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From Navier-Stokes to Einstein

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Abstract

We show by explicit construction that for every solution of the incompressible Navier-Stokes equation in p + 1 dimensions, there is a uniquely associated “dual” solution of the vacuum Einstein equations in p + 2 dimensions. The dual geometry has an intrinsically flat timelike boundary segment Σc whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a “near-horizon” limit in which Σc becomes highly accelerated. The near-horizon expansion in gravity is shown to be mathematically equivalent to the hydrodynamic expansion in fluid dynamics, and the Einstein equation reduces to the incompressible Navier-Stokes equation. For p = 2, we show that the full dual geometry is algebraically special Petrov type II. The construction is a mathematically precise realization of suggestions of a holographic duality relating fluids and horizons which began with the membrane paradigm in the 70’s and resurfaced recently in studies of the AdS/CFT correspondence.

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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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  4. Andrew Strominger
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Correspondence to Cynthia Keeler.

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ArXiv ePrint: 1101.2451

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Bredberg, I., Keeler, C., Lysov, V. et al. From Navier-Stokes to Einstein. J. High Energ. Phys. 2012, 146 (2012). https://doi.org/10.1007/JHEP07(2012)146

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