Comments on the Non-Commutative Description of Classical Gravity

TL;DR

This paper introduces a one-parameter family of classical gravity models using noncommutative tetrads, which retain standard metric properties and are physically equivalent to Einstein's general relativity.

Contribution

It presents a novel noncommutative tetrad formulation of classical gravity that preserves the usual metric and curvature structures, showing physical equivalence to Einstein's theory.

Findings

Noncommutative tetrads obey exotic commutation relations.

Metric sector remains commutative and matches Einstein's expressions.

The theory is physically equivalent to standard general relativity.

Abstract

We find a one-parameter family of Lagrangian descriptions for classical general relativity in terms of tetrads which are not c-numbers. Rather, they obey exotic commutation relations. These noncommutative properties drop out in the metric sector of the theory, where the Christoffel symbols and the Riemann tensor are ordinary commuting objects and they are given by the usual expression in terms of the metric tensor. Although the metric tensor is not a c-number, we argue that all measurements one can make in this theory are associated with c-numbers, and thus that the common invariant sector of our one--parameter family of deformed gauge theories (for the case of zero torsion) is physically equivalent to Einstein's general relativity.