Thermodynamic Interpretation of the Kompanneets-Chernov-Kantowski-Sachs Solutions

TL;DR

The paper shows that Kompaneets--Chernov--Kantowski--Sachs (KCKS) cosmologies can be interpreted as the isentropic evolution of thermodynamic perfect fluids, by embedding them in the T-model framework and applying macroscopic viability conditions for physical realism. It specializes to a generic ideal gas in isentropic evolution, providing explicit expressions for thermodynamic quantities and demonstrating physical viability in wide spacetime regions; it also analyzes γ-law barotropic cases, including a radiation ($\gamma=4/3$) scenario, yielding explicit metric and fluid evolutions. The results reveal that KCKS solutions naturally describe isentropic flows with barotropic closures, including new plane-symmetric Bianchi type I models, and establish a foundation for extending these interpretations to broader symmetry classes. Overall, the work furnishes a physically meaningful thermodynamic interpretation of KCKS spacetimes and highlights their potential as realistic cosmological models with controlled thermodynamic behavior.

Abstract

The spatially homogeneous perfect fluid solutions by Kompanneets-Chernov-Kantowski-Sachs are interpreted as a thermodynamic perfect fluid in isentropic evolution, namely, the isentropic limit of their non-homogeneous generalizations, the T-models. Some specific solutions that model a generic ideal gas are examined, and the associated thermodynamic variables are obtained. We show that the necessary macroscopic conditions for physical reality are fulfilled in wide spacetime domains. The field equations for a classical ideal gas are established, and the behavior of the solution is analyzed. The models fulfilling a relativistic $γ$-law are also examined, and the solutions for some particular cases are obtained.

Figures (7)

• Figure S1: At the left, evolution of the energy density $\rho$ of the ideal gas models for different values of $Q_0$. If $Q_0 > 0$, it is decreasing and positive everywhere, and if $Q_0 < 0$, it is decreasing and positive for $\tau > \tau_1$. At the right, evolution of the average deceleration parameter $q$; it is positive and tends to $\frac{3}{2} \tilde{\gamma}-1$ for non-vanishing values of $Q_0$.
• Figure S2: At the left, evolution of the variable $\pi = p/\rho$ of the ideal gas models for different values of $Q_0$. At the right, evolution of the square of the speed of sound $\chi(\pi)$. Both hydrodynamic quantities have a similar behavior. If $Q_0 > 0$, they decrease in the interval $]1,\tilde{\gamma}-1[$ , and if $Q_0 < 0$, they increase in the interval $]0,\tilde{\gamma}-1[$ for $\tau > \tau_1$.
• Figure S3: At the left, evolution of the temperature $\Theta$ of the ideal gas models for the different values of $Q_0$. At the right, evolution of the matter density $n$.
• Figure S4: Comparison of the barotropic relation $\rho = \rho (p)$ (left) and its first derivative (right), for a classical ideal gas with $\gamma = 5/3$ and for different KCKS ideal models.
• Figure S5: At the left, comparison of the evolution of the solution $\varphi(t)$ of the differential Equation (\ref{['rel-barotropia-KCKS-GIC-isentropic-k_0']}) with $\gamma = 5/3$ (monoatomic case) and initial condition $\varphi(t_0) = \varphi_0$ for different values of the constant parameter $Q_0$. At the right, the same comparison for the function relating the time coordinate $t = \alpha_2$ with the proper time $\tau$.