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A classification of near-horizon geometries of extremal vacuum black holes

J. Math. Phys. 50, 082502 (2009); doi:10.1063/1.3190480

Published 24 August 2009

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Hari K. Kunduri1 and James Lucietti2
1DAMTP, Centre for Mathematical Sciences, University of Cambridge, Wilbeforce Road, Cambridge CB3 0WA, United Kingdom
2Department of Mathematical Sciences, Centre for Particle Theory, University of Durham, South Road, Durham DH1 3LE, United Kingdom
We consider the near-horizon geometries of extremal, rotating black hole solutions of the vacuum Einstein equations, including a negative cosmological constant, in four and five dimensions. We assume the existence of one rotational symmetry in four dimensions (4D), two commuting rotational symmetries in five dimensions (5D), and in both cases nontoroidal horizon topology. In 4D we determine the most general near-horizon geometry of such a black hole and prove it is the same as the near-horizon limit of the extremal Kerr-AdS4 black hole. In 5D, without a cosmological constant, we determine all possible near-horizon geometries of such black holes. We prove that the only possibilities are one family with a topologically S1×S2 horizon and two distinct families with topologically S3 horizons. The S1×S2 family contains the near-horizon limit of the boosted extremal Kerr string and the extremal vacuum black ring. The first topologically spherical case is identical to the near-horizon limit of two different black hole solutions: the extremal Myers–Perry black hole and the slowly rotating extremal Kaluza–Klein (KK) black hole. The second topologically spherical case contains the near-horizon limit of the fast rotating extremal KK black hole. Finally, in 5D with a negative cosmological constant, we reduce the problem to solving a sixth-order nonlinear ordinary differential equation of one function. This allows us to recover the near-horizon limit of the known, topologically S3, extremal rotating AdS5 black hole. Further, we construct an approximate solution corresponding to the near-horizon geometry of a small, extremal AdS5 black ring. ©2009 American Institute of Physics
History: Received 28 July 2008; accepted 5 July 2009; published 24 August 2009
Permalink: http://link.aip.org/link/?JMAPAQ/50/082502/1
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KEYWORDS and PACS

Keywords
PACS
  • 04.70.-s
    Physics of black holes
  • 97.60.Lf
    Black holes
  • 04.50.Cd
    Kaluza-Klein theories
  • 98.80.Cq
    Particle-theory and field-theory models of the early Universe
  • 95.30.Sf
    Relativity and gravitation in astrophysics
  • 04.20.Jb
    Exact solutions in general relativity
  • YEAR: 2009

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PUBLICATION DATA

ISSN:
0022-2488 (print)   1089-7658 (online)
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REFERENCES (79)
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  77. In fact, a special case of this with one independent rotation parameter has been constructed.63
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