# Trapped surfaces, energy conditions, and horizon avoidance in spherically-symmetric collapse

## TL;DR

This paper investigates conditions under which black holes can be avoided during spherically-symmetric collapse, analyzing the role of energy conditions and apparent horizons in semiclassical gravity.

## Contribution

It derives a sufficient condition on exterior geometry to prevent black hole formation and explores the implications of energy condition violations near the apparent horizon.

## Key findings

- A sufficient condition for horizon avoidance is identified.
- The metric near the apparent horizon violates the null energy condition.
- Test particles generally cross the horizon in finite time unless conditions prevent collapse.

## Abstract

We consider spherically-symmetric black holes in semiclassical gravity. For a collapsing radiating thin shell we derive a sufficient condition on the exterior geometry that ensures that a black hole is not formed. This is also a sufficient condition for an infalling test particle to avoid the apparent horizon of an existing black hole and approach it only within a certain minimal distance. Taking the presence of a trapped region and its outer boundary --- the apparent horizon--- as the defining feature of black holes, we explore the consequences of their finite time of formation according to a distant observer. Assuming regularity of the apparent horizon we obtain the limiting form of the metric and the energy-momentum tensor in its vicinity that violates the null energy condition (NEC). The metric does not satisfy the sufficient condition for horizon avoidance: a thin shell collapses to form a black hole and test particles (unless too slow) cross into it in finite time. However, there may be difficulty in maintaining the expected range of the NEC violation, and stability against perturbations is not assured.