Thermodynamic and Dynamic Stability of Asymptotically Anti-de Sitter Black Holes

Hollands and Wald previously established a criterion for dynamic stability of asymptotically flat black holes with respect to linearized axisymmetric perturbations. They showed that stability is equivalent to positivity of a canonical energy on a certain class of these perturbations. We adapt this work to the asymptotically anti-de Sitter case, and find that the restriction to axisymmetric perturbations is lifted as a consequence of the reflecting nature of spatial infinity. The consideration of non-axisymmetric perturbations allows us to address phenomena such as superradiant instabilities. As in the previous work, the canonical energy can be expressed in terms of second order variations of thermodynamic quantities, thereby establishing a connection between thermodynamic and dynamic stability. We discuss the relationship between negative canonical energy configurations and the presence of a generalized ergosphere.