Deep in the knotted black hole

We consider the transition rate of a freely falling Unruh-DeWitt detector, coupled linearly to a massless scalar quantum field prepared in the Hartle-Hawking-Israel state, as a probe of the interior of a black hole. Specifically, we consider the transition rate of a detector in the spinless Ba\~nados-Teitelboim-Zanelli (BTZ) black hole as it freely falls toward and across the horizon and compare it to the corresponding situation for an $\mathbb{R}\text{P}^{2}$ geon. Both the BTZ black hole and its geon counterpart are quotients of $\text{AdS}_3$ spacetime that are identical exterior to the horizon but have different interior topologies. We find outside the horizon that the rates are qualitatively similar, but with the amplitude in the geon spacetime larger than in the BTZ case. Once the detector crosses the horizon, there are notable distinctions characterized by different discontinuities in the temporal derivative of the response rate. These discontinuities can appear outside the horizon if the detector is switched on at a sufficiently early time, within the past white hole horizon. In general, the detector can act as an `early warning system' that both spots the black hole horizon and discerns its interior topology.