The parametric-manifold approach to canonical gravity
The canonical structure of gravitation in general relativity is investigated. The Einstein-Hilbert action is decomposed with respect to a generic congruence of timelike curves. The non-Riemannian geometry of the curves, considered to be points of a 3 — D differentiate manifold, incorporates time as a parameter in the differential structure of the manifold.
The Lagrangian contains the spatial 3-metric gik , its conformal multiplier f and the connection one-form ω as canonical variables. One of the fields, the redshift factor f, emerges as a secondary entity that interacts with a system consisting of the three-metric g and the connection form ω. We investigate the dynamics of the free (g, ω;) system. We find an extension of the phase-space of the system which is computationally viable. We obtain the Hamiltonian density
H =√g(R (3) − ω k D i H ik −λ ik H ik )
where D i is the Riemannian derivative, H ik and λ ik are canonical coordinates in the extended phase space. The primary constraints and most of the equations for the multiplying functions are reasonably simple in the extended phase space.
New gauge fixings are possible here, though our discussion is fully gauge-independent. Our approach reduces to the ADM formulation of canonical gravity when the congruence of time-like curves is chosen hypersurface-orthogonal.
KeywordsField Equation Poisson Bracket Canonical Variable Multiplier Function Timelike Curve
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