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, which are 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.