Thermodynamics of the Stephani Universes

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Abstract
The consistency of the thermodynamics of the most general class of a conformally flat solution with an irrotational perfect fluid source (the Stephani Universes) is examined herein. For the case when the isometry group has dimension r≥2, the Gibbs–Duhem relation is always integrable, but if r<2 it is only integrable for the particular subclass [containing Friedman–Robertson–Walker (FRW) cosmologies] characterized by r=1 and by admitting a conformal motion parallel to the four-velocity. Explicit forms of the state variables and equations of state linking them are provided. These formal thermodynamic relations are determined up to an arbitrary function of time which reduces to the FRW scale factor in the FRW limit of the solutions. It is shown that a formal identification of this free parameter with a FRW scale factor determined by FRW dynamics leads to an unphysical temperature evolution law. If this parameter is not identified with a FRW scale factor, it is possible to find examples of solutions and formal equations of state complying with suitable energy conditions and reasonable asymptotic behavior and temperature laws.
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