  In this paper we couple noncommutative (NC) vielbein gravity to scalar
fields. Noncommutativity is encoded in a star product between forms, given by
an abelian twist (a twist with commuting vector fields). A geometric
generalization of the Seiberg-Witten map for abelian twists yields an extended
theory of gravity coupled to scalars, where all fields are ordinary
(commutative) fields. The vectors defining the twist can be related to the
scalar fields and their derivatives, and hence acquire dynamics. Higher
derivative corrections to the classical Einstein-Hilbert and Klein-Gordon
actions are organized in successive powers of the noncommutativity parameter
\theta^{AB}.
