  Formulating a dust filled spherically symmetric metric utilizing the 3+1
formalism for general relativity, we show that the metric coefficients are
completely determined by the matter distribution throughout the spacetime.
Furthermore, the metric describes both inhomogeneous dust regions and also
vacuum regions in a single coordinate patch, thus alleviating the need for
complicated matching schemes at the interfaces. In this way, the system is
established as an initial-boundary value problem, which has many benefits for
its numerical evolution. We show the dust part of the metric is equivalent to
the class of Lemaitre-Tolman-Bondi (LTB) metrics under a coordinate
transformation. In this coordinate system, shell crossing singularities (SCS)
are exhibited as fluid shock waves, and we therefore discuss possibilities for
the dynamical extension of shell crossings through the initial point of
formation by borrowing methods from classical fluid dynamics. This paper fills
a void in the present literature associated with these collapse models by fully
developing the formalism in great detail. Furthermore, the applications provide
examples of the benefits of the present model.
