The off-shell black hole

The standard (Euclidean) action principle for the gravitational field implies that for spacetimes with black-hole topology, the opening angle at the horizon and the horizon area are canonical conjugates. It is shown that the opening angle bears the same relation to the horizon area that the time separation bears to the mass at infinity. The dependence of the wavefunction on this new degree of freedom is governed by an extended Wheeler--DeWitt equation. The trace of the Euclidean transition amplitude, which requires a sum over all horizon areas, yields a partition function whose logarithm gives the black-hole entropy.