Strong-field Gravitational Wave Lensing in the Kerr Background

TL;DR

This work investigates strong-field gravitational-wave lensing in Kerr spacetimes, extending prior Schwarzschild analyses to spinning black holes. It develops a Kerr wave-scattering framework using the Mano-Suzuki-Takasugi formalism to compute the strong-field scattering factor and the lensed waveform, including spin-induced polarization effects. The authors show that the strong-field scattering factor does not decay at high frequencies and that spin introduces characteristic waveform modulations, with percent-level mismatches possible for favorable geometries; polarization mixing can further enhance observability. These results provide a unified, wave-optics-based approach for interpreting high-precision GW observations and offer a pathway to probe strong-field gravity and black-hole environments with current and future detectors.

Abstract

Gravitational-wave (GW) lensing can encode valuable information about the properties of the intervening lens, but most existing studies remain restricted to the small-deflection, weak-field regime. To bridge this crucial gap, this work presents the first systematic analysis of strong-field, wave-optical GW lensing by a Kerr black hole (BH), extending recent results for non-rotating lens to the astrophysically more relevant case of spinning-lens. Using the Mano-Suzuki-Takasugi formalism, we compute the strong-field scattering factor and show that the the spin produces characteristic modifications to the lensed waveform, and high-frequency incident radiation is not strongly absorbed by the BH lens, contrary to earlier claims. We further derive explicit expressions for the observed waveform for the general source-lens-observer configuration, showcasing the distortions produced by the scattering and quantifying their departure from the Schwarzschild case. Specializing to on-axis scattering, a mismatch analysis for a GW150914-like source lensed by a Kerr BH of mass $M=10^2~\mathrm{M}_\odot$ situated $100M$ away from the source reveals percent-level deviations from the unscattered wave at scattering angles near $30^\circ$, across a range of lens spin values. The mismatch generally decreases as the scattering angle increases, but this behavior can change substantially when polarization mixing induced by scattering becomes significant. In such cases, components that are absent/suppressed in the direct signal may become appreciable due to scattering effects. For a fixed scattering angle, however, the mismatch shows only a weak dependence on the BH spin in the case of on-axis scattering, which may improve for more general configurations. The framework developed here offers a unified treatment of strong-field GW scattering in Kerr spacetime for interpreting future high-precision GW observations.

Figures (10)

• Figure 1: The source (S) is a coalescing binary, which is situated close to a lensing BH (L) of mass $M$ and spin-angular momentum $J$. A portion of the radiation emitted by the source is scattered ($h^{\mu\nu}_{\rm SW}$) by the lens in the direction of the observer (O). The coordinate system is aligned so that the spin of the lensing BH is along the $z$-axis. The $x$-axis is chosen such that the source and lens lie on the $xz$-plane. The angular coordinates of the observer/detector (O) are denoted as $(\theta_o,\phi_o)$. The scattered wave $h^{\mu\nu}_{\rm SW}$, at the observer can be computed in the framework of BH perturbation theory (BHPT). The radiation from the source can also reach the observer "directly" as $h^{\mu\nu}_{\rm dir}$. The direct component is only weakly affected by the lens for large $\theta_o\sim 30^\circ$.
• Figure 2: Plots showing the variations of the on-axis ($\gamma=0$) absorption cross section ($\sigma_a$) and differential scattering cross section ($d\sigma/d\Omega$) as a function of frequency ($\omega$) for different spin values and $\theta=\pi$. These plots match very well with that of Ref. Dolan:2008kf. Note that the absorption cross section becomes negative for corotating ($a>0$) configurations at low frequencies. This is due to superradiant amplification.
• Figure 3: Plots showing the variations of the on-axis ($\gamma=0$) scattering cross section for different combinations of ($M\omega,\, a$). The plots generated by the series reduction method match very well with that of Ref. Dolan:2008kf. Whereas the plots generated by Cesàro summation (with $\alpha=2,5$) technique systematically tend to disagree especially at low values of $\theta$ and large values of $M\omega$.
• Figure 4: Log-Log plots showing the relative impact on the waveform (SFSF) due to on-axis scattering ($\gamma=0$) for $\theta_o=\phi_o=\pi/6$, $r_{\rm SL}=100M$, $r_{\rm SO}\approx r_{\rm LO}$ and different choices of spin $a$. Spin does not seem to significantly affect the overall scale of the SFSF, but it does contribute appreciably in frequency-modulation. Note that results obtained using Cesàro summation predict a decay at high frequencies, which increases with $\alpha$, in contrast to that obtained via series reduction, which maintains a stable amplitude.
• Figure 5: The source (S) is a quasi-circular binary coalescence of two spinless BHs. The orbital-angular momentum of the source is $\vec{\ell}$, oriented so that it faces away from the lens (note the difference with Fig. \ref{['sec:scatter']}). The lens (L) has mass $M$ and spin-angular momentum $|\vec{J}|=M^2\chi$, where $\chi$ is the spin-parameter $\chi\in(-1,1)$. The direction of the lens spin defines the $\hat{z}$-axis and is chosen to be antiparallel (when $\chi>0$) to $\vec{\ell}$, and thus parallel to the source-lens line. The observer (O) is at an angle $\theta_o$ with respect to $\hat{z}$-axis. The GWs emitted by the source reach the observer in two parts, as a direct component $h^{\mu\nu}_{\rm dir}$ and a scattered component $h^{\mu\nu}_{\rm SW}$ which reach roughly at the same time and interfere.