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

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 $δT^μ_μ= 0$ obtained for the first-order equation before, in contrast to the Eckart choice $u_μδT^{μν} u_ν= 0$ adopted as a matching condition in the literature. We emphasize that the new constraint $δT^μ_μ= 0$ can be compatible with the phenomenological derivation of hydrodynamics based on the second law of thermodynamics.