Does General Relativity explain inertia?

[Nugatory]

I didn't mean to imply non-conservation of angular momentum. If you point out to me where I said that, then I will edit in a correction.

Here:

Now if we hold the cylinder in a fixed location and bring it up to an extremely rapid spin...

It wasn’t spinning, we started it spinning, its angular momentum changed.

 

[MikeGomez]

Here:

It wasn’t spinning, we started it spinning, its angular momentum changed.

I couldn't figure out how to edit the post, but let's say for example that it were spun up by using rocket engines attached to it. In that case the change in angular momentum is balanced by considering the momentum of the exhaust of the engines.

But from the reference frame of the bucket (and not being able to look out and see if the dense rotating disk is there or not) is there an experiment that can determine whether the water in the bucket rises due it spinning, or due to 'frame-dragging' inertial effects of the rotating disk around it?

 

[PeterDonis]

This...

a large, extremely dense, hollow cylinder, fixed at a location relative to the distant stars? The walls of the cylinder would be very thick, and the hollow in the center would be small but large enough to suspend a bucket of water, which itself would be held fixed and not rotating with respect to the distant stars.

Now if we hold the cylinder in a fixed location and bring it up to an extremely rapid spin

...is a model.

let's say for example that it were spun up by using rocket engines attached to it. In that case the change in angular momentum is balanced by considering the momentum of the exhaust of the engines.

Not just momentum, angular momentum; the rocket exhaust acquires an equal in magnitude and opposite in sign angular momentum to the cylinder. So the total angular momentum remains zero. And hence there is no frame dragging by the cylinder, because the opposing frame dragging by the rocket exhaust cancels it out.

 

[PeterDonis]

I couldn't figure out how to edit the post, but let's say for example that it were spun up by using rocket engines attached to it. In that case the change in angular momentum is balanced by considering the momentum of the exhaust of the engines.

But from the reference frame of the bucket (and not being able to look out and see if the dense rotating disk is there or not) is there an experiment that can determine whether the water in the bucket rises due it spinning, or due to 'frame-dragging' inertial effects of the rotating disk around it?

You can test whether the bucket is rotating locally by comparing its motion with the motion of a set of gyroscopes that move along the same worldline as the bucket, and seeing whether the bucket rotates with respect to the gyroscopes.

The effect that is called "frame dragging" is a difference between that local definition of "rotating" vs. "non-rotating" and a non-local definition using some distant object (such as the distant stars) as a reference. So there is never any local way to test for "frame dragging". It is always a matter of comparing local observations with non-local ones. For example, Gravity Probe B, the satellite that detected frame dragging due to the Earth's rotation, was non-rotating by the local gyroscope criterion above.

In the case of a cylinder that is rotating for all time (so the total angular momentum of the spacetime is nonzero for all time), the frame dragging effect of the cylinder would be observable as the bucket not rotating locally with respect to gyroscopes as described above, but rotating with respect to the distant stars.