  Typical stars in the Milky Way galaxy have velocities of hundreds of
kilometres per second and experience gravitational accelerations of $\sim
10^{-10}$ m s$^{-2}$, resulting in velocity changes of a few centimetres per
second over a decade. Measurements of these accelerations would permit direct
tests of the applicability of Newtonian dynamics on kiloparsec length scales
and could reveal significant small scale inhomogeneities within the galaxy, as
well increasing the sensitivity of measurements of the overall mass
distribution of the galaxy. Noting that a reasonable extrapolation of progress
in exoplanet hunting spectrographs suggests that centimetre per second level
precision will be attainable in the coming decade(s), we explore the
possibilities such measurements would create. We consider possible confounding
effects, including apparent accelerations induced by stellar motion and reflex
velocities from planetary systems, along with possible strategies for their
mitigation. If these issues can be satisfactorily addressed it will be possible
to use high precision measurements of changing stellar velocities to perform a
"blind search" for dark matter, make direct tests of theories of non-Newtonian
gravitational dynamics, detect local inhomogeneities in the dark matter
density, and greatly improve measurements of the overall properties of the
galaxy.
