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From Navier-Stokes to Einstein. (English) Zbl 1397.83044

Summary: 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.

MSC:

83C55 Macroscopic interaction of the gravitational field with matter (hydrodynamics, etc.)
83C57 Black holes
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
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References:

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