Classical instability of Kerr-AdS black holes and the issue of final state

TL;DR

This work investigates the classical stability of Kerr-AdS black holes under gravitational perturbations, extending prior results that small Kerr-AdS holes are scalar-unstable to gravity via a Teukolsky-based analysis. By performing near-region and far-region (AdS) analyses and matching, it derives the instability condition and growth rates, showing that small four-dimensional Kerr-AdS black holes are gravitationally unstable in the superradiant regime. The study identifies the endpoint of this instability as a Kerr-AdS black hole whose boundary is conformal to a rotating Einstein universe with angular velocity $\Omega=1/\ell$, corresponding to the critical radius $r_+^{\rm c}=\sqrt{a\ell}$; the endpoint is expected to be slightly oblate and in equilibrium with exterior radiation. These results deepen the AdS stability8 context, suggest analogous behavior in higher dimensions, and hint at new bulk solutions that realize the rotating-boundary endpoint geometry.

Abstract

It is now established that small Kerr-Anti-de Sitter (Kerr-AdS) black holes are unstable against scalar perturbations, via superradiant amplification mechanism. We show that small Kerr-AdS black holes are also unstable against gravitational perturbations and we compute the features of this instability. We also describe with great detail the evolution of this instability. In particular, we identify its endpoint state. It corresponds to a Kerr-AdS black hole whose boundary is an Einstein universe rotating with the light velocity. This black hole is expected to be slightly oblate and to co-exist in equilibrium with a certain amount of outside radiation.