  The propagation of high-frequency gravitational waves can be analyzed using
the geometrical optics approximation. In the case of large but finite
frequencies, the geometrical optics approximation is no longer accurate, and
polarization-dependent corrections at first order in wavelength modify the
propagation of gravitational waves via a spin-orbit coupling mechanism. We
present a covariant derivation from first principles of effective ray equations
describing the propagation of polarized gravitational waves, up to first-order
terms in wavelength, on arbitrary spacetime backgrounds. The effective ray
equations describe a gravitational spin Hall effect for gravitational waves and
are of the same form as those describing the gravitational spin Hall effect of
light, derived from Maxwell's equations.
