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Classical background independence is reflected in Lagrangian general relativity through covariance under the full diffeomorphism group. We show how this independence can be maintained in a Hamilton-Jacobi approach that does not accord special privilege to any geometric structure. Intrinsic space-time curvature-based coordinates grant equal status to all geometric backgrounds. They play an essential role as a starting point for inequivalent semiclassical quantizations. The scheme calls into question Wheeler's geometrodynamical approach and the associated Wheeler-DeWitt equation in which 3-metrics are featured geometrical objects. The formalism deals with variables that are manifestly invariant under the full diffeomorphism group. Yet, perhaps paradoxically, the liberty in selecting intrinsic coordinates is precisely as broad as is the original diffeomorphism freedom. We show how various ideas from the past five decades concerning the true degrees of freedom of general relativity can be interpreted in light of this new constrained Hamiltonian description. In particular, we show how the Kuchař multi-fingered time approach can be understood as a means of introducing full four-dimensional diffeomorphism invariants. Every choice of new phase space variables yields new Einstein-Hamilton-Jacobi constraining relations, and corresponding intrinsic Schrödinger equations. We show how to implement this freedom by canonical transformation of the intrinsic Hamiltonian. We also reinterpret and rectify significant work by Dittrich on the construction of 'Dirac observables.' [ABSTRACT FROM AUTHOR]

The conventional group of four-dimensional diffeomorphisms is not realizeable as a canonical transformation group in phase space. Yet there is a larger field-dependent symmetry transformation group which does faithfully reproduce 4-D diffeomorphism symmetries. Some properties of this group were first explored by Bergmann and Komar. More recently the group has been analyzed from the perspective of projectability under the Legendre map. Time translation is not a realizeable symmetry, and is therefore distinct from diffeomorphism-induced symmetries. This issue is explored further in this paper. It is shown that time is not "frozen". Indeed, time-like diffeomorphism invariants must be time-dependent. Intrinsic coordinates of the type proposed by Bergmann and Komar are used to construct invariants. Lapse and shift variables are retained as canonical variables in this approach, and therefore will be subject to quantum fluctuations in an eventual quantum theory. Concepts and constructions are illustrated using the relativistic classical and quantum free particle. In this example concrete time-dependent invariants are displayed and fluctuation in proper time is manifest. [ABSTRACT FROM AUTHOR]