Apparent horizons in the quasi-spherical Szekeres models

TL;DR

This paper explores different definitions of apparent horizons in quasispherical Szekeres models, introducing concepts like absolute apparent horizon and light collapse region, and examines their properties and relations through explicit examples.

Contribution

It extends the understanding of apparent horizons from Lemaître--Tolman models to Szekeres models, proposing new horizon concepts and analyzing their properties.

Findings

AH and AAH coincide in L--T models.

In Szekeres models, AH differs from AAH but they are related.

An observer inside the AH may not be inside the AAH for some time.

Abstract

The notion of an apparent horizon (AH) in a collapsing object can be carried over from the Lema\^{\i}tre -- Tolman (L--T) to the quasispherical Szekeres models in three ways: 1. Literally by the definition -- the AH is the boundary of the region, in which every bundle of null geodesics has negative expansion scalar. 2. As the locus, at which null lines that are as nearly radial as possible are turned toward decreasing areal radius $R$. These lines are in general nongeodesic. The name "absolute apparent horizon" (AAH) is proposed for this locus. 3. As the boundary of a region, where null \textit{geodesics} are turned toward decreasing $R$. The name "light collapse region" (LCR) is proposed for this region (which is 3-dimensional in every space of constant $t$); its boundary coincides with the AAH. The AH and AAH coincide in the L--T models. In the quasispherical Szekeres models, the AH is different from (but not disjoint with) the AAH. Properties of the AAH and LCR are investigated, and the relations between the AAH and the AH are illustrated with diagrams using an explicit example of a Szekeres metric. It turns out that an observer who is already within the AH is, for some time, not yet within the AAH. Nevertheless, no light signal can be sent through the AH from the inside. The analogue of the AAH for massive particles is also considered.