Relativistic thermo-hydrodynamics and conservation laws in continua with thermal inertia

General relativistic Lagrangian is given which yields the effect of heat flux q and nonequilibrium stress γ in the energy-momentum tensor G of an extended reversible fluid exhibiting thermal inertia. Extended Hamilton principle is applied that admits thermal degrees of freedom (entropy four-flux) and yields G as the tensor source of the gravitational field. For the instantaneous properties of this field it is inessential whether the origins of heat flux q and of nonequilibrium stress γ in G are reversible or not, which is the property that makes G directly applicable to real dissipative fluids.