  In fluid dynamics, the Magnus effect is the force perpendicular to the motion
of a spinning object as it moves through a medium. In general relativity, an
analogous effect exists for a spinning compact object moving through matter,
purely as a result of gravitational interactions. In this work we consider a
Kerr black hole moving at relativistic velocities through scalar dark matter
that is at rest. We simulate the system numerically and extract the total
spin-curvature force on the black hole perpendicular to its motion. We confirm
that the force scales linearly with the dimensionless spin parameter $a/M$ of
the black hole up to $a/M = 0.99$, and measure its dependence on the speed $v$
of the black hole in the range $0.1 \le v \le 0.55$ for a fixed spin. Compared
to previous analytic work applicable at small $v$, higher-order corrections in
the velocity are found to be important: the total force is nonzero, and the
dependence is not linear in $v$. We find that in all cases the total force is
in the opposite direction to the hydrodynamical analogue, although at low
speeds it appears to approach the expectation that the Weyl and Magnus
components cancel. Spin-curvature effects may leave an imprint on gravitational
wave signals from extreme mass-ratio inspirals, where the secondary black hole
has a nonnegligible spin and moves in the presence of a dark matter cloud. We
hope that our simulations can be used to support and extend the limits of
analytic results, which are necessary to better quantify such effects in the
relativistic regime.
