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Wilsonian approach to fluid/gravity duality

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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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Authors and Affiliations

  1. Center for the Fundamental Laws of Nature, Harvard University, Cambridge, MA, 02138, U.S.A.

    Irene Bredberg, Cynthia Keeler, Vyacheslav Lysov & Andrew Strominger

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  1. Irene Bredberg
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  2. Cynthia Keeler
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  3. Vyacheslav Lysov
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Correspondence to Irene Bredberg.

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ArXiv ePrint: 1006.1902

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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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