Gravitational Lensing of Gravitational Waves: Spin-wave Optics through Black Hole Scattering

TL;DR

This paper develops a rigorous, divergence-free method for modeling gravitational wave scattering by black holes, revealing wavefront distortions and the Poisson spot, and improving upon traditional asymptotic approaches.

Contribution

It introduces a finite-distance, non-asymptotic partial-wave approach for GW scattering by black holes, resolving divergences and accurately capturing strong-field effects.

Findings

Revealed formation of the Poisson spot in GW scattering

Demonstrated limitations of asymptotic methods at small angles

Highlighted discrepancies between scattering and Kirchhoff diffraction at high frequencies

Abstract

Gravitational-wave (GW) scattering in strong gravitational fields is a central problem in GW lensing. Yet, conventional treatments based on asymptotic expansions suffer from divergences and become unreliable near the optical axis. In this work, we present a rigorous calculation of GW scattering by a Schwarzschild black hole (BH) within the BH perturbation theory. By placing the observer at a finite distance and abandoning the asymptotic expansion of radial wave functions, we obtain a well-convergent partial-wave description without invoking any regularization scheme, thereby naturally resolving the divergences of the partial-wave series and the Poisson spot. We numerically computed the scattered GW waveforms by reconstructing the physical $+$ and $\times$ polarizations from the master variables, revealing the formation of the Poisson spot and pronounced wavefront distortions. A systematic comparison with conventional asymptotic approaches shows that they reproduce only qualitative features at large scattering angles and fail in the forward-scattering region. We further compare the frequency-domain transmission factors derived from the scattering formalism with those obtained from the Kirchhoff diffraction integral, finding significant discrepancies at high frequencies due to the latter's neglect of long-range gravitational effects and polarization evolution. Our results establish a stable and physically transparent framework for GW scattering in strong-field regimes and provide a solid foundation for accurate modeling of GW lensing beyond standard approximations.