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We derive the second-order dissipative relativistic hydrodynamic equations ina generic frame with a continuous parameter from the relativistic Boltzmannequation. We present explicitly the relaxation terms in the energy and particleframes. Our results show that the viscosities are frame-independent but therelaxation times are generically frame-dependent. We confirm that thedissipative part of the energy-momentum tensor in the particle frame satisfies$\delta T^\mu_\mu = 0$ obtained for the first-order equation before, incontrast to the Eckart choice $u_\mu \delta T^{\mu\nu} u_\nu = 0$ adopted as amatching condition in the literature. We emphasize that the new constraint$\delta T^\mu_\mu = 0$ can be compatible with the phenomenological derivationof hydrodynamics based on the second law of thermodynamics.

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