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The propagation of gravitational waves can be described in terms of nullgeodesics by using the geometrical optics approximation. However, at large butfinite frequencies the propagation is affected by the spin-orbit couplingcorrections to geometrical optics, known as the gravitational spin Hall effect.Consequently, gravitational waves follow slightly different frequency- andpolarization-dependent trajectories, leading to dispersive and birefringentphenomena. We study the potential for detecting the gravitational spin Halleffect in hierarchical triple black hole systems, consisting of an emittingbinary orbiting a more massive body acting as a gravitational lens. Wecalculate the difference in time of arrival with respect to the geodesicpropagation and find that it follows a simple power-law dependence on frequencywith a fixed exponent. We calculate the gravitational spin Hall-correctedwaveform and its mismatch with respect to the original waveform. The waveformcarries a measurable imprint of the strong gravitational field if the source,lens, and observer are sufficiently aligned, or for generic observers if thesource is close enough to the lens. We present constraints on dispersive timedelays from GWTC-3, translated from limits on Lorentz invariance violation.Finally, we address the sensitivity of current and future ground detectors todispersive lensing. Our results demonstrate that the gravitational spin Halleffect can be detected, providing a novel probe of general relativity and theenvironments of compact binary systems.

From the gates of the abyss: Frequency- and polarization-dependent lensing of gravitational waves in strong gravitational fields