  We introduce a method to compute particle detector transition probability in spacetime regions of general curved spacetimes provided that the curvature is not above a maximum threshold. In particular we use this method to compare the response of two detectors, one in a spherically symmetric gravitational field and the other one in Rindler spacetime to compare the Unruh and Hawking effects: We study the vacuum response of a detector freely falling through a stationary cavity in a Schwarzschild background as compared with the response of an equivalently accelerated detector traveling through an inertial cavity in the absence of curvature. We find that as we set the cavity in further radiuses from the black hole, the thermal radiation measured by the detector approaches the quantity recorded by the detector in Rindler background showing in which way and at what scales the equivalent principle is recovered in the Hawking-Unruh effect. I.e. when the Hawking effect in a Schwarzschild background becomes equivalent to the Unruh effect in Rindler spacetime. 
Cavities in curved spacetimes: the response of particle detectors