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

TL;DR

This paper derives a unified master equation for gravitational perturbations of maximally symmetric black holes in higher dimensions, applicable to various cosmological constants and geometries, and proves stability for certain four-dimensional cases.

Contribution

It introduces a gauge-invariant master equation for higher-dimensional black hole perturbations, extending previous four-dimensional results and applicable to generic Einstein manifolds.

Findings

Unified 2nd-order wave equation for perturbations in higher dimensions

Proved stability and uniqueness of 4D non-extremal black holes with any Lambda

No simple relation between scalar and vector perturbations in higher dimensions

Abstract

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.