From Navier-Stokes to Einstein

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From Navier-Stokes to Einstein

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Title: From Navier-Stokes to Einstein
Author: Bredberg, Irene; Keeler, Cynthia; Lysov, Vyacheslav; Strominger, Andrew E.

Note: Order does not necessarily reflect citation order of authors.

Citation: Bredberg, Irene, Cynthia Keeler, Vyacheslav Lysov, and Andrew E. Strominger. 2012. From Navier-Stokes to Einstein. Journal of High Energy Physics 2012(7): 146.
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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 \(\sum_c\) whose extrinsic curvature is given by the stress tensor of the Navier-Stokes fluid. We consider a “near-horizon” limit in which \(\sum_c\) 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.
Published Version: doi:10.1007/JHEP07(2012)146
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