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

TL;DR

This paper analyzes a derivative-coupling Unruh-DeWitt detector in 1+1 dimensions, demonstrating well-defined transition rates, infrared regularity, and insights into the Hawking-Unruh effect in various spacetime scenarios.

Contribution

It introduces a regulator-free formulation of the transition probability for derivative-coupling detectors and explores their response in different spacetimes relevant to Hawking-Unruh physics.

Findings

Transition rate remains well-defined with sharp switching.

Detector insensitive to infrared ambiguities in massless limit.

Transition rate in Schwarzschild spacetime diverges near the singularity.

Abstract

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 $r^{-3/2}$.