A Higher Dimensional Stationary Rotating Black Hole Must be Axisymmetric

Abstract

A key result in the proof of black hole uniqueness in 4-dimensions is that a stationary black hole that is "rotating"-i.e., is such that the stationary Killing field is not everywhere normal to the horizon-must be axisymmetric. The proof of this result in 4-dimensions relies on the fact that the orbits of the stationary Killing field on the horizon have the property that they must return to the same null geodesic generator of the horizon after a certain period, P . This latter property follows, in turn, from the fact that the cross-sections of the horizon are two-dimensional spheres. However, in spacetimes of dimension greater than 4, it is no longer true that the orbits of the stationary Killing field on the horizon must return to the same null geodesic generator. In this paper, we prove that, nevertheless, a higher dimensional stationary black hole that is rotating must be axisymmetric. No assumptions are made concerning the topology of the horizon cross-sections other than that they are compact. However, we assume that the horizon is non-degenerate and, as in the 4-dimensional proof, that the spacetime is analytic.

Gravitational perturbations of higher dimensional rotating black holes: Tensor perturbations

“…These would correspond to genuinely stationary black holes which are nevertheless nonaxisymmetric. There would be no violation of the theorem that a stationary black hole must be axisymmetric [42] because this theorem assumes that the stationary Killing field is not normal to the event horizon whereas ξ is normal to the horizon of all the black holes we have been discussing. If such black holes exist then it is natural to guess that these should be the new solutions describing the endpoint of the superradiant instability.…”

Uniqueness Theorem for 5-Dimensional Black Holes with Two Axial Killing Fields

“…Thus, for n = 4 we are always in the situation just described. If n = 5, the higher dimensional rigidity theorem [18] also guarantees at least one more axial Killing field, but for a solution with precisely one extra axial Killing field, we would not be in the situation just described if such solutions were to exist. From now on, we take n = 5, and we postulate that the number of axial Killing fields is N = 2.…”

“…Under these conditions, one of the following 2 statements is true: (i) If t a is tangent to the null generators of H then the spacetime must be static [32]. (ii) If t a is not tangent to the null generators of H, then the higher dimensional rigidity theorem [18] states that there exist N additional linear independent, mutually commuting Killing fields ψ a 1 , . .…”

“…Proof: As a result of the rigidity theorem [18], we can find a horizon cross section H which is itself a Riemannian manifold with induced metric q ab , of dimension 3, invariant under the and O x 2 such that each neighborhood is diffeomorphic to a solid tube S 1 × B 2 . We denote the radial coordinates measuring the distance from the origin in each of the discs B 2 by R 1 for the first tubular neighborhood, and by R 2 for the second tubular neighborhood.…”

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

“…The main success of this is it allowed the proof of a uniqueness theorem for asymptotically flat, topologically spherical, superysymmetric black holes in five dimensional ungauged supergravity: the only solution turns out to be BMPV [2,29]. In the gauged case the near-horizon equations are more complicated and a 1 Note that it has been proved that a stationary, non-extremal black hole in all dimensions is necessarily axisymmetric [43,44], i.e. has at least one rotational Killing vector field so the total symmetry is at least R × U (1).…”