One-Loop quantum gravity in the Einstein universe

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We study quantum gravity with the Einstein-Hilbert action including the cosmological constant on the Euclidean Einstein universe S¹× S³. We compute exactly the spectra and the heat kernels of the relevant operators on S³ and use these results to compute the heat trace of the graviton and ghost operators and the exact one-loop effective action on S¹× S³. We show that the system is unstable in the infrared limit due to the presence of the negative modes of the graviton and the ghost operators. We study the thermal properties of the model with the temperature T=(2πa₁)⁻¹ determined by the radius a₁ of the circle S¹. We show that the heat capacity Cᵥ is well defined and behaves like T³ in the high temperature limit and has a singularity of the type (T-Tc)⁻¹, indicating a second-order phase transition, with the critical temperature Tc determined by the cosmological constant Λ and the radius a of the sphere S³. We also discuss some peculiar properties of the model such as the negative heat capacity as well as possible physical applications.