An Analytic Description of Semi-Classical Black-Hole Geometry

TL;DR

This paper analytically models the spacetime geometry during black hole formation and evaporation, revealing that the collapsing shell remains extremely close to the apparent horizon, with implications for semi-classical gravity.

Contribution

It provides an exact analytic solution to the semi-classical Einstein equations for a collapsing shell, elucidating black hole dynamics and horizon proximity in quantum regimes.

Findings

Proper distance between shell and horizon remains Planck-scale after horizon crossing.

Shell position effectively coincides with the apparent horizon in semi-classical gravity.

Analytic description covers black hole formation to evaporation stages.

Abstract

We study analytically the spacetime geometry of the black-hole formation and evaporation. As a simplest model of the collapse, we consider a spherical thin shell, and take the back-reaction from the negative energy of the quantum vacuum state. For definiteness, we will focus on quantum effects of s-waves. We obtain an analytic solution of the semi-classical Einstein equation for this model, that provides an overall description of the black hole geometry form the formation to evaporation. As an application of this result, we find its interesting implication that, after the collapsing shell enters the apparent horizon, the proper distance between the shell and the horizon remains as small as the Planck length even when the difference in their areal radii is of the same order as the Schwarzschild radius. The position of the shell would be regarded as the same place to the apparent horizon in the semi-classical regime of gravity.