Universality of the hydrodynamic limit in AdS/CFT and the membrane paradigm

We show that at the level of linear response the low-frequency limit of a strongly coupled field theory at finite temperature is determined by the horizon geometry of its gravity dual, i.e., by the membrane paradigm fluid of classical black hole mechanics. Thus, generic boundary theory transport coefficients can be expressed in terms of geometric quantities evaluated at the horizon. When applied to the stress tensor this gives a simple, general proof of the universality of the shear viscosity in terms of the universality of gravitational couplings, and when applied to a conserved current it gives a new general formula for the conductivity. Away from the low-frequency limit the behavior of the boundary theory fluid is no longer fully captured by the horizon fluid even within the derivative expansion; instead, we find a nontrivial evolution from the horizon to the boundary. We derive flow equations governing this evolution and apply them to the simple examples of charge and momentum diffusion.