## Abstract

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-AdS(4) 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 S^1 x S^2 horizon and two distinct families with topologically S^3 horizons. The S^1 x S^2 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 S^3, extremal rotating AdS(5) black hole. Further, we construct an approximate solution corresponding to the near-horizon geometry of a small, extremal AdS(5) black ring.

Original language | English |
---|---|

Article number | 082502 |

Pages (from-to) | - |

Number of pages | 41 |

Journal | Journal of mathematical physics |

Volume | 50 |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 2009 |

## Keywords

- BLACK HOLES
- Exact solutions
- EXTREMAL BLACK-HOLES
- cosmology
- Einstein field equations
- geometry
- Klauza-Klein theory
- topology