Initial data and spherical dust collapse

By considering families of radial null geodesics, we study the subsets of initial data that lead to naked singularities and black holes in inhomogeneous spherical dust collapse. We introduce the notion of central homogeneity for spherical dust collapse and prove that, for the occurrence of naked singularities, the initial data set must in general be centrally homogeneous. Even though mathematically this indicates that naked singularities are in general unstable, we show that centrally inhomogeneous perturbations in the initial data are not physically reasonable. This provides an example of the fact that instability in this context deduced with respect to general perturbations can become stabilized once the class of perturbations is restricted to be physical. This is a potentially important point to bear in mind in the general debates regarding the generic presence of naked singularities in gravitational collapse and in more general debates concerning the questions of genericity in general relativity.