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

[PeterDonis]

That surely depends upon your material model, though.

I don't think there can be any material model that has the wheel having the same shape in the relativistic as in the non-relativistic case. Relativity imposes limits on the properties of all materials. And relativistic kinematics imposes limits on what can and cannot be held constant as the spin rate of a wheel is varied; in particular, there is no Born rigid way to spin up a wheel from lower to higher frequency.

 

[PeterDonis]

The use of the same $N$ invalidates his analysis in one frame or other.

Not for the same relative speed (low or high). The issue is that $N$ cannot be the same for both relative speeds. But if we hold the relative speed fixed, then $N$, the number of times a given point on the wheel contacts the road from start to finish, is frame invariant.

 

[A.T.]

I assume gears mesh at any speed if they mesh at low speed, because in road IFR the contact point of whell-Gear is at zero speed, and in the car IFR it is at same speed than track, so any contraction should be the same.

Let's assume you build the gear already rotating at a given rate from tooth-segments that you accelerate tangentially and connect. The number of teeth you can fit on a circumference of a given radius depends on the rotation rate, because with greater tangential speed the segments are more length contracted.

So if you want your rack to mesh with the gear (no deformation of the gear-teeth), then you must build different wheels with a different number of teeth for any speed.

But if you use the same gear and enforce a constant radius, then the circumference will deform differently at different speeds, and it won't mesh with the rack. Not because their kinetic length contractions are different at the contact, but because the gear teeth are actually physically deformed, as is measurable with stain gauges.

 

[Dale]

@LikenTs one other thing to recognize is that any contradiction between the slow scenario and the fast scenario will not be due to relativity because relativity does not govern the difference between the two scenarios. The fast and slow scenarios are related by material laws of elasticity and stress and strain. So any inconsistency would point to your laws of elasticity being incorrect (probably non-relativistic).

In the fast scenario, relativity governs the relationship between the axle frame and the road frame. If you found a contradiction there, the same scenario in two different frames, then that would be a relativity paradox.

Since you have said that you don’t want to resort to theories of elasticity then I would recommend the following approach. Simply specify the geometry in the fast scenario and don’t even consider the slow scenario. Analyze the fast scenario in the axle and ground frames and check for any contradictions that interest you.

 

[Nugatory]

Since you have said that you don’t want to resort to theories of elasticity then I would recommend the following approach. Simply specify the geometry in the fast scenario and don’t even consider the slow scenario. Analyze the fast scenario in the axle and ground frames and check for any contradictions that interest you.

@LikenTs This is really good advice.... And another step that you might want to consider is to ignore the wheel, instead consider a device that puts a drop of paint on the rail once every second (according to the frame in which the car is at rest) - this is basically what you've been expecting to get out of one turn of the rotating wheel, but allows you to avoid the seriously non-trivial analysis of the wheel. Once you have this relativistic analysis under your belt, you will be much better positioned to take on the wheel.

 

[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.