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

TL;DR

This paper proves that naked singularities can form from gravitational collapse of Type I matter fields and that such formations are stable under small initial data changes, challenging the notion of their genericity.

Contribution

It demonstrates the existence and stability of naked singularities in Type I matter collapse with arbitrary initial data, extending understanding of gravitational collapse outcomes.

Findings

Naked singularities can form from a wide set of initial conditions.

Such singularities are stable under small perturbations of initial data.

The formation of naked singularities is not a generic outcome in gravitational collapse.

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.