paper

Stability of Black Holes and Black Branes

arXiv:1201.0463 · doi:10.1007/s00220-012-1638-1

Abstract

We establish a new criterion for the dynamical stability of black holes in spacetime dimensions in general relativity with respect to axisymmetric perturbations: Dynamical stability is equivalent to the positivity of the canonical energy, $\E$, 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. This is shown by proving that---apart from pure gauge perturbations and perturbations towards other stationary black holes---$\E$ is nondegenerate on and that, for axisymmetric perturbations, $\E$ has positive flux properties at both infinity and the horizon. We further show that $\E$ is related to the second order variations of mass, angular momentum, and horizon area by $\E = δ^2 M - \sum_A Ω_A δ^2 J_A - \fracκ{8π} δ^2 A$, 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 for any black brane corresponding to a thermodynamically unstable black hole, sufficiently long wavelength perturbations can be found with $\E < 0$ and vanishing linearized ADM quantities. Thus, all black branes corresponding to thermodynmically unstable black holes are dynamically unstable, as conjectured by Gubser and Mitra. We also prove that positivity of $\E$ on 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.

54 pages, Latex, 2 figures, v2: Anzatz for momentum in proof of Gubser-Mitra conjecture corrected; factor of 2 in symplectic form corrected; several typos in formulas corrected; v3: revised argument concerning horizon gauge condition on p. 10; typos corrected and several minor changes; reference added; v4: formula (86) for \E corrected, footnote added

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