In this chapter we explain the gauge/gravity duality [1–3], which is a motivation for studying black hole solutions in various numbers of dimensions. The gauge/gravity duality is an equality between two theories. On one hand we have a quantum field theory in d spacetime dimensions. On the other hand we have a gravity theory on a (d + 1)-dimensional spacetime that has an asymptotic boundary which is d-dimensional. It is also sometimes called AdS/CFT, because the simplest examples involve anti-de Sitter spaces and conformal field theories. It is often called gauge/string duality, because the gravity theories are string theories and the quantum field theories are gauge theories. It is also referred to as “holography” because one is describing a (d + 1)-dimensional gravity theory in terms of a lower-dimensional system, in a way that is reminiscent of an optical hologram, which stores a three-dimensional image on a two-dimensional photographic plate. This duality is called a “conjecture”, but by now there is considerable evidence that it is correct. In addition, there are some derivations based on physical arguments.

The simplest example involves an anti-de Sitter spacetime. So, let us start by describing this spacetime in some detail. Anti-de Sitter is the simplest solution of Einstein's equations with a negative cosmological constant. It is the Lorentzian analogue of hyperbolic space, which was historically the first example of a non-Euclidean geometry.