One-Loop quantum gravity in the Einstein universe

TL;DR

This paper computes the one-loop quantum gravity effects on the Euclidean Einstein universe, revealing instability in the infrared, thermal phase transitions, and peculiar thermodynamic properties.

Contribution

It provides exact spectra, heat kernels, and the one-loop effective action for quantum gravity on $S^1\times S^3$, highlighting stability issues and phase transition behavior.

Findings

Infrared instability due to negative modes

Heat capacity behaves as $\sim T^3$ at high temperature

Presence of a second-order phase transition at critical temperature

Abstract

We study quantum gravity with the Einstein-Hilbert action including the cosmological constant on the Euclidean Einstein universe $S^1\times S^3$. We compute exactly the spectra and the heat kernels of the relevant operators on $S^3$ and use these results to compute the heat trace of the graviton and ghost operators and the exact one-loop effective action on $S^1\times S^3$. 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 $T=(2\pi a_1)^{-1}$ determined by the radius $a_1$ of the circle $S^1$. We show that the heat capacity $C_v$ is well defined and behaves like $\sim T^3$ in the high temperature limit and has a singularity of the type $\sim (T-T_c)^{-1}$, indicating a second-order phase transition, with the critical temperature $T_c$ determined by the cosmological constant $\Lambda$ and the radius $a$ of the sphere $S^3$. We also discuss some peculiar properties of the model such as the negative heat capacity as well as possible physical applications.