Hypersurface foliation approach to renormalization of ADM formulation of gravity

Abstract

We carry out ADM splitting in the Lagrangian formulation and establish a procedure in which (almost) all of the unphysical components of the metric are removed by using the 4D diffeomorphism and the measure-zero 3D symmetry. The procedure introduces a constraint that corresponds to the Hamiltonian constraint of the Hamiltonian formulation, and its solution implies that the 4D dynamics admits an effective description through 3D hypersurface physics. As far as we can see, our procedure implies potential renormalizability of {the ADM formulation of} 4D Einstein gravity for which a complete gauge-fixing in the ADM formulation and hypersurface foliation of geometry are the key elements. If true, this implies that the alleged unrenormalizability of 4D Einstein gravity may be due to the presence of the unphysical fields. The procedure can straightforwardly be applied to quantization around a flat background; the Schwarzschild case seems more subtle. We discuss a potential limitation of the procedure when applying it to explicit time-dependent backgrounds.

29 pages, 3 figures, expanded for clarity, refs added, the version to appear in EPJC

References in corpus (6)

• 3+1 Formalism and Bases of Numerical Relativity
• Lagrangian constraints and renormalization of 4D gravity
• Foliation, jet bundle and quantization of Einstein gravity
• Indication for unsmooth horizon induced by quantum gravity interaction
• Quantization of gravity through hypersurface foliation
• Reduction of BTZ spacetime to hypersurfaces of foliation

Cited by in corpus (18)

• Holographic quantization of gravity in a black hole background
• Quantum-induced trans-Planckian energy near horizon
• Foliation, jet bundle and quantization of Einstein gravity
• One-loop renormalization of a gravity-scalar system
• Quantization of gravity through hypersurface foliation
• Quantum-corrected geometry of horizon vicinity
• Boundary dynamics in gravitational theories
• Foliation-based quantization and black hole information
• Foliation-based approach to quantum gravity and applications to astrophysics
• Reduction of gravity-matter and dS gravity to hypersurface
• 4D covariance of holographic quantization of Einstein gravity
• Quantum gravitational effects on boundary
• Revisit of renormalization of Einstein-Maxwell theory at one-loop
• Quantization of gravity and finite temperature effects
• Quantum-gravitational trans-Planckian energy of a time-dependent black hole
• Non-Cauchy surface foliations and their expected connection to quantum gravity theories
• Coarse-graining of the Einstein-Hilbert Action rewritten by the Fisher information metric
• Truncation of Einstein equations through Gravitational Foliation