From Navier-Stokes to Einstein

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.