  Canonical vacuum gravity is expressed in generally-covariant form in order
that spacetime diffeomorphisms be represented within its equal-time phase
space. In accordance with the principle of general covariance, the time mapping
${\T}: {\yman} \to {\rman}$ and the space mapping ${\X}: {\yman} \to {\xman}$
that define the Dirac-ADM foliation are incorporated into the framework of the
Hilbert variational principle. The resulting canonical action encompasses all
individual Dirac-ADM actions, corresponding to different choices of foliating
vacuum spacetimes by spacelike hypersurfaces. In this framework, spacetime
observables, namely, dynamical variables that are invariant under spacetime
diffeomorphisms, are not necessarily invariant under the deformations of the
mappings $\T$ and $\X$, nor are they constants of the motion. Dirac observables
form only a subset of spacetime observables that are invariant under the
transformations of $\T$ and $\X$ and do not evolve in time. The conventional
interpretation of the canonical theory, due to Bergmann and Dirac, can be
recovered only by postulating that the transformations of the reference system
$({\T},{\X})$ have no measurable consequences. If this postulate is not deemed
necessary, covariant canonical gravity admits no classical problem of time.
