Relating Noncommutative SO(2,3)☆ Gravity to the Lorentz-Violating Standard-Model Extension

We consider a model of noncommutative gravity that is based on a spacetime with broken local SO(2,3)(star) symmetry. We show that the torsion-free version of this model is contained within the framework of the Lorentz-violating Standard-Model Extension (SME). We analyze in detail the relation between the torsion-free, quadratic limits of the broken SO(2,3)(*) model and the Standard-Model Extension. As part of the analysis, we construct the relevant geometric quantities to quadratic order in the metric perturbation around a flat background.