Source of the post
Can gravitational lensing be correlated with galactic rotation, both being caused by Dark Matter, rather than two separate observations of phenomenon?
No, galaxies rotate too slowly, and rotation doesn't affect space-time in the correct way. So rotation curves and gravitational lensing are independent measurements of the mass distribution -- one from deflection of light rays in curved space-time, and the other from Kepler's Laws for orbiting masses.
The rotation of mass (its angular momentum) does in fact modify the gravitational field. But instead of making the lensing stronger (as if more mass was there), it causes the space to be whirled around, called "frame dragging". You might be familiar with it in the context of spinning black holes dragging the space-time around them. This frame dragging effect occurs around any rotating mass, and has even been measured in the space-time around Earth
If you compute the angular momentum of a rotating galaxy, or even a massive star, it is quite large. Often it is so large that the object would need to shed some angular momentum in order to be able to collapse into a black hole. But these objects are so extended in size compared to their Schwarzschild radii, that the rotation's effect on the space-time is very small. Anything which is not extremely massive, compact, and rapidly rotating (such as a spinning black hole or millisecond pulsar) will have very weak frame dragging.
To give a very rough idea, consider that the frame dragging speed at some distance from the center of a galaxy cannot be faster than the speed the galaxy is rotating. Spiral galaxies turn out to be in approximately rigid body rotation, and for the Milky Way this is about 250km/s. Assume (very incorrectly) that the frame dragging velocity is also this fast. Then in the time it takes for a light ray to pass across the Milky Way (30kpc ~ 100k years), the frame dragging would have shifted it by just 25 parsec. This is barely resolvable over intergalactic distances, and far too small to explain the observed gravitational lensing by galaxies or clusters. This is also severely overestimating the effect the rotation could possibly have!
But what if the rotation of galaxies actually is somehow generating the observed lensing that we're attributing to dark matter, in some way general relativity doesn't predict or explain? If that's the case, then we should easily notice the discrepancy around stars and planets. Like galaxies, the Sun also has an angular momentum which would make it nearly extremally rotating if collapsed into a black hole. Yet the lensing we observe around it is consistent with the predictions from Schwarzschild (non-rotating) geometry.