Onset and decay of the 1+1 Hawking-Unruh effect: what the derivative-coupling detector saw

We study an Unruh-DeWitt particle detector that is coupled to the proper time derivative of a real scalar field in 1+1 spacetime dimensions. Working within first-order perturbation theory, we cast the transition probability into a regulator-free form, and we show that the transition rate remains well defined in the limit of sharp switching. The detector is insensitive to the infrared ambiguity when the field becomes massless, and we verify explicitly the regularity of the massless limit for a static detector in Minkowski half-space. We then consider a massless field for two scenarios of interest for the Hawking-Unruh effect: an inertial detector in Minkowski spacetime with an exponentially receding mirror, and an inertial detector in $(1+1)$-dimensional Schwarzschild spacetime, in the Hartle-Hawking-Israel and Unruh vacua. In the mirror spacetime the transition rate traces the onset of an energy flux from the mirror, with the expected Planckian late time asymptotics. In the Schwarzschild spacetime the transition rate of a detector that falls in from infinity gradually loses thermality, diverging near the singularity proportionally to $r^{-3/2}$.