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Tolman-Ehrenfest-Klein Law in non-Riemannian geometries

Heat always flows from hotter to a colder temperature until thermal equilibrium be finally restored in agreement with the usual (zeroth, first and second) laws of thermodynamics. However, Tolman and Ehrenfest demonstrated that the relation between inertia and weight uniting all forms of energy in the framework of general relativity implies that the standard equilibrium condition is violated in order to maintain the validity of the first and second law of thermodynamics. Here we demonstrate that the thermal equilibrium condition for a static self-gravitating fluid, besides being violated, is also heavily dependent on the underlying spacetime geometry (whether Riemannian or non-Riemannian). As a particular example, a new equilibrium condition is deduced for a large class of Weyl and f(R) type gravity theories. Such results suggest that experiments based on the foundations of the heat theory (thermal sector) may also be used for confronting gravity theories and prospect the intrinsic geometric nature of the spacetime structure.