Noncommutative gravity and the relevance of the θ-constant deformation

The breaking of diffeomorphism invariance in the Moyal-Weyl (θ-constant) noncommutative (NC) space-time is a well-known and a long-standing problem. It makes the construction of NC gravity models and interpretation of their results very difficult. In order to solve this problem in this letter we construct a NC gravity action based on the NC gauge group and the Seiberg-Witten expansion. The NC equations of motion show that the noncommutativity plays the role of a source for the curvature and/or torsion. Finally, we calculate the NC corrections to the Minkowski space-time and show that in the presence of noncommutativity the Minkowski space-time becomes curved, but remains torsion-free. More importantly, we show that the coordinate system we are using is given by the Fermi normal coordinates; the NC deformation is constant in this particular reference system. The breaking of diffeomorphism invariance is understood as a consequence of working in a preferred reference system. In an arbitrary reference system, the NC deformation is obtained by an appropriate coordinate transformation.