Using the Fedosov theory of deformation quantization of an endomorphism bundle we construct several models of pure geometric, deformed vacuum gravity, corresponding to an arbitrary symplectic noncommutativity tensor. Deformations of Einstein-Hilbert and Palatini actions are investigated. Coordinate covariant field equations are derived up to the second order of the deformation parameter. For some models they are solved and explicit corrections to an arbitrary Ricci-flat metric are pointed out. The relation to the theory of the Seiberg-Witten map is also studied and the correspondence to the spacetime noncommutativity described by the Fedosov $*$-product of functions is explained.

• Received 19 November 2010

DOI:https://doi.org/10.1103/PhysRevD.84.065005