Abstract
Abstract The holographic interpretation of the hydrodynamic entropy current is developed for the case of hydrodynamics with a conserved charge. This is carried out within a framework developed in earlier work [1, 2], which showed how to associate entropy currents with horizons in the dual geometry. The entropy current defined by the event horizon in the dual bulk geometry is calculated. It is also shown that to second order in the gradient expansion the dual geometry possesses a unique Weyl-invariant apparent horizon which also defines an admissable entropy current. At first order both currents coincide with the result obtained on the basis of a purely hydrodynamic analysis [3].
Highlights
That the entropy is still proportional to the area, but perhaps not to the area of the event horizon itself, but of a hypersurface which asymptotes to it at late times [1, 2, 7, 8]
The holographic interpretation of the hydrodynamic entropy current is developed for the case of hydrodynamics with a conserved charge
The ambiguity in the definition of the entropy current in relativistic second order hydrodynamics was recently connected with the issue of dynamical black hole boundaries [1]
Summary
To describe hydrodynamics with a conserved current a Maxwell field in the bulk is required [16, 17, 20]. The form of the metric (2.7) is strongly restricted by the Weyl-invariance of the bulk theory [24]: the functions Vμ are of unit Weyl weight and Gμν are Weyl invariant They can be expressed as linear combinations of independent Lorentz vectors and tensors built out of b, q, uμ, hμν and their derivatives, order by order in the gradient expansion. The metric given above is a solution of the Einstein-Maxwell equations with negative cosmological constant up to second order in gradients, provided that b, q and uμ satisfy the equations of hydrodynamics, i.e. the equations of covariant conservation of the energymomentum tensor and charge current obtained from (2.11) and (2.12) by holographic renormalization [25]. Explicit formulae [20] can be found in appendix (C)
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