Conformal Gravity on Noncommutative Spacetime

Conformal gravity on noncommutative spacetime is considered in this paper. The presupposed gravity action consists of the Brans-Dicke gravity action with a special prefactor of the term, where the Ricci scalar couples to the scalar field, to maintain local conformal invariance and the Weyl gravity action. The commutation relations between the coordinates defining the noncommutative geometry are assumed to be of canonical shape. Based on the moyal star product, products of fields depending on the noncommutative coordinates are replaced by generalized expressions containing the usual fields and depending on the noncommutativity parameter. To maintain invariance under local conformal transformations with the gauge parameter depending on noncommutative coordinates, the fields have to be mapped to generalized fields by using Seiberg-Witten maps. According to the moyal star product and the thus induced Seiberg-Witten maps the generalized conformal gravity action is formulated and the corresponding field equations are derived.