  Recently, it has been suggested that the phenomenology of flat rotation
curves observed at large radii in the equatorial plane of disk galaxies can be
explained as a manifestation of General Relativity instead of the effect of
Dark Matter halos. In this paper, by using the well known weak field, low
velocity gravitomagnetic formulation of GR, the expected rotation curves in GR
are rigorously obtained for purely baryonic disk models with realistic density
profiles, and compared with the predictions of newtonian gravity for the same
disks in absence of Dark Matter. As expected, the resulting rotation curves are
indistinguishable, with GR corrections at all radii of the order of
$v^2/c^2\approx 10^{-6}$. Next, the gravitomagnetic Jeans equations for
two-integral stellar systems are derived, and then solved for the
Miyamoto-Nagai disk model, showing that finite-thickness effects do not change
the previous conclusions. Therefore, the observed phenomenology of galactic
rotation curves at large radii requires Dark Matter in GR exactly as in
newtonian gravity, unless the cases here explored are reconsidered in the full
GR framework with substantially different results (with the surprising
consequence that the weak field approximation of GR cannot be applied to the
study of rotating systems in the weak field regime). In the paper, the
mathematical framework is described in detail, so that the present study can be
extended to other disk models, or to elliptical galaxies (where Dark Matter is
also required in newtonian gravity, but their rotational support can be much
less than in disk galaxies).
