Frame Dragging Rate: Mass Proportionality?

[e2m2a]

TL;DR

What is the relation between the rate of frame dragging and mass source?

Curious about frame dragging. Is the rate at which frame dragging occurs proportional to mass? For example, when the Earth rotates it has been confirmed space drags with the earth. I assume the rate is very small compared to the rotation rate of the earth. However, if the mass of the Earth were to increase such that we have a rotating black hole, does the rate at which the space rotates approach the rate of the rotating black hole?

 

[PeterDonis]

Is the rate at which frame dragging occurs proportional to mass?

Not directly. It's proportional to the angular momentum, as shown in the formula here:

https://en.wikipedia.org/wiki/Lense–Thirring_precession#Precession

The extra correction term basically says that the further you are off the equatorial plane of the rotating body, the smaller the effect. Many discussions ignore this aspect and implicitly assume that an orbit in the equatorial plane is what is being discussed.

If the mass of the Earth were to increase such that we have a rotating black hole, does the rate at which the space rotates approach the rate of the rotating black hole?

A black hole's angular momentum works like any other angular momentum; it's not different because it's a black hole. However, the rate of frame dragging varies as the inverse cube of the distance from the center of the body (as shown in the formula linked to above), and with a body like the Earth it's impossible to get very close to the center, in GR terms (i.e., distance from the center of the same order of magnitude as the mass of the body), because the body itself is in the way. The reason frame dragging can get much stronger around a black hole is that you can get a lot closer to the center and still be outside the body itself.