Abstract
We explicitly construct all stationary, non-static, extremal near horizon geometries in D dimensions that satisfy the vacuum Einstein equations, and that have D−3 commuting rotational symmetries. Our work generalizes [arXiv:0806.2051] by Kunduri and Lucietti, where such a classification had been given in D = 4,5. But our method is different from theirs and relies on a matrix formulation of the Einstein equations. Unlike their method, this matrix formulation works for any dimension. The metrics that we find come in three families, with horizon topology S 2 × T D-4, or S 3 × T D-5, or quotients thereof. Our metrics depend on two discrete parameters specifying the topology type, as well as (D − 2)(D − 3)/2 continuous parameters. Not all of our metrics in D ≥ 6 seem to arise as the near-horizon limits of known black hole solutions.
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Communicated by Piotr T. Chrusciel.
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Hollands, S., Ishibashi, A. All Vacuum Near Horizon Geometries in D-dimensions with (D − 3) Commuting Rotational Symmetries. Ann. Henri Poincaré 10, 1537–1557 (2010). https://doi.org/10.1007/s00023-010-0022-y
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DOI: https://doi.org/10.1007/s00023-010-0022-y