  Over the course of several years, stars trace helical trajectories as they
traverse across the sky due to the combined effects of proper motion and
parallax. It is well known that the gravitational pull of an unseen companion
can cause deviations to these tracks. Several studies have pointed out that the
astrometric mission Gaia will be able to identify a slew of new exoplanets,
stellar binaries, and compact object companions with orbital periods as short
as tens of days to as long as Gaia's lifetime. Here, we use mock astrometric
observations to demonstrate that Gaia can identify and characterize black hole
companions to luminous stars with orbital periods longer than Gaia's lifetime.
Such astrometric binaries have orbital periods too long to exhibit complete
orbits, and instead are identified through curvature in their characteristic
helical paths. By simultaneously measuring the radius of this curvature and the
orbital velocity, constraints can be placed on the underlying orbit. We
quantify the precision with which Gaia can measure orbital accelerations and
apply that to model predictions for the population of black holes orbiting
stars in the stellar neighborhood. Although orbital degeneracies imply that
many of the accelerations induced by hidden black holes could also be explained
by faint low-mass stars, we discuss how the nature of certain putative black
hole companions can be confirmed with high confidence using Gaia data alone.
