Stability of Black Holes and Black Branes

TL;DR

This paper establishes a new criterion linking the dynamical stability of black holes in higher dimensions to the positivity of a canonical energy, connecting it to thermodynamic stability and confirming the Gubser-Mitra conjecture for black branes.

Contribution

It introduces a new stability criterion based on canonical energy, relates it to thermodynamic stability, and proves the Gubser-Mitra conjecture for black branes.

Findings

Dynamical stability is equivalent to positivity of canonical energy on a specific solution space.

Thermodynamic instability of black holes implies dynamical instability for corresponding black branes.

Positivity of canonical energy is equivalent to satisfying a local Penrose inequality.

Abstract

We establish a new criterion for the dynamical stability of black holes in $D \geq 4$ spacetime dimensions in general relativity with respect to axisymmetric perturbations: Dynamical stability is equivalent to the positivity of the canonical energy, $\E$, on a subspace, $\mathcal T$, of linearized solutions that have vanishing linearized ADM mass, momentum, and angular momentum at infinity and satisfy certain gauge conditions at the horizon. This is shown by proving that---apart from pure gauge perturbations and perturbations towards other stationary black holes---$\E$ is nondegenerate on $\mathcal T$ and that, for axisymmetric perturbations, $\E$ has positive flux properties at both infinity and the horizon. We further show that $\E$ is related to the second order variations of mass, angular momentum, and horizon area by $\E = \delta^2 M - \sum_A \Omega_A \delta^2 J_A - \frac{\kappa}{8\pi} \delta^2 A$, thereby establishing a close connection between dynamical stability and thermodynamic stability. Thermodynamic instability of a family of black holes need not imply dynamical instability because the perturbations towards other members of the family will not, in general, have vanishing linearized ADM mass and/or angular momentum. However, we prove that for any black brane corresponding to a thermodynamically unstable black hole, sufficiently long wavelength perturbations can be found with $\E < 0$ and vanishing linearized ADM quantities. Thus, all black branes corresponding to thermodynmically unstable black holes are dynamically unstable, as conjectured by Gubser and Mitra. We also prove that positivity of $\E$ on $\mathcal T$ is equivalent to the satisfaction of a "local Penrose inequality," thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability.