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.

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.

References in corpus (1)

• Stable First-order Particle-frame Relativistic Hydrodynamics for Dissipative Systems

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