4D diffeomorphisms in canonical gravity and abelian deformations

TL;DR

This paper analyzes how 4D spacetime diffeomorphisms affect spatial quantities in canonical gravity, revealing alternative choices for deformations and introducing an abelian generator related to true time evolution, with implications for quantization.

Contribution

It introduces a general formalism to study 4D diffeomorphisms in canonical gravity, revealing alternative deformation choices and a natural abelian generator linked to time evolution.

Findings

Identifies conditions for ADM-Dirac constraints

Discovers alternative deformation choices beyond the ADM Hamiltonian

Introduces an abelian generator related to true time evolution

Abstract

A careful study of the induced transformations on spatial quantities due to 4-dimensional spacetime diffeomorphisms in the canonical formulation of general relativity is undertaken. Use of a general formalism, which indicates the role of the embedding variables in a transparent manner, allows us to analyse the effect of 4-dimensional diffeomorphisms more generally than is possible in the standard ADM approach. This analysis clearly indicates the assumptions which are necessary in order to obtain the ADM-Dirac constraints, and furthermore shows that there are choices, other than the ADM hamiltonian constraint, that one can make for the deformations in the ``timelike'' direction. In particular an abelian generator closely related to true time evolution appears very naturally in this framework. This generator, its relation to other abelian scalars discovered recently, and the possibilities it provides for a group theoretic quantisation of gravity are discussed.