  A supermassive black hole moving through a field of stars will
gravitationally scatter the stars, inducing a backreaction force on the black
hole known as dynamical friction. In Newtonian gravity, the axisymmetry of the
system about the black hole's velocity $\mathbf{v}$ implies that the dynamical
friction must be anti-parallel to $\mathbf{v}$. However, in general relativity
the black hole's spin $\mathbf{S}$ need not be parallel to $\mathbf{v}$,
breaking the axisymmetry of the system and generating a new component of
dynamical friction similar to the Lorentz force $\mathbf{F} = q\mathbf{v}
\times \mathbf{B}$ experienced by a particle with charge $q$ moving in a
magnetic field $\mathbf{B}$. We call this new force gravitomagnetic dynamical
friction and calculate its magnitude for a spinning black hole moving through a
field of stars with Maxwellian velocity dispersion $\sigma$, assuming that both
$v$ and $\sigma$ are much less than the speed of light $c$. We use
post-Newtonian equations of motion accurate to $\mathcal{O}(v^3/c^3)$ needed to
capture the effect of spin-orbit coupling and also include direct stellar
capture by the black hole's event horizon. Gravitomagnetic dynamical friction
will cause a black hole with uniform speed to spiral about the direction of its
spin, similar to a charged particle spiraling about a magnetic field line, and
will exert a torque on a supermassive black hole orbiting a galactic center,
causing the angular momentum of this orbit to slowly precess about the
black-hole spin. As this effect is suppressed by a factor $(\sigma/c)^2$ in
nonrelativistic systems, we expect it to be negligible in most astrophysical
contexts but provide this calculation for its theoretical interest and
potential application to relativistic systems.
