Einstein's Equations and a Cosmology with Finite Matter

We discuss various space-time metrics which are compatible with Einstein's equations and a previously suggested cosmology with a finite total mass. In this alternative cosmology the matter density was postulated to be a spatial delta function at the time of the big bang thereafter diffusing outward with constant total mass. This proposal explores a departure from standard assumptions that the big bang occurred everywhere at once or was just one of an infinite number of previous and later transitions.