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 2nd-order wave equation in a two-dimensional static
spacetime for a gauge-invariant master variable, irrespective of the mode of
perturbations. Our formulation applies to the case of vanishing as well as
non-vanishing cosmological constant Lambda. The sign of the sectional curvature
K of each spatial section of equipotential surfaces is also kept general. In
the four-dimensional Schwarzschild background, this 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 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 Lambda. We also point out that there
exists no simple relation between scalar-type and vector-type perturbations in
higher dimensions, unlike in four dimensions. Although we only treat maximally
symmetric black holes in the present paper, the final master equations are
valid even when the hirozon geometry is described by a generic Einstein
manifold.Comment: 22 pages in the PTP TeX style, no figure. The published versio