  The standard geometrodynamics is transformed into a theory of conformal
geometrodynamics by extending the ADM phase space for canonical general
relativity to that consisting of York's mean exterior curvature time, conformal
three-metric and their momenta. Accordingly, an additional constraint is
introduced, called the conformal constraint. In terms of the new canonical
variables, a diffeomorphism constraint is derived from the original momentum
constraint. The Hamiltonian constraint then takes a new form. It turns out to
be the sum of an expression that previously appeared in the literature and
extra terms quadratic in the conformal constraint. The complete set of the
conformal, diffeomorphism and Hamiltonian constraints are shown to be of first
class through the explicit construction of their Poisson brackets. The extended
algebra of constraints has as subalgebras the Dirac algebra for the
deformations and Lie algebra for the conformorphism transformations of the
spatial hypersurface. This is followed by a discussion of potential
implications of the presented theory on the Dirac constraint quantization of
general relativity. An argument is made to support the use of the York time in
formulating the unitary functional evolution of quantum gravity. Finally, the
prospect of future work is briefly outlined.
