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

General Relativity and Quantum Cosmology;Cosmology and Nongalactic Astrophysics;Astrophysics of Galaxies;High Energy Astrophysical Phenomena