Dynamic and Thermodynamic Stability of Black Holes and Black Branes

Abstract

I describe recent work with Stefan Hollands that establishes a new criterion for the dynamical stability of black holes in $D \ge 4$ spacetime dimensions in general relativity with respect to axisymmetric perturbations: Dynamical stability is equivalent to the positivity of the canonical energy, $\fancyscript{E}$, on a subspace of linearized solutions that have vanishing linearized ADM mass, momentum, and angular momentum at infinity and satisfy certain gauge conditions at the horizon. We further show that $\fancyscript{E}$ is related to the second order variations of mass, angular momentum, and horizon area by $\fancyscript{E} = \delta ^2 M - \sum _i \varOmega _i \delta ^2 J_i - (\kappa /8\pi ) \delta ^2 A$, 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 all black branes corresponding to thermodynamically unstable black holes are dynamically unstable, as conjectured by Gubser and Mitra. We also prove that positivity of $\fancyscript{E}$ 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.