The Incompressible Non-Relativistic Navier-Stokes Equation from Gravity

TL;DR

This paper explores how relativistic hydrodynamics simplifies to the incompressible Navier-Stokes equations under certain limits, revealing new symmetries and providing gravity dual descriptions, with implications for turbulence.

Contribution

It uncovers a new conformal symmetry structure of the Navier-Stokes equations and connects them to gravity duals via holography, extending understanding of fluid dynamics in a relativistic context.

Findings

Relativistic hydrodynamics reduce to Navier-Stokes equations in a specific limit.

Discovered a new conformal symmetry structure of Navier-Stokes equations.

Provided gravity dual descriptions for solutions of the forced Navier-Stokes equations.

Abstract

We note that the equations of relativistic hydrodynamics reduce to the incompressible Navier-Stokes equations in a particular scaling limit. In this limit boundary metric fluctuations of the underlying relativistic system turn into a forcing function identical to the action of a background electromagnetic field on the effectively charged fluid. We demonstrate that special conformal symmetries of the parent relativistic theory descend to `accelerated boost' symmetries of the Navier-Stokes equations, uncovering a possibly new conformal symmetry structure of these equations. Applying our scaling limit to holographically induced fluid dynamics, we find gravity dual descriptions of an arbitrary solution of the forced non-relativistic incompressible Navier-Stokes equations. In the holographic context we also find a simple forced steady state shear solution to the Navier-Stokes equations, and demonstrate that this solution turns unstable at high enough Reynolds numbers, indicating a possible eventual transition to turbulence.