Noncommutative gravity at second order via Seiberg-Witten map

We develop a general strategy to express noncommutative actions in terms of commutative ones by using a recently developed geometric generalization of the Seiberg-Witten map between noncommutative and commutative fields. We apply this general scheme to the noncommutative vierbein gravity action and provide a Seiberg-Witten differential equation for the action itself as well as a recursive solution at all orders in the noncommutativity parameter θ. We thus express the action at order θn⁺2 in terms of noncommutative fields of order at most θn⁺1 and, iterating the procedure, in terms of noncommutative fields of order at most θn. This in particular provides the explicit expression of the action at order θ2 in terms of the usual commutative spin connection and vierbein fields. The result is an extended gravity action on commutative spacetime that is manifestly invariant under local Lorentz rotations and general coordinate transformations.