  Goncalves et al. derived a non-relativistic limit of the generalised Dirac
Hamiltonian in the presence of a gravitational wave, using the exact
Foldy-Wouthuysen transformation. This gave rise to the intriguing notion that
spin-precession may occur even in the absence of a magnetic field. We argue
that this effect is not physical as it is the result of a gauge-variant term
that was an artefact of a flawed application of the exact Foldy-Wouthuysen
transformation. In this paper we derive the correct non-relativistic limit of
the generalised Dirac Hamiltonian in the presence of a gravitational wave,
using both the exact and standard Foldy-Wouthuysen transformation. We show that
both transformations consistently produce a Hamiltonian where all terms are
gauge-invariant. Unfortunately however, we also show that this means the novel
spin-precession effect does not exist.
