On the convexity of Relativistic Hydrodynamics

TL;DR

This paper investigates the convexity conditions of the relativistic hydrodynamic equations for perfect fluids, linking them to the fundamental derivative of the equation of state, and recovers classical limits.

Contribution

It derives the specific conditions under which the relativistic hydrodynamic system is convex, extending classical results to relativistic regimes.

Findings

Convexity conditions expressed via the fundamental derivative.

Classical limit of the conditions recovered.

Provides criteria for hyperbolicity and convexity in relativistic fluids.

Abstract

The relativistic hydrodynamic system of equations for a perfect fluid obeying a causal equation of state is hyperbolic (Anile 1989 {\it Relativistic Fluids and Magneto-Fluids} (Cambridge: Cambridge University Press)). In this report, we derive the conditions for this system to be convex in terms of the fundamental derivative of the equation of state (Menikoff and Plohr 1989 {\it Rev. Mod. Phys.} {\bf 61} 75). The classical limit is recovered.