Perturbations and Stability of Static Black Holes in Higher Dimensions

TL;DR

This paper reviews the formalism and equations for analyzing perturbations and stability of static black holes in higher dimensions, including various types of perturbations and their implications for black hole stability.

Contribution

It introduces a gauge-invariant formalism for perturbations in higher-dimensional static black holes and derives master equations applicable to various black hole solutions.

Findings

Derived a master equation for tensor-type perturbations.

Established decoupled master equations for vector and scalar perturbations.

Reviewed stability analysis of higher-dimensional charged black holes with cosmological constant.

Abstract

In this chapter we consider perturbations and stability of higher dimensional black holes focusing on the static background case. We first review a gauge-invariant formalism for linear perturbations in a fairly generic class of (m+n)-dimensional spacetimes with a warped product metric, including black hole geometry. We classify perturbations of such a background into three types, the tensor, vector and scalar-type, according to their tensorial behaviour on the n-dimensional part of the background spacetime, and for each type of perturbations, we introduce a set of manifestly gauge invariant variables. We then introduce harmonic tensors and write down the equations of motion for the expansion coefficients of the gauge invariant perturbation variables in terms of the harmonics. In particular, for the tensor-type perturbations a single master equation is obtained in the (m+n)-dimensional background, which is applicable for perturbation analysis of not only static black holes but also some class of rotating black holes as well as black-branes. For the vector and scalar type, we derive a set of decoupled master equations when the background is a (2+n)-dimensional static black hole in the Einstein-Maxwell theory with a cosmological constant. As an application of the master equations, we review the stability analysis of higher dimensional charged static black holes with a cosmological constant. We also briefly review the recent results of a generalisation of the perturbation formulae presented here and stability analysis to static black holes in generic Lovelock theory.