Noncommutative gravity

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 θμν. In this case, the spacetime symmetry is restricted to volume-preserving diffeomorphisms which also preserve θμν. Another possibility is an extension of the Kontsevich product to curved spacetime. In both cases, the noncommutative product is nonassociative. We find the order θ2 noncommutative correction to the Newtonian potential in the case of a covariantly constant θμν. It is still of the form 1/r plus an angle-dependent piece. The coupling to matter gives rise to a propagator which is θ dependent. © 2006 IOP Publishing Ltd.