Non-equilibrium dynamics and $AdS_4$ Robinson-Trautman

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 ${\cal H}^+$.