Shell crossings and the Tolman model

We consider the problem of shell crossings and regular maxima in the Tolman model. The necessary and sufficient conditions which guarantee no shell crossings will arise in Tolman Models are derived, and we show explicitly that a Tolman model (in general, with a surface layer) may contain both elliptic and hyperbolic regions without developing any shell crossings and without the hyperbolic regions recollapsing. This finding is contrary to the recent hypothesis of Zel’dovich and Grishchuck. We also show that the properties that distinguish shell crossings from more serious singularities in spherical symmetry are independent of the equation of state. Subject heading: Cosmology I