Tall rotating cylinder near a black hole

[WannabeNewton]

The proper angular velocity of the disks will stay the same, the coordinate angular velocities will be different.

What is your definition of the "proper angular velocity" of a given disk? Are you, for example, representing the rotating disk by a time-like congruence and referring to the twist/vorticity vector of the congruence as the "proper angular velocity"? I have personally never seen the term "proper angular velocity" before which is why I'm asking. Cheers.

 

[pervect]

What is your definition of the "proper angular velocity" of a given disk? Are you, for example, representing the rotating disk by a time-like congruence and referring to the twist/vorticity vector of the congruence as the "proper angular velocity"? I have personally never seen the term "proper angular velocity" before which is why I'm asking. Cheers.

I don't think I've seen it used before either, but it would just be radians per unit of proper time. To be complete I need to add that it's proper time measured at the disks center.

 

[jartsa]

I'm walking at constant proper speed, balancing a tall Born rigid vertical rod on my head. The terrain consists of uphills and downhills, the slope is always 45%.

If this stroll happens in an uniform gravity field, the altitude changes will cause the coordinate velocity of myself and the coordinate velocity of the upper end of the rod to change the same amount. So the rod stays balanced.

In a non-uniform gravity field, like the gravity field of the Earth, the coordinate velocity of the upper end of the rod does not change enough to keep the rod balanced.
To keep the rod vertical, a force that causes an acceleration a1-a2 must push the upper end of the rod.
a1 = coordinate acceleration of lower end
a2 = coordinate acceleration of the upper end

 

[jartsa]

I'm walking at constant proper speed, balancing a tall Born rigid vertical rod on my head. The terrain consists of uphills and downhills, the slope is always 45%.

If this stroll happens in an uniform gravity field, the altitude changes will cause the coordinate velocity of myself and the coordinate velocity of the upper end of the rod to change the same amount. So the rod stays balanced.

In a non-uniform gravity field, like the gravity field of the Earth, the coordinate velocity of the upper end of the rod does not change enough to keep the rod balanced.
To keep the rod vertical, a force that causes an acceleration a1-a2 must push the upper end of the rod.
a1 = coordinate acceleration of lower end
a2 = coordinate acceleration of the upper end

So let's continue the analysis:

The aforementioned force pushes the upper end of the rod a time t, so the momentum change is F*t. From the momentum change we can easily calculate the change of angular momentum relative to some point.

So the change of (angular) momentum of a lifted vertical Born rigid pole, whose lower end moves at constant proper horizontal velocity, seems to be non-problematic: The (angular) momentum relative to a static point decreases.

Now let's consider a person sliding on a frictionless surface, lifting a pole, keeping it vertical.
The person's proper velocity relative to a static point increases. (Because the person is providing the force F)See, we are slowly getting closer to the original problem. But there's one error I have made: I said that if the gravity field is uniform, then a lifted and horizontally moving vertical rod stays balanced without any extra force. That is incorrect for the following reason:

If an object is moving at 1 % of local speed of light, and we lift that object upwards, then it moves at 1% of the new local speed of light, which is a different coordinate speed, so the object's coordinate speed tends to increase, so if it's a tall object, it tends to tilt forwards.

 

[yuiop]

My intuition is rather similar to Peters. If we have a stack of thin, spinning disks, with bearings between them so tha they don't affect each other's motion, if we put them on an Einstein's elevator, when the elevator starts accelerting the disks will naturally all spin at different rates. In the accelerated frame this will be seen as being due to "gravitational time dilation", the top disks will spin slower. So the disks won't all stay lined up.

The proper angular velocity of the disks will stay the same, the coordinate angular velocities will be different.

Agree.

If we try to make the disks all move a the same coordinate angular rate so that they stay "lined up", say by threading rods of some sort through them to keep them aligned, the rods will apply torques to keep the disks all spinning at the same (coordinate) rate. This means the proper rotational rate of the disks will change, and this change will violate Born rigidity.

So we're basically back to the usual paradoxes in trying to define "rigid rods"(the rods which we thread through the disks to keep them rotating at the same rate) which don't have any satistfactory relativistic definition.

Rather than use rods to apply torque, let's individually accelerate the rotation rate of the lower discs so that they all have the same coordinate angular velocity but different proper angular velocities. The discs remain separated by perfect bearings and so are independent. Although there will be stresses during the realignment, the final result (at constant altitude) will be stable, with all discs having the same coordinate velocity. (I am for now ignoring the Herglotz-Noether effect which I have branched off to a separate thread.) Now when we lower this assembly further into the gravitational well, the coordinate angular velocities will change relative to each other, but the proper angular velocities will remain constant. Agree?

Now since the proper angular velocities remain constant we might think that no actual physical change has occurred. However, consider this thought experiment using differential gears:

In the above assembly, the two large Sun gears rotate about the vertical axis. The 4 smaller planet gears only rotate about their own axes if the two Sun gears rotate at different rates. We could attach sensors to the planet gears that transmit the rotation rate of the planet gears about their own axes. Initially the Sun gears are set to rotate clockwise (as seen from above) at the same coordinate rate as each other (as measured by any observer stationary relative to the gravitational field) and the planet gears are not rotating about their own axes.

Now we lower the assembly further into the gravitational well and we should observe that the planet wheels start rotating clockwise (as seen from the centre of the assembly) about their own axes, as the lower Sun wheel slows down relative to the upper Sun wheel. When we raise the assembly back up to its initial altitude, the planet gears once again stop rotating about their own axes. If we continue to raise the assembly higher than its initial altitude, then the planet wheels start to rotate anticlockwise. Agree?

From a mechanical design point of view, the Sun wheels could be larger and thicker relative to the planet wheels, so they act as flywheels and rotating very slowly (e.g. around 2 rpm). Since friction is unavoidable in this relatively complex mechanism, motors could be attached individually to the the top and bottom Sun wheels, that are set up to maintain constant proper angular velocity of the Sun wheel they are attached to. These proper angular velocities of the Sun wheel will not be the same as each other when there is a non-negligible difference in gravitational acceleration between the top and bottom (when the Sun wheels have the same coordinate velocity). It may help to stack a series of these assemblies on top of each other to obtain a greater difference in gravitational potential between the top and bottom.