  The Gaia mission offers a new opportunity to search for the low frequency
gravitational wave background using astrometric measurements. In this paper,
the astrometric effect of gravitational waves is reviewed, with a particular
focus on the effect of non-Einsteinian gravitational wave polarizations. A
stochastic gravitational wave background generates a correlated vector field of
astrometric deflections on the sky. A convenient decomposition for the
correlation matrix is introduced, enabling it to be calculated for all possible
gravitational wave polarizations and compared to the redshift correlations from
the pulsar-timing literature; in the case of a GR background of transverse
traceless gravitational waves, this also allows us to identify an astrometric
analog of the famous Hellings-Downs curve. Finally, the cross-correlation
between the redshift and astrometric signal is also calculated; this may form
the basis for future joint pulsar-timing and astrometry searches for
arbitrarily polarized gravitational wave backgrounds.
