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

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.