Noncommutative gravity

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 θ μν . In this case, the spacetime symmetry is restricted to volume-preserving diffeomorphisms which also preserve θ μν . Another possibility is an extension of the Kontsevich product to curved spacetime. In both cases, the noncommutative product is nonassociative. We find the order θ 2 noncommutative correction to the Newtonian potential in the case of a covariantly constant θ μν . 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.