  Wave packets propagating in inhomogeneous media experience a coupling between
internal and external degrees of freedom and, as a consequence, follow
spin-dependent trajectories. These are known as spin Hall effects, which are
well known in optics and condensed matter physics. Similarly, the gravitational
spin Hall effect is expected to affect the propagation of gravitational waves
on curved spacetimes. In this general-relativistic setup, the curvature of
spacetime acts as impurities in a semiconductor or inhomogeneities in an
optical medium, leading to a frequency- and polarization-dependent propagation
of wave packets. In this letter, we study this effect for strong-field lensed
gravitational waves generated in hierarchical triple black hole systems in
which a stellar-mass binary merges near a more massive black hole. We calculate
how the gravitational spin Hall effect modifies the gravitational waveforms and
show its potential for experimental observation. If detected, these effects
will bear profound implications for astrophysics and tests of general
relativity.
