I Does General Relativity explain inertia?

[mitochan]

Isn't is possible to instead imagine a large, extremely dense, hollow cylinder, fixed at a location relative to the distant stars?

The size of the radius must be less than $\frac{c}{\omega}$ where $\omega$ is angular velocity of the spinning cylinder.
Like as Kerr BHs do, the cylinder drag spacetime to cause centrifugal-like force in the cylinder.

 

[Sagittarius A-Star]

The only one I know of is the Godel universe.

Gödel calculated also a second rotating universe model:

Less well known solutions of Gödel's exhibit both rotation and Hubble expansion and have other qualities of his first model, but traveling into the past is not possible.

Source:
https://en.wikipedia.org/wiki/Gödel_metric#Cosmological_interpretation

see also:

15.5 The electromagnetic analogy in linearized GR
...
Nevertheless, it is easy to see why Thirring’s result was a victory for the Machians. The distant universe, considered as a rotating mass shell relative to a static Earth (never mind its necessarily superluminal transverse velocity!) might similarly, but this time fully, drag the local inertial frame at the Earth along with it. Already in 1913 Einstein had preliminarily obtained the above results on the rotating shell and reported them enthusiastically to an aging and unresponsive Mach.

Source (Google books preview with Firefox):
RELATIVITY -SPECIAL, GENERAL, AND COSMOLOGICAL - Wolfgang Rindler

 

[PeterDonis]

Isn't is possible to instead imagine 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

This model, as you state it, violates conservation of angular momentum. If the cylinder and the distant stars are the only stress-energy in the universe, and they start out not rotating, then the spacetime as a whole has zero angular momentum and will not show any frame dragging anywhere.

In such a model, the only way to start the cylinder spinning is to have something else spin the other way, so the total angular momentum remains zero. And in such a model, there will still be no frame dragging, because the frame dragging induced by the cylinder will be canceled by the frame dragging induced by whatever we had to make spin the other way to conserve angular momentum.

One can, however, construct a model in which the total angular momentum is nonzero at all times, and contains a cylinder which is always rotating, and distant stars which are not rotating (here "rotating" and "not rotating" are defined with respect to the asymptotically flat "rest frame" at infinity of the total center of mass). In such a model, yes, there will be frame dragging effects inside the cylinder--and also outside it.

 

[MikeGomez]

This model, as you state it,

I don't recall stating a model, but if you say I did then I must have.

violates conservation of angular momentum.

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.

One can, however, construct a model in which the total angular momentum is nonzero at all times, and contains a cylinder which is always rotating, and distant stars which are not rotating (here "rotating" and "not rotating" are defined with respect to the asymptotically flat "rest frame" at infinity of the total center of mass).

Yes.

In such a model, yes, there will be frame dragging effects inside the cylinder--and also outside it.

 

[MikeGomez]

Is it fair to say that general relativity explains the force of gravity as an inertial force but does not explain the origin of inertia itself?

I think that is accurate. Although, as Feynman famously said "physics describes the behavior of nature, it doesn't answer 'why' questions", the question on the origin of inertia is still interesting for some people.

 

[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.