Non-commutative Einstein equations and Seiberg-Witten map

The Seiberg--Witten map is a powerful tool in non-commutative field theory, and it has been recently obtained in the literature for gravity itself, to first order in non-commutativity. This paper, relying upon the pure-gravity form of the action functional considered in Ref. 2, studies the expansion to first order of the non-commutative Einstein equations, and whether the Seiberg--Witten map can lead to a solution of such equations when the underlying classical geometry is Schwarzschild.
Comment: 6 and 1/2 pages, based on talk prepared for the Friedmann Seminar, May-June 2011. In the final version, the presentation has been improved, including a better notation