Polarization of Long-Wavelength Gravitational Waves by Rotating Black Holes

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 aω. 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 aω. The two ingredients are combined to give explicit expressions for the helicity-conserving and helicityreversing amplitudes, valid in the long-wavelength limit. PACS numbers: 04.30.-w, 04.30.Db, 04.30.Nk, 04.70.Bw Polarization of Long-Wavelength Gravitational Waves by Rotating Black Holes 2