### Abstract

Introduction As discussed in the first chapter, black holes in four dimensions satisfy remarkable uniqueness properties. Of fundamental importance is the classical result of Carter, Hawking and Robinson that the Kerr solution, which is characterized by its mass <italic>M</italic> and angular momentum <italic>J</italic>, is the unique four-dimensional asymptotically flat stationary (i.e., steady state) solution to the vacuum Einstein equations. A basic step in the proof is Hawking's theorem on the topology of black holes [2], which asserts that, for such black hole spacetimes, cross sections of the event horizon are necessarily spherical, i.e., they are topologically 2-spheres. In short, for conventional black holes in four dimensions, the horizon topology is spherical. However, as we saw in the previous chapter, in higher dimensions black hole horizons need not have spherical topology. With the remarkable discovery by Emparan and Reall [3] of the black ring solution, with its S^{1} × S^{2} horizon topology, the question naturally arose as to what, if any, are the restrictions on horizon topology in higher-dimensional black holes. This issue was addressed in a paper of the author and Rick Schoen [4], in which we obtained a generalization of Hawking's theorem to higher dimensions. This generalization is discussed in sections 7.4 and 7.5. In preparation for that, we review Hawking's black hole topology theorem in section 7.2 and introduce some basic background material on marginally trapped surfaces in section 7.3.

Original language | English (US) |
---|---|

Title of host publication | Black Holes in Higher Dimensions |

Publisher | Cambridge University Press |

Pages | 159-179 |

Number of pages | 21 |

ISBN (Print) | 9781139004176, 9781107013452 |

DOIs | |

State | Published - Jan 1 2012 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)

### Cite this

*Black Holes in Higher Dimensions*(pp. 159-179). Cambridge University Press. https://doi.org/10.1017/CBO9781139004176.008

**Constraints on the topology of higher-dimensional black holes.** / Galloway, Gregory J.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Black Holes in Higher Dimensions.*Cambridge University Press, pp. 159-179. https://doi.org/10.1017/CBO9781139004176.008

}

TY - CHAP

T1 - Constraints on the topology of higher-dimensional black holes

AU - Galloway, Gregory J

PY - 2012/1/1

Y1 - 2012/1/1

N2 - Introduction As discussed in the first chapter, black holes in four dimensions satisfy remarkable uniqueness properties. Of fundamental importance is the classical result of Carter, Hawking and Robinson that the Kerr solution, which is characterized by its mass M and angular momentum J, is the unique four-dimensional asymptotically flat stationary (i.e., steady state) solution to the vacuum Einstein equations. A basic step in the proof is Hawking's theorem on the topology of black holes [2], which asserts that, for such black hole spacetimes, cross sections of the event horizon are necessarily spherical, i.e., they are topologically 2-spheres. In short, for conventional black holes in four dimensions, the horizon topology is spherical. However, as we saw in the previous chapter, in higher dimensions black hole horizons need not have spherical topology. With the remarkable discovery by Emparan and Reall [3] of the black ring solution, with its S1 × S2 horizon topology, the question naturally arose as to what, if any, are the restrictions on horizon topology in higher-dimensional black holes. This issue was addressed in a paper of the author and Rick Schoen [4], in which we obtained a generalization of Hawking's theorem to higher dimensions. This generalization is discussed in sections 7.4 and 7.5. In preparation for that, we review Hawking's black hole topology theorem in section 7.2 and introduce some basic background material on marginally trapped surfaces in section 7.3.

AB - Introduction As discussed in the first chapter, black holes in four dimensions satisfy remarkable uniqueness properties. Of fundamental importance is the classical result of Carter, Hawking and Robinson that the Kerr solution, which is characterized by its mass M and angular momentum J, is the unique four-dimensional asymptotically flat stationary (i.e., steady state) solution to the vacuum Einstein equations. A basic step in the proof is Hawking's theorem on the topology of black holes [2], which asserts that, for such black hole spacetimes, cross sections of the event horizon are necessarily spherical, i.e., they are topologically 2-spheres. In short, for conventional black holes in four dimensions, the horizon topology is spherical. However, as we saw in the previous chapter, in higher dimensions black hole horizons need not have spherical topology. With the remarkable discovery by Emparan and Reall [3] of the black ring solution, with its S1 × S2 horizon topology, the question naturally arose as to what, if any, are the restrictions on horizon topology in higher-dimensional black holes. This issue was addressed in a paper of the author and Rick Schoen [4], in which we obtained a generalization of Hawking's theorem to higher dimensions. This generalization is discussed in sections 7.4 and 7.5. In preparation for that, we review Hawking's black hole topology theorem in section 7.2 and introduce some basic background material on marginally trapped surfaces in section 7.3.

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U2 - 10.1017/CBO9781139004176.008

DO - 10.1017/CBO9781139004176.008

M3 - Chapter

AN - SCOPUS:84884795985

SN - 9781139004176

SN - 9781107013452

SP - 159

EP - 179

BT - Black Holes in Higher Dimensions

PB - Cambridge University Press

ER -