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

TL;DR

This work shows that the long-wavelength, finite-temperature response of strongly coupled field theories with gravity duals is governed by the black hole horizon via the membrane paradigm. By matching horizon data to boundary retarded correlators, it derives horizon-based expressions for transport coefficients, including the universal η/s = 1/(4π) and a general DC conductivity formula, while introducing flow equations that describe how boundary responses emerge from horizon physics at finite frequency. The analysis clarifies when horizon data suffices (low frequency) and when full bulk geometry is required (finite frequency), applying these ideas to charge and momentum diffusion. The results illuminate the geometric origin of universality in holographic transport and provide practical tools for computing boundary responses from horizon physics.

Abstract

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.