Laws of thermodynamic equilibrium through relativistic thermodynamics

Using Israel-Stewart formalism for the description of thermodynamics of an arbitrary relativistic fluid we propose generalization of Tolman-Ehrenfest relation and Klein's law on a general background spacetime. The first relation is a consequence of thermal equilibrium only of a non-viscous fluid, while the latter one is reflection of only no-diffusion condition with or without viscosity. Interestingly, both the relations are obtained independently through the imposition of respective equilibrium conditions and also can be valid in presence of particular dissipative processes in the fluid. For a spacetime with a global timelike Killing vector, these two general relations boil down to standard forms.