Fluid description of gravity on a timelike cut-off surface: beyond Navier-Stokes equation

TL;DR

This paper explores how gravity can be described as a fluid on a timelike surface, extending the fluid/gravity correspondence beyond the Navier-Stokes equation to include Damour Navier-Stokes equations and alternative metric perturbations.

Contribution

It introduces a novel approach to derive the Damour Navier-Stokes equation on a timelike surface and links hydrodynamic and near-horizon expansions in gravity.

Findings

Gravity admits both NS and DNS descriptions on timelike surfaces.

Including specific metric modes corresponds to Einstein's equations with matter.

Hydrodynamic and near-horizon expansions are equivalent in this context.

Abstract

Over the past few decades, a host of theoretical evidence have surfaced that suggest a connection between theories of gravity and Navier-Stokes (NS) equation of fluid dynamics. It emerges out that gravity theory can be treated as some kind of fluid on a particular surface. Motivated by the work carried out by Bredberg et al (JHEP 1207, 146 (2012)) \cite{Bredberg:2011jq}, our paper focuses on including certain modes to the metric which are consistent with the so called hydrodynamic scaling and discuss the consequences, one of which appear in the form of Damour Navier Stokes (DNS) equation with the incompressibility condition. We also present an alternative route to the results by considering the metric as a perturbative expansion in the hydrodynamic scaling parameter $\epsilon$ and with a specific gauge choice, thus modifying the metric. It is observed that the inclusion of certain modes in the metric corresponds to the solution of Einstein's equations in presence of a particular type of matter in the spacetime. This analysis reveals that gravity has both the NS and DNS description not only on a null surface, but also on a timelike surface. So far we are aware of, this analysis is the first attempt to illuminate the possibility of presenting the gravity dual of DNS equation on a timelike surface. In addition, an equivalence between the hydrodynamic expansion and the near-horizon expansion has also been studied in the present context.