Extrinsic curvature as a reference fluid in canonical gravity.

Abstract

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 Schr\"odinger 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 $T(X)$ is described by a Lagrangian ${L}^{T}$ whose field equations imply that the value of $T$ at $X$ is the mean extrinsic curvature $K$ of a $K =\mathrm{const}$ hypersurface passing through $X$. By adding ${L}^{T}$ to the Hilbert Lagrangian ${L}^{G}$, the "extrinsic time field" $T(X)$ is coupled to gravity. Its energy-momentum tensor has the structure of a perfect fluid (the reference fluid) which satisfies weak energy conditions. The canonical analysis of the total action is complicated by the changing rank of the Hessian. When a hypersurface $X(x)$ is transverse to the $K =\mathrm{const}$ foliation, this rank is higher, and one obtains only the standard super-Hamiltonian and supermomentum constraints. At stationary points of $T(x)$, the hypersurface becomes tangent to a $K =\mathrm{const}$ leaf, the rank of the Hessian gets lower, and more constraints arise. At transverse points, the super-Hamiltonian constraint can be solved with respect to the momentum $P(x)$ which is canonically conjugate to the time function $T(x)$. In this form, the constraint leads to a functional Schr\"odinger equation. As one approaches a stationary point of $T(x)$, additional constraints arise. They ultimately invalidate the functional Schr\"odinger equation. If the stationary points fill a region, some constraints become second class and must be eliminated before quantization. On the $K =\mathrm{const}$ foliation itself, such elimination leads to a reduced system of ${3\ensuremath{\infty}}^{3}+1$ first-class constraints: the ${3\ensuremath{\infty}}^{3}$ supermomentum constraints, and a single Hamiltonian constraint describing evolution along the $K =\mathrm{const}$ foliation. The constraint quantization yields an ordinary Schr\"odinger equation for the conformal three-geometry. For a critical value of the coupling, the second-class constraints further proliferate, and the system becomes either inconsistent or dynamically frozen.