Noncommutative $SO(2\text{,}3{)}_{\star }$ gravity: Noncommutativity as a source of curvature and torsion

Noncommutative (NC) gravity is constructed on the canonical noncommutative (Moyal-Weyl) spacetime as a noncommutative $SO(2,3{)}_{\star }$ gauge theory. The NC gravity action consists of three different terms: the first term is of Mac-Dowell Mansouri type, while the other two are generalizations of the Einstein-Hilbert action and the cosmological constant term. The expanded NC gravity action is then calculated using the Seiberg-Witten (SW) map and the expansion is done up second order in the deformation parameter. We analyze in details the low energy sector of the full model. We calculate the equations of motion, discuss their general properties and present one solution: the NC correction to Minkowski spacetime. Using this solution, we explain breaking of the diffeomorphism symmetry as a consequence of working in a particular coordinate system given by the Fermi normal coordinates.