Hamiltonian thermodynamics of the Schwarzschild black hole

TL;DR

This paper adapts Kuchar's analysis of Schwarzschild black hole geometrodynamics to the exterior region with a timelike boundary, deriving a Hamiltonian and partition function that match Euclidean methods and elucidate black hole entropy.

Contribution

It introduces a Hamiltonian formulation for Schwarzschild black holes with boundaries, linking Lorentzian and Euclidean approaches and extending analysis to the RP3 geon topology.

Findings

Hamiltonian contains boundary and bifurcation sphere terms

Partition function matches Euclidean path integral results

Bifurcation sphere term relates to black hole entropy

Abstract

Kucha\v{r} has recently given a detailed analysis of the classical and quantum geometrodynamics of the Kruskal extension of the Schwarzschild black hole. In this paper we adapt Kucha\v{r}'s analysis to the exterior region of a Schwarzschild black hole with a timelike boundary. The reduced Lorentzian Hamiltonian is shown to contain two independent terms, one from the timelike boundary and the other from the bifurcation two-sphere. After quantizing the theory, a thermodynamical partition function is obtained by analytically continuing the Lorentzian time evolution operator to imaginary time and taking the trace. This partition function is in agreement with the partition function obtained from the Euclidean path integral method; in particular, the bifurcation two-sphere term in the Lorentzian Hamiltonian gives rise to the black hole entropy in a way that is related to the Euclidean variational problem. We also outline how Kucha\v{r}'s analysis of the Kruskal spacetime can be adapted to the $\RPthree$ geon, which is a maximal extension of the Schwarzschild black hole with $\RPthree \setminus \{p\}$ spatial topology and just one asymptotically flat region.