Wilsonian Approach to Fluid/Gravity Duality

TL;DR

This paper explores the fluid/gravity duality by analyzing gravitational fluctuations with a finite cutoff, deriving a cutoff-dependent diffusion constant, and connecting membrane paradigm results with AdS/CFT through RG flow.

Contribution

It introduces a unified RG flow framework linking membrane paradigm and AdS/CFT results for fluid dynamics on cutoff surfaces in black hole geometries.

Findings

Derived a cutoff-dependent diffusion constant D(r_c).

Reproduced known AdS/CFT results at infinite cutoff.

Connected membrane paradigm results with RG flow.

Abstract

The problem of gravitational fluctuations confined inside a finite cutoff at radius $r=r_c$ outside the horizon in a general class of black hole geometries is considered. Consistent boundary conditions at both the cutoff surface and the horizon are found and the resulting modes analyzed. For general cutoff $r_c$ the dispersion relation is shown at long wavelengths to be that of a linearized Navier-Stokes fluid living on the cutoff surface. A cutoff-dependent line-integral formula for the diffusion constant $D(r_c)$ is derived. The dependence on $r_c$ is interpreted as renormalization group (RG) flow in the fluid. Taking the cutoff to infinity in an asymptotically AdS context, the formula for $D(\infty)$ reproduces as a special case well-known results derived using AdS/CFT. Taking the cutoff to the horizon, the effective speed of sound goes to infinity, the fluid becomes incompressible and the Navier-Stokes dispersion relation becomes exact. The resulting universal formula for the diffusion constant $D(horizon)$ reproduces old results from the membrane paradigm. Hence the old membrane paradigm results and new AdS/CFT results are related by RG flow. RG flow-invariance of the viscosity to entropy ratio $\eta /s$ is shown to follow from the first law of thermodynamics together with isentropy of radial evolution in classical gravity. The ratio is expected to run when quantum gravitational corrections are included.