On Thermodynamical Kluitenberg Theory in General Relativity

In this paper, we introduce Kluitenberg's formulation of non-equilibrium thermodynamics with internal variables in the context of a Riemannian space, as required by Einstein's general relativity. Using the formulation of the second law of thermodynamics in general coordinates with a pseudo-Euclidean metric, we derive a Levi-Civita-like energy tensor and propose a generalization of the second law within a Riemannian space, in agreement with Tolman's approach. In addition, we determine the expression for the entropy density in a general Riemannian space and identify the new variables upon which it depends. This allows us to deduce, within this framework, the equilibrium inelastic and viscous stress tensors as well as the entropy production. These expressions are consistent with the principle of general covariance and Einstein's equivalence principle.