  Within the framework of noncommutative (NC) deformation of gauge field theory
by twist, we first rederive the NC scalar and gauge field model from our
previous papers and then generalize it to the second order in the
Seiberg-Witten (SW) map. It turns out that SW expansion is finite and that it
ceases at the second order in the deformation parameter, ultimately giving rise
to the equation of motion for the scalar field that is nonperturbative and
exact at the same order. As a further step, we show that the effective metric
put forth and constructed in our previous work satisfies the equations of
Einstein-Maxwell gravity, but only within the first order of deformation and
when the gauge field is fixed by the Coulomb potential of the charged black
hole. Thus obtained NC deformation of the Reissner--Nordstr\"om (RN) metric
appears to have an additional off-diagonal element which scales linearly with a
deformation parameter. We analyze various properties of this metric.
