Relativistic Barotropic Fluids: A Godunov–Boillat Formulation for their Dynamics and a Discussion of Two Special Classes

Abstract

A relativistic fluid is called barotropic if its internal energy $${\rho}$$ and its pressure p are one-to-one related; it is called isentropic if $${\rho}$$ and p are both one-to-one related to the fluid’s particle number density n. Deriving Godunov variables and 4-potentials for the relativistic Euler equations of perfect barotropic fluids, this paper first pursues ideas on symmetric hyperbolicity going back to Godunov and Boillat that Ruggeri and Strumia have elaborated as their theory of convex covariant density systems. The associated additional balance law (not a conservation law in the presence of shock waves) has different interpretations for different fluids. Among all barotropic fluids, we notably distinguish those for which this extra equation coincides with the second law of thermodynamics. We characterize these ‘thermobarotropic’ fluids, a class which comprises in particular the so-called pure radiation gas used in cosmology. The paper also shows that isentropic fluids are not thermobarotropic and that, like in the non-relativistic case, they cannot simultaneously satisfy the conservation laws of mass, momentum, and energy, as soon as shock waves occur. Curiously, relativistic isentropic flows which conserve energy and momentum generate mass across shock waves; this is a counterpart of the well-known fact that across shock waves, non-relativistic isentropic flows which conserve mass and momentum dissipate energy.