Stability of Black Holes and Black Branes

TL;DR

This work introduces a dynamical stability criterion for D≥4 black holes and black branes in vacuum GR based on the canonical energy ${\mathcal{E}}$, defined from the Lagrangian and Noether structures. By restricting to axisymmetric perturbations and imposing horizon gauge conditions, the authors show ${\mathcal{E}}$ is conserved and expressible in terms of second-order variations of mass, angular momenta, and horizon area, linking dynamical stability to a local Penrose inequality. A robust Hilbert-space framework clarifies degeneracies and yields a variational principle: positive ${\mathcal{E}}$ on perturbations with vanishing ADM charges implies stability, while negative ${\mathcal{E}}$ signals instability. The results establish the Gubser-Mitra conjecture for black branes and provide a precise thermodynamic counterpart: thermodynamic instability implies dynamical instability in the long-wavelength brane limit, with the local Penrose inequality providing a necessary-and-sufficient stability criterion for axisymmetric perturbations.

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 $\E = δ^2 M - \sum_A Ω_A δ^2 J_A - \fracκ{8π} δ^2 A$, 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.

Theorems & Definitions (14)

• Lemma 1
• Lemma 2
• Proposition 1
• Proposition 2
• Proposition 3
• Definition 2.1
• Proposition 4
• Theorem 1
• Lemma 3
• Proposition 5