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

TL;DR

This work develops a rigorous, finite-distance BH perturbation framework to compute gravitational-wave scattering by a Schwarzschild black hole, avoiding the divergences of the traditional asymptotic partial-wave approach. By solving the Regge–Wheeler and Zerilli equations without the large-radius expansion and reconstructing the metric, it yields well-behaved wavefields for the two physical GW polarizations (+ and ×) and reveals a Poisson spot and wavefront distortions near the optical axis. The study compares exact results with conventional asymptotic methods and Kirchhoff diffraction, finding that the latter reproduce only broad features at large scattering angles and fail in the forward region due to neglect of long-range gravity and polarization evolution. The work establishes a robust framework for GW lensing in strong-field regimes and highlights significant quantitative differences from standard diffraction formalisms, guiding future template development for GW lensing observations and extending to Kerr spacetimes.

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.

Figures (8)

• Figure 1: Validation of the asymptotic expansion for the normalized radial function $\widehat{u}_{\ell}^{(+)}(k,r)$, with $k=1.0/M$.
• Figure 2: Convergence of the partial-wave series for $\tilde{\Psi}_4$ at selected frequencies $k$ and scattering angles $\theta$. The observer is located at $r=60.0M$.
• Figure 3: Wave fields of the $+$ and $\times$ modes for scattered GWs. The black and gray regions represent the BH event horizon and light ring, respectively.
• Figure 4: Diffraction patterns of the $+$ and $\times$ modes for scattered GWs, compared with results obtained using the conventional asymptotic approach (see Appendix \ref{['app:conventional_computations']}).
• Figure 5: Comparison of transmission factors from BH scattering (dots) and Kirchhoff integral (dashed lines) in the near-axis region.