# Emergence of Apparent Horizon in Gravitational Collapse

## TL;DR

This paper demonstrates the formation of an apparent horizon during gravitational collapse by constructing a smooth, spacelike marginally outer trapped tube emerging from a point, confirming a conjecture in black hole thermodynamics.

## Contribution

It provides a rigorous construction of an apparent horizon in non-spherical gravitational collapse, extending previous results and confirming Ashtekar's conjecture.

## Key findings

- Existence of a smooth, spacelike apparent horizon emerging from a point.
- Construction of a sequence of MOTS with areas tending to zero.
- Confirmation of a conjecture in black hole thermodynamics.

## Abstract

We solve Einstein vacuum equations in a spacetime region up to the "center" of gravitational collapse. Within this region, we construct a sequence of marginally outer trapped surfaces (MOTS) with areas going to zero. These MOTS form a marginally outer trapped tube (apparent horizon). It emerges from a point and is smooth (except at that point) and spacelike. In the proof we employ a scale critical trapped surface formation criterion established by An and Luk and a new type of quasilinear elliptic equation is studied. The main conclusion of this paper proves a conjecture of Ashtekar on black hole thermodynamics. And the spacetimes constructed here could also be viewed as (non-spherically symmetric) generalizations of the well-known Vaidya spacetime.