Is the shell-focusing singularity of Szekeres space-time visible?

TL;DR

This paper investigates the conditions under which the shell-focusing singularity in Szekeres space-time can be visible, challenging previous assumptions by showing that radial null geodesics do not generally exist, and providing criteria for singularity visibility.

Contribution

It demonstrates that radial null geodesics are generally absent in Szekeres space-time, clarifies the symmetry conditions for their existence, and offers criteria for singularity visibility based on initial data.

Findings

Radial null geodesics do not generally exist in Szekeres space-time.

Existence of a radial geodesic implies axial symmetry; two such geodesics imply spherical symmetry.

Conditions on initial data can lead to visibility or censorship of the singularity.

Abstract

The visibility of the shell-focusing singularity in Szekeres space-time - which represents quasi-spherical dust collapse - has been studied on numerous occasions in the context of the cosmic censorship conjecture. The various results derived have assumed that there exist radial null geodesics in the space-time. We show that such geodesics do not exist in general, and so previous results on the visibility of the singularity are not generally valid. More precisely, we show that the existence of a radial geodesic in Szekeres space-time implies that the space-time is axially symmetric, with the geodesic along the polar direction (i.e. along the axis of symmetry). If there is a second non-parallel radial geodesic, then the space-time is spherically symmetric, and so is a Lema\^{\i}tre-Tolman-Bondi (LTB) space-time. For the case of the polar geodesic in an axially symmetric Szekeres space-time, we give conditions on the free functions (i.e. initial data) of the space-time which lead to visibility of the singularity along this direction. Likewise, we give a sufficient condition for censorship of the singularity. We point out the complications involved in addressing the question of visibility of the singularity both for non-radial null geodesics in the axially symmetric case and in the general (non-axially symmetric) case, and suggest a possible approach.