Perturbative Uniqueness of Black Holes near the Static Limit in All Dimensions
Hideo Kodama
TL;DR
This work proves a perturbative uniqueness result for higher-dimensional black holes near the static limit by employing a gauge-invariant perturbation framework that classifies perturbations into tensor, vector, and scalar types. It shows that, for horizons with constant curvature, stationary perturbations that are regular at the horizon and decay (or remain bounded) at infinity are only the trivial mass-variation perturbations (scalar-type) and a finite set of exceptional vector perturbations representing rotation; topological black holes obey an analogous statement. The analysis uses horizon regularity and precise asymptotic fall-off conditions across asymptotically flat, de Sitter, and anti-de Sitter cases, with integral identities in the de Sitter and AdS contexts to rule out non-exceptional perturbations. The results imply perturbative uniqueness of the Myers–Perry and Gibbons–Lü–Page–Pope solutions near the static limit under these boundary conditions, while noting possible non-uniqueness if the horizon geometry is a generic Einstein space with non-constant curvature and the AdS boundary conditions are relaxed.
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.