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

### 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-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 horizon and two distinct families with topologically horizons. The 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 , extremal rotating black hole. Further, we construct an approximate solution corresponding to the near-horizon geometry of a small, extremal black ring.

© 2009 American Institute of Physics

Received 28 July 2008
Accepted 05 July 2009
Published online 24 August 2009

Acknowledgments:
H.K.K. and J.L. are supported by STFC. We thank Pau Figueras, Mukund Rangamani, and Harvey Reall for reading a draft of the manuscript and making useful comments. H.K.K. would like to thank Eric Woolgar for helpful discussions concerning the black hole topology theorems.

Article outline:

I. INTRODUCTION
II. SUMMARY OF MAIN RESULTS
A. Vacuum near-horizon geometries in including a negative cosmological constant
B. Vacuum near-horizon geometries in
C. Vacuum near-horizon geometries in with a negative cosmological constant
III. VACUUM NEAR-HORIZON EQUATIONS
A. Cohomogeneity-1 near-horizon geometries
IV. Four Dimensions
A. Global analysis
V. Five Dimensions
A. Near-horizon equations
B. A class of near-horizon geometries with horizons
1. Homogeneous horizon
2. Inhomogeneous horizon
3. Global analysis of inhomogeneous horizon
C. All Ricci flat solutions with compact horizons
1. Inhomogeneous horizon
2. Inhomogeneous horizon
D. Near-horizon geometry of a “small” extremal black ring?
VI. DISCUSSION

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/content/aip/journal/jmp/50/8/10.1063/1.3190480

2009-08-24

2016-08-28

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