The shadow of a collapsing dark star

TL;DR

This paper investigates how the shadow of a black hole develops over time during gravitational collapse, providing analytical insights for different observer perspectives.

Contribution

It offers the first analytical calculation of the black hole shadow's evolution during collapse, considering both static and infalling observers.

Findings

Shadow size increases over time during collapse

Different observer trajectories affect shadow perception

Provides analytical formulas for shadow evolution

Abstract

The shadow of a black hole is usually calculated, either analytically or numerically, on the assumption that the black hole is eternal, i.e., that it existed for all time. Here we ask the question of how this shadow comes about in the course of time when a black hole is formed by gravitational collapse. To that end we consider a star that is spherically symmetric, dark and non-transparent and we assume that it begins, at some instant of time, to collapse in free fall like a ball of dust. We analytically calculate the dependence on time of the angular radius of the shadow, first for a static observer who is watching the collapse from a certain distance and then for an observer who is falling towards the centre following the collapsing star.