  We derive the asymptotic properties of the mMKG system (Maxwell coupled with
a massive Klein-Gordon scalar field), in the exterior of the domain of
influence of a compact set. This complements the previous well known results,
restricted to compactly supported initial conditions, based on the so called
hyperboloidal method. That method takes advantage of the commutation properties
of the Maxwell and Klein Gordon with the generators of the Poincar\'e group to
resolve the difficulties caused by the fact that they have, separately,
different asymptotic properties. Though the hyperboloidal method is very robust
and applies well to other related systems it has the well known drawback that
it requires compactly supported data. In this paper we remove this limitation
based on a further extension of the vector-field method adapted to the exterior
region. Our method applies, in particular, to nontrivial charges. The full
problem could then be treated by patching together the new estimates in the
exterior with the hyperboloidal ones in the interior. This purely physical
space approach introduced here maintains the robust properties of the old
method and can thus be applied to other situations such as the coupled Einstein
Klein-Gordon equation.
