The relativistic fluid dual to vacuum Einstein gravity

TL;DR

This paper constructs a higher-dimensional Ricci-flat spacetime dual to a relativistic fluid, analyzing its properties through gradient expansions and identifying key transport coefficients and entropy currents.

Contribution

It provides a detailed holographic construction of a relativistic fluid dual to vacuum Einstein gravity, including high-order gradient expansion and entropy current analysis.

Findings

Explicit second-order metric construction in gradient expansion

Identification of six second-order transport coefficients

Verification of non-negative divergence of the entropy current

Abstract

We present a construction of a (d+2)-dimensional Ricci-flat metric corresponding to a (d+1)-dimensional relativistic fluid, representing holographically the hydrodynamic regime of a (putative) dual theory. We show how to obtain the metric to arbitrarily high order using a relativistic gradient expansion, and explicitly carry out the computation to second order. The fluid has zero energy density in equilibrium, which implies incompressibility at first order in gradients, and its stress tensor (both at and away from equilibrium) satisfies a quadratic constraint, which determines its energy density away from equilibrium. The entire dynamics to second order is encoded in one first order and six second order transport coefficients, which we compute. We classify entropy currents with non-negative divergence at second order in relativistic gradients. We then verify that the entropy current obtained by pulling back to the fluid surface the area form at the null horizon indeed has a non-negative divergence. We show that there are distinct near-horizon scaling limits that are equivalent either to the relativistic gradient expansion we discuss here, or to the non-relativistic expansion associated with the Navier-Stokes equations discussed in previous works. The latter expansion may be recovered from the present relativistic expansion upon taking a specific non-relativistic limit.