Distance between collapsing matter and trapping horizon in evaporating black holes

TL;DR

This paper analyzes the proper distance between collapsing matter and the trapping horizon in evaporating black holes, showing it remains extremely small, on the order of the Planck length, at key stages of evaporation.

Contribution

It provides a quantitative bound on the distance between matter and the apparent horizon in evaporating black holes under realistic energy conditions.

Findings

Distance is bounded by Planck scale at Page time.

Distance remains small during evaporation process.

Proper distance cannot exceed order n^{3/2} times Planck length.

Abstract

Assuming that the vacuum energy-momentum tensor is not exceptionally large, we consider 4D evaporating black holes with spherical symmetry and evaluate the proper distance $\Delta L$ between the time-like apparent horizon and the surface of the collapsing matter after it has entered the apparent horizon. We show that $\Delta L$ can never be larger than $\mathcal{O}(n^{3/2}\ell_p)$ when the black hole is evaporated to $1/n$ of its initial mass, as long as $n \ll a^{2/3}/\ell_p^{2/3}$ (where $a$ is the Schwarzschild radius and $\ell_p$ is the Planck length). For example, the distance between the matter and the apparent horizon must be Planckian at the Page time.