Rotation of a vertical cylinder in a gravitational field

[yuiop]

Consider the above rotating vertically aligned cylinder in Schwarzschild geometry.

Would anyone agree or disagree with the following analysis?

If a local observer at the highest end of the cylinder measures that end of the cylinder to be (say) 2 rpm, then another local observer will measure the lower end of the cylinder to be rotating faster (10 rpm in this case at the radii given in the sketch).Assume the cylinder is in equilibrium and rotating due to its own angular momentum on ideal friction free bearings).

If motors are attached to both ends of the cylinder and the cylinder in made to rotate at 2 rpm at the top and the bottom as measured by local clocks, then there will be cumulative sheer stress on the cylinder and it will eventually break.

If a flat disc initially located near the top of the cylinder is made to rotate at 2 rpm (assume the disc is an ideal flywheel without friction) and then transported slowly down to the low end of the cylinder, the disc will still be rotating at 2 rpm as measured by local clocks and will be rotating slower than the lower end of the cylinder.

From the point of view of any stationary observer, the rate of rotation of the transported disc will appear to slow down as it is slowly lowered deeper into the gravitational well.

If the cylinder was initially much higher up and then slowly lowered past a shell observer that remains at a given Schwarzschild radius, the part of the cylinder nearest the observer will appear to progressively slow down.

 

[bcrowell]

From the point of view of any stationary observer, the rate of rotation of the transported disc will appear to slow down as it is slowly lowered deeper into the gravitational well.

Everything seemed fine to me except this. I don't think the stationary observer can determine the rate of rotation of the disc when the disc is somewhere else, unless you specify some method for transmitting signals.

 

[PeterDonis]

I see a few unstated assumptions that I think it would be beneficial to make explicit:

(1) I assume you intend for the cylinder to be "hovering" such that each end remains at the fixed radius given. That means the cylinder needs to be suspended with a rocket or something equivalent--in fact, you may need multiple rockets (see next item).

(2) I assume you intend for there to be zero stress inside the cylinder (i.e., no part of the cylinder exerts any net force on any other part). The reason I assume this is that I'm not sure condition (3) below can be satisfied without it (see next item). For this to be true, the cylinder cannot be supported by a single rocket at its bottom end, unless you also idealize it to have zero mass, which doesn't seem consistent since it's traveling on a timelike worldline. With a cylinder having non-zero mass, the weight of the cylinder will cause non-zero internal stress unless each individual segment of the cylinder is supported by its own rocket, with the rocket thrust gradually decreasing from bottom to top, so the whole thing is in a state of Born rigid acceleration. For an idealized thought experiment I don't have a problem with this, but I think it needs to be explicit.

(3) I assume (or rather, I deduce from your statement of the problem) that the rotation of the cylinder is intended to be such that its entire motion is coherent with respect to Schwarzschild coordinate time. That is, if I paint a radial line (no "twist" around the cylinder) from top to bottom of the cylinder, all portions of that radial line remain radially aligned, so that they all pass a given radial reference line at the same Schwarzschild coordinate time on each revolution. This is the necessary condition for the "proper" rotation rates to vary with altitude as you describe. However, I don't think this condition can be maintained unless there is zero net internal stress in the cylinder--no part can exert any force on any other part. If there were net internal forces anywhere, the cylinder would not be in equilibrium and the radial alignment described above would not be maintained.

You may ask, what about purely radial stress, such as the weight of the cylinder itself? The problem is that I'm not sure a stress that is purely radial at one radius (say, the bottom of the cylinder) will still be purely radial at another radius (say, the top of the cylinder), because of the difference in metric coefficients with radius. I'd have to look at how the stress-energy tensor of the cylinder would transform as the radius changed.

 

[yuiop]

Everything seemed fine to me except this. I don't think the stationary observer can determine the rate of rotation of the disc when the disc is somewhere else, unless you specify some method for transmitting signals.

I was thinking of having some sort of electrical contact on the disc that sparks when it passes a similar contact on a non rotating cage enclosing the the disc. This should spark once per revolution and give an indication of the rotation speed of the disc to the distant observer. Of course the descent speed will cause a Doppler error on the signals so the transport speed would have to be very slow or alternatively the disc can be brought to rest periodically at different heights to observe its rotation rate. You could of course just paint a mark on the disc and use a very good telescope.

I see a few unstated assumptions that I think it would be beneficial to make explicit:

(1) I assume you intend for the cylinder to be "hovering" such that each end remains at the fixed radius given. That means the cylinder needs to be suspended with a rocket or something equivalent--in fact, you may need multiple rockets (see next item).

Yes.

(2) I assume you intend for there to be zero stress inside the cylinder (i.e., no part of the cylinder exerts any net force on any other part). The reason I assume this is that I'm not sure condition (3) below can be satisfied without it (see next item). For this to be true, the cylinder cannot be supported by a single rocket at its bottom end, unless you also idealize it to have zero mass, which doesn't seem consistent since it's traveling on a timelike worldline. With a cylinder having non-zero mass, the weight of the cylinder will cause non-zero internal stress unless each individual segment of the cylinder is supported by its own rocket, with the rocket thrust gradually decreasing from bottom to top, so the whole thing is in a state of Born rigid acceleration. For an idealized thought experiment I don't have a problem with this, but I think it needs to be explicit.

I don't have any problem with an idealised Born rigid multiple rocket arrangement, but possibly that is not even required.

(3) I assume (or rather, I deduce from your statement of the problem) that the rotation of the cylinder is intended to be such that its entire motion is coherent with respect to Schwarzschild coordinate time.

I am not sure this is an absolute requirement. A mark on the top circular face of the cylinder and another mark on the lower circular face of the cylinder should suffice to compare rotation rates of the top and bottom faces of the cylinder without the top and bottom marks being aligned. For example consider a purely Newtonian experiment with a motor drive, a disc connected directly to the motor and a second disc suspended below the first by an elastic band. There will be torsion and stress in the elastic band, but an equilibrium will be achieved where eventually the top disc and the lower disc will be rotating at exactly the same rate despite the non-rigidity and stress present in the elastic band and the discs will not necessarily be aligned. However, the torsion stress may be a problem in the lowered cylinder experiment - see below.

That is, if I paint a radial line (no "twist" around the cylinder) from top to bottom of the cylinder, all portions of that radial line remain radially aligned, so that they all pass a given radial reference line at the same Schwarzschild coordinate time on each revolution. This is the necessary condition for the "proper" rotation rates to vary with altitude as you describe. However, I don't think this condition can be maintained unless there is zero net internal stress in the cylinder--no part can exert any force on any other part. If there were net internal forces anywhere, the cylinder would not be in equilibrium and the radial alignment described above would not be maintained.

I think this is only required for the experiment where the stationary observer observes the slow down of the local part of the cylinder as it is lowered past the observer, where any spiralling of the reference line would distort the measurements. This can be circumvented by periodically bring the cylinder to rest to take the measurements, in which case any spiralling of the reference line will not matter.

You may ask, what about purely radial stress, such as the weight of the cylinder itself? The problem is that I'm not sure a stress that is purely radial at one radius (say, the bottom of the cylinder) will still be purely radial at another radius (say, the top of the cylinder), because of the difference in metric coefficients with radius. I'd have to look at how the stress-energy tensor of the cylinder would transform as the radius changed.

I am not sure how significant this complication is. However, this sort of complication can arise in other measurements over extended distances in a gravitational field. For example if we try to measure vertical/radial ruler distances we have to take into account that if the ruler is supported at the top it will stretch and if it is supported at the bottom it will compress. A stretched or compressed ruler is not a very good measuring device and the experiment has to be carefully designed to allow for this aspect.

I don't think any of the issues are insurmountable for an idealised in-principle thought experiment.

 

[yuiop]

An interesting extension to the original experiment occurs to me. Let us imagine we have a similar rotating cylinder with its lower end at r=2.020202M and its upper end at r = 1000000M. What will a stationary observer that remains at r=2.020202M observer if the cylinder is released and allowed to free fall?

One point of view is that the lower end of the cylinder will stop rotating when it arrives at the event horizon, but will this halt in rotation be transmitted up the cylinder so that our stationary observer outside the event horizon sees the upper part of the cylinder stop rotating? I think not. Mechanical forces are transmitted up the cylinder at the speed of sound in the cylinder material while the lower parts are (locally) approaching the speed of light. Secondly, as far as the stationary observer is concerned, it takes an infinite amount of his local time for the lower end of the cylinder to fall the short distance from r=2.020202M to r=2M. The top of the cylinder will arrive at his location long before the lower part reaches the event horizon from his point of view and the cylinder will appear to be subject to extreme length contraction. So it seems we are unable to use the long rotating cylinder to "probe" what is happening at the event horizon. Any thoughts?

 

[Passionflower]

Would anyone agree or disagree with the following analysis?

So I read correctly that you state that the vertically stationary cylinder increases its rotation speed as measured by a local stationary observer for lower values of r?

Could you show the mathematics? To me it is not intuitively clear (which likely says something about my lack of good intuition).

 

[PeterDonis]

One point of view is that the lower end of the cylinder will stop rotating when it arrives at the event horizon, but will this halt in rotation be transmitted up the cylinder so that our stationary observer outside the event horizon sees the upper part of the cylinder stop rotating? I think not.

I think you're correct that no signal can be transmitted up the cylinder if the lower end is at (or below) the horizon, since any signal sent from the lower end of the cylinder upward is limited to the speed of light, and signals emitted radially outward at the horizon traveling at the speed of light stay at the horizon forever--they never get to any larger radius.

However, I don't agree that the "point of view" that says the cylinder will stop rotating when it arrives at the horizon is correct. The "hovering" viewpoint, which you describe further on in your post (in the quote I'll give below), does not cover the horizon, because the coordinate system on which it is based, exterior Schwarzschild coordinates, becomes singular there. To analyze what happens at the horizon you have to switch to a viewpoint that covers it, such as the viewpoint of an infalling observer (e.g., Painleve coordinates), and from such a viewpoint, the cylinder does *not* stop rotating at the horizon. See below.

Mechanical forces are transmitted up the cylinder at the speed of sound in the cylinder material while the lower parts are (locally) approaching the speed of light.

This is more or less the "hovering" viewpoint of what I stated above, that signals emitted outward at the horizon never get to any larger radius--so as one gets closer and closer to the horizon, it takes longer and longer for signals to get out, approaching the limit of "infinitely long" at the horizon itself. (We can idealize the cylinder as having the maximum possible stiffness allowed by relativity, so that the speed of sound in the cylinder equals the speed of light; this allows us to assume that mechanical forces are transmitted at the speed of light in the cylinder, but no faster.)

Secondly, as far as the stationary observer is concerned, it takes an infinite amount of his local time for the lower end of the cylinder to fall the short distance from r=2.020202M to r=2M. The top of the cylinder will arrive at his location long before the lower part reaches the event horizon from his point of view and the cylinder will appear to be subject to extreme length contraction. So it seems we are unable to use the long rotating cylinder to "probe" what is happening at the event horizon. Any thoughts?

This is true, but again, it does *not* imply that the cylinder stops rotating at the horizon; it only implies that the "hovering" viewpoint can't cover the horizon. From the viewpoint of an infalling observer riding alongside the cylinder and free-falling with it across the horizon, the cylinder keeps rotating at the same rate the whole time; in fact, the infalling observer does *not* see the "time dilation" effect that the hovering observer sees--both ends of the cylinder are always rotating at the same rate as far as the infalling observer is concerned. (Note that this involves another unstated assumption: that tidal gravity can be neglected over the length of the cylinder, which means the black hole has to have a large enough mass for tidal gravity to be negligible over the range of radial coordinates that are covered by the cylinder at any instant of the cylinder's proper time. If tidal effects can't be neglected, they will either stretch the cylinder or cause internal stresses in it, both of which may change how it behaves.)

 

[yuiop]

So I read correctly that you state that the vertically stationary cylinder increases its rotation speed as measured by a local stationary observer for lower values of r?

Could you show the mathematics? To me it is not intuitively clear (which likely says something about my lack of good intuition).

Using units of c=M=G=1:

At r = 2.666666M the time dilation factor is 1/sqrt(1-2/2.666666) = 2

At r = 2.020202M the time dilation factor is 1/sqrt(1-2/2.020202) = 10

This implies that clocks at the lower radius are running 5 times slower than the higher radius. The important interpretational aspect is that you believe that the clocks lower down are "really" running slower than the clocks higher up in absolute terms. To the Schwarzschild observer at infinity, the cylinder is rotating at 1 rpm. To the observer at r=2.666666M the entire cylinder is rotating at 2 rpm simply because his clock is running 2 times slower than the clock at infinity. To the observer at r=2.020202M the entire cylinder is rotating at 10 rpm again due to the slow running of his clock. Any given observer sees the top and the bottom of the cylinder rotating at the same rate, but different observers will disagree on what that rate is. I think you will agree that in order for the cylinder not to be torn apart, there must be a sense in absolute terms that the top and the bottom of the cylinder are rotating at the same speed, because any difference in rotation rates is cumulative and eventually ties the cylinder in knots. The difficult part is proving when and how all parts are rotating at the same "absolute" rate and which observer if any measures this.

Now I am not sure how you go about proving that clocks lower down actually run slower in absolute terms but if you take that the view that apparent slow running of the clocks lower down is just a coordinate illusion, then the difference is more than just philosophical, because the outcome of the predictions for this experiment are different if you take the the alternative interpretation (slow running clocks is a coordinate illusion). The interpretation outlined in this thread is that the slow down of clocks lower down according to the observer at infinity is a real effect and not just a coordinate artefact. The nice thing about this thought experiment is that it really exposes the true nature of gravitational time dilation. Ideally it would be nice to come up with a thought experiment that produces a contradiction if you take the alternative "coordinate artefact" interpretation.

By the way, do you agree with these predictions: ?

An observer lower down has a machine gun that fires bullets once per second (according to his local clock) up towards the higher observer. The bullets arrive at a rate of one every 5 seconds according to the clock of the observer at the higher level. The upper observer has a similar machine gun that fires bullets once per second according to his local clock. These bullets arrive at the lower observer at a rate of 5 bullets every second according to the clock of the lower observer. (If these two observers were having a gun fight, the lower observer would be at a severe disadvantage in terms of damage per second.) Agree? (The coordinate artefact interpretation would predict that the bullets arrive at one per second according to local clocks at both the top and bottom.)