Introduction

In this chapter we will study a particular long-wavelength limit of Einstein's equations with a negative cosmological constant in d + 1 dimensions. In such a limit we find that Einstein's equations reduce to the equations of fluid dynamics (relativistic generalizations of the famous Navier–Stokes equations) in d dimensions. While the motivation for our study lies within the AdS/CFT correspondence of string theory, the fluid/gravity correspondence stands on its own and can be viewed as a map between two classic dynamical systems.

Prelude: CFT stress tensor dynamics from gravity

An important consequence of the AdS/CFT correspondence (see Chapter 12) is that the dynamics of the stress(–energy–momentum) tensor in a large class of d dimensional strongly coupled quantum field theories is governed by the dynamics of Einstein's equations with negative cosmological constant in d + 1 dimensions. To begin with, we shall try to provide the reader with some intuition for this statement and argue that searching for a tractable corner of this connection leads one naturally to the fluid/gravity correspondence.

In its most familiar example, the AdS/CFT correspondence asserts that SU (N) N = 4 super Yang–Mills (SYM) theory is dual to type-IIB string theory on AdS5 × S5. It has long been known that in the 't Hooft limit, which involves taking N → ∞ while keeping the coupling λ fixed, the gauge theory becomes effectively classical.