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 4d, two commuting rotational symmetries in 5d and in both cases non-toroidal 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 × S 2 horizon and two distinct families with topologically S 3 horizons. The S 1 × 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 non-linear ODE 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.

On classification of geometries with SO(2,2) symmetry

“…Our interest in these geometries is primarily motivated by the fact that there is a special class of extremal black holes, Extremal Vanishing Horizon (EVH) black holes, where in the near horizon limit lead to such geometries, see [1,[9][10][11][12][13][14][15] for previous analysis of EVH black holes. This should be contrasted with the near horizon limit of usual extremal black holes, where one generically finds an AdS 2 factor, rather than an AdS 3 [16][17][18][19]. The appearance of local AdS 3 factor in the EVH case can be attributed to the fact that for EVH black holes the co-dimension two horizon surface of the black hole horizon has the peculiar property that its area vanishes due to the presence of a vanishing one-cycle on the horizon [20].…”

NR/HEP: roadmap for the future

“…(See Refs. [158,159] for results in this direction.) Can this be used to classify extreme black holes?…”

superparticle near horizon of extreme Kerr–Newman–AdS–dS black hole

“…A large class of extreme black holes in four and five dimensions exhibits SO(2, 1) symmetry in the near horizon limit [1,2]. The most interesting example of such a kind is the Kerr black hole in four dimensions [3].…”