1.3.4 The orbit-space geometry near Killing horizons . 47 1.3.5 Near-horizon geometry . 48 1.3.6 Asymptotically flat stationary metrics . 52 1.3.7 Domains of outer communications, event horizons . 54 1.4 Extensions............................... 55 1.4.1 Distinct extensions . 55 1.4.2 Inextendibility . 56 1.4.3 Uniqueness of a class of extensions . 58 1.5 TheReissner-Nordstr¨ommetrics . 61 1.6 TheKerrmetric ........................... 63 1.6.1 Non-degenerate solutions (a2 <m2): Bifurcate horizons . 72 1.6.2 Surface gravity, thermodynamical identities . 76 1.6.3 Carter’stimemachine . 78 1.6.4 Extreme case a2 = m2: horizon, near-horizon geometry, cylindricalends. .. .. .. .. .. .. 79 iii iv CONTENTS 1.6.5 TheErnstmapfortheKerrmetric . 81 1.6.6 Theorbitspacemetric. 82 1.6.7 Kerr-Schild coordinates . 83 1.6.8 Dain coordinates . 83 1.7 Majumdar-Papapetrou multi black holes . 84 1.7.1 Adding bifurcation surfaces . 88 1.8 The Kerr-de Sitter/Kerr-anti-de Sitter metric . 89 1.8.1 Asymptoticbehavior. 92 2 Emparan-Reall “black rings” 95 2.1 x ξ ,ξ .............................. 97 ∈ { 1 2} 2.2 Signature ............................... 98 2.3 y = ξ1 ................................. 99 2.4 Asymptoticflatness. 99 2.5 y ................................102 → ±∞ 2.6 Ergoregion...............................103 2.7 Blackring...............................104 2.8 Somefurtherproperties . .105 2.9 A Kruskal-Szekeres type extension . 110 2.10 Globalstructure . .113 2.10.1 The event horizon has S2 S1 R topology . 114 × × 2.10.2 Inextendibility at z = ξF , maximality . 115 2.10.3 Conformal infinity I ....................116 2.10.4 Uniqueness and non-uniqueness of extensions . 117 2.10.5 Other coordinate systems . 118 3 Rasheed’s Kaluza-Klein black holes 121 3.1 Rasheed’smetrics. .121 3.2 Zerosofthedenominators . .123 3.3 Regularity at the outer Killing horizon ............126 H+ 3.4 Asymptoticbehaviour . .127 4 Diagrams, extensions 129 4.1 Causality for a class of bloc-diagonal metrics . 129 4.1.1 Riemannian aspects . 130 4.1.2 Causality ...........................132 4.2 Thebuildingblocs . .. .. .. .. .. .. .. .133 4.2.1 Two-dimensional Minkowski spacetime . 133 4.2.2 Higher dimensional Minkowski spacetime . 135 1 4.2.3 F − divergingatbothends . .137 4.2.4 F 1 diverging at one end only . 139 R − 4.2.5 Generalised Kottler metrics with Λ < 0 and m = 0 . 140 R 4.3 Putting things together . 141 4.3.1 Four-blocsgluing . .141 4.3.2 Two-blocsgluing . .146 4.4 Generalrules .............................147 4.5 Black holes / white holes . 149 CONTENTS v 4.6 Birminghammetrics . .150 4.6.1 Cylindrical solutions . 151 4.6.2 Naked singularities . 152 4.6.3 Spatially periodic time-symmetric initial data . 153 4.6.4 Killinghorizons. .153 4.6.5 Curvature...........................154 4.6.6 The Euclidean Schwarzschild - anti de Sitter metric . 158 4.7 Projectiondiagrams . .162 4.7.1 Thedefinition.. .. .. .. .. .. .. .163 4.7.2 Simplestexamples . .165 4.7.3 TheKerrmetrics. .167 4.7.4 The Kerr-Newman metrics . 176 4.7.5 The Kerr - de Sitter metrics . 177 4.7.6 The Kerr-Newman - de Sitter metrics . 181 4.7.7 The Kerr-Newman - anti de Sitter metrics . 184 4.7.8 The Emparan-Reall metrics . 186 4.7.9 The Pomeransky-Senkov metrics . 191 4.8 Black holes and Cauchy horizons . 193 4.8.1 Kerr-Newman-(a)dS-type and Pomeransky-Senkov-type models.............................194 4.8.2 Taub-NUT metrics . 196 5 Alternative approaches 201 5.1 The standard approach and its shortcomings . 201 5.2 BlackholeswithoutScri . .205 5.2.1 Naive black holes . 206 5.2.2 Quasi-local black holes . 208 6 Dynamical black holes: the Robinson-Trautman metrics 213 6.1 Robinson–Trautman spacetimes. 213 6.1.1 m> 0.............................216 6.1.2 m< 0.............................219 6.1.3 Λ =0 .............................220 6 II Background Material 225 A Pseudo-Riemannian geometry 227 A.1 Manifolds ...............................227 A.2 Scalarfunctions. .228 A.3 Vectorfields..............................228 A.3.1 Liebracket ..........................231 A.4 Covectors ...............................231 A.5 Bilinear maps, two-covariant tensors . 233 A.6 Tensorproducts.. .. .. .. .. .. .. .. .234 A.6.1 Contractions . 236 A.7 Raising and lowering of indices . 236 vi CONTENTS A.8 TheLiederivative . .. .. .. .. .. .. .. .238 A.8.1 A pedestrian approach . 238 A.8.2 The geometric approach . 241 A.9 Covariant derivatives . 247 A.9.1 Functions ...........................248 A.9.2 Vectors ............................249 A.9.3 Transformation law . 250 A.9.4 Torsion ............................251 A.9.5 Covectors ...........................251 A.9.6 Higher order tensors . 253 A.10 The Levi-Civita connection . 253 A.10.1 Geodesics and Christoffel symbols . 255 A.11 “Local inertial coordinates” . 256 A.12Curvature ...............................258 A.12.1 Bianchi identities . 262 A.12.2 Pair interchange symmetry . 265 A.12.3 Summmary for the Levi-Civita connection . 267 A.12.4 Curvature of product metrics . 268 A.12.5 An identity for the Riemann tensor . 269 A.13Geodesics ...............................270 A.14 Geodesic deviation (Jacobi equation) . 272 A.15Exterioralgebra . .274 A.16 Submanifolds, integration, and Stokes’ theorem . 278 A.16.1Hypersurfaces. .279 A.17Oddforms(densities) . .282 A.18Movingframes ............................283 A.19LovelockTheorems . .292 A.19.1 Lovelock Lagrangeans . 293 A.19.2 Lovelock tensors . 296 A.20Cliffordalgebras . .299 A.20.1 Eigenvalues of γ-matrices .. .. .. .. .. .304 A.21 Killing vectors and isometries . 305 A.21.1 Killing vectors . 306 A.21.2 Analyticity of isometries . 310 A.21.3 The structure of isometry groups of asymptotically flat spacetimes ..........................310 A.21.4 Killing vectors vs. isometry groups . 311 A.22Nullhyperplanes . .313 A.23Thegeometryofnullhypersurfaces . 316 A.24 Elements of causality theory . 319 B A collection of identities 321 B.1 ADMnotation ............................321 B.2 Somecommutators . .321 B.3 Bianchi identities . 322 B.4 Linearisations. .. .. .. .. .. .. .. .. .322 B.5 Warpedproducts . .. .. .. .. .. .. .. .322 CONTENTS vii B.6 Hypersurfaces.............................323 B.7 Conformaltransformations . .323 B.8 Laplaciansontensors . .324 B.9 Stationarymetrics . .325 Bibliography 327 Part I Black holes 1 Chapter 1 An introduction to black holes Black holes belong to the most fascinating objects predicted by Einstein’s theory of gravitation. Although they have been studied for years,1 they still attract tremendous attention in the physics and astrophysics literature. It turns out that several field theories are known to possess solutions which exhibit black hole properties: The “standard” gravitational ones which, according to our current pos- • tulates, are black holes for all classical fields. The “dumb holes”, which are the sonic counterparts of black holes, first • discussed by Unruh [269]. The “optical” ones – the black-hole counterparts arising in the theory of • moving dielectric media, or in non-linear electrodynamics [186, 221]. The “numerical black holes” – objects constructed by numerical general • relativists. (An even longer list of models and submodels can be found in [13].) In this work we shall discuss various aspects of the above. The reader is referred to [34, 105, 155, 162, 232, 272] and references therein for a review of quantum aspects of black holes. Insightful animations of journeys in a black hole spacetime can be found at http://jilawww.colorado.edu/~ajsh/insidebh/schw.html. We start with a short review of the observational status of black holes in astrophysics. 1.1 Black holes as astrophysical objects When a star runs out of nuclear fuel, it must find ways to fight gravity. Current physics predicts that dead stars with masses up to the Chandrasekhar limit, MmcH = 1.4M , become white dwarfs, where electron degeneracy supplies the ⊙ 1The reader is referred to the introduction to [48] for an excellent concise review of the history of the concept of a black hole, and to [47, 160] for more detailed ones. 3 4 CHAPTER 1. AN INTRODUCTION TO BLACK HOLES necessary pressure. Above the Chandrasekhar limit 1.4M , and up to a second ⊙ mass limit, MNS,max 2 3M , dead stars are expected to become neutron ∼ − ⊙ stars, where neutron degeneracy pressure holds them up. If a dead star has a mass M > MNS,max, there is no known force that can hold the star up. What we have then is a black hole. While there is growing evidence that black holes do indeed exist in astro- physical objects, and that alternative explanations for the observations dis- cussed below seem less convincing, it should be borne in mind that no undis- puted evidence of occurrence of black holes has been presented so far. The flagship black hole candidate used to be Cygnus X-1, known and studied for years (cf., e.g., [48, 226]), and it still remains a strong one. Table 1.12 lists a series of further strong black hole candidates in X-ray binary systems; Mc is mass of the compact object and M is that of its optical companion; some ∗ other candidates, as well as references, can be found in [40, 203, 212, 215]. The binaries have been divided into two families: the High Mass X-ray Binaries (HMXB), where the companion star is of (relatively) high mass, and the Low Mass X-ray Binaries (LMXB), where the companion is typically below a solar mass.
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