An Intrinsic Coordinate Reference Frame Procedure I: Tensorial Canonical Weyl Scalars

Abstract: Canonical quantization of gravity in general relativity is greatly simplified by the artificial decomposition of space and time into a 3+1 formalism. Such a simplification may appear to come at the cost of general covariance. This requires tangential and perpendicular infinitesimal diffeomorphisms generated by the symmetry group under the Legendre transformation of the given action. This gauge generator, along with the fact that Weyl curvature scalars may act as ``intrinsic coordinates" (or a dynamical reference frame) which depend only on the spatial metric $g_{ab}$ and the conjugate momenta $p^{cd}$, allow for an alternative approach to canonical quantization of gravity. In this paper we present the tensorial solution of the set of Weyl scalars in terms of canonical phase-space variables.