Bulletin of the American Physical Society
APS April Meeting 2013
Volume 58, Number 4
Saturday–Tuesday, April 13–16, 2013; Denver, Colorado
Session C5: Invited Session: Recent Developments in Mathematical Relativity |
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Sponsoring Units: GGR Chair: James Isenberg, University of Oregon Room: Governor's Square 14 |
Saturday, April 13, 2013 1:30PM - 2:06PM |
C5.00001: On the dynamic formation of trapped surfaces in vacuum Invited Speaker: Sergiu Klainerman |
Saturday, April 13, 2013 2:06PM - 2:42PM |
C5.00002: The black hole stability problem Invited Speaker: Mihalis Dafermos |
Saturday, April 13, 2013 2:42PM - 3:18PM |
C5.00003: Dynamic and Thermodynamic Stability of Black Holes Invited Speaker: Robert Wald 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: Dynamical 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 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 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. [Preview Abstract] |
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