Is it true that frame-dragging (as applied to galactic rotation curves) goes as second order in $v/c$ rather than $(GM/Rc²)(v/R)$?

Question

I have seen various people (see e.g. comments here) dismissing this article using the argument that GR frame-dragging is second order in $v/c$ and therefore insignificant because $v << c$ for galactic rotation. However, I read in Zee's "Einstein gravity in a nutshell" that the induced angular velocity (i.e. the component of angular velocity with no angular momentum) goes as $ω \sim (GM/Rc²)(v/R)$, which appears 1st order in $v/c$.

Given the effective value of $(GM/Rc²)$ for a rotating disk is likely to be much larger than that of a sphere of comparable radius and mass, can Ludwig's work can be dismissed without running full GR numerical simulations on the galaxies Ludwig considers?

Answer 1

As has been pointed out in the comments, since the gravitational field strength (GM/r^2) is equal to the centripetal acceleration (v^2/r) for test masses (before any additional angular velocity resulting from frame-dragging is taken into account): $(GM/Rc²) \sim v^2/c^2 $

This means that if the gravitational field was strong enough for frame-dragging to be important, the galaxy would have orbiting masses with relativistically significant rotational velocities, irrespective of the mass distribution of the galaxy.