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

Over the horizon: distinguishing the Schwarzschild spacetime and the $\mathbb{RP}^3$ spacetime using an Unruh-DeWitt detector