Birth of a timelike naked singularity

We investigate the causal structure of the Harada-Iguchi-Nakao (HIN)’s exact solution in detail, which describes the dynamical formation of a naked singularity in the collapse of a regular spherical cluster of counterrotating particles. There are three kinds of radial null geodesics in the HIN spacetime. One is the regular null geodesics and the other two are the null geodesics which terminate at the singularity. The central massless singularity is a timelike naked singularity and satisfies the strong curvature condition along the null geodesics except for the instant of singularity formation. The cluster dynamically asymptotes to the singular static Einstein cluster in which the centrifugal force is balanced with gravity. The HIN solution provides an interesting example which demonstrates that collisionless particles invoke a timelike naked singularity.