Thermodynamics of Einstein static Universe with boundary

Abstract: The de Sitter state and the static Einstein Universe are unique states that have a constant scalar Ricci curvature ${\cal R}$. It was shown earlier that such a unique symmetry of the de Sitter state leads to special thermodynamic properties of this state, which are determined by the local temperature $T=1/(\pi R)$, where $R$ is the radius of the cosmological horizon. Then, what happens in the static Universe? We consider the original Einstein Universe, i.e. the so-called spherical Universe. It is half of the elliptical Universe $R\times S^3$, which was introduced later. Formally, the original Einstein Universe has a boundary at $r=R$. Here we consider the boundary at $r=R$ as a surface that connects the Einstein Universe with the thermal environment. In this realization of the Einstein Universe, it is characterized by a local temperature $T=1/(\pi R)$, which is analogous to the local temperature of the de Sitter state. Or, conversely, the temperature of environment heat bath determines the radius of the Universe, $R=1/\pi T$. The thermodynamics of the bounded Einstein Universe is also analogous to the thermodynamics of the de Sitter state. In particular, the entropy of the bounded Universe satisfies the holographic relation, $S=A/4G$, where $A=2\pi^2R^2$ is the area of the boundary. This shows that in thermodynamics the physical boundary of the static Einstein Universe plays the same role as the cosmological de Sitter horizon. In this Universe, Zeldovich's stiff matter is also preferred.