Hamiltonian Gravity and Noncommutative Geometry

TL;DR

This paper develops a framework combining Hamiltonian gravity with noncommutative geometry, expressing the gravitational action in terms of a foliated spacetime with noncommutative spatial geometry, simplifying the constraint structure.

Contribution

It introduces a novel approach to formulating Hamiltonian gravity using time-dependent noncommutative geometry, eliminating the need for an extraneous shift vector.

Findings

Vector constraint derived without shift vector

Expressed gravitational action in noncommutative geometric variables

Provides a new perspective on foliated spacetime in quantum gravity

Abstract

A version of foliated spacetime is constructed in which the spatial geometry is described as a time dependent noncommutative geometry. The ADM version of the gravitational action is expressed in terms of these variables. It is shown that the vector constraint is obtained without the need for an extraneous shift vector in the action.