Stability of higher-dimensional Schwarzschild black holes

We investigate the classical stability of higher-dimensional Schwarzschild black holes with respect to linear perturbations in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes. This formalism was recently developed by the present authors. The perturbations are classified into three types, those of tensor, vector and scalar modes, according to their tensorial behaviour on the spherical section of the background metric. The vector- and scalar-type modes correspond, respectively, to the axial and polar modes in the four-dimensional case. We show that for each mode of the perturbations, the spatial derivative part of the master equation is a positive, self-adjoint operator in the L-2-Hilbert space, and hence that the master equation for each type of perturbation has no normalisable negative modes that would correspond to unstable solutions. In the same Schwarzschild background, we also analyse static perturbations of the scalar mode and show that there exists no static perturbation that is regular everywhere outside the event horizon and is well behaved at the spatial infinity. This confirms the uniqueness of the higher-dimensional spherically symmetric, static vacuum black hole, within the perturbation framework. Our strategy for treating the stability problem is also applicable to other higher-dimensional maximally symmetric black holes with a non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable with respect to tensor and vector perturbations.