  Starting from a realistic One-Boson-Exchange model of the nucleon nucleon
interaction the relativistic mean field for nucleons is determined within the
Dirac Brueckner Hartree Fock approach for finite nuclei. The matrix elements of
the axial charge operator evaluated for the solutions of the Dirac equation
with this selfenergy are investigated. These matrix elements are enhanced with
respect to the equivalent non relativistic ones obtained from the solutions of
the Schr\"odinger equation with the non relativistic equivalent potential. The
present results confirm at a qualitative level the results for the axial charge
renormalization obtained with perturbative approaches. However, the results
obtained differ in size from those of the perturbative approach and are nucleus
and state dependent.
