  Flat rotation curves in disk galaxies represent the main evidence for large
amounts of surrounding dark matter. Despite of the difficulty in identifying
the dark matter contribution to the total mass density in our Galaxy, stellar
kinematics, as tracer of gravitational potential, is the most reliable
observable for gauging different matter components. This work tests the
flatness of the MW rotation curve with a simple general relativistic model
suitable to represent the geometry of a disk as a stationary axisymmetric dust
metric at a sufficiently large distance from a central body. Circular
velocities of unprecedented accuracy were derived from the Gaia DR2 data for a
carefully selected sample of disk stars. We then fit these velocities to both
the classical, i.e. including a dark matter halo, rotation curve model and a
relativistic analogue, as derived form the solution of Einstein's equation. The
GR-compliant MW rotational curve model results statistically indistinguishable
from its state-of-the-art DM analogue. This supports our ansatz that a
stationary and axisymmetric galaxy-scale metric could "fill the gap" in a
baryons-only Milky Way, suggestive of star orbits dragged along the background
geometry. We confirmed that geometry is a manifestation of gravity according to
the Einstein theory, in particular the weak gravitational effect due to the
off-diagonal term of the metric could mimic for a "DM-like" effect in the
observed flatness of the MW rotation curve. In the context of Local Cosmology,
our findings are suggestive of a Galaxy phase-space as the exterior
gravitational field of a Kerr-like source (inner rotating bulge) without the
need of extra-matter.
