General Relativity pp 429-526 | Cite as

# Black Holes

## Abstract

This extended chapter of almost hundred pages on black hole physics is a central part of the book. We begin with Israel’s original demonstration of the statement that a static black hole solution of Einstein’s vacuum equation has to be spherically symmetric, and is thus a Schwarzschild black hole. Then a detailed derivation of the Kerr solution is given, that makes efficient use of Cartan’s calculus of differential forms. The properties of this rotating black hole solution, that ranks among the most important solutions of Einstein’s (vacuum) equations, are carefully analysed, and it is shown in a more general setting what is behind the results. Later, this is also used in derivations of the four laws of black hole dynamics, which are formally closely related to the main laws of thermodynamics. A section is devoted on accretion tori around Kerr black holes. Furthermore, we present the evidence for black holes in some X-ray binary systems and for supermassive black holes in galactic centres.

## Keywords

Black Hole Event Horizon Active Galactic Nucleus Kerr Black Hole Null Hypersurface## References

## Textbooks on General Relativity: Selection of (Graduate) Textbooks

- 9.R.M. Wald,
*General Relativity*(University of Chicago Press, Chicago, 1984) zbMATHGoogle Scholar - 10.G. Ellis, S. Hawking,
*The Large Scale Structure of Space-Time*(Cambridge University Press, Cambridge, 1973) zbMATHGoogle Scholar - 15.C.W. Misner, K.S. Thorne, J.A. Wheeler,
*Gravitation*(Freeman, New York, 1973) Google Scholar - 16.N. Straumann,
*General Relativity and Relativistic Astrophysics*. Texts and Monographs in Physics (Springer, Berlin, 1984) CrossRefGoogle Scholar

## Textbooks on General Physics and Astrophysics

- 33.V.I. Arnold,
*Mathematical Methods of Classical Mechanics*. Graduate Texts in Mathematics, vol. 60 (Springer, Berlin, 1989) Google Scholar

## Mathematical Tools: Modern Treatments of Differential Geometry for Physicists

- 39.R. Abraham, J.E. Marsden,
*Foundations of Mechanics*, 2nd edn. (Benjamin, Elmsford, 1978) zbMATHGoogle Scholar

## Mathematical Tools: Selection of Mathematical Books

- 46.B. O’Neill,
*Semi-Riemannian Geometry with Applications to Relativity*(Academic Press, San Diego, 1983) zbMATHGoogle Scholar - 59.M.E. Taylor,
*Partial Differential Equations*(Springer, New York, 1996) CrossRefGoogle Scholar

## Research Articles, Reviews and Specialized Texts: Chapter 4

- 140.

## Research Articles, Reviews and Specialized Texts: Chapter 7

- 243.R. Schoen, S.-T. Yau, Commun. Math. Phys.
**90**, 575 (1983) MathSciNetADSzbMATHCrossRefGoogle Scholar

## Research Articles, Reviews and Specialized Texts: Chapter 8

- 244.W. Israel, Phys. Rev.
**164**, 1776 (1967) ADSCrossRefGoogle Scholar - 245.W. Israel, Commun. Math. Phys.
**8**, 245 (1968) MathSciNetADSCrossRefGoogle Scholar - 246.M. Heusler,
*Black Hole Uniqueness Theorems*(Cambridge University Press, Cambridge, 1996) zbMATHCrossRefGoogle Scholar - 247.M. Heusler, Stationary black holes: uniqueness and beyond. Living Rev. Relativ.
**1**, 6 (1998). http://www.livingreviews.org MathSciNetADSGoogle Scholar - 248.M.S. Volkov, D.V. Gal’tsov, Phys. Rep.
**319**, 1–83 (1999) MathSciNetADSCrossRefGoogle Scholar - 249.R.P. Kerr, Phys. Rev. Lett.
**11**, 237 (1963) MathSciNetADSzbMATHCrossRefGoogle Scholar - 250.E.T. Newman, A.I. Janis, J. Math. Phys.
**6**, 915 (1965) MathSciNetADSzbMATHCrossRefGoogle Scholar - 251.E.T. Newman et al., J. Math. Phys.
**6**, 918 (1965) ADSCrossRefGoogle Scholar - 252.G.C. Debney et al., J. Math. Phys.
**10**, 1842 (1969) MathSciNetADSCrossRefGoogle Scholar - 253.G. Neugebauer, R. Meinel, J. Math. Phys.
**44**, 3407 (2003) MathSciNetADSzbMATHCrossRefGoogle Scholar - 254.R. Meinel, arXiv:1108.4854
- 255.R.D. Blandford, R.L. Znajek, Mon. Not. R. Astron. Soc.
**179**, 433 (1977) ADSGoogle Scholar - 256.K.S. Thorne, R.H. Price, D.A. MacDonald,
*Black Holes: The Membrane Paradigm*(Yale University Press, New Haven, 1986) Google Scholar - 257.N. Straumann, The membrane model of black holes and applications, in
*Black Holes: Theory and Observation*, ed. by F.W. Hehl, C. Kiefer, R.J.K. Metzler (Springer, Berlin, 1998) Google Scholar - 258.N. Straumann, Energy extraction from black holes, in
*Recent Developments in Gravitation and Cosmology*, ed. by A. Macias, C. Lämmerzahl, A. Camacho. AIP Conference Proceedings, vol. 977 (American Institute of Physics, New York, 2008). arXiv:0709.3895 Google Scholar - 259.B. Carter, Commun. Math. Phys.
**10**, 280 (1968) zbMATHGoogle Scholar - 260.J.M. Bardeen, W.H. Press, S.A. Teukolsky, Astrophys. J.
**178**, 347 (1972) ADSCrossRefGoogle Scholar - 261.M. Heusler, N. Straumann, Class. Quantum Gravity
**10**, 1299 (1993) MathSciNetADSCrossRefGoogle Scholar - 262.M. Heusler, N. Straumann, Phys. Lett. B
**315**, 55 (1993) MathSciNetADSCrossRefGoogle Scholar - 263.V. Iyer, R.M. Wald, Phys. Rev. D
**50**, 846 (1994) MathSciNetADSCrossRefGoogle Scholar - 264.
- 265.R.M. Wald, Black holes and thermodynamics, in
*Black Holes and Relativistic Stars*, ed. by R.M. Wald (University of Chicago Press, Chicago, 1998) Google Scholar - 266.B. Carter, J. Math. Phys.
**10**, 70 (1969) ADSzbMATHCrossRefGoogle Scholar - 267.B. Carter, Black hole equilibrium states, in
*Black Holes*, ed. by C. DeWitt, B.S. DeWitt (Gordon & Breach, New York, 1973) Google Scholar - 268.D.R. Gies, C.T. Bolton, Astrophys. J.
**304**, 371 (1986) ADSCrossRefGoogle Scholar - 269.A.P. Cowley et al., Astrophys. J.
**272**, 118 (1983) ADSCrossRefGoogle Scholar - 270.J.E. McClintock, R.A. Remillard, Astrophys. J.
**308**, 110 (1986) ADSCrossRefGoogle Scholar - 271.J.A. Orosz, Inventory of black hole binaries, in
*IAU Symposium No. 212: A Massive Star Odyssey, from Main Sequence to Supernova*, Lanzarote, 2002, ed. by K.A. van der Hucht, A. Herraro, C. Esteban (Astronomical Society of the Pacific, San Francisco, 2003). astro-ph/0209041 Google Scholar - 272.R.M. Wagner et al., Astrophys. J.
**556**, 42 (2001) ADSCrossRefGoogle Scholar - 273.J.E. McClintock et al., Astrophys. J.
**551**, L147 (2001) ADSCrossRefGoogle Scholar - 274.M. Miyoshi, J. Moran, J. Herrnstein, L. Greenhill, N. Nakai, P. Diamond, M. Inone, Nature
**373**, 127 (1995) ADSCrossRefGoogle Scholar - 275.
- 276.M. Kozlowski, M. Jaroszynski, M.A. Abramowicz, Astron. Astrophys.
**63**, 209 (1978) MathSciNetADSzbMATHGoogle Scholar - 277.M. Sikore, M. Jaroszynski, M.A. Abramowicz, Astron. Astrophys.
**63**, 221 (1978) ADSGoogle Scholar - 278.D. Giulini, J. Math. Phys.
**39**, 6603 (1998) MathSciNetADSzbMATHCrossRefGoogle Scholar - 279.P.T. Chrusciel, E. Delay, G. Galloway, R. Howard, Ann. Inst. Henri Poincaré
**2**, 109 (2001) MathSciNetzbMATHCrossRefGoogle Scholar - 280.P.T. Chrusciel, Helv. Phys. Acta
**69**, 529 (1996) MathSciNetADSzbMATHGoogle Scholar - 281.J.H. Krolik,
*Active Galactic Nuclei*(Princeton University Press, Princeton, 1999) Google Scholar - 282.Ch.J. Willott, R.J. McLure, M. Jarvis, Astrophys. J.
**587**, L15 (2003) ADSCrossRefGoogle Scholar - 283.R. Penrose, Structure of spacetime, in
*Battelle Rencontres: 1967 Lectures in Mathematics and Physics*, ed. by C. DeWitt, J.A. Wheeler (Benjamin, New York, 1968) Google Scholar - 284.R. Genzel et al., Nature
**425**, 934 (2003) ADSCrossRefGoogle Scholar - 285.M.J. Valtonen et al., Nature
**452**, 851 (2008) ADSCrossRefGoogle Scholar