Black holes, Achilles and the original tortoise coordinate

TL;DR

This paper introduces a simple toy model using a Zeno time coordinate to help students understand why objects can fall into black holes despite the 'frozen star' perspective, without complex metrics.

Contribution

It presents a novel educational toy model based on Zeno's Achilles and the tortoise story to clarify black hole horizon concepts for beginners.

Findings

The toy model demonstrates how objects can cross the horizon in finite proper time.

It provides an accessible teaching tool without requiring advanced metric knowledge.

The model clarifies misconceptions about the 'frozen star' phenomenon.

Abstract

The "frozen star" picture of black holes, based on the fact that an outside observer will never see a collapsing sphere shrink sufficiently to form a black hole horizon, was a historical obstacle to the understanding of black holes, and continues to be a stumbling point for students trying to understand horizons. Promoting the discrete steps in Zeno's original story of Achilles and the tortoise to a "Zeno time coordinate" provides a quantitative toy model that allows students to understand why the "frozen star" phenomenon does not mean that objects cannot fall into a black hole. The toy model can be used for teaching about this particular feature of black holes in an introductory setting that does not introduce the Schwarzschild metric and its tortoise coordinates.