Paddlewheels in a gravity well and angular momentum

[Q-reeus]

OK, we are agreed that instantaneous angular momentum of a mass at the edge of a flywheel can look weird in both SR and GR and that in SR overall magnitude of the angular momentum is invariant. I would contend the overall magnitude is also invariant in GR in both the vertical and horizontal case, so we differ here.

Here is an observation you might like to think about, that seems obvious to me but I have never seen mentioned in these forums or anywhere else for that matter (so I may be wrong). It is the analogue between horizontal and vertical measurements in GR (Schwarzschild coordinates) and transverse and longitudinal measurements in SR.

Longitudinal length contraction in SR: L = L₀ / gamma
Vertical length contraction in GR: L = L₀ / Gamma

Transverse length contraction in SR: L = L₀
Horizontal length contraction in GR: L = L₀

Longitudinal time dilation in SR: T = T₀ * gamma
Vertical time dilation in GR: T = T₀ * Gamma

Transverse time dilation in SR: T = T0 * gamma
Horizontal time dilation in GR: T = T0 * Gamma

Longitudinal acceleration in SR: a = a₀ / gamma^3
Vertical acceleration in GR: a = a₀ / Gamma^3

Transverse acceleration in SR: a = a₀ / gamma
Horizontal acceleration in GR: a = a₀ / Gamma

Longitudinal angular momentum in SR: L = L₀
Vertical angular momentum in GR: L = L₀

Transverse angular momentum in SR: L = L₀
Horizontal angular momentum in GR: L = L₀

Longitudinal angular KE in SR: KE = KE₀ / gamma
Vertical angular KE in GR: KE = KE₀ /Gamma

Transverse angular KE in SR: KE₀ / gamma
Horizontal angular KE in GR: KE₀ / Gamma

Where:

gamma = sqrt(1-v^2/c^2)

and

Gamma = sqrt(1-2GM/(rc^2))

In each case there is an analogue between Longitudinal measurements in SR and Vertical measurements in GR and between Transverse measurement in SR and Horizontal measurements in GR and all we have to do is replace gamma with Gamma. We can even obtain Gamma directly from gamma by replacing v in gamma with the local escape velocity at r, so that v = sqrt(2GM/r).

These analogous relationships have to be used with caution as there are some differences. The coordinate speed of light appear to change in GR but in not SR. The coordinate time for a light signal to go from the back of a rocket to the front is longer than the return trip in SR, while the time for a signal to go up is the same as the time to go down in GR. Having said that, with due care, we can use these analogous relationships to provide some insight into what happens in the GR case by considering the analogous SR case.

Q-reeus: Agreed it looks weird with a large translational boost in SR setting. But the magnitude overall is an invariant there, whereas in GR it's not imo.

I should qualify that by saying instead 'it's not imo always true that coordinate measured angular momentum is conserved in GR via standard SC's'.

Looking at your list yuiop, I assume all the '0' subscripted quantities are meant to be the proper values, so would expect in most cases there multiplication, not division by the relevant gamma/Gamma factors [exceptions being the last two sets involving angular KE in SR]. Anyway, when I get some real time, will look more closely at the twin masses flywheel in SC's and see if I got it wrong there or not (transverse spin axis case). Recall though you only claimed back then there was merely an average conservation of L over a complete rotation cycle, which creates problems as the implied coordinate observed fluctuations would never be locally observed. Perhaps though we have switched perspectives to some extent!

 

[yuiop]

Recall though you only claimed back then there was merely an average conservation of L over a complete rotation cycle, which creates problems as the implied coordinate observed fluctuations would never be locally observed. Perhaps though we have switched perspectives to some extent!

It is the same in SR. For a rolling wheel, the rim speed and centripetal forces are equal at every point around the rim in the rest frame of the axle, but in the road rest frame, the part in contact with the road is stationary and the part at the top is going at twice the speed of the axle implying, asymmetrical centrifugal forces and fluctuating angular momentum as a particle goes around the wheel, but it all works out in the end ;) Wheels still manage to roll without bouncing and averaged angular momentum over complete cycles is much more intuitive.

 

[jartsa]

This is not correct. Let's say the rocket is initially at rest in the Earth rest frame and the body of the rocket is spinning at 100 rpm around its long axis. At the rear of the rocket is a non rotating propulsion and crew section that is not rotating connected by a bearing to the rest of the rocket. Now the rocket is launched to say 0.8c parallel to its long axis, relative to the Earth, then in the Earth rest frame the rocket will be rotating at 60 rpm, but due to time dilation the body of the rocket will still be rotating at 100 rpm as measured by the crew in the non spinning section of the rocket.

This is a non-spinning rocket that consists of a non-spinning propulsion section and a non-spinning crew section, carrying a spinning flywheel. (the spinning section is the flywheel)

Now I agree that flywheels experience normal velocity time dilation.

Qreeus and I agree that the spinning of an axially expanding spinning thing does not experience gravitational time dilation.

We can convert the aforementioned thing into a rocket, by saying that the expansion is violent.

Qreeus, do we agree, that spinning rate is unchanged for a distant observer in this violent expansion case too?

 

[Q-reeus]

It is the same in SR. For a rolling wheel, the rim speed and centripetal forces are equal at every point around the rim in the rest frame of the axle, but in the road rest frame, the part in contact with the road is stationary and the part at the top is going at twice the speed of the axle implying, asymmetrical centrifugal forces and fluctuating angular momentum as a particle goes around the wheel, but it all works out in the end ;) Wheels still manage to roll without bouncing and averaged angular momentum over complete cycles is much more intuitive.

What - just G-R (Galilean Relativity) to worry about?! Book-keeping gets awkward for sure, and if analyzing the motion of a single particle there is periodic exchange of L between vehicle and road. The complete cure is to consider any two equally massive particles oppositely placed about the axle. There is not even a perceived net fluctuation in L, and without checking I will just assert the same holds in SR case also (otherwise a sudden crash stop at the 'wrong' angular orientation and hey presto we have gained/lost net L). Still to get round to redoing the flywheel in gravity well scene, but can say for now we got it wrong in principle before - consistency requires use of coordinate speeds normalized wrt coordinate light speeds, then used within the full SR expressions for KE and p etc. More anon.

 

[Q-reeus]

Qreeus and I agree that the spinning of an axially expanding spinning thing does not experience gravitational time dilation.

Not quite. As per earlier thread, considered as a tube of matter uniformly expanding axially, a local observer at some fixed radius from spin axis will be in a decreasing mass density region, and so redshift/clock-rate is changing. Locally perceived as increasing spin rate.

We can convert the aforementioned thing into a rocket, by saying that the expansion is violent.

The picture I now have is of 'a spinning rocket accelerating in a snug guide tube', and expelled mass/gas now spins at the same rate as the rocket.

Qreeus, do we agree, that spinning rate is unchanged for a distant observer in this violent expansion case too?

If my picture is accurate, yes to the extent relative velocity is nonrelativistic. Otherwise the situation is modulated by SR effects as per #28 - slower and continually slowing rotation for receding/broadside relative motion, speeding up within a narrowing angular range of approaching relative motion. You may find this link to other links interesting to follow: http://casa.colorado.edu/~ajsh/sr/srfs.html

 

[jartsa]

Not quite. As per earlier thread, considered as a tube of matter uniformly expanding axially, a local observer at some fixed radius from spin axis will be in a decreasing mass density region, and so redshift/clock-rate is changing. Locally perceived as increasing spin rate.

The picture I now have is of 'a spinning rocket accelerating in a snug guide tube', and expelled mass/gas now spins at the same rate as the rocket.

If my picture is accurate, yes to the extent relative velocity is nonrelativistic. Otherwise the situation is modulated by SR effects as per #28 - slower and continually slowing rotation for receding/broadside relative motion, speeding up within a narrowing angular range of approaching relative motion. You may find this link to other links interesting to follow: http://casa.colorado.edu/~ajsh/sr/srfs.html

How about if we consider a spinning, laser beams emitting disk, in a straight, light reflecting tube. Light that is emitted from the middle of the disk travels throuh the tube at speed of light. Light emitted from the rim of the disk travels through the tube slower than speed of light.

(the spinning axis and tube's symmetry axis are parallel, and all laser beams are parallel to tube's symmery axis when the laser sources are at rest)

The light going though the tube must carry angular momentum that's what's slowing it down.

Next thing to do here might be to replace lasers with electron guns ... then we might conclude that the electron population's spinning rate is unchanged when the electrons are accelerated by the electron guns ... But first we should agree about the laser light case.

 

[Q-reeus]

How about if we consider a spinning, laser beams emitting disk, in a straight, light reflecting tube. Light that is emitted from the middle of the disk travels throuh the tube at speed of light. Light emitted from the rim of the disk travels through the tube slower than speed of light.

(the spinning axis and tube's symmetry axis are parallel, and all laser beams are parallel to tube's symmery axis when the laser sources are at rest)

Getting better with describing a scenario - that much is good.

The light going though the tube must carry angular momentum that's what's slowing it down.

Are you sure? In the frame of one of the peripheral lasers on the spinning disk, which direction is the back-reaction force from that laser pointing? How does that force transform into the frame we see it from? Try here for some clues: http://en.wikipedia.org/wiki/Special_relativity#Force So do you find the disk to be changing angular momentum? Then what does conservation of angular momentum imply about the emitted light going down the tube? How about emitted frequency (and any frequency spreading going on)?
[EDIT: But as yourself, and yuiop in #38 describe, there is spiral motion. So absence of torque is not the only factor to consider - here there is dispersal of angular momentum generated prior to emission.]

Next thing to do here might be to replace lasers with electron guns ... then we might conclude that the electron population's spinning rate is unchanged when the electrons are accelerated by the electron guns ... But first we should agree about the laser light case.

No doubt fun to keep bringing up new scenarios, but maybe best to get some good grounding in the basics and then you will be able to answer all this kind of thing yourself jartsa. Sorry to disappoint but apart from being no expert myself, can't really keep this thing up too long.

 

[yuiop]

How about if we consider a spinning, laser beams emitting disk, in a straight, light reflecting tube. Light that is emitted from the middle of the disk travels through the tube at speed of light. Light emitted from the rim of the disk travels through the tube slower than speed of light.

The light from the rim of the disk only appears to be traveling slower than the speed of light relative to the tube, because it bouncing of the side of the tube walls in a spiral fashion. I guess you know that, as that is presumably why you specified a light reflecting tube. I guess you also know that if the light coming of the back of the disc forms a cylinder shape parallel to the tube when the disc is not spinning, that it will form shape when the disc is spinning.

(the spinning axis and tube's symmetry axis are parallel, and all laser beams are parallel to tube's symmery axis when the laser sources are at rest)

But not when the disc is spinning.

The light going though the tube must carry angular momentum that's what's slowing it down.

Again, only slow going down the tube in a broad sense, because of the spiral path taken by the peripheral light beams. In between reflections of the side of the tube, each photon is traveling in a straight line and traveling at the speed of light. Agree?

 

[jartsa]

The light from the rim of the disk only appears to be traveling slower than the speed of light relative to the tube, because it bouncing of the side of the tube walls in a spiral fashion. I guess you know that, as that is presumably why you specified a light reflecting tube. I guess you also know that if the light coming of the back of the disc forms a cylinder shape parallel to the tube when the disc is not spinning, that it will form shape when the disc is spinning.
But not when the disc is spinning.
Again, only slow going down the tube in a broad sense, because of the spiral path taken by the peripheral light beams. In between reflections of the side of the tube, each photon is traveling in a straight line and traveling at the speed of light. Agree?

Yeah.

And when the light is spiralling ahead, it makes one round in same time as the disk made one round, when the light left the disk. I'm talking about peripheral light beam here, which has no room to expand sideways. The spiral formed by the beam has no room to expand, I mean.

 

[jartsa]

Let us consider a very simple scenario: Beings living on a spinning planet can't use the spinning of the planet as an energy source. I guess we are all familiar with this.

So we know that bouncing up and down on the pole of a spinning planet does not cause any frictional heating, that would be caused by the planet and the bouncing object having different spinning rates.

So therefore: spinning motion is immune to gravitational time dilation and velocity time dilation.

 

[ZapperZ]

Locked, pending moderation.

Zz.