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

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?

• Look, I know that the popular articles on this idea were really enthusiastic, and a lot of non-physicists are excited, but I promise you that frame dragging is a well-understood effect and it's clearly not big enough. And Zee's result is exactly in agreement with what I said. Since $v/R$ is the actual angular velocity, and $GM/R^2 \sim v^2$ by the usual expression for centripetal force, your result just says $\omega_{\mathrm{ind}} \sim (v^2/c^2) \omega$. And even if it was only suppressed by one power of $v/c$, that's still $10^{-3}$, which is nowhere near big enough. Jul 31, 2021 at 4:50
• You may ask, but then does that imply every journalist who wrote about this, and every programmer who upvoted it on Hacker News, every single one of those people was completely misguided? Yes. Actual knowledge is rare. Almost everything is done by BSing on the spot. People willing to pontificate on general relativity outnumber those who have read a single textbook on the subject by at least a thousand to one. Jul 31, 2021 at 4:52
• To add to @knzhou's comments, the burden of proof is not on people doing simulations to stop what they are doing and check the claim of one paper. If Ludwig is convinced that he has found a way around the conventional wisdom that frame dragging is too small to explain rotation curves (narrator: he hasn't), then the burden is on Ludwig to explain (with convincing evidence) why his critics are wrong. This burden of proof is an unspoken rule in science but crucial; no one would get anything done if every bold claim had to be rigorously checked by the community. Jul 31, 2021 at 5:00
• "This does not inspire confidence". Opinions? Everyone has got one! Seriously though, most GR effects become insignificant beyond about 100 Schwarzschild radii. Jul 31, 2021 at 11:29
• Thanks, that makes sense. Jul 31, 2021 at 22:23

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): $$(GM/Rc²) \sim v^2/c^2$$