Relativistic Gravitational Collapse of a Cylindrical Shell of Dust

TL;DR

This paper investigates the gravitational collapse of a cylindrical dust shell, revealing that it forms a naked singularity which can be extended beyond, indicating weak gravity effects and the potential for a complete spacetime.

Contribution

It introduces a boundary condition at the naked singularity allowing a unique extension of spacetime beyond it, demonstrating the singularity's weak nature and the possibility of a complete spacetime.

Findings

Naked singularity forms on the symmetry axis during collapse.

The extended spacetime is complete for almost all geodesics.

The naked singularity is weak, with the metric being $C^{1-}$ at that point.

Abstract

The gravitational collapse of a thick cylindrical shell of dust matter is investigated. It is found that a spacetime singularity forms on the symmetry axis and that it is necessarily naked, i.e., observable in principle. We propose a physically reasonable boundary condition at this naked singularity to construct the solution including its causal future. This boundary condition enables us to construct the unique continuation of spacetime beyond the naked singularity and ensures that the dust shell passes through the naked singularity. When the cylindrical shell leaves its symmetry axis away, the naked singularity disappears, and regularity is recovered. We construct numerical solutions with this feature. This result implies that the gravity produced by a thick cylindrical shell of dust is too weak to bind the shell even if it engenders the formation of a curvature singularity which is so strong as to satisfy the limiting focusing condition. For this reason, this naked singularity is very weak in the extended spacetime; the metric tensor is $C^{1-}$ even at the naked singularity, and the extended spacetime is complete for almost all geodesics. This feature is also seen for singular hypersurfaces. Such an extended spacetime can be regarded as phenomenological in the sense that it is valid if the relevant microphysics length scale is sufficiently small compared to the scale of interest.