Explain the oscillatory decay of helicity-dependent centroid displacement

Determine the origin of the oscillatory behavior in the decay of the centroid displacement between images of clockwise- and counterclockwise-rotating electric dipoles as the dipole frequency or its distance from the Schwarzschild black hole increases.

Background

The paper studies the optical Magnus or gravitational spin Hall effect by comparing images produced by rotating electric dipoles with opposite helicities near a Schwarzschild black hole. The centroid displacement between the two images is expected to decrease as the inverse dipole frequency and, in a separate setup, as the inverse distance from the black hole. Numerical results support these scaling laws but also show oscillations around the predicted decay.

The authors note that the oscillations may be related to black-hole resonances, but do not establish their physical origin. Understanding this behavior would clarify how finite-frequency scattering, resonance excitation, and helicity-dependent propagation contribute to wave-optical images.

References

The difference tends to $0$ as $1/\Omega_{\rm d}$ and $1/r_0$, respectively; the oscillating nature of the decay is not yet well understood.

— Wave optical imaging of an oscillating electric dipole orbiting a black hole  (2609.10689 - Nazaré et al., 9 Sep 2026) in Section 3.2, Fig. 6 caption and surrounding discussion