Polarization of Long-Wavelength Gravitational Waves by Rotating Black Holes

TL;DR

This paper calculates how rotating black holes differentially scatter long-wavelength gravitational waves based on their helicity, revealing that rotation induces partial polarization in initially unpolarized waves.

Contribution

It provides explicit formulas for scattering amplitudes considering black hole rotation, using low-frequency solutions and spheroidal harmonic expansions, advancing understanding of gravitational wave interactions with rotating black holes.

Findings

Black hole rotation causes differential scattering of gravitational wave helicities.

A partial polarization is induced in unpolarized incident waves due to rotation.

Explicit expressions for scattering amplitudes are derived in the long-wavelength limit.

Abstract

The scattering cross section for a long-wavelength planar gravitational wave impinging upon a rotating black hole is calculated, for the special case in which the direction of incidence is aligned with the rotation axis. We show that black hole rotation leads to a term in the cross section that is proportional to $a\omega$. Hence, contrary to some claims, co-rotating and counter-rotating helicities are scattered differently, and a partial polarization is induced in an unpolarized incident wave. The scattering amplitudes are found via partial wave series. To compute the series, two ingredients are required: phase shifts and spin-weighted spheroidal harmonics. We show that the phase shifts may be found from low-frequency solutions of the radial Teukolsky equation derived by Mano, Suzuki and Takasugi. The spheroidal harmonics may be expanded in spherical harmonics; we present expansions accurate to second order in $a \omega$. The two ingredients are combined to give explicit expressions for the helicity-conserving and helicity-reversing amplitudes, valid in the long-wavelength limit.