Is there a Dark Matter Paradox?

Question

According to this preprint, The Importance of Being Symmetric: Flat Rotation Curves from Exact Axisymmetric Static Vacuum Spacetimes, spiral galaxies possess flat rotation curves without assuming Dark Matter. But this is in contrast to the power spectrum of the microwave background where the second and the third peak are being considered to proof the existence of Dark Matter.

But if true, isn't it a paradoxal situation such as Dark Matter exists and thus forms a halo around spiral galaxies not required, however, because the rotation curves are flat without it?

What do you think about that? I appreciate any suggestions.

Answer 1

I have not read the article in detail (we are not peer review), but you should read it with extreme skepticism. There is a long history of works wrongly claiming that general relativity can explain flat galactic rotation curves where Newtonian gravity cannot. From the conclusion of an earlier article, Costa et al. (2023):

In a growing number of works, based both on linearized theory [127, 131] and exact models [13–15, 129, 130], it has been asserted that gravitomagnetic effects can have a significant impact [13, 15, 127, 131], or even totally account for the galactic flat rotation curves [14, 129, 130]. In the framework of a weak field slow motion approximation, this has been shown to be impossible, and such claims addressed, in [125, 128, 132]. It has however been argued [13, 15, 129, 130] that in the exact theory this is possible, due to nonlinear effects not captured in linearized theory (and not manifest “locally” [13, 129, 130]), basing such claims on the BG model, or variants of it. Our exact analysis shows such claims to be also unfounded, the conclusion extending to the akin models in [15, 129, 130] (where the same misguided choice of reference observers is made). The exact approach proves also useful in determining the nature of the sources: we were able to easily identify a pair of oppositely charged rods of gravitomagnetic monopoles as the source of the gravitomagnetic field $\vec H$; in the far field region, however, this field is, to leading (i.e., dipole) order, indistinguishable from that of a spinning source.

In my view, these works are all fundamentally misguided because galaxies are deep in the Newtonian limit, having potential depths of order $10^{-6}c^2$ and velocities of order $10^{-3}c$. We know that general relativity reduces to Newtonian gravity in this limit. There is no reason for general relativity to make different predictions from Newtonian gravity in this context.

As an aside, I enjoyed the following passage from Costa et al. (2023):

As it replaced Newtonian mechanics as the state of the art theory of gravitation, general relativity brought along equations allowing for a more precise description of gravitational phenomena, at the cost, however, of a mathematical complexity effectively preventing their full use in actual astrophysical scenarios. Since then, two fields have evolved parallelly in a largely separate way: one, finding and mathematically characterizing exact solutions of the Einstein field equations [1–3], which in most cases do not correspond to realistic physical systems (or whose physical significance is not totally clear); and the other, approximate methods such as post-Newtonian theory, for actual astrophysics.

It is a useful point to make here because laypeople and students tend to focus on the exact solutions to a much greater degree than researchers do. The focus of astrophysics research is to understand the systems that actually arise in the Universe, using approximation methods and numerical calculations. This is largely disjoint from the effort of coming up with exact solutions to the Einstein field equations, which are almost never of astrophysical relevance.

Answer 2

I agree with KDP that cosmology is in crisis at the moment. How this crisis will be solved, with or without Dark Matter is not clear.

However, I have very serious doubts about the solution proposed in the preprint you link to in the top of your question. It supposes an axisymmetric universe.

I have no doubt that the calculations done in that paper for a single galaxy are correct.

But they can be correct only for those galaxies the axis of rotation of which are in the direction of the axis of symmetry of the Universe. But clearly, galaxies have axes of rotation in all directions. And the rotation curves do not depend on the direction of the axis of rotation. So this paper is not relevant to cosmology.

Answer 3

I don't think this paper reveals anything new at all. It is well known in Newtonian physics that a disc-shaped mass distribution can yield a flat(tish) rotation curve and galaxies are perfectly well-described by Newtonian physics (see Sten's answer). Crucially, that flatness only persists whilst a significant fraction of the mass is still exterior to the test particle. In other words, for a disk with an exponentially decaying density with radius, the rotation curve is indeed flat in the region of 1-2 exponential scale lengths. Once beyond that, then the rotation curve falls in a pseudo-Keplerian way as expected.

The difficulty is that the density of visible matter in spiral galaxies like the Milky Way has an exponential radial decay length of 3 kpc, whilst the rotation curve is flat to >30 kpc (e.g., Bhattacharjee et al. 2014). The rotation curve is measured using tracers such as halo star, globular closters, gas clouds and satellite galaxies.

Thus to explain the rotation curve with a disk, you need the disk to have a density that decays with an exponential scale lengths of 15 kpc or more - at least 5 times that of the visible matter density.

What this means is that once beyond a few kpc, the disk density would need to be dominated by non-luminous mass, a.k.a "dark matter". Thus, no problem has been solved at all.

Worse, because we know what the mass density is in the disk from studying the motion of stars perpendicular to the disk, we know that the mass density in the visible disk is not dominated by dark matter, but by the visible stars and gas.