  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.
