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.