Introduction

Consistent theories of quantum gravity in spacetimes that asymptote to anti-de Sitter (AdS) spacetime are equivalent to quantum field theories defined on the conformal boundary of the spacetime [1]. A pedagogical discussion of this “holographic correspondence” may be found in [2, 3] and in Chapter 12 of this volume. While some deeper questions arising from the correspondence remain to be understood from first principles, the conceptual “Gestalt switch” involved in viewing physical processes simultaneously from a gravitational and a field-theoretic perspective has provided an invaluable source of physical intuition as well as computational power. In particular, in the large-N limit of quantum field theories, to be recalled shortly, the gravitational description becomes weakly curved and the tools of general relativity may be harnessed.

This chapter will be concerned with black holes in four-dimensional asymptotically AdS spacetimes. By focusing on charged planar black holes, we will establish an interface with the rich phenomenology of (2+1)-dimensional quantum field theories that has been widely studied in condensed matter physics. Planarity of the horizon will translate into the statement that the dual quantum field theory propagates on a background Minkowski spacetime in 2+1 dimensions. The perhaps more familiar spherical foliation of asymptotically AdS spacetimes would correspond to considering quantum field theories on a spatial sphere; this complicates the field-theoretic physics by introducing a scale, the radius of the sphere, and also does not correspond to a situation of significant interest in condensed matter physics at present.