Naked singularities and the hoop conjecture: An analytic exploration.

Abstract

The formation of a naked singularity during the collapse of a finite object would pose a serious difficulty for the theory of general relativity. The hoop conjecture suggests that this possibility will never happen provided the object is sufficiently compact (\ensuremath{\lesssim}M) in all of its spatial dimensions. But what about the collapse of a long, nonrotating, prolate object to a thin spindle? Such collapse leads to a strong singularity in Newtonian gravitation. Here we construct an analytic sequence of momentarily static, prolate, and oblate dust spheroids in full general relativity. In the limit of large eccentricity the solutions all become singular. However, when the spheroids are sufficiently large there are no apparent horizons, lending support to the hoop conjecture. These solutions thus suggest that naked singularities with matter could possibly form in asymptotically flat spacetimes.