Galilean fluids from non-relativistic gravity

TL;DR

This paper develops a non-relativistic gravity framework with AdS solutions, deriving a Galilean fluid correspondence and simplifying the equations of motion, thus extending the fluid/gravity duality to non-relativistic settings.

Contribution

It introduces a non-relativistic gravity theory with AdS solutions, establishes a fluid/gravity correspondence for Galilean fluids, and simplifies the equations of motion for NRG.

Findings

Derived non-relativistic AdS brane solutions from NRG.

Confirmed the boundary energy-momentum tensor describes a Galilean fluid.

Presented simplified equations of motion for non-relativistic gravity.

Abstract

The $1/c$-expansion of general relativity appropriately sourced by matter can be used to derive an action principle for Newtonian gravity. The gravitational part of this action is known as non-relativistic gravity (NRG). It is possible to source NRG differently and in such a way that one can construct solutions that are not described by Newtonian gravity (as they do not admit a notion of absolute time). It is possible to include a negative cosmological constant such that NRG admits a non-relativistic AdS solution. This non-relativistic AdS vacuum has Killing vectors that form the Galilean conformal algebra and a boundary that admits a conformal class of Newton-Cartan geometries. This begs the question of whether there exists an analogue of the fluid/gravity correspondence for NRG. In this paper we derive a non-relativistic AdS brane solution of NRG and confirm that it corresponds to the $1/c^2$-expansion of the AdS black brane geometry. We perform a Galilean boost of the non-relativistic AdS brane and derive the associated boundary energy-momentum tensor. We then show that this is the energy-momentum tensor of a massless Galilean fluid and explain how this is linked to the conformal isometries of the boundary. Along the way, we also present several new results for the theory of non-relativistic gravity itself. In particular we present a rewriting that greatly shortens and simplifies the equations of motion of the NRG action.