From a General Relativity point of view Gravity is given as the result of spacetime curvature interacting with energy-mass density. To get to the Newtonian limit one needs to take
- a) Non-relativistic speeds
- b) Weak gravitational fields.
Now, I understand that to have a more complete understanding of the trajectory of Mercury, one needs to study it's interaction with Sun from a GR point of view. Then one concludes to the famous Advance of the Perihelion, for which I quote from from Wikipedia:
In 1859, the French mathematician and astronomer Urbain Le Verrier reported that the slow precession of Mercury's orbit around the Sun could not be completely explained by Newtonian mechanics and perturbations by the known planets. He suggested, among possible explanations, that another planet (or perhaps instead a series of smaller 'corpuscules') might exist in an orbit even closer to the Sun than that of Mercury, to account for this perturbation. (Other explanations considered included a slight oblateness of the Sun.) The success of the search for Neptune based on its perturbations of the orbit of Uranus led astronomers to place faith in this possible explanation, and the hypothetical planet was named Vulcan, but no such planet was ever found. The perihelion precession of Mercury is 5,600 arcseconds (1.5556°) per century relative to Earth, or 574.10±0.65 arcseconds per century relative to the inertial ICFR. Newtonian mechanics, taking into account all the effects from the other planets, predicts a precession of 5,557 arcseconds (1.5436°) per century. In the early 20th century, Albert Einstein's general theory of relativity provided the explanation for the observed precession. The effect is small: just 42.98 arcseconds per century for Mercury; it therefore requires a little over twelve million orbits for a full excess turn. Similar, but much smaller, effects exist for other Solar System bodies: 8.62 arcseconds per century for Venus, 3.84 for Earth, 1.35 for Mars, and 10.05 for 1566 Icarus.”
What I would like to understand is this: One of the reason I heard in class for assuming dark-matter is that given the Newtonian relationship for the speed of the stars in function with the centrifugal force, we show that the galaxy has not enough mass to maintain it' s shape. But, why in the first place use Newtonian mechanics and not GR. I understand that we are at the low speed limit, but what about the weak gravitational limit?
So, is there a certain procedure from which we can show that the gravity interactions of the Galaxy do not need a GR approach? If we need GR to fully understand the trajectory of Mercury, how can it be that we don' t need GR for the shape of the Galaxy and the understanding of the reasons holding it together? Thanks.