Dynamic and Thermodynamic Stability of Black Holes and Black Branes

          @misc{ pirsa_PIRSA:13060014,
            doi = {10.48660/13060014},
            url = {https://pirsa.org/13060014},
            author = {Wald, Robert},
            keywords = {Strong Gravity},
            language = {en},
            title = {Dynamic and Thermodynamic Stability of Black Holes and Black Branes},
            publisher = {Perimeter Institute},
            year = {2013},
            month = {jun},
            note = {PIRSA:13060014 see, \url{https://pirsa.org}}
          }
          

Abstract

I describe recent 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: Dynamic stability is equivalent to the 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.  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
thermodynmically unstable black holes are dynamically unstable, as conjectured by Gubser and Mitra. 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.