General Relativity as Classical Limit of Evolutionary Quantum Gravity

We analyze the dynamics of the gravitational field when the covariance is restricted to a synchronous gauge. In the spirit of the Noether theorem, we determine the conservation law associated to the Lagrangian invariance and we outline that a non-vanishing behavior of the Hamiltonian comes out. We then interpret such resulting non-zero ``energy'' of the gravitational field in terms of a dust fluid. This new matter contribution is co-moving to the slicing and it accounts for the ``materialization'' of a synchronous reference from the corresponding gauge condition. Further, we analyze the quantum dynamics of a generic inhomogeneous Universe as described by this evolutionary scheme, asymptotically to the singularity. We show how the phenomenology of such a model overlaps the corresponding Wheeler-DeWitt picture. Finally, we study the possibility of a Schr\"odinger dynamics of the gravitational field as a consequence of the correspondence inferred between the ensemble dynamics of stochastic systems and the WKB limit of their quantum evolution. We demonstrate that the time dependence of the ensemble distribution is associated with the first order correction in $\hbar$ to the WKB expansion of the energy spectrum.