NON-COMMUTATIVE DIFFEOMORPHISM INVARIANT GRAVITY

Seiberg-Witten maps of vierbein and spin-connection in the vector representation of the gauged Lorentz group SO(1, 3) are used to construct a non-commutative version of the Hilbert-Einstein Lagrangian. By imposing the vierbein postulate and by demanding a torsion free geometry the non-commutative Lagrangian can be extended to become diffeomorphism invariant, provided that the non-commmutativity parameters form the components of a covariantly constant bivector and the spacetime has a (2 + 2)-decomposable structure.