Extrinsic curvature as a reference fluid in canonical gravity

An introduction of embedding variables as physical fields locked to the metric by coordinate conditions helps to turn the quantum constraints into a many-fingered time Schroedinger equation. An attempt is made to generalize this process from noncanonical (Gaussian and harmonic) coordinate conditions to a canonical coordinate condition (the constant mean extrinsic curvature slicing). The dynamics of a scalar field {ital T}({ital X}) is described by a Lagrangian {ital L}{sup {ital T}} whose field equations imply that the value of {ital T} at {ital X} is the mean extrinsic curvature {ital K} of a {ital K}=const hypersurface passing through {ital X}. By adding {ital L}{sup {ital T}} to the Hilbert Lagrangian {ital L}{sup {ital G}}, the extrinsic time field'' {ital T}({ital X}) is coupled to gravity. Its energy-momentum tensor has the structure of a perfect fluid (the reference fluid) which satisfies weak energy conditions.