Action integrals and partition functions in quantum gravity

TLDR

The action for a gravitational field can be evaluated on a complexified spacetime section that avoids singularities, yielding finite, purely imaginary values that can be interpreted as probabilities for finding such metrics in the vacuum or as contributions to the partition function of a grand canonical ensemble. The study aims to evaluate the entropy of Kerr–Newman and de Sitter metrics using this action approach. The authors compute the action on complexified spacetime sections and relate it to the horizon area to evaluate the entropy of the metrics. The finite, purely imaginary actions for Kerr–Newman and de Sitter metrics correspond to probabilities or partition function contributions, and the entropy derived equals one quarter the horizon area, agreeing with previous results, while stationary systems without horizons possess no gravitational entropy.

Abstract

One can evaluate the action for a gravitational field on a section of the complexified spacetime which avoids the singularities. In this manner we obtain finite, purely imaginary values for the actions of the Kerr-Newman solutions and de Sitter space. One interpretation of these values is that they give the probabilities for finding such metrics in the vacuum state. Another interpretation is that they give the contribution of that metric to the partition function for a grand canonical ensemble at a certain temperature, angular momentum, and charge. We use this approach to evaluate the entropy of these metrics and find that it is always equal to one quarter the area of the event horizon in fundamental units. This agrees with previous derivations by completely different methods. In the case of a stationary system such as a star with no event horizon, the gravitational field has no entropy.