  In our previous work we have constructed a model of noncommutative (NC)
gravity based on $SO(2,3)_\star$ gauge symmetry. In this paper we extend the
model by adding matter fields: fermions and a $U(1)$ gauge field. Using the
enveloping algebra approach and the Seiberg-Witten map we construct actions for
these matter fields and expand the actions up to first order in the
noncommutativity (deformation) parameter. Unlike in the case of pure NC
gravity, first non-vanishing NC corrections are linear in the noncommutativity
parameter. In the flat space-time limit we obtain a non-standard NC
Electrodynamics. Finally, we discuss effects of noncommutativity on
relativistic Landau levels of an electron in a constant background magnetic
field and in addition we calculate the induced NC magnetic dipole moment of the
electron.
