Extrinsic curvature as a reference fluid in canonical gravity

Karel V. Kuchař
Phys. Rev. D 45, 4443 – Published 15 June 1992
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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 Schrödinger 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 LT whose field equations imply that the value of T at X is the mean extrinsic curvature K of a K =const hypersurface passing through X. By adding LT to the Hilbert Lagrangian LG, 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 =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 =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 Schrödinger equation. As one approaches a stationary point of T(x), additional constraints arise. They ultimately invalidate the functional Schrödinger equation. If the stationary points fill a region, some constraints become second class and must be eliminated before quantization. On the K =const foliation itself, such elimination leads to a reduced system of 33+1 first-class constraints: the 33 supermomentum constraints, and a single Hamiltonian constraint describing evolution along the K =const foliation. The constraint quantization yields an ordinary Schrödinger 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.

  • Received 16 January 1992

DOI:https://doi.org/10.1103/PhysRevD.45.4443

©1992 American Physical Society

Authors & Affiliations

Karel V. Kuchař*

  • Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland
  • Institute for Theoretical Physics, University of Vienna, Boltzmanngasse 5, A-1090 Vienna, Austria
  • Department of Theoretical Physics, Imperial College, Exhibition Road, London SW7 2BZ, United Kingdom

  • *On sabbatical leave from Department of Physics, University of Utah, Salt Lake City, UT 84112.

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Vol. 45, Iss. 12 — 15 June 1992

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