Will a spherical mass be set in motion by a spherical shell rotating around it?

In summary, in general relativity, rotation of mass can create framedragging effects, just like linear motion. This is due to the off-diagonal components in the mass-energy-momentum tensor. This means that around Bonnor beams and rotating masses, there will be framedragging. However, it is not possible to have a perfectly spherical rotating shell due to the effects of self-gravity. This means that the rotation of a spherical mass would not cause it to start rotating, as it would already be in the shape dictated by its self-gravity before any other effects could manifest. Further research may be needed to fully understand the implications of this.
  • #1
JandeWandelaar
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In general relativity, rotation of mass gives rise to framedraging effects, just like linear motion does, because of the off-diagonal components in the mass-energy-momentum tensor. So around Bonnor beams there is framedragging, as well around a rotating mass.

Now imagine a spherical rotating shell with a spherical mass in it center. We would expect the mass to start rotating, but it also seems it needs a kind of "handle" for the effect to "get a grip" on the ball. Say two diametrically positioned extra pieces of mass attached to the ball on the equator.

I'm not sure if this is so or not.
 
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  • #2
A (fully symmetrical) spherical shell has no gravitational influence on what is inside it.
 
  • #3
But a rotating spherical shell is not a spherically symmetric gravitational source, since angular momentum varies with latitude.

I don't know the answer to the question, but I don't think it can be dismissed on trivial grounds. I think the answer may be to solve it numerically and see. I recall @pervect had looked at rotating rings a bit, which may be relevant.
 
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  • #4
JandeWandelaar said:
Now imagine a spherical rotating shell
There is no such thing. A rotating shell will not be spherical. A spherical shell will not be rotating.
 
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  • #5
That's true. I meant almost spherical.
 
  • #6
PeterDonis said:
There is no such thing. A rotating shell will not be spherical. A spherical shell will not be rotating.
Can't we assume an ellipsoid shell rotating, so it becomes spherical?
 
  • #7
PeterDonis said:
There is no such thing. A rotating shell will not be spherical. A spherical shell will not be rotating.
Just to be clear, do you mean that it's impossible to find a spacelike surface with the geometry of a sphere in a "rotating" axisymmetric spacetime? Or that a material surface with that spatial geometry will not be a spherically symmetric source due to its rotation?
 
  • #8
PeterDonis said:
There is no such thing. A rotating shell will not be spherical. A spherical shell will not be rotating.
Can't we assume an ellipsoid shell rotating, so it becomes spherical?
 
  • #9
JandeWandelaar said:
Can't we assume an ellipsoid shell rotating, so it becomes spherical?
No. A non-rotating shell that is not spherical will be pulled into a spherical shape by its self-gravity. A rotating shell that is spherical will be pulled into an ellipsoidal shape by its self-gravity.

Objects whose self-gravity is negligible can maintain other shapes (for example, asteroids often have highly irregular shapes), but such objects also have negligible frame dragging (negligible by orders of magnitude more than their self-gravity), so they aren't suitable objects for this thread's discussion anyway. Any object that has enough frame dragging to be useful for the scenario you propose also has enough self-gravity for what I said at the start of this post to be true.
 
  • #10
Ibix said:
Just to be clear, do you mean that it's impossible to find a spacelike surface with the geometry of a sphere in a "rotating" axisymmetric spacetime? Or that a material surface with that spatial geometry will not be a spherically symmetric source due to its rotation?
See my post #9.
 
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  • #11
PeterDonis said:
See my post #9.
PeterDonis said:
No. A non-rotating shell that is not spherical will be pulled into a spherical shape by its self-gravity. A rotating shell that is spherical will be pulled into an ellipsoidal shape by its self-gravity.

Objects whose self-gravity is negligible can maintain other shapes (for example, asteroids often have highly irregular shapes), but such objects also have negligible frame dragging (negligible by orders of magnitude more than their self-gravity), so they aren't suitable objects for this thread's discussion anyway. Any object that has enough frame dragging to be useful for the scenario you propose also has enough self-gravity for what I said at the start of this post to be true.
The effect might be negligible, say for an ellipsoid shell, but there is enough time to make the effects appear. I'm not sure though if a spherical mass will be set in motion.
 
  • #12
JandeWandelaar said:
The effect might be negligible, say for an ellipsoid shell, but there is enough time to make the effects appear.
You're missing the point. The self-gravity effect that forces a rotating shell to be an ellipsoid and a non-rotating shell to be spherical is stronger, by many orders of magnitude, than the frame dragging effect that would cause other objects to rotate due to the shell's rotation. So long before any effects on any other objects would appear, the shell itself would be in the shape that its rotation (or lack thereof) forces it to be in by its self-gravity.
 
  • #13
PeterDonis said:
You're missing the point. The self-gravity effect that forces a rotating shell to be an ellipsoid and a non-rotating shell to be spherical is stronger, by many orders of magnitude, than the frame dragging effect that would cause other objects to rotate due to the shell's rotation. So long before any effects on any other objects would appear, the shell itself would be in the shape that its rotation (or lack thereof) forces it to be in by its self-gravity.
Yes, that's clear. But what effect the rotating shell has on an object inside at rest?
 
  • #14
JandeWandelaar said:
Yes, that's clear.
Ok, good.

JandeWandelaar said:
what effect the rotating shell has on an object inside at rest?
Frame dragging due to a rotating shell can cause an object inside the shell to rotate.
 
  • #15
PeterDonis said:
Ok, good.Frame dragging due to a rotating shell can cause an object inside the shell to rotate.
I'm not sure if a spherical mass gets accelerated though.
 
  • #16
JandeWandelaar said:
I'm not sure if a spherical mass gets accelerated though.
Accelerated how?
 
  • #17
PeterDonis said:
Accelerated how?
Angular. Is seems that if the mass is spherical, the framedragging can't "get a grip".
 
  • #18
JandeWandelaar said:
Angular. Is seems that if the mass is spherical, the framedragging can't "get a grip".
Whgat does that even mean?

You've been told several times that the setup you seek is impossible. We're in a "Yeah, but if you could way February..."

I am assuming, since this is at I-level, you understand the basics of GR. IKn GR, the source is not mass, but the stress-energy tensor. A rotating sphere has, by virtue of its rotation, a stress-energy tensor with substantial terms other than T00. I am assuming you know that.

The fact that T00 might be spherically symmetric does not mean T is. Indeed, by positing a situation ahat is not spherically symmetric but axially symmetric, imposing a spherical symmetry makes no sense.

You want to make a spherical approximation, and then ask about the effect of non-spherical terms. That is, you want the non T00 terms to simultaneously small enough to ignore and large enough to have an effect. You can't have it both ways.
 
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  • #19
Vanadium 50 said:
Whgat does that even mean?

You've been told several times that the setup you seek is impossible. We're in a "Yeah, but if you could way February..."

I am assuming, since this is at I-level, you understand the basics of GR. IKn GR, the source is not mass, but the stress-energy tensor. A rotating sphere has, by virtue of its rotation, a stress-energy tensor with substantial terms other than T00. I am assuming you know that.

The fact that T00 might be spherically symmetric does not mean T is. Indeed, by positing a situation ahat is not spherically symmetric but axially symmetric, imposing a spherical symmetry makes no sense.

You want to make a spherical approximation, and then ask about the effect of non-spherical terms. That si, you want the non T00 terms to simultaneously small enough to ignore and large enough to have an effect. You can't have it both ways.

Suppose there is a cylindrically symmetric, rotating shell. In the center we have a massive sphere of uniform mass, at rest. I have the feeling it won't start rotating, unless we, say, attach two diametrically opposed pieces of mass on the equator of the spherical mass, which can provide a torque.
 
  • #20
JandeWandelaar said:
I have the feeling
Good for you?

You seem to be asking us to calculate any scenarioo you come up with. Usually we ask that people show some effort first.
 
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  • #21
Vanadium 50 said:
Good for you?

You seem to be asking us to calculate any scenarioo you come up with. Usually we ask that people show some effort first.
 
  • #22
I have found the answer. It appears the calculation has been done by Thirring. But thanks anyway.
 
  • #23
JandeWandelaar said:
Angular. Is seems that if the mass is spherical, the framedragging can't "get a grip".
You're thinking of it wrong. Frame dragging is not a "force", any more than gravity itself is a force in GR. Frame dragging is a property of the geometry of spacetime: it describes what happens in the spacetime around a rotating object, or inside a rotating shell, to objects that are in free fall, unaffected by any forces. So it doesn't need to "get a grip" on any object.
 
  • #24
Yes, I know. But how can a spherical mass start rotating? It can't. I found a link to a nice paper by Thirring. A jewel! Here.

it's in German but I think it says, in math, that the sphere can't get in motion. Only when already in motion it seems to accelerate. If we put two small pieces of mass on the sphere then the sphere might have acceleration from rest.
 
  • #25
JandeWandelaar said:
how can a spherical mass start rotating?
Again you are thinking of it wrong. The spherical mass doesn't have to "start rotating". It just needs to free fall. Frame dragging means that, as it free falls, it will rotate relative to an observer at infinity. But it will not feel any force.

JandeWandelaar said:
it's in German but I think it says, in math, that the sphere can't get in motion. Only when already in motion it seems to accelerate.
As far as I can tell, Thirring is talking here about the kind of frame dragging observed in the motion of an object in a free-fall orbit in the space outside of a rotating body--for example, the kind that was tested experimentally by Gravity Probe B.

However, the possible effects of frame dragging are much more general than this.

JandeWandelaar said:
If we put two small pieces of mass on the sphere then the sphere might have acceleration from rest.
This is speculation on your part, which is out of bounds for discussion here. (It is also speculation based on your erroneous idea that frame dragging is a "force" that has to "get a grip" on an object. I strongly suggest that you discard this erroneous idea.)
 
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  • #26
PeterDonis said:
Again you are thinking of it wrong. The spherical mass doesn't have to "start rotating". It just needs to free fall. Frame dragging means that, as it free falls, it will rotate relative to an observer at infinity. But it will not feel any force.As far as I can tell, Thirring is talking here about the kind of frame dragging observed in the motion of an object in a free-fall orbit in the space outside of a rotating body--for example, the kind that was tested experimentally by Gravity Probe B.

However, the possible effects of frame dragging are much more general than this.This is speculation on your part, which is out of bounds for discussion here. (It is also speculation based on your erroneous idea that frame dragging is a "force" that has to "get a grip" on an object. I strongly suggest that you discard this erroneous idea.)

I don't see it as a force. I spoke of free fall. The article treats a mass inside a slowly rotating shell. There is no torque to let the mass move from rest, it appears. How can a ball start rotating freely? Thanks for your help. It's clear now.
 
  • #27
JandeWandelaar said:
How can a ball start rotating freely?
Because the rules of geometry are different here - straight lines are no longer the "natural" unaccelerated motion. In a spacetime with frame dragging the "natural" motion is to rotate with respect to distant observers. You would have to apply a force to the ball to stop it doing so.
 
  • #28
Ibix said:
Because the rules of geometry are different here - straight lines are no longer the "natural" unaccelerated motion. In a spacetime with frame dragging the "natural" motion is to rotate with respect to distant observers. You would have to apply a force to the ball to stop it doing so.
But how do you apply force to a sphere?
 
  • #29
JandeWandelaar said:
But how do you apply force to a sphere?
Why does it matter? Your original question was if it would rotate. Assuming there's frame dragging (which I don't think we've proved yet), the answer is yes.
 
  • #30
To stop it from accelerating you have to apply force. But you have to attach things on the surface for that. I think though that the answer is given in the linked article. From rest a z-axis symmetrical mass can't start rotating by a massive -axi symmetric shell
 
  • #31
So we have a spherical, or possibly cylindrical mass in rotation, with a spherical mass inside, or possibly outside it. Do I have this right?
 
  • #32
Vanadium 50 said:
So we have a spherical, or possibly cylindrical mass in rotation, with a spherical mass inside, or possibly outside it. Do I have this right?
Yes. A spherical mass can't be set in motion if its in the center of the "spherical" shell. If the spgerical mass doesn't reside in the center, it will start to move around.
 
  • #33
JandeWandelaar said:
Yes. A spherical mass can't be set in motion if its in the center of the "spherical" shell. If the spgerical mass doesn't reside in the center, it will start to move around.
From memory, the Lense-Thirring metric isn't valid on the rotation axis. IIRC, it's an approximation valid in the limit of small angular momentum and outside the mass. So if that's what you are relying on for your conclusion, it's not valid.
 
  • #34
Ibix said:
From memory, the Lense-Thirring metric isn't valid on the rotation axis. IIRC, it's an approximation valid in the limit of small angular momentum and outside the mass. So if that's what you are relying on for your conclusion, it's not valid.
Not sure what you mean by what I rely on. I rely on my intuition that a spherical mass inside the shell won't start to rotate. I think that only if you put small pieces of mass on it, at the equator, it will start rotating.
 
  • #35
JandeWandelaar said:
I rely on. I rely on my intuition
We know. And your intuition was developed in a non-relativistic world. That's why we calculate. It appears that you have moved away from asking a question to pushing your point of view: that you intuition is somehow superior to the calculations of people who can do them. Don't go there. Ask questions, sure - convince us that you';re right based on intution? Nope. This is science.

JandeWandelaar said:
Yes.
You were presented multiply incompatible options. Which one do you mean? "Yes" is not an acceptable answer.

Furthermore, your intuition is wrong. Consider two concentric spherical shells. Replace the outer shell with a swarm of satellites in circular orbits. This is a zillion Gravity Probe B experiments. Now, set the inner shell spinning. Frame dragging will cause each and every satellite to precess.

From the point of view of a distant observer, the outer shell has gained angular momentum. Where did it comes from? The only place possible is the inner shell. And since angular momentum is conserved (you can add "as measured by a distant observer in Minkowski space" if you like) they inner sphere must slow down. So unquestionably the outer shell exerts a torque on the inner shell.

(This torque is, of course, tiny compared to the torque in launching the satellite swarm in the first place, and for planet-sized objects, the moments of inertia are many, many orders of magnitude larger than the satellite swarm)

My question is did you not even look up the Wikipedia article before coming here and advocating your own wrong point of view? Or did you read it and pretend you didn't because it doesn't support it? I know neither option is particularly positive - I can't help that.

Finally, the question you initially posed is in the section "Lense-Thirring Effect inside a rotating shell" in the article https://en.wikipedia.org/wiki/Frame-dragging;
 
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