Entropy current in conformal hydrodynamics

In recent work [1, 2], 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.