Thermodynamic class II Szekeres–Szafron solutions. Singular models

Abstract

A family of parabolic Szekeres-Szafron class II solutions in local thermal equilibrium is studied and their associated thermodynamics are obtained. The subfamily with the hydrodynamic behavior of a generic ideal gas (defined by the equation of state p = knΘ) results to be an inhomogeneous generalization of flat FLRW γ-law models. Three significative interpretations that follow on from the choice of three specific thermodynamic schemes are analyzed in depth. First, the generic ideal gas in local thermal equilibrium; this interpretation leads to an inhomogeneous temperature Θ. Second, the thermodynamics with homogeneous temperature considered by Lima and Tiomno (CQG 6 1989). And third, a new model having exactly the homogeneous temperature of the FLRW limit. It is shown that the three models above fulfill the necessary macroscopic requirements for physical reality (positivity of matter density and temperature, energy conditions and compressibility conditions) in wide domains of the spacetime.

Relativistic hydrodynamic approach to the classical ideal gas

“…A first result in this direction presented in section VIII shows that the only CIG spherically symmetric solutions in geodesic motion and Y ′ = 0 are the above-mentioned FLRW models (proposition 9). A similar limited result seems to derive from the study of the Szekeres-Szafron solutions that model a CIG [22]. One can overcome this situation in looking for exact solutions that approximate a CIG at low temperatures.…”

“…For the Poisson gases the expressions of the matter density n and the specific internal energy ǫ are more generic than in the case of a CIG. Thus, the second of the positivity constraint P in (22) does not hold necessarily, and then the second compressibility constraint H given by (25) applies for 1 < γ ≤ 2. Therefore:…”

“…Work in progress reveals this fact. Indeed, elsewhere [22] we have studied class II Szekeres-Szafron models in local thermal equilibrium and, although we have found physically realistic solutions, none of them represents a classical ideal gas. A similar situation occurs in analyzing the thermodynamic meaning of the Stephani universes, a task we undertook years ago [7]: there are no exact classical ideal gas models.…”

“…Moreover these models fulfill the compressibility conditions in a wide range of the interval [0, 1]. Nevertheless, the class II Szekeres-Szafron metrics that are ideal gas models only approximate a classical ideal gas at zero order [22].…”

Relativistic kinematic approach to the classical ideal gas

“…If we substitute in the fundamental system (1) a generic equation of state for a particular one, corresponding to a specific perfect fluid, we can state the restricted inverse and direct problems. In [2] we have solved these problems for the set of generic ideal gases, and this study has been applied to physically interpret some already known perfect fluid solutions of the Einstein equation [3] [4] [5].…”