Rigorous calculation of scalar scattering in Schwarzschild background: the convergence of partial-wave series and Poisson spot

TL;DR

The paper tackles two key pathologies in scalar scattering on Schwarzschild backgrounds—divergent partial-wave series and a Poisson-spot divergence near the optical axis—showing they arise from the use of asymptotic radial expansions. By computing the scattered field at finite radius and avoiding large-$kr$ asymptotics, the authors obtain convergent partial-wave sums for both Newtonian and Regge–Wheeler potentials, with a natural truncation $\ell_{\max}\sim kr$. They present exact scalar solutions in the Newtonian case via paraboloidal coordinates and via partial waves, and extend the framework to Schwarzschild BH scattering, computing radial functions, phase shifts, and the full scattered waveform while verifying energy conservation and consistency with Kirchhoff diffraction in near-axis regions. The work thus provides a rigorous, finite-radius full-wave treatment of BH wave scattering, enabling accurate wave-optics lensing analyses and informing gravitational-wave lensing templates, especially near the optical axis where previous methods failed.

Abstract

Black hole (BH) perturbation theory and the scattering models provide a powerful framework for studying gravitational lensing at the wave-optics level. However, conventional calculations encountered two issues: the divergence of the partial-wave series and the divergence of the Poisson spot near the optical axis. These issues hinder the accurate calculation of lensed waveforms and the study of polarization and wave characteristics in the lensing process, especially near the optical axis. This work demonstrates that both divergences stem from the asymptotic expansion of the radial wave function. By computing the scattered wave function at finite radii and avoiding the asymptotic expansion, we naturally obtain convergent results. We compute scalar waves scattered by (1) a weak-gravity body with Newtonian potential and (2) a Schwarzschild BH with Regge-Wheeler potential. In both cases, we analyze the convergence of the partial-wave series and present finite-luminosity diffraction patterns, with a bright Poisson spot. The above calculations are compared with the Kirchhoff diffraction integral in the near-axis regions and give consistent results. Our investigations provide a foundation for studying gravitational wave scattering by BHs and understanding lensing at the wave-optics level.

Figures (15)

• Figure 1: The summation and convergence of Eq. (\ref{['eq-2:plane-wave-spherical-harmonics-expansion']}) for a monochromatic planar scalar wave, with $kr=60.0$ and $\theta=\{0,\pi/6,\pi/3,\pi/2\}$. The truncation is $\ell_{\max}\sim kr$ approximately.
• Figure 2: Comparison between the spherical Bessel function $j_{\ell}(z)$ and its large-$kr$ expansion for low and high $\ell$ modes. The region we focus on is around $z\sim60.0$. The low-$\ell$ modes (e.g., $\ell=0,1,2,3\ll 60.0$) are well approximated by their large-$kr$ expansion. But this is invalid for the high-$\ell$ modes (e.g., $\ell=60,61,62,63\sim 60.0$).
• Figure 3: Scattered monochromatic scalar wave fields calculated through Eq. (\ref{['eq-3:scattering-wave-function-paraboloidal']}) with $k=\{0.5,1.0,1.5,2.0\}/M$. The abscissa ($x$-axis) and ordinate ($z$-axis) in these figures are in units of BH mass. The red crosses in the figures represent the position of the scatterer.
• Figure 4: The convergence of Eq. (\ref{['eq-3:scattering-wave-function-PW']}) and its consistency with paraboloidal-coordinates solution (\ref{['eq-3:scattering-wave-function-paraboloidal']}). These calculations are implemented for $k=1.0/M$ and at radius $r=60.0M$, therefore the PWS truncation is approximately $\ell_{\max}\sim kr\sim60$. The horizontal dashed line means the values evaluated from Eq. (\ref{['eq-3:scattering-wave-function-paraboloidal']}).
• Figure 5: Comparison between the radial function $\tilde{R}_{\ell}(k,r)$, involving the Kummer hypergeometric function, and its large-$kr$ approximation, which is valid/invalid around $kr\sim60.0$ for the low/high-$\ell$ modes.