Abstract
The Robinson-Trautman space-times provide solutions of Einstein's equations with negative cosmological constant, which settle to $AdS_4$ Schwarzschild black hole at late times. Via gauge/gravity duality they should describe a system out of equilibrium that evolves towards thermalization. We show that the area of the past apparent horizon of these space-times satisfies a generalized Penrose inequality and we formulate as well as provide evidence for a suitable generalization of Thorne's hoop conjecture. We also compute the holographic energy-momentum tensor and deduce its late time behavior. It turns out that the complete non-equilibrium process on the boundary is governed by Calabi's flow on $S^2$. Upon linearization, only special modes that arise as supersymmetric zero energy states of an associated supersymmetric quantum mechanics problem contribute to the solution. We find that each pole of radiation has an effective viscosity given by the eigenvalues of the Laplace operator on $S^2$ and there is an apparent violation of the KSS bound on $\eta / s$ for the low lying harmonics of large $AdS_4$ black holes. These modes, however, do not satisfy Dirichlet boundary conditions, they are out-going and they do not appear to have a Kruskal extension across the future horizon ${\cal H}^+$.
Highlights
Solutions approaching the AdS black hole at late times
What is perhaps less known is the remarkable fact that the effective quantum mechanics of all four-dimensional black holes is supersymmetric in that the Hamiltonian can be written as the square of a supercharge
The system relaxes to equilibrium by radiating the excess energy, which escapes to null infinity, and settles to a spherically symmetric configuration provided by the Schwarzschild solution
Summary
The Robinson-Trautman metrics provide an exact class of radiative solutions of Einstein equations, which are available for all values of the cosmological constant Λ, including, in particular, Λ < 0. This class exists in four space-time dimensions and has been thought to describe the effect of spherical gravitational waves emitted by bounded sources. Such metrics do not capture the general features of gravitational radiation, they are often interesting to consider in detail at linear and non-linear level. Higher dimensional generalizations of Robinson-Trautman metrics have been considered in the literature, but they do not seem to support non-trivial radiative solutions. We summarize some basic facts about them that will be used later
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