From phys.org and other sources

Why do most cosmologists insist on believing that Riemannian geometry is the final word in gravitation theory? Certainly it is the correct post-Newtonian approximation to a complete theory of classical gravitation, but that most assuredly does not mean it is the final word. Indeed, there are many examples of observations which suggest the opposite -- that it is merely the first step along a long path.

The first example is the deflection of stellar light. In 1919, Eddington showed that there was such an effect, during his eclipse expedition. What was not possible, however, was to show that the observed effect was precisely that predicted by Einstein. Oh, it was tantalizingly close, to be sure, determining the exact magnitude of the effect depends highly on where the actual size and shape of the sun -- that is, where is the exact location of the edge of the solar disc, and what is the oblateness of the rotating sun?

Next is the precession of the planetary perihelia. Calculations of these is not an easy task, because a) Newtonian perturbations dwarf all other effects by at least an order of magnitude, and b) the magnitude of the effect now also includes the oblateness of the planet itself, which turns out to be far more important than solar oblateness. Every planet began as a ball of molten rock, so there is no reason to believe any of them are even close to being spherical, even if they are balls of solid rock today.

I have not done active research for a great many years, so the latest figures I have for both of these are quite old. In 1982, for example, the difference between the predicted and observed relativistic correction to Mercury's precession gave only a 3-σ confidence. Experimental verification of a predicted result requires 5 or better. So, I have to ask if anyone knows of recent results that give a better confidence?

But we are not done. Enter galactic rotation, stage right. General relativity alone is quite incapable of describing the rotations of the vast majority of galaxies. What was the "final" answer to that problem? Dark matter. This is an ad hoc addition to gravitation theory for the sole purpose of explaining all those pesky problematic galactic rotations -- but there are a few galaxies out there for which relativity alone is sufficient to describe their rotation. In my mind, dark matter is nothing but a deus ex machina hauled out to make the script fit the plot -- or, as Einstein himself once put it, "drilling in places where drilling is easy."

This is already getting very long, so I am not going to describe alternative theories, but rather direct you to a couple of places where such theories have been proposed. Notably, these include the work of John Moffat and Joao Maguiejo who independently derived a theory in which the speed of light is variable^[1]. Of all of these proposals, my favourite at this time is the scalar-tensor-vector theory first proposed by Moffat and others in 2006. On the cosmic scale, it appears to explain such things as galactic rotations and gravitational lensing without the need for dark matter, while, on the small scale, eg. our solar system, it predicts no observable deviation from special relativity.

1. It is interesting to note that Magueijo's paper was published in 1998, in the same journal which only 5 years earlier, had rejected Moffat's original paper.

Interesting.

Here is the original article that the CNN story was based on:

https://news.mit.edu/2024/exotic-black-holes-could-be-dark-matter-byproduct-0606

and the published paper:

https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.132.231402