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

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.

Comments: 8 pages, 2 figures. The significance of clear definition of local rest frames for a viscous relativistic fluid is emphasized as an unsolved problem. The relaxation equations are corrected by adding the vorticity and acceleration terms, which were missed in the previous version.