  A recent analysis of data from the ESA Gaia mission demonstrated that the
kinematics of stars in the Milky Way can be modelled without invoking the
presence of dark matter whatsoever. Indeed, the higher-than-Keplerian
velocities observed in outer stars can be ascribed to the properties of the
general relativistic (GR) metric assumed to describe the Galaxy. Here we
generalize the concept, and derive the most general exact GR model for a
stationary axi-symmetric dynamically cold dust structure.
  We explicitly show how deviations from the commonly adopted Newtonian
dynamics are indeed manifest even at low velocities and low densities. We
provide for the first time a detailed description of the frequency shift
experienced by photons travelling from any emission site within an external
disc galaxy to the observer, relating the outcome of the shift measurement to
the gravitational properties of the galaxy. Finally, we devise a novel,
groundbreaking observational test potentially able to fully characterize the GR
metric under the minimal set of assumptions mentioned above. The proposed
experiment exploits the effects non-diagonal GR terms have on the frequency
shift of photons, ultimately providing a test to evaluate whether dark matter
is actually required by disc galaxy kinematics.
