  We show that a particle detector can distinguish the $\mathbb{RP}^3$ geon
from the Schwarzschild black hole, even though they differ only by a
topological identification beyond the event horizon. This shows that the
detector can read out information about the non-local structure even when
separated from the non-locality by an event horizon. Our analysis of the
dependence of the transition on the detector gap is novel, and in principle
presents an interesting observational signal.
