Thermodynamics of Einstein static Universe with boundary

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 $T=1/(πR)$, 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 $T=1/(πR)$, 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, $R=1/πT$. 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 $A=2π^2R^2$ 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.

💡 Research Summary

The paper investigates the thermodynamic properties of a bounded static Einstein universe and demonstrates that they are essentially identical to those of the de Sitter state. The authors begin by recalling that both the de Sitter spacetime and the static Einstein universe (specifically the half‑space (R\times S^{3})) possess a constant scalar Ricci curvature ({\cal R}). This constancy allows the curvature to act as a thermodynamic variable, analogous to a conserved charge, and to define a local temperature that is fixed throughout the spacetime. In de Sitter space the temperature is (T=H/\pi) (with (H) the Hubble parameter), which is twice the usual Gibbons–Hawking temperature.

To construct a comparable situation for the Einstein universe, the authors consider only one hemisphere of the three‑sphere, introducing a physical boundary at (r=R). This boundary is treated as a surface that couples the interior to an external heat bath. The area of the boundary is (A=2\pi^{2}R^{2}). By imposing the “gravitational neutrality” condition (T^{\mu\nu}=0) (the sum of matter, vacuum‑energy and gravitational stress‑energy tensors vanishes), they derive equilibrium relations for the energy densities and pressures. The gravitational component has an equation‑of‑state parameter (w_{R}=-1/3). For a generic matter component with equation‑of‑state (p_{M}=w_{M}\rho_{M}) the equilibrium conditions lead to explicit expressions for (\rho_{M}) and (\rho_{\Lambda}) in terms of the radius (R). In the special case (w_{M}=1) (Zeldovich stiff matter) the vacuum energy density becomes (\rho_{\Lambda}=2\rho_{M}=1/(4\pi G R^{2})).

A key result is the identification of a local temperature associated with the bounded Einstein universe: \