A very successful and important application of gauge/gravity duality has emerged in the context of hydrodynamics. In generalisation of the dynamics of fluids, the term hydrodynamics generically refers to an effective field theory describing long-range, low-energy fluctuations about equilibrium.

Recently, experimental evidence has accumulated that the quark–gluon plasma observed in heavy-ion collision experiments is best described by a strongly coupled relativistic fluid, rather than by a gas of weakly interacting particles. Strongly coupled fluids are intrinsically difficult to describe by standard methods. This explains the success of applying gauge/gravity duality to this area of physics. In particular, gauge/gravity duality has made predictions of universal values of certain transport coefficients in strongly coupled fluids. The most famous example of this is the ratio of shear viscosity over entropy density, which takes a very small value. Beyond these results, gauge/gravity duality has provided a fresh look at relativistic hydrodynamics, for which many new non-trivial properties have been uncovered using the fluid/gravity correspondence.

We will describe these results in some detail. The starting point is to introduce linear response theory and Green's functions which respect the causal structure. Then we move on to an introduction to relativistic hydrodynamics. We consider the energy-momentum tensor and a conserved current and their dissipative contributions in an expansion in derivatives of fluctuations. We define the associated first-order transport coefficients and subsequently relate them to the retarded Green's function by virtue of appropriate Green–Kubo relations. This provides a link between macroscopic hydrodynamic properties and microscopic physics as described by the Green's functions. Using gauge/gravity duality methods to evaluate the relevant Green's functions, we compute the charge diffusion constant and the shear viscosity.