Scale invariant solutions in relativistic hydrodynamics

Abstract

The goal of this work is to describe the scale invariant solutions of a typical relativistic hydrodynamic model. We shall take as representative model an Israel-Stewart framework, where the energy-momentum conservation laws for a conformal invariant fluid are supplemented by a Cattaneo-Maxwell equation for its viscous energy-momentum tensor. In these models the viscous energy-momentum tensor relaxes to its Landau-Lifshitz value on a finite time scale. We assume the parameters of the model depend on the speed of light $c$ in such a way that as $c\to \infty$ the fluid becomes an incompressible fluid obeying the Navier-Stokes equations. We seek the scale invariant solutions for this model and find that for finite $c$ there are two basic patterns, one which reproduces Kolmogorov turbulence when $c\to \infty$, and another whose damping rate diverges in that limit. We point out the scaling relations that allow the latter flow pattern to sustain an entropy cascade.