  We study the generally covariant theory governing an isotropic spacetime
region with uniform energy density. Gibbons, Hawking and York showed that
fixing the induced boundary metric yields a well-posed variational problem.
However, as we demonstrate, fixing the boundary metric violates general
covariance and allows the mass of a back hole to vary. This observation has
dramatic consequences for path integrals: A sum over spacetimes with fixed
boundary metrics is a sum over classically distinct black holes. Instead, we
merely demand that coordinates exist such that the metric at the boundary is
the Schwarzschild-(A)dS metric of fixed mass M and two-sphere radius R. We
derive the action that yields a well-posed variational problem for these
physical boundary conditions. The action vanishes for all stationary and
isotropic spacetimes. A vanishing action implies that both a Schwarzschild
black hole and pure de Sitter space each have one unique semiclassical state.
Our results provide a novel and radically conservative approach to several
long-standing issues in quantum gravity, such as the wavefunction of the
universe, the black hole information paradox, vacuum decay rates and the
measure problem of eternal inflation.
