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 δTμμ=0 obtained for the first-order equation before, in contrast to the Eckart choice uμδTμνuν=0 adopted as a matching condition in the literature. We emphasize that the new constraint δTμμ=0 can be compatible with the phenomenological derivation of hydrodynamics based on the second law of thermodynamics.