From Petrov-Einstein to Navier-Stokes

TL;DR

This paper demonstrates how Einstein gravity near a hypersurface can be reduced to fluid dynamics described by the Navier-Stokes equations, revealing a deep connection between gravity and fluid mechanics.

Contribution

It establishes a novel link between Einstein geometry with Petrov type I conditions and the derivation of Navier-Stokes equations from gravitational constraints.

Findings

Einstein constraints reduce to Navier-Stokes equations in a specific limit.

Petrov type I condition constrains extrinsic curvature to fluid variables.

The approach provides a geometric interpretation of fluid dynamics within gravity.

Abstract

We consider a p+1-dimensional timelike hypersurface \Sigma_c embedded with a flat induced metric in a p+2-dimensional Einstein geometry. It is shown that imposing a Petrov type I condition on the geometry reduces the degrees of freedom in the extrinsic curvature of \Sigma_c to those of a fluid in \Sigma_c. Moreover, expanding around a limit in which the mean curvature of the embedding diverges, the leading-order Einstein constraint equations on \Sigma_c are shown to reduce to the non-linear incompressible Navier-Stokes equation for a fluid moving in \Sigma_c.