Stability of Higher-Dimensional Schwarzschild Black Holes

TL;DR

This paper proves the classical stability of higher-dimensional Schwarzschild black holes against linear perturbations using a gauge-invariant formalism, showing no unstable modes exist and confirming the uniqueness of these black holes.

Contribution

The authors develop a gauge-invariant framework to analyze perturbations of higher-dimensional black holes, demonstrating their stability and uniqueness within this formalism.

Findings

All perturbation modes do not admit negative, normalisable solutions, indicating stability.

No static, regular perturbations exist outside the horizon, confirming black hole uniqueness.

The stability results extend to higher-dimensional black holes with cosmological constant.

Abstract

We investigate the classical stability of the higher-dimensional Schwarzschild black holes against linear perturbations, in the framework of a gauge-invariant formalism for gravitational perturbations of maximally symmetric black holes, recently developed by the authors. The perturbations are classified into the tensor, vector, and scalar-type modes according to their tensorial behaviour on the spherical section of the background metric, where the last two modes correspond respectively to the axial- and the polar-mode in the four-dimensional situation. We show that, for each mode of the perturbations, the spatial derivative part of the master equation is a positive, self-adjoint operator in the $L^2$-Hilbert space, hence that the master equation for each tensorial type of perturbations does not admit normalisable negative-modes which would describe unstable solutions. On the same Schwarzschild background, we also analyse the static perturbation of the scalar mode, and show that there exists no static perturbation which is regular everywhere outside the event horizon and well-behaved at spatial infinity. This checks the uniqueness of the higher-dimensional spherically symmetric, static, vacuum black hole, within the perturbation framework. Our strategy for the stability problem is also applicable to the other higher-dimensional maximally symmetric black holes with non-vanishing cosmological constant. We show that all possible types of maximally symmetric black holes (thus, including the higher-dimensional Schwarzschild-de Sitter and Schwarzschild-anti-de Sitter black holes) are stable against the tensor and the vector perturbations.