Stability of Black Holes and Black Branes

By S. Hollands, R. Wald. 2012

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We establish a new criterion for the dynamical stability of black holes in D ≥ 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, $${\mathcal{T}}$$, 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—$${\mathcal{E}}$$ is nondegenerate on $${\mathcal{T}}$$ and that, for axisymmetric perturbations, $${\mathcal{E}}$$ has positive flux properties at both infinity and the horizon. We further show that $${\mathcal{E}}$$ is related to the second order variations of mass, angular momentum, and horizon area by $${\mathcal{E} = \delta^2 M -\sum_A \Omega_A \delta^2 J_A - \frac{\kappa}{8\pi}\delta^2 A}$$, 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 for any black brane corresponding to a thermodynamically unstable black hole, sufficiently long wavelength perturbations can be found with $${\mathcal{E} < 0}$$ 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 $${\mathcal{E}}$$ on $${\mathcal{T}}$$ 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. Although we restrict our considerations in this paper to vacuum general relativity, most of the results of this paper are derived using general Lagrangian and Hamiltonian methods and therefore can be straightforwardly generalized to allow for the presence of matter fields and/or to the case of an arbitrary diffeomorphism covariant gravitational action.

Subfields: Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/79d0da0281714d8e5e9b65e090152d4542234651

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Citations in the corpus

Thermodynamics of Rotating Black Holes and Black Rings: Phase Transitions and Thermodynamic Volume (2014, unknown): “, [80–84].”
Local subsystems in gauge theory and gravity (2016, unknown): “Explicitly, it is given by (see [43]):”
The linear stability of the Schwarzschild solution to gravitational perturbations (2016, background): “It is here already that we shall first 6Here one should mention that the original Kay–Wald [39] approach to boundedness on Schwarzschild obtained pointwise estimates for ϕ directly from this degenerate energy, applied to an auxilliary solution ϕ̃ such that ∂tϕ̃ = ϕ.”
Nonlinear gravity from entanglement in conformal field theories (2017, methodology): “Thinking of this gravitational theory as a system described by Hamiltonian mechanics, it turns out that the flow defined by the Lie derivative LξA can be understood as the flow on phase space associated with some corresponding Hamiltonian function HξA [16, 29].”