Master equations for perturbations of generalised static black holes with charge in higher dimensions

TL;DR

This work generalizes the perturbation theory of static black holes to include charge in higher dimensions with Einstein-horizon geometries. By constructing gauge-invariant master variables, the authors reduce the coupled Einstein-Maxwell perturbations to two decoupled Schrödinger-type equations for each perturbation type (tensor, vector, scalar), with explicit source terms suitable for emission calculations. They derive concrete master equations for both generic and exceptional vector and scalar modes, and provide detailed stability analyses using S-deformation techniques across different horizon curvatures and cosmological constants, identifying broad regions of stability while highlighting open regimes (notably some scalar cases in higher dimensions). The explicit potentials and source structures enable practical computations of gravitational and electromagnetic emissions from higher-dimensional charged black holes and pave the way for applications to AdS/CFT and singular-source perturbations such as higher-dimensional C-metrics.

Abstract

We extend the formulation for perturbations of maximally symmetric black holes in higher dimensions developed by the present authors in a previous paper (hep-th/0305147) to a charged black hole background whose horizon is described by an Einstein manifold. For charged black holes, perturbations of electromagnetic fields are coupled to the vector and scalar modes of metric perturbations non-trivially. We show that by taking appropriate combinations of gauge-invariant variables for these perturbations, the perturbation equations for the Einstein-Maxwell system are reduced to two decoupled second-order wave equations describing the behaviour of the electromagnetic mode and the gravitational mode, for any value of the cosmological constant. These wave equations are transformed into Schrödinger-type ODEs through a Fourier transformation with respect to time. Using these equations, we investigate the stability of generalised black holes with charge. We also give explicit expressions for the source terms of these master equations with application to the emission problem of gravitational waves in mind.

Figures (4)

• Figure 1: The parameter ranges in which the spacetime contains a regular black hole (the shaded regions).
• Figure 2: Ranges of the value $x_H$.
• Figure 3: Examples of $V_{S+}$ for $K=1$ and $\lambda<0$.
• Figure 4: Examples of $V_{S-}$.