Theoretical Aspects of Direct Waves in Kerr Black Holes: Pole-Splitting Method for Ringdown Analysis

Abstract

We formulate the theoretical aspects of direct waves (DWs) in the case of extreme-mass merger. A DW is a source-driven waveform characterized by a complex frequency ${\omega }_{\mathrm{G}}$, which reflects the orbital motion of the particle in the vicinity of the black hole, including a part of the orbit inside the ergoregion: its real part is governed by frame dragging and its imaginary part by the redshift of the source. Using the Green's function technique, we derive the source-driven frequency ${\omega }_{\mathrm{G}}$, describe its screening by the potential barrier, and discuss its relation to dynamically excited quasinormal modes (QNMs). We then introduce a pole-splitting method that divides the full waveform into a QNM-pole sector and a non-QNM sector. Unlike QNM filtering, which multiplies the waveform spectrum by a filter function and thereby deforms it through a frequency-dependent time shift (i.e., group delay), our pole-splitting method merely divides the transfer function into pole and non-pole parts, separating the full waveform. Simulating a quasi-circular plunge into a Kerr black hole with medium and rapid spins, we find that the frequency and decay rate of the non-pole sector in the dominant mode, $\ell =m=2$, evolve consistently with ${\omega }_{\mathrm{G}}$-or with its screened counterpart ${\omega }_{\mathrm{screen}}$-establishing the DW as a probe of the ergoregion and of the redshift effect around a black hole.

17 pages, 8 figures. Sec. II F and all related discussion have been removed because the analysis in that section was incorrect. The main results and conclusions are unaffected