The Microcanonical Functional Integral. I. The Gravitational Field

TL;DR

This paper formulates the gravitational field in a finite region as a microcanonical system, deriving thermodynamic properties and partition functions for black holes using functional integrals, bridging gravity and statistical mechanics.

Contribution

It introduces a novel functional integral approach to describe gravitational systems as microcanonical ensembles, connecting boundary data with black hole thermodynamics.

Findings

 the density of states relates to black hole horizon area

Derived thermodynamic laws for stationary black holes

Connected gravitational functional integrals with Feynman's quantum mechanics

Abstract

The gravitational field in a spatially finite region is described as a microcanonical system. The density of states $\nu$ is expressed formally as a functional integral over Lorentzian metrics and is a functional of the geometrical boundary data that are fixed in the corresponding action. These boundary data are the thermodynamical extensive variables, including the energy and angular momentum of the system. When the boundary data are chosen such that the system is described semiclassically by {\it any} real stationary axisymmetric black hole, then in this same approximation $\ln\nu$ is shown to equal 1/4 the area of the black hole event horizon. The canonical and grand canonical partition functions are obtained by integral transforms of $\nu$ that lead to "imaginary time" functional integrals. A general form of the first law of thermodynamics for stationary black holes is derived. For the simpler case of nonrelativistic mechanics, the density of states is expressed as a real-time functional integral and then used to deduce Feynman's imaginary-time functional integral for the canonical partition function.