Static, stationary, and inertial Unruh-DeWitt detectors on the BTZ black hole

We examine an Unruh-DeWitt particle detector coupled to a scalar field in three-dimensional curved spacetime. We first obtain a regulator-free expression for the transition probability in an arbitrary Hadamard state, working within first-order perturbation theory and assuming smooth switching, and we show that both the transition probability and the instantaneous transition rate remain well defined in the sharp switching limit. We then analyze a detector coupled to a massless conformally coupled field in the Hartle-Hawking vacua on the Banados-Teitelboim-Zanelli black hole, under both transparent and reflective boundary conditions at the infinity. A selection of stationary and freely falling detector trajectories are examined, including the corotating trajectories, for which the response is shown to be thermal. Analytic results in a number of asymptotic regimes, including those of large and small mass, are complemented by numerical results in the interpolating regimes. The boundary condition at infinity is seen to have a significant effect on the transition rate.