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 $\delta T^\mu_\mu = 0$ 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.