Closed trapping horizons without singularity

TL;DR

This paper investigates models of regular, non-singular black holes formed by collapsing null shells, analyzing trapping horizons and energy conditions, and highlighting fundamental issues with current models related to energy condition violations.

Contribution

It provides a general framework for the existence of singularity-free trapping horizons and explicitly models Hawking radiation effects in such black holes.

Findings

Models necessarily violate the null energy condition in non-compact regions.

Explicit metric form for regular black holes with Hawking radiation.

Identification of flaws in existing models regarding energy conditions.

Abstract

In gravitational collapse leading to black hole formation, trapping horizons typically develop inside the contracting matter. Classically, an ingoing trapping horizon moves towards the centre where it reaches a curvature singularity, while an outgoing horizon moves towards the surface of the star where it becomes an isolated, null horizon. However, strong quantum effects at high curvature close to the centre could modify the classical picture substantially, e.g. by deflecting the ingoing horizon to larger radii, until it eventually reunites with the outgoing horizon. We here analyse some existing models of regular "black holes" of finite lifespan formed out of ingoing null shells collapsing from $\mathscr{I}^-$, after giving general conditions for the existence of (singularity-free) closed trapping horizons. We study the energy-momentum tensor of such models by solving Einstein's equations in reverse and give an explicit form of the metric to model a Hawking radiation reaching $\mathscr{I}^+$. A major flaw of the models aiming at describing the formation of black holes (with a Vaidya limit on $\mathscr{I}^-$) as well as their evaporation is finally exhibited: they necessarily violate the null energy condition up to $\mathscr{I}^-$, i.e. in a non-compact region of spacetime.