  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.
