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Action for the Dirac spinor field coupled to gravity on noncommutative (NC)Moyal-Weyl space-time is obtained without prior knowledge of the metric tensor.We emphasise gauge origins of gravity (i.e. metric structure) and itsinteraction with fermions by demonstrating that a classical action invariantunder $SO(2,3)$ gauge transformations can be exactly reduced to the Diracaction in curved space-time after breaking the original symmetry down to thelocal Lorentz $SO(1,3)$ symmetry. The commutative, $SO(2,3)$ invariant actioncan be straightforwardly deformed via Moyal-Weyl $\star$-product to its NC$SO(2,3)_\star$ invariant version which can be expanded perturbatively in thepowers of the deformation parameter using the Seiberg-Witten map. Thegravity-matter couplings in the expansion arise as an effect of the gaugesymmetry breaking. We calculate in detail the first order NC correction to theclassical Dirac action in curved space-time and show that it does not vanish.This significant feature of the presented model enables us to potentiallyobserve the NC effects already at the lowest perturbative order. Moreover, NCeffects are apparent even in the flat space-time limit. We analyse NCmodification of the Dirac equation, Feynman propagator and dispersion relationfor electrons in Minkowski space-time.

Dirac field and gravity in NC $SO(2,3)_\star$ model