Seminar

I describe work with with Stefan Hollands that establishes a new criterion for the dynamical stability of black holes in $D \geq 4$ spacetime dimensions in general relativity with respect to axisymmetric perturbations: Positivity of the canonical energy, $\mathcal E$, 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 implies mode stability. Conversely, failure of positivity of $\mathcal E$ on this subspace implies the existence of perturbations that cannot asymptotically approach a stationary perturbation. We further show that $\mathcal E$ is related to the second order variations of mass, angular momentum, and horizon area by $\mathcal E = \delta^2 M - \sum_i \Omega_i \delta^2 J_i - (\kappa/8\pi) \delta^2 A$, thereby establishing a close connection between dynamic stability and thermodynamic stability. Thermodynamic instability of a family of black holes need not imply dynamic 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. We also prove that positivity of $\mathcal E$ 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.