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(∞) 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 \( \frac{\eta }{s} \) 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.
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Bredberg, I., Keeler, C., Lysov, V. et al. Wilsonian approach to fluid/gravity duality. J. High Energ. Phys. 2011, 141 (2011). https://doi.org/10.1007/JHEP03(2011)141
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DOI: https://doi.org/10.1007/JHEP03(2011)141