Extrinsic time formalism for the Euclidean black hole in a box

We return to the problem of Euclidean black holes in thermodynamical equilibrium with radiation, under the conditions of the statistical mechanical canonical ensemble. This time we use Kuchař’s ‘‘extrinsic time’’ formalism in order to select the set of true dynamical variables. With this set of canonical variables, {R̃,${\mathit{P}}_{\mathit{R}\mathrm{̃}}$,M,${\mathit{P}}_{\mathit{M}}$}, and York’s proposal for the boundary and ‘‘horizon’’ terms, we compute a classically consistent total action for the Euclidean black hole. We show explicitly that the ‘‘zero-loop’’ partition function derived from this total action is identical to the one already in the literature of this area. © 1996 The American Physical Society.