Entropy Current in Conformal Hydrodynamics

TL;DR

This work develops a Weyl-covariant formulation of conformal hydrodynamics to second order and constructs an entropy current consistent with the holographically determined energy-momentum tensor for N=4 SYM. By enforcing the second law, it derives a specific dissipative entropy current and relates its production rate to second-order transport coefficients, then applies the construction to N=4 SYM to obtain an explicit Js^λ and entropy production formula. It shows that N=4 SYM lies outside the traditional Israel–Stewart framework due to nonzero shear–shear and Weyl-curvature couplings, thereby providing a generalized approach to causal, conformal hydrodynamics. The results bridge holographic fluid dynamics with relativistic hydrodynamics and point toward future gravity-based validation and extensions to charged plasmas and phenomenology in heavy-ion collisions.

Abstract

In recent work (arXiv:0712.2456, arXiv:0712.2451) the energy-momentum tensor for the N=4 SYM fluid was computed up to second derivative terms using holographic methods. The aim of this note is to propose an entropy current (accurate up to second derivative terms) consistent with this energy-momentum tensor and to explicate its relation with the existing theories of relativistic hydrodynamics. In order to achieve this, we first develop a Weyl-covariant formalism which simplifies the study of conformal hydrodynamics. This naturally leads us to a proposal for the entropy current of an arbitrary conformal fluid in any spacetime (with d>3). In particular, this proposal translates into a definite expression for the entropy flux in the case of N=4 SYM fluid. We conclude this note by comparing the formalism presented here with the conventional Israel-Stewart formalism.