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It is well known that dark matter was introduced to explain the orbits of stars in a galaxy. My main question is this: Were these calculations done using Newton or Einstein?

People I ask tell me that there's no difference because of small velocities and weak gravity, but my concern is that Newton cannot be right for large distances. In fact, if a star orbits the center of a galaxy in a distance of, say, 10.000 light years, then the star "sees" the gravitational center in the direction where it was 10.000 years ago. So for a circular orbit, the acceleration vector is not perpendicular to the velocity velocity, from which I would conclude that the speed of the star must be higher than in Newton's theory.

I wonder whether someone with more experience than I could run a simulation and tell me whether this makes any difference.

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    $\begingroup$ You're basically asking about modified Newtonian dynamics (MOND, see Wiki), which has its own problems. $\endgroup$ – Kyle Kanos Aug 8 '17 at 15:11
  • $\begingroup$ @Kyle: I'm interested in the predictions by general relativity; I only used the Newtonian language in order to make my point. $\endgroup$ – Franz Lemmermeyer Aug 8 '17 at 15:46
  • $\begingroup$ So something like this? $\endgroup$ – Kyle Kanos Aug 8 '17 at 15:49
  • $\begingroup$ In fact, if a star orbits the center of a galaxy in a distance of, say, 10.000 light years, then the star "sees" the gravitational center in the direction where it was 10.000 years ago. Not true. See Carlip, xxx.lanl.gov/abs/gr-qc/9909087v2 $\endgroup$ – user4552 Aug 8 '17 at 15:57
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    $\begingroup$ You are pointing at something interesting: finite speed of gravitation would also destroy earth orbit around the sun. This was noted by Laplace and he concluded the speed of gravitation is much larger than the speed of light. Apparently if you include Lorentz-transformation (contraction) this effect is counterbalanced, I was told once. I dont fully understand this, but basically finite speed of interaction+ special relativity seems to give the same as plain Newtonian physics. $\endgroup$ – lalala Aug 8 '17 at 19:58
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enter image description here

This is one of the set of images from this paper below.

Abstract:

The ΛCDM standard cosmological model is strongly supported by multiple lines of evidence, particularly from observations at large scales such as the CMB and large scale structure. There are some indications, however, of problems at smaller scales. An alternative to the CDM approach is to modify the gravitational force, as exemplified by the MOdified Newtonian Dynamics (MOND) idea. While evidence suggests MOND cannot account for dynamics at all scales without dark matter, it has been successful at galactic scales. Due to the complexity of the theory, however, most tests of MOND have extended no further than using a simple scaling relation to determine rotation curves or velocity dispersions. Therefore, to test the concept more thoroughly we require numerical simulations. We discuss the development and testing of a new N-body solver, using two distinct formulations of MOND, that is incorporated into the RAMSES code. The theory of MOND as a modification of Newtonian gravity is briefly summarised. We then show how it is implemented in the code, providing an example of an idealised test case and future applications.

My apologies for not being able to provide you with a direct link, my android tablet will not copy the link,but if you search for:

Numerical simulations of Modified Newtonian Dynamics G N Candlish , R Smith, and M Fellhauer, you should find it easily.

One possible drawback to this simulation is that the paper is 5 years old, and the number of elements plotted may be less than is possible today. The authors do point out the limitations of their method.

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Mercury is in an orbit around the Sun of radius around 50 million km (167 light seconds), with a period of 88 days.

So the ratio of radius to period in similar time units is $2.2\times 10^{-5}$. The effects of General Relativity compared with Newtonian gravity are very small, but of course measurable.

A star in orbit in our Galaxy at a radius of 10,000 light years will have an orbital period of about 100 million years if the rotation curve is flat. The ratio here is $10^{-4}$. Thus if this were the critical factor, perhaps one would also expect minor perturbations of the same order as the corrections to Mercury's orbit around the Sun.

But I don't think that is the critical factor. What matters is the curvature of space, the importance of which can be roughly assessed by how far $GM/Rc^2$ departs from zero.

For Mercury around the Sun, this ratio is $3\times 10^{-8}$. For a star in orbit in the Galaxy, where the orbit perhaps encloses a few $10^{10}$ solar masses, the ratio is a few $10^{-7}$.

So yes, I would expect GR to slightly modify the orbital dynamics with respect to Newtonian gravity, but not at any level that would even be measurable using current techniques to derive rotation curves.

BTW If you abandon Newtonian gravity in favour of GR then you also abandon the idea of gravity as a force. So there is no issue of a force vector not being perpendicular to the velocity.

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