Car, wheels and Lorentz contraction of the road: Is this a Paradox?

[LikenTs]

Right - but the road will be length contracted in the driver's frame, so the same number of teeth on the same radius wheel will not mesh with the shorter pitch of the teeth on the fast moving road.

OK. I see your point (And @Dale's, @A.T.'s,...)
But then, Ehrenfest Paradox is false. In hub of wheel IFR there is no any perimeter length contraction. At any rotation speed perimeter is 2πR, and pitch of teeth keeps constant. In this IFR there is absolutely no difference in the wheel from its state of rest, except that it is spinning. However, at the point of contact wheel-track, at same location-time, track, and pitches between teeth, are contracted. This difference can only arise because track moves straight while wheel slice at that point is rotating. If there is a paradox it is this one. So they cannot mesh. Analogy gears -track vs rolling wheels-road is invalid, and the only solution is discrepancy about N in low and high speed scenarios. All this, I think, is what SR states.

And despite everything, I still have the intuition that N is conserved between the experiments at low and high speed and that gear and gear-track mesh at any speed. So, I am outside of SR, I suspect. And it is very well tested experimentally. I don't know if there is any experiment that clearly indicates that N is not conserved at different speeds. Interestingly, in the case that the wheel undergoes a Lorentz contraction of its radius when it rotates, it would always hold that gear and track mesh and N is conserved at any speed. Contraction of radius is something that seems to agree in some obscure way with SR. Wheel rim shrinks as apparently expected due to shrinkage of radius.

 

[Dale]

And despite everything, I still have the intuition that N is conserved between the experiments at low and high speed and that gear and gear-track mesh at any speed.

Well, you will have to keep aware of that. It takes much more effort to correct bad intuition than it does to build good intuition.

I don't know if there is any experiment that clearly indicates that N is not conserved at different speeds.

With existing materials you will get an expansion of the wheel even at non-relativistic velocities and it will break long before relativistic effects become relevant. So the non-conservation of N is not a relativistic issue.

 

[LikenTs]

Well, you will have to keep aware of that. It takes much more effort to correct bad intuition than it does to build good intuition.

Letś imagine a long gear-track, and a car with gear wheels, progressively picking up speed and pausing sometimes to be an IFR, and at a point the car starts to vibrate because the wheels don't mesh well with the track. It is awful. It seems to contradict the principle that in proper space and time, in an intertial frame, with car windows closed, you cannot know your relative speed to anything, but in this case you can know your velocity by frame vibrations. And we are not talking about characteristics of the materials or elasticity, which would be understandable, but due to principles of space-time transformation.

 

[PeroK]

It seems to contradict the principle that in proper space and time, in an intertial frame, with car windows closed, you cannot know your relative speed to anything

There's no such principle. The principle of relativity says you cannot determine an absolute speed, relative to some fundamental reference frame.

 

[Nugatory]

It seems to contradict the principle that in proper space and time, in an intertial frame, with car windows closed, you cannot know your relative speed to anything, but in this case you can know your velocity by frame vibrations

How so? The car is constantly being buffeted by the forces from its impact with the bumps that you have designed into the road. Because the car is being subjected to these forces it is not moving inertially, and non-inertial motion is readily detected without reference to anything outside. Basically you've just designed a sort of clumsy accelerometer into the system.

(As an aside, your use of the phrase "in an inertial frame" suggests you have a common misunderstanding of what a frame is. Remember, everything is always in all frames all the time and we always do all our calculations using whichever frame happens to be most convenient, and what matters is not being "in an inertial frame", but rather "moving inertially")

 

[Dale]

It seems to contradict the principle that in proper space and time, in an intertial frame, with car windows closed, you cannot know your relative speed to anything

This isn’t a principle. I think you mean the principle that you cannot know your absolute speed. You certainly can know relative speeds.

And we are not talking about characteristics of the materials or elasticity, which would be understandable, but due to principles of space-time transformation.

Again, you cannot avoid talking about the material characteristics in this scenario

 

[PeroK]

Here's a nice view of this "paradox". Imagine a number of equal length objects that form a regular polygon when at rest. The polygon has a certain size.

Now, if we get all the objects moving with a common speed in the direction of their length and contrive to get them instantaneously to form a polygon, then this polygon will be smaller.

 

[Sagittarius A-Star]

And we are not talking about characteristics of the materials or elasticity, which would be understandable

The elasticity plays a role, because in your scenario, stress is created in the rotating gear by centrifugal force and by relativistic effects. In the rotating reference frame of the gear, the circumference of a circle around the origin with radius $R$ is $U > 2\pi R$, as measured with short rigid rods.

Einstein and general relativity
The rotating disc and its connection with rigidity was also an important thought experiment for Albert Einstein in developing general relativity.[4] He referred to it in several publications in 1912, 1916, 1917, 1922 and drew the insight from it, that the geometry of the disc becomes non-Euclidean for a co-rotating observer. Einstein wrote (1922):[5]

66ff: Imagine a circle drawn about the origin in the x'y' plane of K' and a diameter of this circle. Imagine, further, that we have given a large number of rigid rods, all equal to each other. We suppose these laid in series along the periphery and the diameter of the circle, at rest relatively to K'. If U is the number of these rods along the periphery, D the number along the diameter, then, if K' does not rotate relatively to K, we shall have $U/D=\pi$ . But if K' rotates we get a different result. Suppose that at a definite time t of K we determine the ends of all the rods. With respect to K all the rods upon the periphery experience the Lorentz contraction, but the rods upon the diameter do not experience this contraction (along their lengths!). It therefore follows that $U/D>\pi$ .

Source:
https://en.wikipedia.org/wiki/Ehrenfest_paradox

 

[LikenTs]

There's no such principle. The principle of relativity says you cannot determine an absolute speed, relative to some fundamental reference frame.

I mean the principle that all physical laws should be the same in every inertial frame of reference. If your are accelerated in a train you can measure it because the bowl of soup is overflowing, but if train is at constant speed, in an indoor lab, without communication with the outside, you cannot know your speed, whether absolute or relative, through experiments. In the style of the rocket of the equivalence principle of GR.

How so? The car is constantly being buffeted by the forces from its impact with the bumps that you have designed into the road. Because the car is being subjected to these forces it is not moving inertially, and non-inertial motion is readily detected without reference to anything outside. Basically you've just designed a sort of clumsy accelerometer into the system.

There is no impact with bumps if you are not accelerated, wheels keeps rolling by inertia, in perfect synchrony with the gear-rack moving at constant speed backwards. Ideally there is no friction in the axes, and we can think that there is no gravity either because we are in free space.

When you hit the throttle you push the track back, when you release the throttle it's like the track doesn't exist and you move freely. Except if you start to have high speed on the track, and start to get collisions due to a coordinate transformation between IFRs. Then you somehow determine that you have gone too far by increasing your relative speed.
´

Again, you cannot avoid talking about the material characteristics in this scenario

But the Lorentz transformation has nothing to do with objects, it's about geometry of events between different observers. Rigid solids are not possible because they involve instantaneous interactions. And in ideal experiments on SR I do not think it is necessary to resort to resistance of materials or thermodynamics.

 

[Dale]

But the Lorentz transformation has nothing to do with objects

Agreed. I have mentioned this a couple of times already, but perhaps you will notice it this time:

The fast and the slow scenarios are not related to each other by a Lotentz transform.

Rigid solids are not possible because they involve instantaneous interactions.

But Born rigid motion is possible in some circumstances. Angular acceleration is not one of those.

And in ideal experiments on SR I do not think it is necessary to resort to resistance of materials

It is when you don’t have Born rigid motion. In those cases strain is unavoidable. Most thought experiments use only Born rigid motion precisely to avoid these issues and allow an ideal analysis.

 

[A.T.]

And despite everything, I still have the intuition that N is conserved between the experiments at low and high speed and that gear and gear-track mesh at any speed.

All so called "paradoxes" in relativity are a result of wrong assumptions based on intuition.

In the context of Bell's spaceship paradox and Ehrenfest paradox (which are linear and circular versions of the same paradox) these are key things to keep mind:

1) Relativistic length contraction relates lengths in different frames at the same time, not lengths at different time points (like before and after acceleration). The intuition that accelerated objects must shrink is based on the additional assumption that their proper length stays constant. But if the proper length changes, then it is possible to change speed of object while keeping its length constant. This is what happens to the rope in Bell's paradox and to the rim in Eherenfest paradox.

2) Relativistic length contraction by itself is just a coordinate effect and cannot be measured in a frame invariant manner with strain gauges, etc. Only in combination with additional boundary conditions (keep the rocket spacing constant, keep the wheel radius constant), it can imply changes in proper length, which are physically deforming the material, extending telescopic struts, etc (frame invariant measures).

3) There are avoidable and unavoidable physical deformations (proper-geometry changes). In thought experiments we can postulate external body forces, which provide the necessary acceleration and support to any part of the body, so that internal stresses would seem unnecessary. But even under this idealized assumptions, there can be unavoidable physical deformations if we try to enforce certain geometric boundary conditions.

 

[Ibix]

This difference can only arise because track moves straight while wheel slice at that point is rotating.

Imagine that I tie a piece of elastic between my finger and thumb. It has unstressed rest length $L$. I see you coming towards me with Lorentz factor $\gamma$ and stretch the elastic out by a factor of $\gamma$. You then measure a length-contracted length of $\gamma L/\gamma=L$. Is that a paradox? Or am I just messing with you by stretching the elastic in a particular way so it looks (naively) like length contraction didn't happen?

Similarly, the rim of the wheel is stressed. Its "natural" length when spinning is less than $2\pi R$, but it cannot contract because it can't compress the rest of the material of the disc, so it is under stress and stretched (assuming the disc doesn't bow into a dish here). If it doesn't bow, you could attach a lot of little rulers to the rim, each one by a single nail at its center point. If the total rest length of the rulers is $2\pi R$ you will find that there are one or more gaps between the rulers because they aren't forced to stretch in a way that hides their length contraction.

 

[Nugatory]

There is no impact with bumps if you are not accelerated, wheels keeps rolling by inertia,

You said the car is “vibrating”. That is back and forth motion, therefore not inertial and will be measurable with an accelerometer. (In fact, calculating the constant speed of a subway car from an accelerometer trace was a lab exercise in my first year physics class)

 

[PeroK]

I mean the principle that all physical laws should be the same in every inertial frame of reference.

Okay, but we are not talking about that in this thread. We are talking about a vehicle moving relative to a surface, using some sort of interlocking gearing system.

if train is at constant speed, in an indoor lab, without communication with the outside, you cannot know your speed, whether absolute or relative, through experiments.

First, I've never heard of a principle that you cannot measure relative speed. The measurement and symmetry of relative speed is a vital building block of SR - even though it's not often highlighted.

The key word here, however, is communication. If a train is rattling along on uneven tracks, then that is communication. Similarly, the mismatch in the gears provides communication of the relative speed.

In general, you need to be careful making your own interpretation of things like the principle of relativity and the equivalence principle. If you try to push these things beyond their precise formulation, then you will lead yourself astray.

It may be worth pointing out that, after nearly 50 posts, the efforts of the numerous advisors have not been totally successful in helping you understand the physics. You still seem more tempted to let your own ideas lead you astray.

 

[LikenTs]

Imagine that I tie a piece of elastic between my finger and thumb. It has unstressed rest length $L$. I see you coming towards me with Lorentz factor $\gamma$ and stretch the elastic out by a factor of $\gamma$. You then measure a length-contracted length of $\gamma L/\gamma=L$. Is that a paradox? Or am I just messing with you by stretching the elastic in a particular way so it looks (naively) like length contraction didn't happen?

Similarly, the rim of the wheel is stressed. Its "natural" length when spinning is less than $2\pi R$, but it cannot contract because it can't compress the rest of the material of the disc, so it is under stress and stretched (assuming the disc doesn't bow into a dish here). If it doesn't bow, you could attach a lot of little rulers to the rim, each one by a single nail at its center point. If the total rest length of the rulers is $2\pi R$ you will find that there are one or more gaps between the rulers because they aren't forced to stretch in a way that hides their length contraction.

I have been reading more about the Ehrenfest paradox and there is also another interpretation, that all contour rules are in the same circle of simultaneity with respect to the central inertial observer and there would not be Lorentz contraction. While in a rod that moves straight the clocks at their ends show a delay.

In this way it is not necessary to imagine that there is a Lorentz force, capable of causing stress in an object. @A.T. also mentions Bell's paradox. The interpretation would be that the rope is not broken by Lorentz forces but because the rockets are separating in the proper reference frame.

The key word here, however, is communication. If a train is rattling along on uneven tracks, then that is communication. Similarly, the mismatch in the gears provides communication of the relative speed.

But the lab in train would not need communication, it would just start to notice accelerations and slowdowns, caused by misalignment of gears, in turn caused not by a real force but by Lorentz contractions. Yes, I accept that it may not be a strict violation of the principle but it seems to violate it in spirit.

It may be worth pointing out that, after nearly 50 posts, the efforts of the numerous advisors have not been totally successful in helping you understand the physics. You still seem more tempted to let your own ideas lead you astray.

No, I'm learning a lot and I appreciate everyone's input. Not just me but anyone who in the future sees this thread about the relativistic gears problem. And always acknowledging if my own ideas fail miserably or I'm off the mark.

 

[Dale]

In this way it is not necessary to imagine that there is a Lorentz force, capable of causing stress in an object. @A.T. also mentions Bell's paradox. The interpretation would be that the rope is not broken by Lorentz forces but because the rockets are separating in the proper reference frame.

The Lorentz force is something completely different and totally unrelated to anything we have been discussing in this thread.

in turn caused not by a real force but by Lorentz contractions.

This is false and has been explained to you multiple times. This is very frustrating. I have told you repeatedly that these are actual mechanical deformations measurable with strain gauges and must be analyzed using material laws like elasticity. Do you think elastic forces are not real forces? Do you think centripetal forces are not real forces?

You have also been told repeatedly that the fast and slow scenarios are not related by a Lorentz transform, so how can you ascribe the vibrations to length contraction?

 

[A.T.]

I have been reading more about the Ehrenfest paradox and there is also another interpretation, that all contour rules are in the same circle of simultaneity with respect to the central inertial observer and there would not be Lorentz contraction. While in a rod that moves straight the clocks at their ends show a delay.

"I have read somewhere" is not a valid source. Every piece of the rim has different length in the frame where it moves tangentially compared to a frame where it is at rest (that is Lorentz contraction). Simultaneity is completely irrelevant here, because you can spin the wheel for a long time at constant rate, and measure it's constant proper-geometry with rulers attached to it, without any use of clocks.

@A.T. also mentions Bell's paradox. The interpretation would be that the rope is not broken by Lorentz forces but because the rockets are separating in the proper reference frame.

Different frames can invoke different "reasons" for the breaking of the rope, they just have to agree that it breaks. In the frame where the accelerating rope keeps a constant length the reason provided could be that the force fields between atoms of the rope are contracting and cannot span the full length anymore.

Yes, I accept that it may not be a strict violation a the principle but it seems to violate it in spirit.

Violation of what? There is nothing forbidding the detection of relative velocities.

 

[Sagittarius A-Star]

But then, Ehrenfest Paradox is false. In hub of wheel IFR there is no any perimeter length contraction. At any rotation speed perimeter is 2πR, and pitch of teeth keeps constant. In this IFR there is absolutely no difference in the wheel from its state of rest, except that it is spinning.

Do you mean by "wheel IFR" a non-rotating frame, in which the center of the rotating wheel is at rest?
To say "In hub of wheel IFR" does not specify a frame, only a location, but length contraction is frame-dependent. Which reference frame do you mean?

 

[PeroK]

But the lab in train would not need communication, it would just start to notice accelerations and slowdowns, caused by misalignment of gears, in turn caused not by a real force but by Lorentz contractions. Yes, I accept that it may not be a strict violation of the principle but it seems to violate it in spirit.

It should be clear that vibrations from the road are every bit as much communication as light coming through the window!

This is what I'm talking about. We tell you something and you just argue that it's not the case. You can't learn like that.

 

[Nugatory]

also mentions Bell's paradox. The interpretation would be that the rope is not broken by Lorentz forces but because the rockets are separating in the proper reference frame.

What is this "proper" reference frame of which you speak? There is nothing that makes one reference frame more or less proper or preferred than any other, and no matter which frame we use when analyzing a problem all frames will agree about all the physical facts.

In Bell's spaceship paradox, the physical fact is that stresses build up in the rope until it breaks; if we were to attach a stress gauge to the rope it would show the stresses and all frames will agree about the existence of these stresses. In some frames this is explained as the ships moving apart while the rope maintains a constant length, in others it is explained as the rope contracting while the ships maintain the same separation. Neither description is more "proper" than the other.

 

[Ibix]

I have been reading more about the Ehrenfest paradox and there is also another interpretation, that all contour rules are in the same circle of simultaneity with respect to the central inertial observer and there would not be Lorentz contraction. While in a rod that moves straight the clocks at their ends show a delay.

That sounds like nonsense, I'm afraid. At best it is a horribly badly mangled attempt at paraphrasing something about different coordinate systems. I'm guessing you asked ChatGPT to summarise information about the Ehrenfest Paradox and it's made its usual physics jargon stew from it.

In this way it is not necessary to imagine that there is a Lorentz force, capable of causing stress in an object.

Nobody said there was any such thing. The Lorentz force is the force on a charged particle due to electromagnetic fields - it has nothing to do with Lorentz transforms. All of the forces we are talking about here are elastic and centrifugal/centripetal forces.

The interpretation would be that the rope is not broken by Lorentz forces but because the rockets are separating in the proper reference frame.

As @Nugatory has pointed out, this is also nothing to do with the Lorentz force (which is still nothing to do with Lorentz contraction). The string breaks from good old fashioned elastic stress.

 

[PeterDonis]

In Bell's spaceship paradox, the physical fact is that stresses build up in the rope until it breaks; if we were to attach a stress gauge to the rope it would show the stresses and all frames will agree about the existence of these stresses. In some frames this is explained as the ships moving apart while the rope maintains a constant length, in others it is explained as the rope contracting while the ships maintain the same separation. Neither description is more "proper" than the other.

We can even give a frame-independent, invariant explanation for why the stresses build up: that the expansion scalar of the congruence of worldlines describing the rope is positive. The expansion scalar is an invariant and will have the same value no matter what frame it is computed in.

 

[A.T.]

In this way it is not necessary to imagine that there is a Lorentz force, capable of causing stress in an object.

As others already noted, Lorentz force has a different meaning.

Also note that you can avoid the stresses by using a compliant structure made of telescopic struts connected with joints and external forces to provide the locally needed acceleration. But you cannot avoid some change of proper geometry, which will be measured locally by the telescopic parts and joints as a frame invariant fact.

 

[Sagittarius A-Star]

I have been reading more about the Ehrenfest paradox and there is also another interpretation, that all contour rules are in the same circle of simultaneity with respect to the central inertial observer and there would not be Lorentz contraction. While in a rod that moves straight the clocks at their ends show a delay.

Can you please provide a link to were you read this?

 

[LikenTs]

Can you please provide a link to were you read this?

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber

Some excerpts:

Observers anywhere in the rotating frame and observers in the non-rotating frame all agree on simultaneity.

Although frames agree on simultaneity, it can be shown that standard clocks in each run at different rates. (Note that two clocks running at different rates can nonetheless both agree on simultaneity, i.e., that no time elapsed off either one between two events.) Time dilation does occur, but it is not symmetric, i.e., rotating and non-rotating observers agree that time dilation occurs on the disk relative to the stationary frame.

The Lorentz contraction is a direct result of non-agreement in simultaneity between frames. If there is agreement in simultaneity, there is no Lorentz contraction.

There is simply no kinematic imperative for the rods to try to contract. No tension arises in the disk as it is spun up, and no relativistically induced rupturing occurs.

Rods in inertial frames with velocities equal to the tangent velocities at a given disk radius can not be used to measure the circumference, since the ”surrogate frames postulate” for equivalence of inertial and non-inertial standard rods is not valid for the rotating frame, and is generally invalid for any non-time-orthogonal frame, and doing so leads to a discontinuity in time.

 

[Dale]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber

Unfortunately, Foundations of Physics Letters is not a recognized professional scientific journal. It is not listed in the Clarivate master journal list. This article is not a valid reference per the forum rules. In particular, it has several problems which explain why it was not able to be published in any reputable journal.

One is the claim “special relativity is restricted to inertial systems”. That is clearly false and even Einstein’s seminal 1905 paper included an analysis of a non-inertial system. Even non-inertial reference frames are part of special relativity. The math used to transform between arbitrary coordinates is well known and is not anathema to SR.

His 4 postulates of general relativity are non-recognizable.

He states “these results should be expected”, as though they were not. They were expected and predicted by Einstein who encouraged Michelson and Gale in their efforts on this topic.

His entire section 2.1 entirely misrepresents both the scientific community’s understanding of these topics as well as the compatibility of these concepts with special relativity.

His phrase “heretofore seemingly sacrosanct” is designed to convey the impression that he is challenging a religious community’s doctrine. In fact, the coordinate system he uses is completely standard and used routinely by the scientific community.

Most importantly for this discussion, the claim “No tension arises in the disk as it is spun up” is completely unjustified. The angular velocity in the entire paper is always a fixed $\omega$. The author never even considers angular acceleration and so can make no claims about what happens as a disk is “spun up”. And nowhere is a single calculation about the expansion tensor performed which is what would be required to claim that there is no material strain. So neither the “no tension arises” nor the “as it is spun up” parts of the claim are supported by any of the math anywhere in the paper.

In short, this is not a valid source. You should not use it as support for any posts here at PF.

 

[Sagittarius A-Star]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber
Some excerpts:
...
The Lorentz contraction is a direct result of non-agreement in simultaneity between frames. If there is agreement in simultaneity, there is no Lorentz contraction.

Thank you for providing the link instead of only writing "I have been reading more about the Ehrenfest paradox and there is also another interpretation ...". However, as @Dale wrote, this is not a valid source.

R. Klauber's argument contains a logical flaw.

3.2.2 No Lorentz Contraction.
The Lorentz contraction is a direct result of non-agreement in simultaneity between frames. If there is agreement in simultaneity, there is no Lorentz contraction. To show this we need one additional, presumably inviolable, postulate. That is,

3. The proper spacetime length of any path is invariant under any transformation, i.e., it is the same for all observers.

Hence, for two frames in relative motion (notation should be obvious)
$$(∆s)^2 = − c^2(∆t)^2 + (∆l)^2 = − c^2(∆t^′)^2 + (∆l^′)^2\ \ \ \ \ (5)$$
For a rod at rest in the primed frame, an observer in the unprimed frame sees that rod such that its endpoints are events which for him occur simultaneously, i.e., $∆t = 0$. But in the primed system those events are not, according to standard relativity theory, simultaneous and $∆t^′ \neq 0$. This means $∆l \neq ∆l^′$, and results in Lorentz contraction [27]. If, however, the same two events could also appear simultaneous in the primed system, then $∆t^′ = 0$ and $∆l$ must equal $∆l^′$.

His equation (5) is based on both coordinate systems (primed and unprimed) being time-orthogonal. But for $(∆t = 0) \Rightarrow (∆t^′ = 0)$ he uses a non-time-orthogonal coordinate system (clock synchronization based on anisotropic one-way-speed of light) along the rim of the rotating disk. That is logically inconsistent.

 

[Ibix]

Most importantly for this discussion, the claim “No tension arises in the disk as it is spun up” is completely unjustified. The angular velocity in the entire paper is always a fixed $\omega$. The author never even considers angular acceleration and so can make no claims about what happens as a disk is “spun up”. And nowhere is a single calculation about the expansion tensor performed which is what would be required to claim that there is no material strain. So neither the “no tension arises” nor the “as it is spun up” parts of the claim are supported by any of the math anywhere in the paper.

As a matter of curiosity I set up a simple "constant angular acceleration" model and computed the expansion scalar. Consider a disk that is initially at rest in some inertial frame. At time $t=0$ it starts to spin with (in this frame) constant angular acceleration $\dot\omega$, so at time $t$ all points on the disc have the same angular velocity $\dot\omega t$. This is not a particularly plausible acceleration model in a relativistic case (as discussed above) but it's easy to work with.

Noting that the three velocity magnitude of each point on the disc is $\dot\omega tr$ then in polar coordinates ($ds^2=-dt^2+dr^2+r^2d\phi^2+dz^2$, $c=1$) the four-velocity field is $$X^a=\left(
\frac{1}{\sqrt{1-\dot\omega^2t^2r^2}},
0,
\frac{\dot\omega t}{\sqrt{1-\dot\omega^2t^2r^2}},
0\right)$$where I have used $X^a$ and a -+++ sign convention to be consistent with the notation in the Wikipedia page on congruences. Grinding through the maths there leads to the expansion scalar$$
\theta = \frac{\dot\omega^2r^2t}{\left(1-\dot\omega^2t^2r^2\right)^{3/2}}
$$Note that there's an assumption that $X^a$ is timelike so this is only defined for $\left(\dot\omega tr\right)^2<1$, which is a technical way of saying that the rim of the disc must be travelling slower than light. It also means that the division by zero at $\dot\omega tr=1$ is in the region where this expression isn't valid. This expansion scalar is manifestly positive for positive $t$, so two nearby points on the rim of the disc will indeed see the distance between them grow. (It being negative for negative $t$ is also expected since extending this congruence to negative times would represent the disc slowing its spin at that time.) Or, to put it another way, short rulers along the rim of the disc will indeed separate from one another, contrary to the claim in the paper.

Edit: in fact, the entire expansion tensor is interesting. In these coordinates the only non-zero components are $\theta_{tt}$, $\theta_{\phi t}$, $\theta_{t\phi}$ and $\theta_{\phi\phi}$, which shows that there's only expansion/contraction in the time and tangential directions - nothing in the radial or vertical directions. This is exactly what you'd expect from the naive momentarily co-moving inertial frame analysis.

Here is a Maxima batch file to compute all of the above.

/* Compute the expansion scalar of a simple "constant angular acceleration" case in flat */
/* spacetime. Procedure follows Wikipedia link below, including an assumption of -+++    */
/* metric signature.                                                                     */ 
/* https://en.wikipedia.org/wiki/Congruence_(general_relativity)#Kinematical_description */

load(ctensor);
load(eigen);

/* Minkwoski metric, -+++, cylindrical polars */
dim: 4;
ct_coords: [t, r, phi, z];
lg: matrix([-1, 0, 0,   0],
           [ 0, 1, 0,   0],
           [ 0, 0, r^2, 0],
           [ 0, 0, 0,   1]);

/* Compute the inverse metric, ug, and the Christoffel symbols, mcs. */
/* NB: mcs[i,j,k] is \Gamma^k_{ij}.                                  */
cmetric();
christof(true);

/* Some useful functions for later */
tensorProduct(u,v) := block(
  /* Takes two one-index tensors u^a and v^b (expressed as matrices) and */
  /* returns u^a v^b. Either or both tensors may have lower indices -    */
  /* the resulting tensor has the implied indices.                       */
  [i, j, ans],
  ans: zeromatrix(4,4),
  for i: 1 thru 4 do block(
    for j: 1 thru 4 do block(
      ans[i, j]: u[i][1] * v[j][1]
    )
  ),
  ans
);
covdif(x) := block(
  /* Computes the covariant derivative of the lower-index tensor x_b, */
  /* (expressed as a matrix). That is, this returns \nabla_a x_b.     */
  [i, j, k, ans],
  ans: zeromatrix(4,4),
  for i: 1 thru 4 do block (
  	for j: 1 thru 4 do block (
  		ans[i, j]: diff(x[i][1], ct_coords[j]),
  		for k: 1 thru 4 do block (
  			ans[i, j]: ans[i, j] - mcs[j, i, k] * x[k][1]
  		)
  	)
  ),
  ans
);
symmetrise(T) := block (
  /* Returns the symmetric part of a two-index tensor T^{ab} or T_{ab}. */
  (T + transpose(T)) / 2
);

/* Construct the "accelerating rotation" congruence, which represents a */
/* three-velocity field with magnitude (dotomega * t) * r.              */
v: dotomega * t * r;
gamma: 1/sqrt(1-v^2);
X: columnvector([gamma, 0, dphi, 0]);
assume(dotomega^2*r^2*t^2 < 1);
solve(-1 = transpose(X).lg.X, dphi);
X: substitute([%[2]], X);

/* Compute the projection operator h^a{}_b and the symmetrised covariant */
/* derivative of X_a, \nabla_{(a}X_{b)}.                                */
hul: ug.lg + tensorProduct(X, lg.X);
sdXll: symmetrise(covdif(lg.X));

/* Compute the expansion tensor */
theta_ll: zeromatrix(4,4);
for a: 1 thru 4 do block(
  for b: 1 thru 4 do block(
    for m: 1 thru 4 do block(
      for n: 1 thru 4 do block(
        theta_ll[a, b]: theta_ll[a, b] + hul[m, a]*hul[n, b] * sdXll[m, n]
      )
    )
  )
);

/* Compute the mixed-index version of the expansion tensor and its trace, */
/* the expansion scalar, \theta.                                          */
theta_ul: ug.theta_ll;
theta:0;
for a: 1 thru 4 do block(
  theta: theta + theta_ul[a, a]
);
theta: ratsimp(theta);

 

[Ibix]

R. Klauber's argument for this contains a logical flaw.

And his "postulate 3" is unnecessary since it is a direct consequence of the way the interval is defined.

 

[A.T.]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber

Some excerpts:

Rods in inertial frames with velocities equal to the tangent velocities at a given disk radius can not be used to measure the circumference, since the ”surrogate frames postulate” for equivalence of inertial and non-inertial standard rods is not valid for the rotating frame, and is generally invalid for any non-time-orthogonal frame, and doing so leads to a discontinuity in time.

Given that the centripetal acceleration at the rim can be made arbitrarily small (by increasing the radius), I wonder what Klauber thinks happens to tangentially moving inertial rods, when they are given even the smallest amount of centripetal acceleration. Do they suddenly expand to their proper length?

That would be a problematic discontinuity, unlike the one he worries about stemming from mere conventions of simultaneity.

 

[Sagittarius A-Star]

New Perspectives on the Relativistically Rotating Disk and Non-time-orthogonal Reference Frames Robert D. Klauber

He writes:

4.1 Rotating Frame Metric and Transformations

This coordinate transformation, where upper case coordinates represent the inertial frame
K, lower case denote the rotating frame k, and the axis of rotation is coincident with both the
Zand z axes, is
$cT = ct \ \ \ \ \ (8a)$
$R = r \ \ \ \ \ (8b)$
$Φ = φ + ωt \ \ \ \ \ (8c)$
$Z = z \ \ \ \ \ (8d)$
...
The transformation (8) seems Galilean in nature, rather than relativistic, and if it is valid
(as most researchers today feel that it is), we should not be surprised to find the disk exhibiting
at least some Galilean characteristics.

Maybe he took his transformation (8) from the Wikipedia article about Born coordinates, "Transforming to the Born chart":
https://en.wikipedia.org/wiki/Born_coordinates#Transforming_to_the_Born_chart

I'm not familiar with Born coordinates. But in the Wikipedia article about it, they continue with a complete relativistic transformation to a Langevin observer frame on the rim of the rotating disk, while R. Kolb only adds later in his chapter 4.3.1 a $\gamma$-factor for time dilation.

R. Klauber also refers to F. Selleri, but doesn't use the Selleri-transformation (=modified Lorentz transformation without the term for "relativity of simultaneity") for a non-time-orthogonal frame on the rim of the rotating disk. Instead he uses a wrong "shortcut".

 

[Ibix]

Grinding through the maths there leads to the expansion scalar$$
\theta = \frac{\dot\omega^2r^2t}{\left(1-\dot\omega^2t^2r^2\right)^{3/2}}
$$

Replying to myself, sorry, but one other comment on this is that in this case $\theta=\frac{d}{dt}\gamma$ where $\gamma$ is the Lorentz gamma factor associated with the velocity $\dot\omega tr$ - i.e., the linear velocity of a point at radius $r$ at a time $t$ into the acceleration. The interpretation of $\theta$ is that it's the fractional increase in volume of a small region - and since we've established that the expansion is one dimensional, this is saying that the rate of growth of that 1d stretch is equal to the rate of growth of $\gamma$, exactly as you'd expect if rulers were length contracting exactly like in linear acceleration.

This case is a bit special to this particular congruence, I think, but it's a rather nice result.

 

[Sagittarius A-Star]

Interestingly, in the case that the wheel undergoes a Lorentz contraction of its radius when it rotates, it would always hold that gear and track mesh and N is conserved at any speed. Contraction of radius is something that seems to agree in some obscure way with SR. Wheel rim shrinks as apparently expected due to shrinkage of radius.

• If you allow in a thought experiment the rim (circumference) to freely contract while spinning-up, i.e. by having a disk made of very elastic rubber, and only the circumference is made of a thin layer of metal and ignoring centrifugal force, then the conclusion in this statement is correct, except the word "obscure".
• If you force in another thought experiment the rim (circumference) to not contract while spinning-up, i.e. by having only the rim of a wheel made of rubber and ignoring centrifugal force, then obviously gear and track don't mesh and N is not conserved at any speed. In this case the rim would get internal mechanical stress (for the same reason as the thread in the Bell spaceships paradox). This case also agrees with SR.
  • 

Source of the picture:
https://www.spacetimetravel.org/tompkins/7

If you are interested in understanding SR, I recommend not to waste your time with the paper of R. Klauber, because it describes a different physics than SR, by introducing a strange combination of Galilei transformation and time dilation. SR was never refuted by any experiment.

 

[A.T.]

Interestingly, in the case that the wheel undergoes a Lorentz contraction of its radius when it rotates, it would always hold that gear and track mesh and N is conserved at any speed.

The shrinking of the radius is not Lorentz contraction, which happens only in the tangential direction. But you can allow the radius to shrink physically (e.g. proper length change of telescoping radial beams), to preserve the proper length of the rim. That would indeed allow to the gears to mesh at any speed, while preserving the number of teeth and thus revolutions per given proper length of track.

 

[Sagittarius A-Star]

But then, Ehrenfest Paradox is false. In hub of wheel IFR there is no any perimeter length contraction. At any rotation speed perimeter is 2πR, and pitch of teeth keeps constant. In this IFR there is absolutely no difference in the wheel from its state of rest, except that it is spinning. However, at the point of contact wheel-track, at same location-time, track, and pitches between teeth, are contracted. This difference can only arise because track moves straight while wheel slice at that point is rotating. If there is a paradox it is this one.

I can't comment on this because I don't understand your text, and you did not answer my related question in posting #53.