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 \$\Sigma\_c\$ whose extrinsic curvature is given by the stress tensor of
the Navier-Stokes fluid. We consider a "near-horizon" limit in which \$\Sigma\_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.
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