  In an isotropic and homogeneous Hubble expansion, all transverse motion is
peculiar. Like the radial peculiar velocities of galaxies, transverse peculiar
velocities are a means to trace the density of matter that does not rely on
light tracing mass. Unlike radial peculiar velocity measurements that require
precise redshift-independent distances in order to distinguish between the
Hubble expansion and the observed redshift, transverse peculiar velocities can
be measured using redshifts alone as a proxy for distance. Extragalactic proper
motions can therefore directly measure peculiar velocities and probe the matter
power spectrum. Here we develop two-point transverse velocity correlation
statistics and demonstrate their dependence on the matter power spectrum. We
predict the power in these correlation statistics as a function of the physical
separation, angular separation, and distance of pairs of galaxies and
demonstrate that the effect of large scale structure on transverse motions is
best measured for pairs of objects with comoving separations less than about 50
Mpc. Transverse peculiar velocities induced by large scale structure should be
observable as proper motions using long baseline radio interferometry or
space-based optical astrometry.
