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We carry out ADM splitting in the Lagrangian formulation and establish aprocedure in which (almost) all of the unphysical components of the metric areremoved by using the 4D diffeomorphism and the measure-zero 3D symmetry. Theprocedure introduces a constraint that corresponds to the Hamiltonianconstraint of the Hamiltonian formulation, and its solution implies that the 4Ddynamics admits an effective description through 3D hypersurface physics. Asfar as we can see, our procedure implies potential renormalizability of {theADM formulation of} 4D Einstein gravity for which a complete gauge-fixing inthe ADM formulation and hypersurface foliation of geometry are the keyelements. If true, this implies that the alleged unrenormalizability of 4DEinstein gravity may be due to the presence of the unphysical fields. Theprocedure can straightforwardly be applied to quantization around a flatbackground; the Schwarzschild case seems more subtle. We discuss a potentiallimitation of the procedure when applying it to explicit time-dependentbackgrounds.

Hypersurface foliation approach to renormalization of ADM formulation of gravity