# Constant plane hypothesis

• yuiop

#### yuiop

During discussions in other threads I have been coming to the conclusion that the idea proposed here may be true.

Consider a plane with constant z coordinates. All measurements of length are made relative to the x and y coordinates that lie on the plane. It seems that all measurements of length, time, velocity and acceleration (linear or angular) that are made entirely within the plane, by observers that remain on the plane, are independent of any linear velocity or acceleration in the z direction.

Extensions:

a) This remains true even if the plane is rotating with constant proper angular velocity.

b) This remains true if the observers on the plane are moving relative to the plane (as long as they remain on the plane).

c) This remains true even if observers are accelerating relative to the plane (as long as they remain on the plane).

d) This remains true even if the observers are undergoing circular motion relative to the plane around an axis orthogonal to the plane.

e) It may be possible to extend the idea to higher derivatives of motion.

Of course angular motion that has tangential velocity parallel to the plane has an angular velocity vector orthogonal to the plane, but for the purposes of this idea, angular motion with tangential velocity that remains in the plane counts as a measurement on the plane.

This hypothesis has far reaching claims and would greatly simplify many relativistic calculations and proofs if true. Can anyone think of any counter examples besides the implication that it is denied by the Hergotz-Noether theorem which is under discussion in other threads currently?

I think it will be an interesting discussion to discuss where some of the extensions fall down, which extensions may be valid and where there are issues, how they can be 'made to work'. I think any conclusions that remain valid (even with additional qualifications) will be make it a useful hypothesis.

This hypothesis can be thought of as extending the accepted clock postulate http://math.ucr.edu/home/baez/physics/Relativity/SR/clock.html in ways that have probably been done instinctively in the past as part of other work, but never as a subject in its own right.

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What do you mean by "that are made entirely within the plane"? An observer of mass ##m## standing on a scale placed on a planar platform at rest in a gravitational field ##g## will see a scale reading of ##mg## kilograms but if the platform is accelerating upwards with magnitude ##a## relative to the gravitational field then the observer will see a scale reading of ##m(g+ a)## kilograms so the scale measurement is clearly not equivalent between the two scenarios.

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What do you mean by "that are made entirely within the plane"? An observer of mass ##m## standing on a scale placed on a planar platform at rest in a gravitational field ##g## will see a scale reading of ##mg## kilograms but if the platform is accelerating upwards with magnitude ##a## relative to the gravitational field then the observer will see a scale reading of ##m(g+ a)## kilograms so the scale measurement is clearly not equivalent between the two scenarios.

I should of made it clear that the observers on the plane will of course measure the proper acceleration of the plane. (I made that clear in another thread where I first mentioned the idea. https://www.physicsforums.com/showpost.php?p=4493051&postcount=62 ) The acceleration vector orthogonal to the plane (a measure of change in the velocity vector orthogonal to the plane) and does not count as a measurement "in the plane" (for our purposes here).

Are you aware that without rotation, any shape 3-d body can be Born rigid accelerated, in any trajectory and pattern of acceleration desired? This is all explicitly allowed by Herglotz-Noether. The only limitation is the size of the object- for any acceleration pattern, there is a maximum size object that can be Born rigidly accelerated. Normally, it is not much of a limitation: for 1 g acceleration, the limitation is about 1 light year.

With rotation, it appears from that other thread, that several of us are convinced (especially by the Epp et.al. paper) that you first linked, that many statements about zero thickness objects in the literature are not correct. These arguments in the literature implicitly add additional constraints beyond Born rigidity, e.g. that the motion must be consistent with being part of a 3-d object; or that shape change in the inertial frame is assumed impermissible.

However, my opinion is that 2-surfaces are not meaningful approximations of any realizable object. And therefore, for rotation, noting but inertial motion is possible preserving Born rigidity.

Your real world plausibility arguments have no weight, because real world objects not rigid at all, let alone Born rigid. There is no problem with accelerating a rotating disk in the real world, because there is no expectation that it remains Born rigid.

b) This remains true if the observers on the plane are moving relative to the plane (as long as they remain on the plane).

But pervect showed that if the observers are moving relative to the plane, and the plane is accelerating, then those observers see the plane as curved, not plane. So their motion does make a difference.

There is no problem with accelerating a rotating disk in the real world, because there is no expectation that it remains Born rigid.

But there is an expectation that its state is reasonably close to stationary, i.e., that any deviations from rigid motion are periodic. If the deviations are non-periodic, i.e., if they build up over time, then the object can't remain intact indefinitely. I'm still not entirely sure that, for example, the motions described in the Lhosa et al. paper have deviations from rigid motion that meet the above requirement.

But there is an expectation that its state is reasonably close to stationary, i.e., that any deviations from rigid motion are periodic. If the deviations are non-periodic, i.e., if they build up over time, then the object can't remain intact indefinitely. I'm still not entirely sure that, for example, the motions described in the Lhosa et al. paper have deviations from rigid motion that meet the above requirement.

Within the bounds of realized experiments, we know we can have flywheels either horizontal or vertical sitting on the ground (or in further motion in vehicles) with no problem. And they continue to operate for long periods of time. Thus we know that there is a substantial domain of practice for which the deviations from theoretical rigidity of any type are insignificant.

As for deviations building over time, have you seen any discussions in the literature suggesting that in any of these rotation situations, there is an expectation growing deviation? I've never seen this even suggested except by you (but I could easily have missed something). Any reference on why to expect this would be useful. Or a clear argument about why to expect it.

I just found another paper (besides Llosa et. al.) addressing near rigid motion (in GR, but SR is obviously a special case). [Note, that while the Epp et.al. paper was very interesting, and with surprising results relative to common claims, it really doesn't model rigid bodies or rigid motion because it says nothing about the interior, even to rule out the interior being a chaotic mess.] Since I just found this, I have no opinion or conclusions from it yet:

http://pubman.mpdl.mpg.de/pubman/item/escidoc:153641:1/component/escidoc:153640/335128.pdf

But pervect showed that if the observers are moving relative to the plane, and the plane is accelerating, then those observers see the plane as curved, not plane. So their motion does make a difference.

I do not disagree that the flat plane with inertial orthogonal motion appears to be a curved surface to observers on the surface. I am also not suggesting that observers would be completely unaware of the orthogonal acceleration of their surface, as there are number of ways they could determine this. Perhaps I should better of used the word surface rather than plane. By "measurements in the plane", I mean measurements by infinitesimal rulers or short radar measurements along the surface, with no short cutting allowed. For example, consider an observer on a rod sliding along the surface, that measures the proper length of his rod to be L when the orthogonal motion of the surface is inertial. Now when the surface has orthogonal acceleration, a "long radar" measurement would suggest that the light beam takes a straight path that is shorter than than the curved length of the rod as measured by very short rulers laid along the rod. To a free falling observer falling towards the surface, the "long radar" light beam appears to rise above the flat surface and a fall back towards it taking a longer curved path than the length of the flat rod (flat to the inertial observer). Anyway, these "long radar" measurements take a route that does not remain on the surface, so are not allowed measurements. The difference between long and short radar measurements is discussed in this article on Born coordinates http://en.wikipedia.org/wiki/Born_coordinates#Radar_distance_in_the_large

Anyway, I am suggesting that the proper length of the sliding rod remains L, whether or not the surface it is sliding on has inertial or accelerating orthogonal motion and that this measurement is made by infinitesimal ideal rulers laid along the (possibly curved) surface. To an observer stationary relative to the surface watching the rod slide past, the length of the rod is simply L*sqrt(1-v2) where v is the velocity of the rod relative to the observer and this is completely independent of the orthogonal acceleration of the surface.

By way of example, I would like to present a worked example based on pervect's sliding and accelerating coordinate transformation: https://www.physicsforums.com/showpost.php?p=4466113&postcount=25

Let us assume that the sliding observer (O at rest in S) is initially (at t=0) moving at 0.8c relative to the inertial observer (o at rest in s). This gives a value for K of 1.333, (the constant proper velocity of the sliding observer). In frame S a test rod is moving at -0.8c and the measured length in S is ΔX=0.6. The (T,X) coordinates are initially (0,0) for the front of the test rod and and (0, 0.6) for the back of the rod. One second later in S the coordinates are (1,-0.8) and (1,-0.2) respectively. When these coordinates are transformed to the s frame, they become (for a value of g=0.5) (t,x) = (0,0) and (0.8, 1) for the front and back initially and later (0.407,0) and (1.502, 1) respectively. We note that the test rod is stationary in s and has a proper length of 1 which does not change over time. The times are not simultaneous, but this does no matter as the rod is stationary in this frame. Changing the value of g does alter the time coordinates in s, but does not affect the x coordinates. We also note that the length contraction of the test rod is exactly what we would expect from inertial motion in Minkowski space. i.e in S the rod appears to be moving at -0.8c and is length contracted by a factor of 0.6 which is the exactly what we expect from the gamma factor with that simple linear relative velocity and we can completely ignore the acceleration of the surface.

Also note that in pervect's coordinates, the surface the observer slides along might appear to be curved, but all points on that curved surface have coordinate z=0, despite the apparent curvature.

I have attached a simple excel spreadsheet that takes T,X sliding pervect coordinates as inputs and transforms them to x,t coordinates. (Only alter the values highlighted in yellow.)

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• pcoords2.xlsx
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In my last wordy post, I could of simply observed that in pervect's sliding rod in an Einstein elevator coordinates, the X and Y transformations are given by:

$$x \left( T,X \right) =KT+X\sqrt { 1+{K}^{2}}$$
$$y \left( Y \right) =Y$$

.. and noted that the transformations in the x,y plane are independent of the acceleration parameter g, (unlike the T and Z transformations). Also note that K is a constant equal to ##(dx/dt)/\sqrt{1-(dx/dt)^2}## where dx/dt is measured at time t=0.

P.S. I have attached an extended and slightly more refined spreadsheet that transforms pervect's accelerating (T,X,Y,Z) coordinates to the inertial (t,x,y,z) coordinates.

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• pcoords3.xlsx
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Within the bounds of realized experiments, we know we can have flywheels either horizontal or vertical sitting on the ground (or in further motion in vehicles) with no problem. And they continue to operate for long periods of time. Thus we know that there is a substantial domain of practice for which the deviations from theoretical rigidity of any type are insignificant.

This might be of interest: 600,000,000 rpm in the Earth's gravitational field, for a man made object. http://www.st-andrews.ac.uk/news/archive/2013/title,224725,en.php

But there is an expectation that its state is reasonably close to stationary, i.e., that any deviations from rigid motion are periodic. If the deviations are non-periodic, i.e., if they build up over time, then the object can't remain intact indefinitely. I'm still not entirely sure that, for example, the motions described in the Lhosa et al. paper have deviations from rigid motion that meet the above requirement.

As for deviations building over time, have you seen any discussions in the literature suggesting that in any of these rotation situations, there is an expectation growing deviation? I've never seen this even suggested except by you (but I could easily have missed something). Any reference on why to expect this would be useful. Or a clear argument about why to expect it.

I guess we have all looked for literature on this and not found anything useful (for the case of behavior of a quasi-rigid rotating disk with nonzero thickness undergoing uniform acceleration in the direction if its spin axis - orthogonal to its major surfaces). So, reading over Peter's analysis in the various threads, the gist of it is convincing to me, which I'll sum up as follows (Peter, as always, tell me where you disagree):

As a first take on a disk with thickness, consider a pair of zero thickness disks with a small separation between them. If both are initially rotating at the same angular speed clockwise in some inertial frame, then begin uniform acceleration, it would appear to an observer in the center of the top disc (the one undergoing less proper acceleration, assuming lines orthogonal to the disk surfaces behave like Born rigid linear acceleration) that the bottom disc starts to spin slowly in the counterclockwise direction. [I assume this top disc observer is rotating with the top disc]. It think the rate of this counterclockwise motion is constant, but obviously, the total shear increases without bound.

To me, obviously, some part of this analysis is wrong as a description of the nature of realistic quasi-rigid disc. I also cannot, so far, go through and see exactly what the Llosa methodology predicts for this - but it least it clearly does make a prediction (reasonable or not). The only issue as that the math is challenging for me (and not only me). This, in contrast to the Epp et.al. methodology which concludes that boundary of such a disk could remain strictly Born rigid, but says nothing, at all, in principle, about the interior.

My intuition about how to proceed is to imagine:

a) we couple these ideal zero thickness discs with springs
b) we accept that we cannot claim a static equilibrium at any time, else this static equilibrium itself violates Herglotz-Noether.

This suggests something like perpetual oscillation between the top and bottom discs mediated by the springs, as Peter was hinting at. If we wanted to try to find something like this in the Llosa formalism, we would look for the possibility of periodicity in the solution, exactly as Peter has been suggesting.

Generally, it seems we're on our own here - I have looked long and hard for any prior work on this particular problem, to no avail.

Are you aware that without rotation, any shape 3-d body can be Born rigid accelerated, in any trajectory and pattern of acceleration desired? This is all explicitly allowed by Herglotz-Noether.
I was not aware of that. Thanks for the info. I assume you mean we can have any pattern of acceleration for the centre of mass (where that can be defined) for the body? For the well studied example of an accelerating rod, the proper acceleration at different points along the rod has to vary as per 1/d and any other "pattern of acceration, e.g equal proper acceleration at all points of the rod, will result in the breakdown of Born rigidity.

However, my opinion is that 2-surfaces are not meaningful approximations of any realizable object. And therefore, for rotation, noting but inertial motion is possible preserving Born rigidity.
I am absolutely sure you are aware of the notion of an infinitesimal quantity. This is a very small slice that is non zero, but so small that any errors are negligible. For example consider a box that contains a gas. The density (p) of this gas is a function of height (h), e.g. p = 2+1/h^2. Now it is possible to calculate the mass of gas contained in this box by integrating the density function. It would be silly to argue that the density of a zero thickness horizontal slice exactly at height h would be infinite because the slice has zero volume. It would be equally silly to argue that the total mass of a zero thickness slice is zero, so the total mass of the gas in the box is zero when we sum up the slices.

Your real world plausibility arguments have no weight, because real world objects not rigid at all, let alone Born rigid. There is no problem with accelerating a rotating disk in the real world, because there is no expectation that it remains Born rigid.
In the original thread I was trying to talk about real world examples, but the thread was derailed because the forum members insisted that I show that is it possible to have rotation in a gravitational field, before they would continue the discussion. Nevertheless, I find the Born rigidity discussions interesting. It would be useful to know the scale of the problem. Is the induced stress a function of angular velocity or radius? How do we minimise it? Can we usefully discuss the gravitational effects on a 1 metre disc rotating at 1 rpm, without introducing significant errors due to ignoring Born rigidity failure?

As a first take on a disk with thickness, consider a pair of zero thickness disks with a small separation between them. If both are initially rotating at the same angular speed clockwise in some inertial frame, then begin uniform acceleration, it would appear to an observer in the center of the top disc (the one undergoing less proper acceleration, assuming lines orthogonal to the disk surfaces behave like Born rigid linear acceleration) that the bottom disc starts to spin slowly in the counterclockwise direction. [I assume this top disc observer is rotating with the top disc]. It think the rate of this counterclockwise motion is constant, but obviously, the total shear increases without bound.
Now this is a positive approach. The obvious solution is to speed up the lower disc so that its rotation rate matches the rotation rate of the upper disc, as measured by the observer on the top disc. When synchronised like this, the two discs could be locked together by balsa wood rods without breaking the rods. To show failure in this scenario, it is necessary to demonstrate that that a single infinitesimal thickness disc cannot maintain constant angular velocity when held at constant altitude in a gravitational field. The Schwarzschild metric shows no evidence of that.

Basically we seem to be in a 'aerodynamics predicts bumblebees can't fly and don't care that they actually do', situation.

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I was not aware of that. Thanks for the info. I assume you mean we can have any pattern of acceleration for the centre of mass (where that can be defined) for the body? For the well studied example of an accelerating rod, the proper acceleration at different points along the rod has to vary as per 1/d and any other "pattern of acceration, e.g equal proper acceleration at all points of the rod, will result in the breakdown of Born rigidity.
Yes, any world line for the COM, as long as the rest of the body moves in just the right way.
I am absolutely sure you are aware of the notion of an infinitesimal quantity. This is a very small slice that is non zero, but so small that any errors are negligible. For example consider a box that contains a gas. The density (p) of this gas is a function of height (h), e.g. p = 2+1/h^2. Now it is possible to calculate the mass of gas contained in this box by integrating the density function. It would be silly to argue that the density of a zero thickness horizontal slice exactly at height h would be infinite because the slice has zero volume. It would be equally silly to argue that the total mass of a zero thickness slice is zero, so the total mass of the gas in the box is zero when we sum up the slices.
But equally true are situations where there is no valid limiting procedure. A 2-surface embedded in 3-space is an example. The limit of of ever thinner disk shaped open sets is ... nothing, the empty set. It is not a 2-surface. I believe this matches the current scenario more - all open sets are bound by Herglotz-Noether. The lower dimensional surface admits motions that cannot by emulated by any open set.
Now this is a positive approach. The obvious solution is to speed up the lower disc so that its rotation rate matches the rotation rate of the upper disc, as measured by the observer on the top disc. When synchronised like this, the two discs could be locked together by balsa wood rods without breaking the rods. To show failure in this scenario, it is necessary to demonstrate that that a single infinitesimal thickness disc cannot maintain constant angular velocity when held at constant altitude in a gravitational field. The Schwarzschild metric shows no evidence of that.

Basically we seem to be in a 'aerodynamics predicts bumblebees can't fly and don't care that they actually do', situation.

No, we're in a situation where something that doesn't exist, but is a mathematical possibility (a Born rigid object), can sometimes be used to approximate real world situations (e.g. pure acceleration), and in other cases cannot. Unfortunately, the real world rotating disc in gravity or acceleration is really hard to analyze with relativistic corrections.

In the original thread I was trying to talk about real world examples, but the thread was derailed because the forum members insisted that I show that is it possible to have rotation in a gravitational field, before they would continue the discussion.

I'm not sure which of the several threads on this topic you're referring to--probably this one?

In any case, I'm not sure you've correctly characterized the issue that has kept this topic going on. Nobody disputes that it is "possible to have rotation in a gravitational field". But we would like to understand *how* GR, as a theory, models that kind of scenario. Saying that it "ought" to work out a certain way, based on your intuition, is *not* the same as showing that it *does* in fact work out that way. (See my comment later in this post in response to your statement about the Schwarzschild metric.)

Basically, we have an apparent conflict between theory and common experience:

(1) Common experience says that you can rotate an object about a vertical axis in a gravitational field without having it torn apart by stresses. That implies that you can subject an object to both rotation and linear acceleration along its rotation axis and have the resulting motion be stationary--i.e., its state remains more or less the same over time; whatever variations there are are periodic and don't build up.

(2) Theory (in the form of the Herglotz-Noether theorem) says that a motion that combines rotation and linear acceleration along the rotation axis can't be a Born rigid motion. Of course there are possible stationary states, in the above sense, that are not perfect Born rigid motions; a real motion will have some variation around the ideal of Born rigidity. However, a stationary motion should average to a Born rigid motion (see below for more on why this is), and the theory appears to be saying that there is no such motion possible for a real stationary motion to average to.

Your response to the above has basically been to say that your intuition says #1 is right. But I don't think anyone is arguing about what intuition says. The problem is that we don't know how to make the theory give the answer that intuition says is the right one, even though this is a scenario that should be well within GR's domain of validity. What theoretical inputs we have so far suggest that the intuitive answer is the wrong one, but as PAllen remarked, it doesn't look like there is much help in the literature on this specific question, so we're basically on our own.

Nevertheless, I find the Born rigidity discussions interesting. It would be useful to know the scale of the problem. Is the induced stress a function of angular velocity or radius?

The closest thing we've seen so far to an answer to this type of question is the Lhosa et al. paper that PAllen linked to in some recent thread or other, which said that the "strain rate" (which is, roughly speaking, the rate at which strain builds up in an object due to failure of Born rigidity) is proportional to the square of the angular velocity. I did a very rough back of the envelope calculation that seemed to show that, for a 1 meter disk rotating at 1 radian per second, the strain would exceed the breaking strength of the strongest material we know in about 4 months. But I'm not sure I understand the math in that paper well enough to know if my calculation was valid.

How do we minimise it?

You can't. Failure of Born rigidity (more precisely, failure of "approximate Born rigidity"--failure of the motion to average, over the long term, to a Born rigid motion) means that eventually the object is torn apart; there's no way around it. (More precisely, there's no way around it without at some point changing the object's motion to one that does average to a Born rigid motion. For example, you can spin up a disc while constraining it to stay at the same radius and within the same plane, as long as you stop spinning it up, and let it stay at a constant angular velocity, before the stresses in the disc have built up to the breaking point.)

For an example of this, consider the Bell Spaceship Paradox. Two spaceships start out mutually at rest in an inertial frame, with their clocks synchronized; then, at time t = 0 in that frame, they both turn on their rockets and accelerate in the same direction with the same proper acceleration. A string hangs between the ships, attached to each ship at one end, and initially slack. What happens to it?

The answer is that the string stretches and eventually breaks; and the fundamental reason *why* it stretches and eventually breaks is that its motion fails to be Born rigid; the congruence of worldlines that the individual small pieces of the string must follow for it to remain attached at both ends has nonzero expansion. The only way to stop the stretching and breaking from happening is to change the string's motion, for example by detaching one end of it from one of the ships, so that it now *is* Born rigid (at least on average). In other words, there is *no* possible motion of the string that both keeps it attached to both ships *and* prevents it from breaking.

In the case we're talking about here, the problem is nonzero shear, not nonzero expansion; but the basic consequence is the same: something builds up over time until it exceeds the strength of the material. See below.

Can we usefully discuss the gravitational effects on a 1 metre disc rotating at 1 rpm, without introducing significant errors due to ignoring Born rigidity failure?

Intuitively, of course we should be able to. But no one here has figured out how to model this mathematically in a way that doesn't make predictions similar to my #2 above.

The obvious solution is to speed up the lower disc so that its rotation rate matches the rotation rate of the upper disc, as measured by the observer on the top disc. When synchronised like this, the two discs could be locked together by balsa wood rods without breaking the rods.

Could it? Remember that in order to spin up the lower disc in a Born rigid manner, its radius must decrease *and* it must bend out of plane. Both of these things will bend the rods. Basically, the rods have to transmit force to the lower disc to speed it up, and that causes stress in the rods.

Also, I don't think the spin-up of the lower disc will be a one-time thing: I think the lower disc will have to continually speed up its rotation in order to keep up with the upper disc. If that's true, then there will be a strain induced *somewhere* in the assembly that will continually increase until it exceeds the breaking strength of the material. This is what PAllen meant by saying that "the shear increases without bound".

Of course, I haven't mathematically modeled this scenario to show how the stresses would build up; as I've said, no one here has figured out how to do that. But as I understand the H-N theorem math from the papers that have been linked to in these threads, it says that the congruence of worldlines that describes two discs with the constraint you have given--that both discs have the same rotation rate as seen by an observer comoving with the center of the top disc--has nonzero shear. And just as in the Bell Spaceship Paradox scenario, this nonzero shear--i.e., the failure of the motion to be Born rigid--will eventually cause the object to be torn apart.

To show failure in this scenario, it is necessary to demonstrate that that a single infinitesimal thickness disc cannot maintain constant angular velocity when held at constant altitude in a gravitational field. The Schwarzschild metric shows no evidence of that.

Really? You can model this scenario mathematically using the Schwarzschild metric? Please show your work.

Actually, what I'm pretty sure you mean here is that your intuition says that, *if* a mathematical model were constructed using the Schwarzschild metric, it would show what you say. But again, nobody is arguing with what intuition says. We are trying to figure out how the theory actually models this stuff, not how we intuitively guess it should model this stuff.

So unless you can back up your claim about the Schwarzschild metric with actual math, I think you should not be making it. That's not forcing you to "show that it's possible"; that's asking you to not say things about the math if you can't actually show the math.

Basically we seem to be in a 'aerodynamics predicts bumblebees can't fly and don't care that they actually do', situation.

Aerodynamics only predicted that bumblebees could not fly when it was using the wrong model. The right model (that bumblebees don't fly the way birds and airplanes do; they fly the way helicopters and hummingbirds do, by brute force) predicts that bumblebees can fly just fine.

The problem is that no one here has been able to come up with the counterpart of the "right" model in this case. We (meaning we here in these threads) understand what intuition and common experience tell us; but we don't at this point understand *how* to reconcile that with what the theory seems to be telling us.

The closest thing we've seen so far to an answer to this type of question is the Lhosa et al. paper that PAllen linked to in some recent thread or other, which said that the "strain rate" (which is, roughly speaking, the rate at which strain builds up in an object due to failure of Born rigidity) is proportional to the square of the angular velocity. I did a very rough back of the envelope calculation that seemed to show that, for a 1 meter disk rotating at 1 radian per second, the strain would exceed the breaking strength of the strongest material we know in about 4 months. But I'm not sure I understand the math in that paper well enough to know if my calculation was valid.
Would you agree that this is NOT even close to what would actually be observed in a lab? This is clearly at odds with what is actually observed, so either the rotation theories are incorrect or our understanding of them is incorrect.

For an example of this, consider the Bell Spaceship Paradox. Two spaceships start out mutually at rest in an inertial frame, with their clocks synchronized; then, at time t = 0 in that frame, they both turn on their rockets and accelerate in the same direction with the same proper acceleration. A string hangs between the ships, attached to each ship at one end, and initially slack. What happens to it?

The answer is that the string stretches and eventually breaks; and the fundamental reason *why* it stretches and eventually breaks is that its motion fails to be Born rigid; the congruence of worldlines that the individual small pieces of the string must follow for it to remain attached at both ends has nonzero expansion. The only way to stop the stretching and breaking from happening is to change the string's motion, for example by detaching one end of it from one of the ships, so that it now *is* Born rigid (at least on average). In other words, there is *no* possible motion of the string that both keeps it attached to both ships *and* prevents it from breaking.
I understand that, but I also understand that if the two ships are accelerating in a pattern that matches those of Rindler observers, that the string would not snap. In other words the impossibility of maintaining Born rigidity in this example depends on insisting on arbitrary constraints.
In the case we're talking about here, the problem is nonzero shear, not nonzero expansion; but the basic consequence is the same: something builds up over time until it exceeds the strength of the material. See below.
Sheer will only occur (between vertically stacked discs) if a single infinitesimal thickness rotating disc experiences a a progressive change in angular velocity or radius, purely as a result of being at constant altitude.
Could it? Remember that in order to spin up the lower disc in a Born rigid manner, its radius must decrease *and* it must bend out of plane. Both of these things will bend the rods. Basically, the rods have to transmit force to the lower disc to speed it up, and that causes stress in the rods.
Remember that the lower disc is spun up to the required velocity by a motor or rockets before* attaching the rods. The rods are not transmitting any force, unless we can show that a single disc cannot spin at constant angular velocity when at constant altitude, See above.
Also, I don't think the spin-up of the lower disc will be a one-time thing: I think the lower disc will have to continually speed up its rotation in order to keep up with the upper disc.{/quote]This was the whole point of my original thread about the tall rotating cylinder in a gravitational field. I do not think it is too difficult to prove mathematically, that the rotation rate of the lower disc does NOT have to alter over time to keep up with the upper disc. I believe this can be demonstrated using the Schwarzschild metric. We take a disc at height r constant rotation rate w1 as measured by a non rotating observer stationary at r and another disc at height r+h with constant rotation rate w2 as measured by a non rotating stationary observer at r+h and I predict that the ratio w1/w2 will remain constant over time.
If that's true, then there will be a strain induced *somewhere* in the assembly that will continually increase until it exceeds the breaking strength of the material. This is what PAllen meant by saying that "the shear increases without bound".
As I said, I think we can easily prove that is not true.
Really? You can model this scenario mathematically using the Schwarzschild metric? Please show your work. So unless you can back up your claim about the Schwarzschild metric with actual math, I think you should not be making it. That's not forcing you to "show that it's possible"; that's asking you to not say things about the math if you can't actually show the math.
I will come back to this, but first we have to establish if we can agree that conservation of momentum applies to objects at constant altitude in the Schwarzschild metric and that the spacetime is essentially Minkowskian in an local enough region.
The problem is that no one here has been able to come up with the counterpart of the "right" model in this case.
That was my point.

This is clearly at odds with what is actually observed, so either the rotation theories are incorrect or our understanding of them is incorrect.

Well, yes, it appears, so, but *how*? Once again, there's no point in just saying "that's not what we observe". The point at issue is not what we observe, but how to reconcile the theory with what we observe.

In other words the impossibility of maintaining Born rigidity in this example depends on insisting on arbitrary constraints.

Well, the constraint of having Rindler observers is just as "arbitrary"; both cases are obviously physically realizable. My point was simply that having a nonzero expansion (or shear) has physical consequences. In this particular case, those consequences can be avoided by changing the motion to one that has zero expansion, and the theory happily provides a description of one.

What we're trying to figure out is, what motion can we find *in the theory* that matches our intuitive picture of "a rotating disk linearly accelerated along its axis of rotation" *and* has nonzero shear (and expansion)? And the problem is that, so far, the theory appears to be saying that it doesn't have one, except in the unphysical case of a disk with exactly zero thickness.

I do not think it is too difficult to prove mathematically, that the rotation rate of the lower disc does NOT have to alter over time to keep up with the upper disc. I believe this can be demonstrated using the Schwarzschild metric.

I would try first using the Rindler metric, since it's simpler, and since that's the case I was discussing anyway.

first we have to establish if we can agree that conservation of momentum applies to objects at constant altitude in the Schwarzschild metric

Conservation of what momentum?

the spacetime is essentially Minkowskian in an local enough region.

Obviously I agree with that, since I've been doing all of my analysis in the analogous flat spacetime case anyway.

I would try first using the Rindler metric, since it's simpler, and since that's the case I was discussing anyway.
OK, let's look at this from a Rindler perspective, with reference to: http://mathpages.com/home/kmath422/kmath422.htm

That link states that the velocity ##v_j## relative to the initial inertial reference frame of the ##j^{th}## particle on a rod with Born acceleration is given by:

##v_j = \tanh(a_j*\tau_j)##

where ##\tau_j## is the proper time that elapses for the particle.

Now if we have two particles on the rod with equal velocity in the initial inertial reference frame, such that the accelerating rod is momentarily at rest with a co-moving inertial reference frame with velocity v relative to the first inertial reference frame, then:

##\tanh(a_1*\tau_1)=v_1##

and

##\tanh(a_2*\tau_2)=v_2##

so when ##v_1 =v_2## we can say:

##\tanh(a_1*\tau_1)=\tanh(a_2*\tau_2)##

so:

$$\frac{\tau_1}{\tau_2} = \frac{a_2}{a_1}$$

Since the proper acceleration is constant for both particles, the ratio of their proper times is also constant. This means that an observer at the leading edge of the rod sees the clock at the trailing edge as ticking with a constant (but lower) frequency and conversely the observer at the trailing edge of the accelerating rod see the clock at the front ticking with a constant (but higher) frequency. If we replace the clocks at the front and back of the rod with frictionless flywheels, they will maintain the same frequency relative to each other, as measured by the accelerating observers. If j=1 represents the leading observer and j=2 the rear observer, then if we speed up the rear flywheel by a factor of ##a_1/a_2## the flywheels will appear to rotate at the same rate as each other according to either observer so the flywheels are synchronised. This synchronisation of the flywheels rotation rates is permanent and does not very over over time so there is no sheer.

Conservation of what momentum?
Coordinate or proper measurement of linear or angular momentum.

Now let's look at circular motion in the Schwarzschild metric. With reference to: http://www.fourmilab.ch/gravitation/orbits/ we see that for an orbiting particle, the orbital velocity is given by:

$$\frac{d\phi}{d\tau} = \frac{L}{r^2}$$

Since we are talking about motion at constant altitude, r=constant and L is the angular momentum at infinity which is also a constant, so the proper orbital velocity ##d\phi/d\tau## must also be constant. The link also gives the relation between the proper time and the coordinate time as:

$$\frac{dt}{d\tau} = \frac{E}{(1-2GM/r^2)}$$

Since E is a constant and since we only considering constant altitude, r is a constant, then it follows that the ratio between coordinate time and proper time is also constant. Therefore the coordinate orbital period of the particle is also constant and the orbital period measured today is the same as the orbital period measured tomorrow or at any other time. Therefore there is nothing mysterious going on the Schwarzschild metric that causes objects at constant altitude to speed up or slow down or ignore conservation of angular momentum. While the orbiting particle is not the exact analogue of a spinning at constant altitude, the example demonstrates that angular momentum is conserved in the Schwarzschild metric and I have never seen anything else suggested.

The Schwarzschild metric also tells us that there is no horizontal length contraction. This means that the length of a horizontal rod as measured by a local observer is the same as the length measured by the observer at infinity. This implies that the radius of a disc spinning around its vertical axis, is no different to that of a disc spinning in flat spacetime.

Now the question I put to you is this. If a non rotating but linearly accelerating observer is comoving with a linearly accelerating rotating disc, such that he measures no change in angular momentum (assuming conservation) and such that he measures no change in radius or mass of the disc, then would an observer riding on the edge of the rotating disc measure changes over time of the disc? Would there be sudden and catastrophic failure of the disc, without any observed external changes to the disc?

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OK, let's look at this from a Rindler perspective

This is for particles that are part of the Rindler congruence, i.e., they are all linearly accelerated only. What we need is an analysis of particles that are linearly accelerated *and* rotating; i.e., their actual proper acceleration is a combination of the linear acceleration and the centripetal acceleration due to rotation.

Now let's look at circular motion in the Schwarzschild metric.

This is the wrong kind of circular motion. What we need is a motion that remains at constant ##r##, but varies in both ##\phi## *and* ##\theta##--i.e., radial acceleration (to remain at constant ##r##) *plus* rotation about the center of the disk, in the tangential (i.e., ##\theta - \phi##) plane. That means the full motion must take into account all three spatial dimensions ("full motion" meaning the motion of both disks, at different altitudes); it's not the same as an orbit in a single spatial plane, which is what your results apply to.

In both of these cases, the point is that you are not actually describing, explicitly, the motion of the particles that are linearly accelerated and rotating. You are simply assuming that the rotation just acts like a "clock" as seen by the observer at the center of the disk, who is only linearly accelerating. But that's supposed to be what's being proved; it can't just be assumed. I know it seems intuitively obvious, but that's the same intuition that is apparently being contradicted by things like the Herglotz-Noether theorem, so we can't rely on it here. We need an actual analytical description of the motion of the rotating particles that explicitly *shows* that their period is constant as seen by the observer at the center of the disk. (That's one objective that the analysis I'm doing in the "spin-off" thread is ultimately aimed at.)

This is for particles that are part of the Rindler congruence, i.e., they are all linearly accelerated only. What we need is an analysis of particles that are linearly accelerated *and* rotating; i.e., their actual proper acceleration is a combination of the linear acceleration and the centripetal acceleration due to rotation.
Obviously you do not agree that a perfect flywheel is effectively a clock or we would not have an issue here. Any metric or transformation I came up with that included linear acceleration along with rotation would assume that the proper angular momentum is conserved, but you require that I prove that first. (Interestingly, I note that pervect assumes that proper angular velocity is constant in his spin off thread.)

This is the wrong kind of circular motion. What we need is a motion that remains at constant ##r##, but varies in both ##\phi## *and* ##\theta##--i.e., radial acceleration (to remain at constant ##r##) *plus* rotation about the center of the disk, in the tangential (i.e., ##\theta - \phi##) plane. That means the full motion must take into account all three spatial dimensions ("full motion" meaning the motion of both disks, at different altitudes); it's not the same as an orbit in a single spatial plane, which is what your results apply to.
I am aware of that. That is why I said
While the orbiting particle is not the exact analogue of a spinning (disc) at constant altitude,...
in my last post. It was meant to be just a clue that under certain circumstances angular momentum is conserved in GR. Now I will try a slightly different example that may be closer to what you require. Refering to the above diagram, consider a particle orbiting at constant ##\varphi## in a small circle around the North pole. For a small enough circle, this will approximate a small flat disc.While describing this circle, r and ##\varphi## remain constant. This implies rotational symmetry for a rotation in the ##\theta## direction. Noether's conservation theorem tells us that angular momentum is conserved if there is rotational symmetry. Therefore, (if I have interpreted Noether correctly) the angular momentum of the disc (or ring) at constant altitude should remain constant over time. Due to the spherical symmetry of the Schwarzschild metric this circle at constant r, does not have to be around the North pole.

We need an actual analytical description of the motion of the rotating particles that explicitly *shows* that their period is constant as seen by the observer at the center of the disk. (That's one objective that the analysis I'm doing in the "spin-off" thread is ultimately aimed at.)
I could do an analysis of pervect's coordinate system for a linearly accelerating 2+1D disc that would show that the period remains constant as measured by the observer at the centre of the disc, but I am pretty sure he assumed that when constructing the transformation system.

Anyway, it seems that the idea that Born rigidity failure between two stacked thin discs is due to sheer stress caused by differential speeds of the two discs, depends on the failure of conservation of angular momentum in an accelerating reference frame. I think we need to start a separate thread on that conservation law.

Anyway, it seems that the idea that Born rigidity failure between two stacked thin discs is due to sheer stress caused by differential speeds of the two discs, depends on the failure of conservation of angular momentum in an accelerating reference frame. I think we need to start a separate thread on that conservation law.

Actually, it follows from conservation of angular momentum. Proper angular momentum is equivalent to what is measured in an instantly colocated, comoving, inertial frame, in SR. This remains constant for each disc. For the top and bottom disk, undergoing orthogonal Rindler motion, each disc perceive's the other as having its inertia and angular speed modified by Rindler g00, conserving angular momentum but producing shear (due to the difference in angular speed).

Any metric or transformation I came up with that included linear acceleration along with rotation would assume that the proper angular momentum is conserved, but you require that I prove that first.

I'm not looking for a metric or transformation; I'm looking for a description of a congruence of worldlines describing a linearly accelerated rotating disk (or a stack of them) that has a zero expansion tensor. If that congruence turns out to have each thin disk at a constant ##z## having a constant angular velocity, fine. But you can't just assume that's the case, and also assume that the congruence has zero expansion. You have to prove that both of those things can simultaneously be true.

(Interestingly, I note that pervect assumes that proper angular velocity is constant in his spin off thread.)

But the congruence of worldlines that pervect described using that assumption does not have a zero expansion tensor; at least, computations so far indicate that it doesn't.

It was meant to be just a clue that under certain circumstances angular momentum is conserved in GR.

Nobody is arguing about that; certainly I'm not.

Now I will try a slightly different example that may be closer to what you require.

Once again, you don't have to prove that angular momentum can be conserved. That's not the problem. The problem is finding a congruence describing a linearly accelerated, rotating disk, that has zero expansion. That is the critical question that we do not have an answer for. Your arguments here about Noether's theorem and angular momentum conservation are fine, but they don't address that critical question. See further comments below.

I could do an analysis of pervect's coordinate system for a linearly accelerating 2+1D disc that would show that the period remains constant as measured by the observer at the centre of the disc, but I am pretty sure he assumed that when constructing the transformation system.

And that isn't relevant to the critical question; see above.

Anyway, it seems that the idea that Born rigidity failure between two stacked thin discs is due to sheer stress caused by differential speeds of the two discs, depends on the failure of conservation of angular momentum in an accelerating reference frame.

No, it depends on the nonexistence of a congruence describing the discs' motion that has zero expansion. I don't see any way of addressing that question with arguments about angular momentum conservation.

First, angular momentum could be perfectly well conserved in an object that is tearing itself apart due to shear stress, as long as the pieces that were flying apart were doing so along appropriate trajectories. (PAllen's post in response to yours gives an example of how the nonzero shear could arise.)

Second, angular momentum conservation might not hold even in an object that *was* moving on a congruence having zero expansion, if the object was accelerated. If the force producing the acceleration is not applied in a way that produces zero torque, then the object's angular momentum will change, even if the overall motion has zero expansion.

Actually, it follows from conservation of angular momentum. Proper angular momentum is equivalent to what is measured in an instantly colocated, comoving, inertial frame, in SR. This remains constant for each disc. For the top and bottom disk, undergoing orthogonal Rindler motion, each disc perceive's the other as having its inertia and angular speed modified by Rindler g00, conserving angular momentum but producing shear (due to the difference in angular speed).
If the top observer measures the proper angular velocity of his disc to be w and the bottom observer also measures the speed of his disc to also be w, then the top observer will indeed perceive the the bottom disc to be turning slower by a factor of g00 and the bottom observer will consider the the top disc to be turning faster by a factor of g00. Now if we perform a "one time operation" of physically speeding the rear disc up by a factor of g00, they will both consider the discs to turning at the same constant rate for the rest of time and their is no sheer. This is the same as speeding up the rear clock by the appropriate factor so that the clock frequencies tick at the same rate according to both observers.

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But pervect showed that if the observers are moving relative to the plane, and the plane is accelerating, then those observers see the plane as curved, not plane. So their motion does make a difference.

I have had another look at the transformations given by Pervect
and there appears to be a number of issues with them. For certain inputs it is possible to get the inertial coordinates progressively going further back in time, while the time is going forward in the rocket. The vertical height of the rocket relative to height in the elevator changes over time. In the inertial coordinates the rocket is banana shaped and the tail and nose of the rocket droop below the floor of the elevator.

Since Pervect does not really show how he obtained the coordinate transformations, I decided to start again with a combination of Rindler and Lorentz transformations. Basically I transform from the Rocket reference frame (T,X,Y,Z) to the accelerating elevator reference frame (t',x',y',z') using the (reverse) Lorentz transformations and then from the elevator frame to the inertial reference frame (t,x,y,z) using Rindler coordinate transformations.

The (reverse) Lorentz transformation: Rocket (T,X,Y,Z) --> Elevator (t',x',y',z')

##t' = \gamma(T + v_0X)##
##x' = \gamma(X + v_0T)##
##y' = Y##
##z' = Z##

where ##v_0## is the horizontal velocity of the sliding rocket in the +x direction, relative to the elevator and inertial observer when ## t=t'=T =x = x' = X = 0## and this relative velocity is constant when measured in the elevator reference frame. The elevator floor is parallel to the x,y plane and the elevator accelerates in the z direction. ##\gamma## is the time dilation factor ##\sqrt{1-v_0^2}##.

The Rindler transformation: Elevator (t',x',y;,z') --> Inertial (t,x,y,z)

##t = (z' + 1/g) \sinh(gt')##
##x = x'##
##y = y'##
##z = (z' + 1/g) \cosh(gt')-1/g##

where g is the constant proper acceleration measured on the floor of the elevator. The ##-1/g## in the expression for z is an offset that ensures that z=z'=0 when t=t'=0.

Combining the above two transformations I get:

Rocket coords (T,X,Y,Z) --> Inertial coords (t,x,y,z)

##t = (Z + 1/g) \sinh(g\gamma (T-v_0X))##
##x = \gamma(X + v_0T) ##
##y = Y##
##z = (Z + 1/g)\cosh(g\gamma(T-v_0X)) - 1/g##

This transformation is much better behaved, with no time reversal and the rocket remains parallel with the elevator floor. At first I thought Pervect was introducing the curvature and rotation because light appears to travel in curves in an accelerated frame and since the human brain has evolved to assume light travels in straight lines the visual effect is that straight lines appear curved. However, there is no reason why the curvature is only there when the rocket is moving relative to the elevator. If we are going to define straight lines as lines parallel to a light path, then the curvature should also be present in the Rindler metric and it is not.

Pervect uses constant proper velocity (K) in his equations and this is related to (##v_0)## in the above equations by ##v_0 = K/\sqrt{1+K^2}##. Substituting this expression into the above transformation gives:

##t = (Z + 1/g) \sinh(gT\sqrt{1+K^2}-KX)##
##x = KT + X\sqrt{1+K^2} ##
##y = Y##
##z = (Z + 1/g)\cosh(gT\sqrt{1+K^2}-KX) - 1/g##

which is more easily compared directly with Pervect's version in the link above.

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In the last post I gave the transformations from the sliding rocket frame to the elevator frame to the inertial frame. This post is for the transformations in the reverse direction from inertial coords (t,x,y,z) to the accelerating elevator coords (t',x',y',z') to the rocket coords (T,X,Y,Z).

The Rindler transformation: Inertial (t,x,y,z) --> Elevator (t',x',y;,z')

##t' = 1/g * \tanh^{-1}\left(t/(z + 1/g)\right)##
##x' = x##
##y' = y##
##z' = \sqrt{(z + 1/g)^2 -t^2} -1/g##

The Lorentz transformation: Elevator (t',x',y',z') --> Rocket (T,X,Y,Z)

##T = \gamma(t' - v_0x')##
##X = \gamma(x'- v_0t')##
##Y=y'##
##Z=z'##

Combining the above two transformations:

Inertial coords (t,x,y,z) --> Rocket coords (T,X,Y,Z)

##T = \gamma(1/g * \tanh^{-1}\left(t/(z + 1/g)\right)- v_0x)##
##X = \gamma(x - v_0/g * \tanh^{-1}\left(t/(z + 1/g)\right)) ##
##Y = y##
##Z = \sqrt{(z + 1/g)^2 -t^2} -1/g##

In terms of the proper velocity K (See previous post), the above transformation gives:

Inertial coords (t,x,y,z) --> Rocket coords (T,X,Y,Z)

##T = \sqrt{1+K^2}/g * \tanh^{-1}\left(t/(z + 1/g)\right)- Kx##
##X = x\sqrt{1+K^2} - K/g * \tanh^{-1}\left(t/(z + 1/g)\right) ##
##Y = y##
##Z = \sqrt{(z + 1/g)^2 -t^2} -1/g##