Stability of Higher-Dimensional Schwarzschild Black Holes

TL;DR

The paper addresses the stability of $(n+2)$-dimensional Schwarzschild black holes under linear gravitational perturbations by employing a gauge-invariant perturbation formalism that yields three master equations—tensor, vector, and scalar types—each of Schrödinger form with a corresponding potential ($V_T$, $V_V$, $V_S$). The authors prove the operator $A = -\frac{d^2}{dr_*^2} + V$ is positive self-adjoint in $L^2$ either directly (tensor) or via transformations that render an effective potential nonnegative (vector and scalar), implying no normalisable negative-energy modes exist. They also show there are no regular static scalar perturbations outside the event horizon, supporting uniqueness of higher-dimensional static vacua, and extend the stability analysis to maximally symmetric black holes with nonzero cosmological constant for tensor and vector perturbations, with discussions on generalised stability criteria via Lichnerowicz eigenvalues. Collectively, the work solidifies the dynamical stability of higher-dimensional Schwarzschild spacetimes and provides a general framework for assessing stability of related black holes in higher dimensions.

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.

Figures (9)

• Figure 1: The potential barriers $V_T$ surrounding $(n+2)$-dimensional Schwarzschild black holes for tensor perturbations of the mode $l=2$. Here, $r_h$ denotes the horizon radius.
• Figure 2: The potential barriers $V_T$ surrounding $(n+2)$-dimensional Schwarzschild-de Sitter black holes for tensor perturbations of the mode $l=2$. The horizontal axis is normalised with respect to the black hole horizon radius $r_h$.
• Figure 3: The potential barriers $V_T$ surrounding $(n+2)$-dimensional Schwarzschild-anti-de Sitter black holes for tensor perturbations of the mode $l=2$.The horizontal axis is normalised with respect to the black hole horizon radius $r_h$.
• Figure 4: The potential barriers $V_V$ surrounding $(n+2)$-dimensional Schwarzschild black holes for vector perturbations of the mode $l=2$. The horizontal axis is normalised with respect to the black hole horizon radius $r_h$.
• Figure 5: The potential barriers $V_V$ surrounding $(n+2)$-dimensional Schwarzschild-de Sitter black holes for vector perturbations of the mode $l=2$. The horizontal axis is normalised with respect to the black hole horizon radius $r_h$.