High-Speed Collapse of a Hollow Sphere of Type I Matter

TL;DR

This paper investigates the collapse dynamics of a hollow sphere of matter with high initial velocity, revealing the critical role of pressure in singularity formation and the conditions under which naked singularities can form.

Contribution

It provides a perturbative analysis of high-speed spherical collapse with specific equations of state, highlighting pressure effects on collapse outcomes and singularity visibility.

Findings

High initial velocity is strongly decelerated by pressure effects.

Negative tangential pressure can lead to naked singularity formation.

Collapse with certain equations of state resembles Vaidya spacetime.

Abstract

In this paper, we study the dynamics of a hollow spherical matter collapsing with very large initial velocity. The spacetime is initially very similar to the Vaidya solution, and the deviations from this background are treated perturbatively. The equations of state for radial pressure $p_{\rm R}=k\rho$ and tangential one $p_{\rm T}=w\rho$ with constant $k$ and $w$ are assumed. We find for the case of equations of state $k< 1$ and $0<w\leq1$ that the initial velocity, which is nearly the speed of light, is strongly decelerated. This result implies that the pressure is essential to the property of singularity formation in gravitational collapse even for initially nearly light-speed collapse. By contrast, in cases with the negative tangential pressure, the present result implies that the central naked singularity similar to that of the Vaidya spacetime can be formed, even though the radial pressure is positive, and the weak, strong and dominant energy conditions hold. Especially, in the case of $w<-(1-k)/4$, the high-speed collapse will produce the spacetime structure very similar to that of the Vaidya spacetime.