On the Thermodynamics of Simple Non-Isentropic Perfect Fluids in General Relativity

TL;DR

This paper investigates the thermodynamic consistency of non-isentropic perfect fluids in general relativity, analyzing integrability conditions of the Gibbs-Duhem relation across various spacetime symmetries and exact solutions.

Contribution

It identifies conditions under which the Gibbs-Duhem relation is integrable for perfect fluids and examines specific solutions, revealing limitations in their thermodynamic consistency.

Findings

Gibbs-Duhem relation is integrable for static and isentropic fluids.

Most Szekeres solutions do not satisfy the integrability condition.

FRW cosmology is a trivial case where integrability holds.

Abstract

We examine the consistency of the thermodynamics of irrotational and non-isentropic perfect fluids complying with matter conservation by looking at the integrability conditions of the Gibbs-Duhem relation. We show that the latter is always integrable for fluids of the following types: (a) static, (b) isentropic (admits a barotropic equation of state), (c) the source of a spacetime for which $r\ge 2$, where $r$ is the dimension of the orbit of the isometry group. This consistency scheme is tested also in two large classes of known exact solutions for which $r< 2$, in general: perfect fluid Szekeres solutions (classes I and II). In none of these cases, the Gibbs-Duhem relation is integrable, in general, though specific particular cases of Szekeres class II (all complying with $r<2$) are identified for which the integrability of this relation can be achieved. We show that Szekeres class I solutions satisfy the integrability conditions only in two trivial cases, namely the spherically symmetric limiting case and the Friedman-Roberson-Walker (FRW) cosmology. Explicit forms of the state variables and equations of state linking them are given explicitly and discussed in relation to the FRW limits of the solutions. We show that fixing free parameters in these solutions by a formal identification with FRW parameters leads, in all cases examined, to unphysical temperature evolution laws, quite unrelated to those of their FRW limiting cosmologies.