From Navier-Stokes To Einstein

TL;DR

This paper constructs a precise duality between solutions of the incompressible Navier-Stokes equations and vacuum Einstein solutions in higher dimensions, revealing a deep connection between fluid dynamics and gravity.

Contribution

It explicitly demonstrates a duality linking Navier-Stokes solutions to Einstein geometries, establishing a mathematical equivalence between hydrodynamic and gravitational near-horizon expansions.

Findings

Dual geometries correspond to Navier-Stokes solutions

Near-horizon limit equates gravity expansion to fluid dynamics

For p=2, the geometry is Petrov type II

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.