It is a well known fact that, in the absence of Dark Matter, the observation
of the rotation curves of galaxies cannot be explained in terms of Newtonian
gravity. Rotation curves become flat in the outer regions, in contrast to what
is expected according to Keplerian motion. Far from the galactic center, the
gravitational field is supposed to be weak enough so we expect to be able to
use Newtonian Gravity; however, even in the weak-field approximation, there are
general relativistic effects without a Newtonian counterpart, such as the
gravitomagnetic effects originating from mass currents. Using the
gravitoelectromagnetic approach to the solution of Einstein equations in the
weak-field and slow-motion approximation, we discuss some simple arguments that
suggest the surprising result that gravitomagnetic effects may have a relevant
role in better understanding the impact of Dark Matter on galactic dynamics. In
addition, treating matter as a fluid of dust, we study the influence of
post-Newtonian effects on the fluid vorticity.

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