Why ghosts don't touch: a tale of two adventurers falling one after another into a black hole

TL;DR

This paper explores the behavior of two observers falling into a Schwarzschild black hole using Kruskal-Szekeres coordinates, clarifying misconceptions about their interactions near the horizon and demonstrating educational value for relativity courses.

Contribution

It extends previous discussions by analyzing the observers' experiences beyond the event horizon using nonsingular coordinates, highlighting the non-physical nature of 'touching ghosts' in black hole physics.

Findings

Observers do not physically touch or interact at the horizon.

Distance and time concepts near the horizon are clarified.

Redshift effects are demonstrated between infallers.

Abstract

The case for the utility of Kruskal-Szekeres coordinates in classroom made by Augousti et al. in this journal (Eur. J. Phys. 33:1--11, 2012) is strengthened by extending their discussion beyond the event horizon of the black hole. Observations made by two adventurers following one another into a Schwarzschild black hole are examined in terms of these nonsingular coordinates. Two scenarios are considered, the first corresponding to one observer following the other closely, the second to a significant distance between the two of them, precluding the existence of a common inertial system. In particular, the concepts of distance and temporal separation near the horizon and the redshift of the first infaller's image as seen by the second are investigated. The results show that the notion of "touching ghosts" does not correspond to the local physics of two observers falling into a black hole. The story line is interesting enough and the mathematical details are sufficiently simple to use the example in a general relativity course, even at the undergraduate level.