Noncommutative Gravity

TL;DR

This paper explores extending noncommutative geometry to curved spacetime, analyzing the effects on gravitational potentials and propagators, with a focus on nonassociative products and symmetry restrictions.

Contribution

It introduces two methods for extending noncommutativity to curved spacetime and calculates the second-order correction to the Newtonian potential.

Findings

Noncommutative correction to Newtonian potential is of order  with angle dependence.

Spacetime symmetry reduces to volume-preserving diffeomorphisms.

Propagator becomes -dependent due to noncommutativity.

Abstract

We consider simple extensions of noncommutativity from flat to curved spacetime. One possibility is to have a generalization of the Moyal product with a covariantly constant noncommutative tensor $\theta^{\mu\nu}$. In this case the spacetime symmetry is restricted to volume preserving diffeomorphisms which also preserve $\theta^{\mu\nu}$. Another possibility is an extension of the Kontsevich product to curved spacetime. In both cases the noncommutative product is nonassociative. We find the the order $\theta^2$ noncommutative correction to the Newtonian potential in the case of a covariantly constant $\theta^{\mu\nu}$. It is still of the form $1/r$ plus an angle dependent piece. The coupling to matter gives rise to a propagator which is $\theta$ dependent.