On the Rotation Curve of Disk Galaxies in General Relativity

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 (GR) instead of the effect of dark matter (DM) 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 DM. As expected, the resulting rotation curves are indistinguishable, with GR corrections at all radii of the order v 2/c 2 ≍ 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 DM 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 this article, the mathematical framework is described in detail, so that the present study can be extended to other disk models, or to elliptical galaxies (where DM is also required in Newtonian gravity, but their rotational support can be much less than in disk galaxies).