The relativistic fluid dual to vacuum Einstein gravity

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.

Holographic reconstruction and renormalization in asymptotically Ricci-flat spacetimes

“…If we then decompose K [0] in terms of the shear tensor σ ab and the mean curvature: 10) it then follows from the Hamiltonian constraint equation (3.1) that: σ ab σ ab = 0. Now, since h [0]ab is positive definite and σ ab Hermitian, then there exists an invertible matrix Q such that (see e.g.…”

“…10 In this way, by finding the unique solution to the initial value problem with such initial data sets, we are able to find the asymptotics of such embeddings. We will begin by developing exact hyperboloidal sets and in section 5 allow deviations of K [0] away from its asymptotic value by considering arbitrary subleading contributions.…”

“…Replacing both conditions in the trace equation σ ab σ ab = 0 implies σ ab = 0 locally and hence everywhere since the equation is tensorial. 10 There is a technical point regarding conformal compactness. If the embedding of some Cauchy surface is conformally compact, then it is possible to show that the surface is also conformally compact (see e.g.…”

“…In order to do so, we need the GaussCodazzi identity: 10) where prime denotes differentiation with respect to z and D i γ jk := 0. If we use our ansatz in (B.10), we find: In this way, the ansatz:…”

“…-7]. A different recent attempt [8][9][10] focuses on subregions of some Ricci-flat spacetime bounded by timelike hypersurfaces and reconstructs the metric of such a region from data belonging to a relativistic fluid describing the hydrodynamic regime of some QFT on the timelike boundary.…”