Galactic Rotation Speeds - Ehrenfest Paradox, Gravitational time dilation, Dark Matter - all of the above?

Question

The observed paths and speeds of objects, part of some distant galaxy, do not match up with speed vs distance curves it seems - the observed speeds are not falling off in fact they're trending as increasing or constant with distance. Wikipedia explains the phenomenon far better - wiki: Galaxy_rotation_curve

Dark matter is one explanation but it isn't without it's faults and flaws, at times raising more questions than being helpful. Particularly when trying to answer what dark matter might be composed of, and that the theory needs fine-tuning for any such model to match observations; from a lay-person's POV the idea is nice but seems to involve some handwaving regarding fine-tuning or composition (specifically what the secondary effects might be, as they may differ between candidate compositions).

With that in mind it occurred to me that galaxies are very large, and rotating. I don't have a clear picture of scale in mind when it comes to this... However it does seem like one (or both) of these two effects may be contributing:

1. Ehrenfest Paradox - I'm not certain it is applicable, but it is remarkable that there are features in common between a galaxy and a relativistically-rotating disk, the latter being the subject of a paradox regarding geometry.
2. Gravitational time dilation - because of the size and mass distribution of a galaxy, to an observer outside the galaxy, the center would gain time slower than the arms / edges; effectively the edges of a galaxy would appear to move faster than they should, when using inner orbital speeds (in a higher gravitational potential) as a reference.

To help understand why I suspect these effects, it is worth noting the sheer breadth of a galaxy, that it has to be measured in kpc (kiloparsecs)... It can take many 1000's of years for light from one side of a galaxy to reach the other (enough time for cumulative time dilation to become a noteworthy factor I would think), suggesting that a galaxy's internal dynamics may be taken for granted in explaining the observed orbital speeds.

In other words it may be the case that locally (in the remote galaxy), these excessive orbital speeds may not be excessive after all.

Which, if any of these effects, are or could be contributing to the discrepancy in expected orbital speeds of objects in a remote galaxy as observed here?

Answer 1

The orbital velocity of the Sun within the Milky Way is around 230 km/sec, and we can use this speed to calculate how big the relativistic effects would be. The relevant quantity is the Lorentz factor, $\gamma$, because the length contraction and time dilation depend on $\gamma$. The equation for $\gamma$ is:

$$ \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} $$

and feeding in $v = 2.3 \times 10^5$ gives us

$$ \gamma \approx 1.0000003 $$

Relativistc effects become significant when $\gamma$ becomes significantly greater than unity, so for the Sun orbiting in the Milky Way any relativistic effects on its orbit are negligably small.

You would get different answers for different galaxies, but the Milky way is fairly typical of the larger spiral galaxies so it seems safe to conclude relativistic effects can't account for the observed rotation curves.

Answer 2

As John Rennie mentioned, a galaxy is not relativistic (speed too low, gravitational field also). So, the Ehrenfest paradox does not apply. On the other hand, if a galaxy compacts itself towards the size of a black hole, there is, indeed, a paradox with relativistic mechanics. See some calculation of my vintage: https://hal.archives-ouvertes.fr/hal-01399780/

Answer 3

Although the time dilation due to velocity is negligible, what might apply is the retarded time. You may be able to modify Jefemenko's electromagnetic equations to apply to a gravitational context. (Jackson or Griffiths should suffice). Spiral galaxies are not smooth disks, so the spiral arm will have moved by the time the gravitational signal reaches a test mass. If the test mass is behind an arm, the gravitational signal will be stronger because the signal left from a nearby point. Whether that is a sufficient alternative to dark matter, I do not know.

Another idea to consider is the gravitoelectromagnetic approximation to general relativity. The B/H field should be insignificant in this context though.