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

We show that for any spacetime dimension greater than or equal to four, the Einstein equations for gravitational perturbations of maximally symmetric vacuum black holes can be reduced to a single 2nd-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 dimension developed by us, and the variable for the final 2nd-order master equation is given by a simple combination of gauge-invariant variables in this formalism. Our formulation applies to the case with non-vanishing cosmological constant Λ as well as to the case Λ = 0. 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 Λ = 0 and K = 1, this master equation for the scalar perturbation coincides with the Zerilli equation for the polar mode and that for the vector perturbation with the Regge-Wheeler equation for the axial mode. Furthermore, in the four-dimensional Schwarzschild-anti-de Sitter background with Λ < 0 and K = 0, 1, our equation coincides with those derived by Cardoso and Lemos recently. As a simple application, we prove the perturbative stability and uniqueness of four-dimensional non-extremal spherically symmetric black holes for any Λ. We also point out that a simple relation between the scalar-type and the vector-type perturbations does not exist in higher dimensions, unlike in four dimension. Although we treat in the present paper only the case in which the horizon geometry is maximally symmetric, the final master equations also hold even when the hirozon geometry is described by a generic Einstein manifold, by an appropriate reinterpretation of the curvature K and the eigenvalues for harmonic tensors.