The Navier-Stokes equation and solution generating symmetries from holography

TL;DR

This paper explores how solution-generating symmetries from spacetime Ehlers transformations can be applied via holography to generate new solutions of the Navier-Stokes equations, revealing symmetries and energy scaling behaviors.

Contribution

It introduces a formalism connecting spacetime Ehlers transformations to solution generation in holographic hydrodynamics, extending symmetry analysis to Navier-Stokes fluids.

Findings

Linear energy scaling from RG flow with zero vorticity

Identification of Z_2 symmetries at fixed viscosity

Application of Ehlers transformations to fluid duals

Abstract

The fluid-gravity correspondence provides us with explicit spacetime metrics that are holographically dual to (non-)relativistic nonlinear hydrodynamics. The vacuum Einstein equations, in the presence of a Killing vector, possess solution-generating symmetries known as spacetime Ehlers transformations. These form a subgroup of the larger generalized Ehlers group acting on spacetimes with arbitrary matter content. We apply this generalized Ehlers group, in the presence of Killing isometries, to vacuum metrics with hydrodynamic duals to develop a formalism for solution-generating transformations of Navier-Stokes fluids. Using this we provide examples of a linear energy scaling from RG flow under vanishing vorticity, and a set of Z_2 symmetries for fixed viscosity.