  We establish the connection between the standard ADM 3+1 treatment of matter
with its characteristic equivalent, in the context of spherical symmetry. The
flux-conservative rendition of the fluid equations are obtained. Considering
adiabatic distributions of perfect fluid, we evolve the system using the
so-called post-quasi-static approximation in radiation coordinates. We obtain
an adiabatic matter evolution in the quasi-static regime or slow motion, which
is not shear-free nor geodesic.
