Second-order Relativistic Hydrodynamic Equations for Viscous Systems; how does the dissipation affect the internal energy?

TL;DR

This paper derives second-order relativistic hydrodynamic equations incorporating dissipation, revealing frame-independent viscosities and frame-dependent relaxation times, and clarifies the internal energy's behavior in viscous systems.

Contribution

It provides a general derivation of second-order dissipative relativistic hydrodynamics with explicit relaxation terms, introducing a new energy-momentum tensor constraint compatible with thermodynamics.

Findings

Viscosities are frame-independent.

Relaxation times are frame-dependent.

Energy-momentum tensor satisfies elta T^mu_ = 0 in the particle frame.

Abstract

We derive the second-order dissipative relativistic hydrodynamic equations in a generic frame with a continuous parameter from the relativistic Boltzmann equation. We present explicitly the relaxation terms in the energy and particle frames. Our results show that the viscosities are frame-independent but the relaxation times are generically frame-dependent. We confirm that the dissipative part of the energy-momentum tensor in the particle frame satisfies $\delta T^\mu_\mu = 0$ obtained for the first-order equation before, in contrast to the Eckart choice $u_\mu \delta T^{\mu\nu} u_\nu = 0$ adopted as a matching condition in the literature. We emphasize that the new constraint $\delta T^\mu_\mu = 0$ can be compatible with the phenomenological derivation of hydrodynamics based on the second law of thermodynamics.