Stationary Anti-de Sitter Black Hole Solutions in and out of Equilibrium

Abstract

Stationary solutions are perhaps the most fundamental objects in any physical theory. In this thesis we numerically obtain a number of stationary black hole solutions to the Einstein-Maxwell equations with a negative cosmological constant. The AdS/CFT correspondence has caused an upsurge in studying anti-de Sitter (AdS) space, and under this duality, the study of classical black holes corresponds to studying thermal states of very strongly coupled conformal field theories (CFTs) — an extremely challenging feat without the aid of holography. Firstly, we consider a braneworld scenario, which is a string theory-inspired mechanism for extra dimensions in our Universe. We present such rotating braneworld black hole solutions, which closely resemble standard Kerr black holes at large scales, supporting the phenomenological viability of the braneworld model. We shall discuss how these solutions are also dual to a four-dimensional black hole solution whose geometry is corrected by the coupling to a strongly coupled CFT with a UV cutoff. Next, we consider a CFT living on a fixed black hole background with a positive cosmological constant. There are two classes of bulk solutions in this case, distinguished by their horizon structure, and corresponding to two different phases of the CFT on the boundary black hole background. One possible bulk solution is static and contains two disconnected Killing horizons, whilst the other is only stationary and contains only a single horizon which is not Killing. We then consider the effect of deforming a CFT by the addition of a background electric field. The dual solution also has a non-Killing horizon, but due to the fact that the flow is generated by the Joule effect rather than a difference in temperatures, we find subtle differences in the horizon structure to other flowing solutions previously obtained. From the dual solution, we compute the conductivity of the field theory. Finally, we present the first examples of static, charged binary black hole solutions. These are held in dynamic equilibrium due to the presence of a background electric field. The solutions run over a surprisingly large parameter space, and some of these parameters do not have clear interpretations in terms of the boundary theory. Indeed, the solutions represent a continuous non-uniqueness for given boundary charges of asymptotically anti-de Sitter black hole solutions in Einstein-Maxwell theory. A common theme will be the ways in which these solutions succeed and fail to be in equilibrium. Many of the solutions possess non-Killing horizons which have classical flow along them, which has a profound impact on the horizon structure. Others possess multiple horizons which may have distinct temperatures or electrostatic potentials. In either case, these features prevent the solutions from being in thermodynamic equilibrium, despite the solutions being stationary. Chapter 1 provides a literature review, whilst Chapter 2 describes the numerical methods used to obtain the solutions of the thesis, many of which are standard. The original work of the thesis is presented in Chapters 3-6, which are each based on a paper (Refs. [1–4], respectively) written in collaboration with Jorge E. Santos.