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 forgravitational perturbations of maximally symmetric vacuum black holes can bereduced to a single 2nd-order wave equation in a two-dimensional staticspacetime for a gauge-invariant master variable, irrespective of the mode ofperturbations. Our formulation applies to the case of vanishing as well asnon-vanishing cosmological constant Lambda. The sign of the sectional curvatureK of each spatial section of equipotential surfaces is also kept general. Inthe four-dimensional Schwarzschild background, this master equation for ascalar perturbation is identical to the Zerilli equation for the polar mode andthe master equation for a vector perturbation is identical to the Regge-Wheelerequation for the axial mode. Furthermore, in the four-dimensionalSchwarzschild-anti-de Sitter background with K=0,1, our equation coincides withthose derived by Cardoso and Lemos recently. As a simple application, we provethe perturbative stability and uniqueness of four-dimensional non-extremalspherically symmetric black holes for any Lambda. We also point out that thereexists no simple relation between scalar-type and vector-type perturbations inhigher dimensions, unlike in four dimensions. Although we only treat maximallysymmetric black holes in the present paper, the final master equations arevalid even when the hirozon geometry is described by a generic Einsteinmanifold.Comment: 22 pages in the PTP TeX style, no figure. The published versio