***The observed motions of galaxies, and of stars on the fringes of galaxies, cannot be explained by general relativity***
Well. It cannot be explained by Newton's theory either. In fact, the astronomical analyses that predict dark matter are not relativistic. Only f=ma is used.
So dark matter has nothing to do with relativity.
Besides. There is no gravity in General Relativity. There is only space dynamics and curvature do to the presence of mass. If we ever are able to detect gravitons, then GR is banished from reality. However, it is a accurate calculational tool in any event.
Moreover, the accelerations on the fringe of galaxies are not low. They are easily measured. They are known.
What is not known are the constituents of dark matter, and the constituents of dark energy - for that matter. I claim that one of the principal constituents of dark matter is the axion super fluid. It is motionless in space according to GUT theory and so it behaves like a giant frictionless BEC (Bose-Einstein Condensate). That sounds like heaven to me.
There are probably other constituents, like the supersymmetric partners.
And then there is dark energy. Nobody has a clue about this one except that it may be related to the vacuum energy. But the vacuum energy calculates to be ten to the 120 power more energetic than dark energy. Not much of a clue. The real problem is why vacuum energy is so tiny on a galactic scale while measuremants of it in the lab (the Casmuir effect) verify its vast strength.