Causal dissipation in the relativistic dynamics of barotropic fluids

Abstract

This paper develops two causal four-field theories of relativistic fluid dynamics, one for viscous fluids that are barotropic and the other for viscous heat-conductive fluids that are “thermo-barotropic,” a property that is newly defined here. While similar to the deduction of a five-field theory in the authors’ article Causal dissipation for the relativistic dynamics of ideal gases [Freistühler, H. and Temple, B., Proc. R. Soc. A 473, 20160729 (2017)], the argumentation is consistently carried out in a way that stays closer to Eckart’s flow frame (than to Landau’s), and the result is directly compared with the four-field theories resulting from early proposals made by Eckart, Lichnerowicz, and Choquet-Bruhat. The key idea is to impose second-order symmetric hyperbolicity in the sense of Hughes, Kato, and Marsden [Arch. Rational Mech. Anal. 63, 273–294 (1976)], in the natural Godunov variables that make the corresponding system of perfect-fluid conservation laws symmetric hyperbolic in the first order sense. The new theory for viscous heat-conductive thermo-barotropic fluids belongs to the Hughes-Kato-Marsden class; the new theory for viscous general barotropic fluids lies on the boundary of that class. As in the five-field theory, the coefficients of bulk viscosity ζ, shear viscosity η, and, in the case of thermo-barotropic fluids, that of heat conductivity χ are free parameters, and in terms of these, the relativistic dissipation tensor is uniquely determined under three conditions which the authors propose as definitive: symmetric hyperbolicity, sharp causality, and first-order equivalence with Eckart—the requirement that the resulting equations be equivalent to the Eckart equations to leading order in ζ, η, and χ. The new theory for general viscous barotropic fluids complements the one given in the abovementioned paper for massive ideal gases. The new theory for viscous heat-conductive thermo-barotropic fluids notably includes the case of pure radiation, providing as one application a quantitative correction to the authors’ previous proposal in Causal dissipation and shock profiles in the relativistic fluid dynamics of pure radiation [Freistühler, H. and Temple, B., Proc. R. Soc. A 470, 20140055 (2014)].