ar X iv : g r - qc / 0 50 10 24 v 1 8 J an 2 00 5 Conformal geometrodynamics

The standard geometrodynamics is reformulated into a theory of conformal geometrodynamics by extending the ADM phase space for canonical general relativity to a phase space consisting of York’s exterior mean curvature time, conformal 3-metric and their momenta. Accordingly, an additional constraint is introduced, called the conformal constraint. In terms of the new canonical variables, a diffeomorphism constraints 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 another expression 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. A formal 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. 1 Overview of the conventional geometrodynamics We start by recapitulating the essence of the standard Dirac-ADM paradigm of canonical gravity [1, 2]. The spacetime is assumed to be globally hyperbolic with compact spatial sectors. Under a 3 + 1 split using arbitrary space-time coordinates (xa, t) the metric is expressed as ds = −Ndt + gab(dx +Ndt)(dx +N dt) (1) in terms of the spatial metric gab, lapse function N and shift vector N a, where a, b, · · · = 1, 2, 3 denote space coordinate indices. These indices may be lowered and raised using gab and its inverse ∗Previous address: Department of Physics, Lancaster University, Lancaster LA1 4YB, England Permanent address from 1 January 2005 As Visiting Fellow