Conformal geometrodynamics: True degrees of freedom in a truly canonical structure

TL;DR

This paper reformulates canonical general relativity into conformal geometrodynamics by extending the phase space with York's mean curvature, leading to a new set of first-class constraints and a potentially clearer path to quantum gravity.

Contribution

It introduces a conformal extension of the ADM phase space with a new constraint, deriving a consistent set of first-class constraints and exploring implications for quantum gravity.

Findings

The conformal, diffeomorphism, and Hamiltonian constraints form a first-class algebra.

The extended algebra includes the Dirac and Lie algebras as subalgebras.

Potential for York time to facilitate unitary quantum evolution.

Abstract

The standard geometrodynamics is transformed into a theory of conformal geometrodynamics by extending the ADM phase space for canonical general relativity to that consisting of York's mean exterior curvature time, conformal three-metric and their momenta. Accordingly, an additional constraint is introduced, called the conformal constraint. In terms of the new canonical variables, a diffeomorphism constraint is derived from the original momentum constraint. The Hamiltonian constraint then takes a new form. It turns out to be the sum of an expression that previously appeared in the literature and extra terms quadratic in the conformal constraint. The complete set of the conformal, diffeomorphism and Hamiltonian constraints are shown to be of first class through the explicit construction of their Poisson brackets. The extended algebra of constraints has as subalgebras the Dirac algebra for the deformations and Lie algebra for the conformorphism transformations of the spatial hypersurface. This is followed by a discussion of potential implications of the presented theory on the Dirac constraint quantization of general relativity. An argument is made to support the use of the York time in formulating the unitary functional evolution of quantum gravity. Finally, the prospect of future work is briefly outlined.