Microcanonical Action and the Entropy of a Rotating Black Hole

TL;DR

This paper develops a microcanonical approach to quantum gravity, deriving a Jacobi action for general relativity, and applies it to a rotating black hole to explore its entropy and density of states.

Contribution

It introduces a microcanonical functional integral framework for quantum gravity and computes the Jacobi action for rotating black holes, linking it to black hole entropy.

Findings

Derived Jacobi's action for general relativity.

Applied the framework to a rotating black hole.

Discussed the relation to black hole entropy.

Abstract

The authors have recently proposed a ``microcanonical functional integral" representation of the density of quantum states of the gravitational field. The phase of this real--time functional integral is determined by a ``microcanonical" or Jacobi action, the extrema of which are classical solutions at fixed total energy, not at fixed total time interval as in Hamilton's action. This approach is fully general but is especially well suited to gravitating systems because for them the total energy can be fixed simply as a boundary condition on the gravitational field. In this paper we describe how to obtain Jacobi's action for general relativity. We evaluate it for a certain complex metric associated with a rotating black hole and discuss the relation of the result to the density of states and to the entropy of the black hole. (Dedicated to Yvonne Choquet-Bruhat in honor of her retirement.)