Polarised Black Holes in AdS

We consider solutions in Einstein-Maxwell theory with a negative cosmological constant that asymptote to global $AdS_{4}$ with conformal boundary $S^{2}\times\mathbb{R}_{t}$. At the sphere at infinity we turn on a space-dependent electrostatic potential, which does not destroy the asymptotic $AdS$ behaviour. For simplicity we focus on the case of a dipolar electrostatic potential. We find two new geometries: (i) an $AdS$ soliton that includes the full backreaction of the electric field on the $AdS$ geometry; (ii) a polarised neutral black hole that is deformed by the electric field, accumulating opposite charges in each hemisphere. For both geometries we study boundary data such as the charge density and the stress tensor. For the black hole we also study the horizon charge density and area, and further verify a Smarr formula. Then we consider this system at finite temperature and compute the Gibbs free energy for both $AdS$ soliton and black hole phases. The corresponding phase diagram generalizes the Hawking-Page phase transition. The $AdS$ soliton dominates the low temperature phase and the black hole the high temperature phase, with a critical temperature that decreases as the external electric field increases. Finally, we consider the simple case of a free charged scalar field on $S^{2}\times\mathbb{R}_{t}$ with conformal coupling. For a field in the $SU(N)$ adjoint representation we compare the phase diagram with the above gravitational system.