We use the Seiberg-Witten map (SW map) to expand noncommutative gravity
coupled to fermions in terms of ordinary commuting fields. The action is
invariant under general coordinate transformations and local Lorentz rotations,
and has the same degrees of freedom as the commutative gravity action. The
expansion is given up to second order in the noncommutativity parameter
{\theta}. A geometric reformulation and generalization of the SW map is
presented that applies to any abelian twist. Compatibility of the map with
hermiticity and charge conjugation conditions is proven. The action is shown to
be real and invariant under charge conjugation at all orders in {\theta}. This
implies the bosonic part of the action to be even in {\theta}, while the
fermionic part is even in {\theta} for Majorana fermions.Comment: 27 pages, LaTeX. Revised version with proof of charge conjugation
symmetry of the NC action and its parity under theta --> - theta (see new
sect. 2.6, sect. 6 and app. B). References added. arXiv admin note:
substantial text overlap with arXiv:0902.381
