A new continuum model for general relativistic viscous heat-conducting media

Abstract

The lack of formulation of macroscopic equations for irreversible dynamics of viscous heat-conducting media compatible with the causality principle of Einstein's special relativity and the Euler-Lagrange structure of general relativity is a long-lasting problem. In this paper, we propose a possible solution to this problem in the framework of SHTC equations. The approach does not rely on postulates of equilibrium irreversible thermodynamics but treats irreversible processes from the non-equilibrium point of view. Thus, each transfer process is characterized by a characteristic velocity of perturbation propagation in the non-equilibrium state, as well as by an intrinsic time/length scale of the dissipative dynamics. The resulting system of governing equations is formulated as a first-order system of hyperbolic equations with relaxation-type irreversible terms. Via a formal asymptotic analysis, we demonstrate that classical transport coefficients such as viscosity, heat conductivity, etc., are recovered in leading terms of our theory as effective transport coefficients. Some numerical examples are presented in order to demonstrate the viability of the approach. This article is part of the theme issue 'Fundamental aspects of nonequilibrium thermodynamics'.

Keywords: causal dissipation; hyperbolicity; non-equilibrium thermodynamics.

Figure 1.

(a) Heat conduction based on the SHTC model. Temperature distribution at t = 0.5 for two different relaxation times τ_h = 5 × 10⁻³ and τ_h = 5 × 10⁻⁴, starting from a temperature jump initially located at x = 0.5. (b) Relativistic Sod shock tube problem solving the SHTC model (4.6) with viscosity and heat conduction with different relaxation times τ_h = τ_sh = 2 × 10⁻³, 2 × 10⁻² and 2 × 10⁻¹. As reference, also the exact solution of the Riemann problem of the ideal relativistic Euler equations (RHD) is shown.