  In this conceptual paper we construct a local version of Shape Dynamics that
is equivalent to General Relativity in the sense that the algebras of Dirac
observables weakly coincide. This allows us to identify Shape Dynamics
observables with General Relativity observables, whose observables can now be
interpreted as particular representative functions of observables of a
conformal theory at fixed York time. An application of the observable
equivalence of General Relativity and Shape Dynamics is to define the
quantization of General Relativity through quantizing Shape Dynamics and using
observable equivalence. We investigate this proposal explicitly for gravity in
2+1 dimensions.
