Stability of Naked Singularity arising in gravitational collapse of Type I matter fields

Abstract

Considering gravitational collapse of Type I matter fields, we prove that, given an arbitrary $C^{2}$- mass function $\textit{M}(r,v)$ and a $C^{1}$- function $h(r,v)$ (through the corresponding $C^{1}$- metric function $\nu(t,r)$), there exist infinitely many choices of energy distribution function $b(r)$ such that the `true' initial data ($\textit{M},h(r,v)$) leads the collapse to the formation of naked singularity. We further prove that the occurrence of such a naked singularity is stable with respect to small changes in the initial data. We remark that though the initial data leading to both black hole and naked singularity form a "big" subset of the true initial data set, their occurrence is not generic. The terms `stability' and `genericity' are appropriately defined following the theory of dynamical systems. The particular case of radial pressure $p_{r}(r)$ has been illustrated in details to get clear picture of how naked singularity is formed and how, it is stable with respect to initial data.