This paper studies a particular piece of gravitational-wave signal that can arise when a small object plunges into a rapidly spinning black hole. The authors call that piece a direct wave (DW). A DW carries a complex frequency set by two near-horizon effects: the real part comes from frame dragging (the black hole’s dragging of nearby motion) and the imaginary part comes from gravitational redshift of the source as it nears the horizon.

The team derives the DW using Green’s-function methods. In frequency space the black-hole response splits into several contributions: residues at quasinormal-mode (QNM) poles (the familiar damped tones of a ringing black hole), a branch-cut integral that gives a late-time power-law tail, and a prompt large-arc piece. By evaluating the source integral carefully they obtain the source-driven complex frequency, which they call ω_G, and they describe how that frequency can be screened by the black hole’s potential barrier associated with the light ring (the region tied to unstable photon orbits).

To separate these pieces in a simulated waveform the authors introduce a pole-splitting method. This method decomposes the transfer function (the linear map from source to signal) into a pole sector and a non-pole sector. The pole sector contains QNMs and may include part of the DW; the non-pole sector is constructed to contain no QNMs and so can reveal non-QNM behavior without the frequency-dependent time shifts introduced by ordinary “QNM filtering.” The paper applies this split to numerical simulations of quasi-circular plunges into Kerr black holes, focusing on the dominant angular mode (ℓ = m = 2).

Their simulations for medium and rapid black-hole spins show that the non-pole sector’s frequency and decay rate follow the predicted source-driven frequency ω_G, or a screened version ω_screen when the source lies inside the potential barrier. This supports the idea that DWs can probe the ergoregion (the zone where matter is forced to co-rotate with the hole) and the redshift near the horizon. The authors also predict that the late-time decay of the DW is governed by a third-order horizon mode, and they discuss technical effects such as pole-skipping at Matsubara modes caused by the transmissivity of the Kerr geometry in the Sasaki–Nakamura formalism.