0801.3471v2.pdf - arXiv:0801.3471v2[hep-th 14 Feb 2008...

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arXiv:0801.3471v2 [hep-th] 14 Feb 2008 Black Holes in Higher Dimensions Roberto Emparan Instituci´o Catalana de Recerca i Estudis Avan¸ cats (ICREA) and Departament de F´ ısica Fonamental, Universitat de Barcelona Marti i Franqu` es 1, E-08028 Barcelona, Spain email: [email protected] Harvey S. Reall Department of Applied Mathematics and Theoretical Physics University of Cambridge, Centre for Mathematical Sciences Wilberforce Road, Cambridge CB3 0WA United Kingdom email: [email protected] Abstract We review black hole solutions of higher-dimensional vacuum gravity, and of higher- dimensional supergravity theories. The discussion of vacuum gravity is pedagogical, with detailed reviews of Myers-Perry solutions, black rings, and solution-generating techniques. We discuss black hole solutions of maximal supergravity theories, including black holes in anti-de Sitter space. General results and open problems are discussed throughout. 1
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Contents 1 Introduction 4 1.1 Why gravity is richer in d > 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Why gravity is more difficult in d > 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2 Scope and organization of this article 8 2.1 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 3 Basic concepts and solutions 9 3.1 Conserved charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 3.2 The Schwarzschild-Tangherlini solution and black p -branes . . . . . . . . . . . . . . . . . . . 11 3.3 Stability of the static black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.4 Gregory-Laflamme instability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4 Myers-Perry solutions 14 4.1 Rotation in a single plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4.2 General solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2.1 Phase space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2.2 Global structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.3 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5 Vacuum solutions in five dimensions 22 5.1 Black rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.1.1 One angular momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 5.1.2 Two angular momenta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2 Stationary axisymmetric solutions with d - 3 rotational symmetries . . . . . . . . . . . . . 28 5.2.1 Weyl solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 5.2.2 General axisymmetric class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 5.3 Multi-black hole solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 5.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 6 Vacuum solutions in more than five dimensions 41 6.1 Approximate solutions from curved thin branes . . . . . . . . . . . . . . . . . . . . . . . . . 41 6.2 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 6.3 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 7 Solutions with a gauge field 45 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 7.2 Topologically spherical black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.2.1 Non-extremal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.2.2 BPS solutions: the BMPV black hole . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.3 Black rings with gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.3.1 Dipole rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 7.3.2 Charged black rings, supersymmetric black rings . . . . . . . . . . . . . . . . . . . . 47 7.4 Solution-generating techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 7.5 Multi-black hole solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2
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8 General results and open problems 49 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8.2 Black hole topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8.3 Uniqueness of static black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 8.4 Stationary black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 8.5 Supersymmetric black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 8.6 Algebraic classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 8.7 Laws of black hole mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 8.8 Hawking radiation and black hole thermodynamics . . . . . . . . . . . . . . . . . . . . . . . 53 8.9 Apparent and isolated horizons, and critical phenomena . . . . . . . . . . . . . . . . . . . . 54 9 Solutions with a cosmological constant 56 9.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 9.2 Schwarzschild-AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 9.3 Stationary vacuum solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 9.4 Gauged supergravity theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 9.5 Static charged solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 9.6 Stationary charged solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 10 Acknowledgments 63 3
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1 Introduction Classical General Relativity in more than four spacetime dimensions has been the subject of in- creasing attention in recent years. Among the reasons why it should be interesting to study this
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