A master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions

We show that in four or more spacetime dimensions, the Einstein equations for gravitational perturbations of maximally symmetric vacuum black holes can be reduced to a single second-order wave equation in a two-dimensional static spacetime, irrespective of the mode of perturbations. Our starting point is the gauge-invariant formalism for perturbations in an arbitrary number of dimensions developed by the present authors, and the variable for the final second-order master equation is given by a simple combination of gauge-invariant variables in this formalism. Our formulation applies to the case of non-vanishing as well as vanishing cosmological constant A. The sign of the sectional curvature K of each spatial section of equipotential surfaces is also kept general. In the four-dimensional Schwarzschild background with Lambda = 0 and K = 1, the master equation for a scalar perturbation is identical to the Zerilli equation for the polar mode and the master equation for a vector perturbation. is identical to the Regge-Wheeler equation for the axial mode. Furthermore, in the four-dimensional Schwarzschild-anti-de Sitter background with Lambda < 0 and K = 0, 1, our equation coincides with those recently derived by Cardoso and Lemos. As a simple application, we prove the perturbative stability and uniqueness of four-dimensional non-extremal spherically symmetric black holes for any Lambda. We also point out that there exists no simple relation between scalar-type and vector-type perturbations in higher dimensions, unlike in four dimension. Although in the present paper we treat only the case in which the horizon geometry is maximally symmetric, the final master equations are valid even when the horizon geometry is described by a generic Einstein manifold, if we employ an appropriate reinterpretation of the curvature K and the eigenvalues for harmonic tensors.