Perturbative Uniqueness of Black Holes near the Static Limit in All Dimensions

TL;DR

This paper proves that near the static limit, higher-dimensional static black holes are uniquely characterized by mass, angular momentum, and cosmological constant, with some exceptions related to horizon geometry and boundary conditions.

Contribution

It establishes perturbative uniqueness of static black holes in higher dimensions near the static limit, extending previous results to various asymptotic geometries.

Findings

No non-trivial stationary perturbations for negative cosmological constant except rotation and parameter variations.

Black hole solutions are parametrized by mass, angular momentum, and cosmological constant near the static limit.

Non-uniqueness may occur for horizons with non-constant curvature or under certain boundary conditions in AdS.

Abstract

The behaviour of stationary gravitational perturbations is studied for generalised static black holes in spacetimes of greater than three dimensions, using the formulation developed by the present author and Ishibashi. For the case in which the horizon has a spatial section with constant curvature, it is proved that irrespective of the value of the cosmological constant, there exists no stationary perturbation that is regular at the horizon(s) and falls off at infinity in the case of negative cosmological constant, except for those corresponding to the stationary rotation of black holes and the variation of the background parameters. This result indicates that regular neutral black hole solutions that are either asymptotically flat, de Sitter or anti-de Sitter can be parametrised by mass, (multiple component) angular momentum and the cosmological constant near the spherically symmetric and static limit. A similar conclusion is obtained for topological black holes. It is also pointed out that this perturbative uniqueness near the static limit may not hold in the case in which the horizon geometry is described by a generic Einstein space with non-constant sectional curvature. Further, non-uniqueness in the asymptotically anti-de Sitter case under a weaker boundary condition at infinity related to the AdS/CFT argument is discussed.