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 ${\theta }^{\mu \nu }$. 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.

11 pages, more references added

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