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 gravi-tational 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, respec-tively, 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 L2-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 perturbatio