Abstract
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.
Highlights
The existence of fluid solutions in gravity is expected/predicted by holography on general grounds
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 dual theory
One expects to find the same feature on the dual gravitational side, i.e., there should exist a bulk solution corresponding to the thermal state, and nearby solutions corresponding to the hydrodynamic regime
Summary
2.2 Integration scheme We start with the zeroth order seed metric (2.7) in the form ds2 = −2puadxadr + [ηab + (1 − θ)uaub]dxadxb, θ = 1 + p2(r − 1). Where we define hab ≡ ηab + uaub, ua ≡ ηabub Weighting derivatives such that ∂r ∼ 1 and ∂a ∼ ǫ, adding a piece gμ(nν) to the metric at order ǫn engenders a change in the Ricci tensor δ Rr(nr ). Allowing for gauge transformations ξ(n)μ at order ǫn, as well as re-definitions δu(n)a(x) and δp(n)(x) of the fluid velocity and pressure, the solution above generalises to gr(nr ) = −2pua∂rξ(n)a, gr(na) = −ua[p∂rξ(n)r − (1 − θ)ub∂rξ(n)b + δp(n)] + ηab∂rξ(n)b − pδu(an), ga(nb) = ga(nb) − uaub[p2ξ(n)r + 2pδp(n)(r − 1)] + 2(1 − θ)u(aδu(bn) ).
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