Trapped surfaces in prolate collapse in the Gibbons-Penrose construction

TL;DR

This paper explores the formation and properties of trapped surfaces in models of prolate and cylindrical null dust shell collapse, providing explicit examples and phase diagrams that relate trapped surface presence to physical parameters.

Contribution

It offers new explicit examples of trapped surfaces in prolate collapse and a phase diagram for trapped surface existence in cylindrical shells, advancing understanding of gravitational collapse.

Findings

Trapped surfaces can exist on shells without existing beforehand in the spacetime.

Presence of trapped surfaces depends on parameters like M/L and m/M, with thresholds identified.

Results align with the hoop conjecture in certain limits.

Abstract

We investigate existence and properties of trapped surfaces in two models of collapsing null dust shells within the Gibbons-Penrose construction. In the first model, the shell is initially a prolate spheroid, and the resulting singularity forms at the ends first (relative to a natural time slicing by flat hyperplanes), in analogy with behavior found in certain prolate collapse examples considered by Shapiro and Teukolsky. We give an explicit example in which trapped surfaces are present on the shell, but none exist prior to the last flat slice, thereby explicitly showing that the absence of trapped surfaces on a particular, natural slicing does not imply an absence of trapped surfaces in the spacetime. We then examine a model considered by Barrabes, Israel and Letelier (BIL) of a cylindrical shell of mass M and length L, with hemispherical endcaps of mass m. We obtain a "phase diagram" for the presence of trapped surfaces on the shell with respect to essential parameters $\lambda \equiv M/L$ and $\mu \equiv m/M$. It is found that no trapped surfaces are present on the shell when $\lambda$ or $\mu$ are sufficiently small. (We are able only to search for trapped surfaces lying on the shell itself.) In the limit $\lambda \to 0$, the existence or nonexistence of trapped surfaces lying within the shell is seen to be in remarkably good accord with the hoop conjecture.