Microcanonical functional integral and entropy for eternal black holes

TL;DR

This paper develops a microcanonical path integral approach for eternal black holes, calculating their entropy and internal degrees of freedom, and finds that the semiclassical entropy is zero, indicating a pure state description.

Contribution

It introduces a novel microcanonical functional integral framework for eternal black holes, including internal degrees of freedom in thermodynamic analysis.

Findings

Semiclassical entropy of the black hole is zero.

Functional integral describes a pure state.

Internal degrees of freedom can be explicitly included.

Abstract

The microcanonical functional integral for an eternal black hole system is considered. This requires computing the microcanonical action for a spatially bounded spacetime region when its two disconnected timelike boundary surfaces are located in different wedges of the Kruskal diagram. The path integral is a sum over Lorentzian geometries and is evaluated semiclassically when its boundary data are chosen such that the system is approximated by any Lorentzian, stationary eternal black hole. This approach opens the possibility of including explicitly the internal degrees of freedom of a physical black hole in path integral descriptions of its thermodynamical properties. If the functional integral is interpreted as the density of states of the system, the corresponding entropy equals ${\cal S} = A_H/4 - A_H/4 = 0$ in the semiclassical approximation, where $A_H$ is the area of the black hole horizon. The functional integral reflects the properties of a pure state. The description of the black hole density of states in terms of the eternal black hole functional integral is also discussed.