On the Rotation Curve of Disk Galaxies in General Relativity

Abstract

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).

Can general relativity play a role in galactic dynamics?

“…If we do not consider the presence of the additional component B HS , a self consistent analysis of the solution of the Eqs. ( 1),( 2) and ( 5) within the mass distribution leads to the results obtained by Ciotti [6], i.e. the smallness of the gravitomagnetic corrections with respect to the Newtonian behaviour.…”

“…In this paper we addressed the issue of the relevance of GR corrections to the galactic rotation curves; in the weak-field and slow-motion approximation which leads to the gravitoelectromagnetic analogy, we pointed out that there are solutions for the gravitomagnetic field that might have a relevant impact on the velocity profile. There is no contradiction with what was recently obtained by Ciotti [6], since Ciotti's results refer to the nonhomogenous solutions (NHS) for the gravitomagnetic potential, that are directly related to its sources, the mass currents, while our statement refers to the homogenous solutions (HS), whose relation with the sources is determined by the equilibrium conditions of the dust fluid. The origin of these HS is strictly related to the general solution of the Einstein's equations for a neutral, stationary and axisymmetric rotating dust [23] and they are present also in the so-called strong gravitomagnetic regime.…”

“…The velocity profiles for the Kuzmin-Toomre model and the exponential disk are in Figure 1. In both cases, we see that the corrections to the Newtonian velocity profile (the curve in the bottom in both plots) are not negligible, unlike what happens for the NHS, where they are in the order of 1 part to 10 6 [6].…”

“…In this context, it is well known that the problem of the flatness of the rotation curves [2,3] is one of the observations supporting the existence of Dark Matter (DM) [4,5]. In particular [6] DM is required at large galactocentric distances, as testified by radio observations in HI gas, while Newtonian models are in agreement with the observed rotation curves over a large fraction of the optical disk; in addition, DM is relevant for stability reasons [7].…”

“…Ciufolini and Wheeler [18], Mashhoon et al [19], Ruggiero and Tartaglia [20], Mashhoon [21], Costa and Natario [22]). The question is then to evaluate if these effects are large enough to actually change the dynamics: according to a recent paper by Ciotti [6], the answer is negative, since the gravitomagnetic corrections within the matter distribution are smaller by a factor 10 −6 with respect to the usually adopted Newtonian profile. These conclusions are based on a very careful and detailed analysis, and do not depend on the specific model considered by the author, since their validity derives from the very relations between the gravitomagnetic field and its sources.…”