Charged and rotating AdS black holes and their CFT duals
S. W. Hawking, H. S. Reall
TL;DR
The paper investigates charged and rotating AdS black holes in $AdS_5\times S^5$ and $AdS_4\times S^7$ and their CFT duals on rotating Einstein universes, focusing on stability, thermodynamics, and the interplay between strong and zero-coupling limits. It demonstrates that classical superradiant instabilities are evaded when the corotating Killing vector remains timelike outside the horizon, and explores how bulk charge and boundary chemical potentials influence phase structure and BE condensation in the weakly coupled CFT compared to strong coupling, with a novel finite-temperature limit where angular velocity approaches the speed of light. The study reveals that divergences in the free CFT and the supergravity action share the same form in this limit and that the canonical $4/3$ ratio between strong- and weak-coupling free energies persists only in certain regimes, while the RNAdS solutions exhibit a distinct AdS-dominated phase structure. Overall, the work clarifies the stability landscape of rotating and charged AdS black holes and highlights qualitative differences between strongly coupled holographic CFTs and their free counterparts on rotating boundaries, enriching our understanding of AdS/CFT thermodynamics under rotation and chemical potentials.
Abstract
Black hole solutions that are asymptotic to $ AdS_5 \times S^5$ or $ AdS_4 \times S^7$ can rotate in two different ways. If the internal sphere rotates then one can obtain a Reissner-Nordstrom-AdS black hole. If the asymptotically AdS space rotates then one can obtain a Kerr-AdS hole. One might expect superradiant scattering to be possible in either of these cases. Superradiant modes reflected off the potential barrier outside the hole would be re-amplified at the horizon, and a classical instability would result. We point out that the existence of a Killing vector field timelike everywhere outside the horizon prevents this from occurring for black holes with negative action. Such black holes are also thermodynamically stable in the grand canonical ensemble. The CFT duals of these black holes correspond to a theory in an Einstein universe with a chemical potential and a theory in a rotating Einstein universe. We study these CFTs in the zero coupling limit. In the first case, Bose-Einstein condensation occurs on the boundary at a critical value of the chemical potential. However the supergravity calculation demonstrates that this is not to be expected at strong coupling. In the second case, we investigate the limit in which the angular velocity of the Einstein universe approaches the speed of light at finite temperature. This is a new limit in which to compare the CFT at strong and weak coupling. We find that the free CFT partition function and supergravity action have the same type of divergence but the usual factor of 4/3 is modified at finite temperature.