Ringdown modulation of acceleration radiation in the Schwarzschild background

Abstract

We derive an analytic first-order description of how Schwarzschild ringdown affects a detector-based detailed-balance diagnostic in a near-horizon, single-mode setting. A freely falling two-level system couples to a cavity-filtered outgoing mode of fixed asymptotic frequency, whose static Schwarzschild response gives geometric photon statistics and a detailed-balance ratio governed by the surface gravity. We perturb this baseline by an even-parity, axisymmetric quadrupolar quasinormal mode and work in ingoing Eddington-Finkelstein coordinates, regular at the future horizon. The perturbation shifts the outgoing eikonal through the double-null contraction of the metric perturbation along the outgoing congruence. After fixing the residual endpoint phase calibration on the cavity worldtube, this redshift-map deformation induces a first-order decaying-oscillatory correction to the detector detailed-balance exponent at the quasinormal frequency. We express the geometric response through a closed boundary formula at the sampling radius and state the adiabatic, narrowband, and linear-response conditions under which the result applies. Detector details, including the gap, switching, and wavepacket profile, enter only through a smooth prefactor, while the ringdown dependence is carried by the quasinormal frequency and calibrated response coefficient. The modulation vanishes in the zero-amplitude, late-time, and stationary quadrupolar limits. The result is not a modification of the Hawking temperature, global Hawking flux, or dynamical horizon thermality, but a controlled correction to an operational detector/cavity detailed-balance observable.

33 pages, 3 figures