The Actual Solution To The Ehrenfest Paradox (Relativistic Spinning Disk)

[LaurieAG]

19 hours ago, Jaro said:

I am almost convinced by the arguments in this post. I say almost because there is one troubling feature that I cannot resolve using the same arguments. And it is this:

If all the clocks on the spinning wheel are synchronized with each other, and they are synchronized (though ticking slower) with clocks in the stationary frame, then the coordinates representing constant time for both rotating and stationary observers are lines in the plane perpendicular to the time coordinate. However, suppose that the short rods the wheel is composed of were to decouple from the spokes and fly out in a straight line. To an observer in the stationary frame, who has not seen these rods be part of a rotating structure a moment go (all he sees is a rod moving in a straight line), the rods would now be Lorenz contracted and two clocks attached to the end of each rod would not be synchronized to him.

So, my issue is the apparently discontinuous transition from one coordinate system in which all observers agree on lengths and the synchronicity of clocks to one where they do not.

Thank you for addressing this point.

 

Hi Jaro,

There is only 1 observer and many emission points so It's easiest to think of the points on a ring/rim and the 'spokes' as the relevant length contraction to a particular point on the ring/rim as distances in the Gron Fig 9 part C plot (this is not a frame) which can be measured in time or distance (traveled by light in that time). Also, all of the points on the ring/wheel frame are synchronized at the axle and this axle is part of the carriage frame which is parallel with the road frame where the observer is situated.

 

[BobbySteeles]

On 7/21/2020 at 3:36 PM, AnssiH said:

I just recently stumbled upon a Quora question regarding Ehrenfest Paradox, and noticed that people were giving completely wrong answers to it.
 
Basically the paradox is this; In terms of Special Relativity, how does spinning disk work in special relativity, if the circumference of the rotating disk undergoes length-contraction (since it's parallel to motion) while its radius does not (since it's perpendicular to motion), and this would imply that [math]\frac{circumference}{diameter}  \neq \pi  [/math].
 
I checked bunch of similar questions of the same topic, and can't find a single person giving the correct answer on Quora. Instead I find all sorts of face-palm inducing nonsense like;

• Only the atoms length contract, but the space between them does not.
• The disk would tear into smaller pieces along the radius to give shorter total circumference.
• The disk would implode under pressure from the shrinking circumference.
• Centrifugal forces would counteract Lorentz contraction.
• There's no strong enough material to build such disk because of Born rigidity and elasticity, thus no paradox.
• You need to use General Relativity to solve the paradox.
• The geometry of the spinning wheel is non-euclidean. Just accept it.

And bunch of other answers going completely off on a tangent on topics like people inside a spinning train setting their clocks. Basically every single answer I can find tells me the author probably holds serious misconceptions about Special Relativity itself.
 
Okay, it's Quora so I shouldn't expect too much, but still I would have expected that at least someone would have given the solution to to something this simple, instead of seeing bunch of people with credentials compete with silly answers. Some of those people are citing their own book about the topic while giving a terrible answer... I mean I'm not that smart, but I solved this problem in my head, while driving. It's really that simple if you actually already understand Special Relativity properly.
 
What really surprised me was when I went on to check how does Wikipedia see this, and it also doesn't explain the proper solution. There is only one passing mention of the correct solution (kind of, possibly, can't really tell) in the "Brief History" section... with no actual explanation. I guess this is why no one in Quora also knows the answer, but still I'm quite dumbfounded to realize that the actual solution is apparently not very well known at all. I can't find any article actually explaining the correct solution.
 
Looking at all the bad answers, it seems to me that that there are few different reasons why most people get this so wrong.
 
One is that many people think about length contraction as something that happens to objects, when more accurately it's what happens to your coordinate system when you change your perspective from one inertial frame to another, and follow Einstein convention for isotropic C. If you think it happens to "objects" because they "move", you might be inclined to bring up stuff like "atoms shrink by space between them does not", and that is completely wrong perspective.
 
Second is that many authors start to analyze realistic materials and Born rigidity, which to some people perhaps seem like a way out of the paradox in some convoluted way. But that is also a complete red herring. The paradox is a thought experiment, and it has got nothing to do with realistic materials. It's about geometry in terms of special relativity, which ought to produce self-consistent results regardless of inertial frame. Solving Ehrenfest paradox by bringing up realistic materials and centrifugal forces is like solving twin paradox with "planet earth cannot produce enough energy to actually run that experiment".
 
Third reason is that a mathematical analysis in the framework of special relativity is easiest to do by making certain approximations, which are exactly the approximations leading into wrong answers. That misleading approximation is the idea of placing a number of straight rods along the circumference of the disk, and this approximation is exactly what will give you wrong answers. That's right, Einstein's own analysis is also flawed for the same reason, even though it led into insights that led into General Relativity.
 
Why that approximation produces a critically wrong answer, and what is the correct answer? I'll explain in a bit...
 
The correct perspective
 
First, just to convince the reader that this problem is in fact fully solvable in terms of Special Relativity without any hocus pocus about elastic materials, please be aware that the frame transformation from one inertial frame to another can be conceived as a sort of rotating / scaling of events in spacetime.
 
Like this;
https://en.wikipedia.org/wiki/File:Lorentz_transform_of_world_line.gif
 
The dots in that animation represent events as plotted on a spacetime diagram, and the "squishing" of the whole structure represents frame transformation from one inertial frame to another. Some events get pushed "towards the future" and some events get pushed "towards the past". Nothing actually happens to "objects" just because we choose to plot them in a different inertial frame; it's just about how we must plot events, if we are to assume isotropic C, and if we are to remain self-consistent in our mapping between frames.
 
It really is a good idea to view Special Relativity simply as self-consistent frame transformation rules, and you start seeing that the whole question of length contraction is not about how different observers "see things", or how they "measure things", or "what happens to objects", but rather about how the universe must be plotted in spacetime diagrams when assuming different notions of simultaneity.
 
In a nutshell, if we switch from one inertial frame representation to another - assuming unique simultaneity to each frame - we must plot the world state "ahead" of us as pushing towards the future (things that had not yet happened in old frame, have already happened in new frame), and conversely the world state "behind" us as pushing towards the past (something that had already happened in old frame, has not yet happened in new frame). Analyzing moving objects like this is what leads into the concept of "length contraction".
 
Since we are effectively molding the spacetime diagram around, but preserving the same exact light-like connections between events (the causality - the order of connected events - remains unchanged), it should be pretty easy for anyone to see that if it is possible to represent a spinning disk as a "set of events" in one frame, and it would have to also transform along with all the other events in self-consistent manner to any different frame without hiccups. From this perspective, the actual question behind the paradox is simple; how would the spinning disk plot onto a spacetime diagram in terms of different notions of simultaneity?
 
Even if you can't instantly figure out the exact solution, you should be able to already convince yourself that there is an exact solution out there which would just mold the (events making up the) spinning disk in consistent manner, along with everything else around the situation. What that exact solution is - let's get to it.

The common error
 
Once the above is understood correctly, next it should be pretty easy to see how the "rigid rods along the circumference" analysis leads you down the wrong path, and at the same time get an glimpse of the correct solution.

• First, imagine a wheel-of-fortune, with pins sticking out from the outer circumference.
• Then we take a spoked wheel (a bicycle wheel), just proper size to snuggly fit inside the pins of the wheel-of-fortune.
• Last, let's enclose the whole two-disk setup inside a box with a snug fit.

The purpose of this setup is to signal us if we are doing something inconsistent with our transformation - if the inner wheel fits inside the larger wheel, and if both wheels fit inside the box in one inertial frame, this must be so in all inertial frames. If it's not, we have performed an error in our analysis.
 
Now let's take two straight rods - A and B - of exactly the same length, and tie rod A between two spokes of the inner wheel, so that both ends of the rod sit exactly on the circumference of the wheel. Don't worry about how good knots you can make - it's just a thought experiment about geometry!
 
Now let's set the inner wheel (and inner wheel only) spinning at a relativistic speed in a lab frame.
 
Since we have rod A spinning along, let's think about what happens if we shoot rod B along an inertial frame so that its path and speed co-incides exactly with the spinning rod A. To make discussion easier, let's look at the setup from the perspective where the rods will co-incide at the "bottom" part of the wheel.
 
At first glance it might seem like those two rods could be setup to become momentarily stationary in relation to each other, and thus their lengths would have to exactly match. But this would be an error. This is essentially the mistake that still exists in most commonly presented solutions (just see the Wikipedia article to find one example).
 
The rod that is attached to the spinning wheel is - obviously - never moving in straight line; it is rotating. It's front end is always moving in different direction than its back end (each end is moving parallel to the part of the circumference it touches). So, the first question is, how would we actually plot this situation in the frame of rod B? Remember, this is just about plotting events in terms of the SR convention of simultaneity.
 
If we plot the external box of the whole setup, in terms of the inertial frame of rod B, it's easy to propose relativistic speeds where the entire box gets plotted as length contracted to shorter length than rod B. The (non-rotating) wheel-of-fortune inside the box must also be mapped inside the box in every frame, and similarly squashed in the direction of motion - snuggly fitting inside the box. And the rotating bicycle wheel must fit also inside the pins of the wheel-of-fortune. It will get plotted also as snuggly fitting inside the wheel-of-fortune. Note though, the spokes will be plotted as curved because it is actively rotating and we are mapping it by a tilted simultaneity plane - this is just the flipside of the coin same coin that makes us map it as squashed.
 
Basically the internal configuration of our setup cannot change based on what inertial frame we map it from - rod A does not suddenly poke through the walls of the box just because we choose to plot the situation in different inertial frame. If we think it does, we are making an error in our analysis, or using invalid frame transformation. Basically it would imply an inconsistent change in the configuration of our system (some objects transforming in different ways than others - clearly invalid)
 
If we investigate a moment where the exact middle points of the rods meet in the same inertial frame, and we choose to plot this in terms of rod B's simultaneity, then the "front" end of rod A (in terms of direction of rotation) has already passed the "front" end of rod B (in terms of direction of motion of rod B in lab frame) some time ago. To be more accurate, since it's attached to a rotating wheel, it is also plotted as curving "upwards", already moving away from rod B. That's because this is essentially a temporal transformation, and we have transformed the world state "ahead" if us towards the "future".
 
And conversely, the world state behind us is plotted as pushing towards the past; the rear ends of the rods have not yet met. And since the rod is constantly rotating, the rear end of rod A is also plotted as curving "upwards", and moving towards rod B.
 
This is why, if you plot down the shape of the spokes of the wheel from the perspective of rod B, the end result looks like this;
 
https://en.wikipedia.org/wiki/File:Relativistic_wheels.gif
 
This is simply a result of plotting the events making up the "supposed world state" as transformed as per the Einstein convention of clock synchronization. A convention for plotting data. Nothing more, nothing less.
 
The error almost everyone makes is that they view length contraction as something actually occurring to objects themselves. This makes them more prone to assume that it is a good enough approximation to just take rod A as momentarily occupying some inertial frame. If you assume this, and repeat the above thought experiment, you end up in a situation where rod A and rod B can share the inertial frame, and thus are plotted as the same length. Since rod B can be plotted as longer than the entire box, sharing a momentary inertial frame would mean that rod A could momentarily become longer than the box encapsulating the whole contraption, poking a hole through it!
 
Also, since it was attached to the spokes of the rotating wheel, you'd have to plot the wheel also as having a larger length between two spokes than can be made to fit inside the box wheel-of-fortune, or inside the box. Obviously this result would mean your analysis is completely invalid, plain and simple.
 
And make no mistake about this - the same error happens no matter how short measurement rods you use. Shorter rods have smaller error, but you can always propose a speed where the error becomes obvious. And with smaller rods there's more of them so the end result of any full analysis is exactly as invalid. Basically you can't have an entire rod in a single inertial frame, while also being attached to the spinning wheel. These are mutually exclusive circumstances.
 
The same error exists in Einstein's co-rotating observer thought experiment, albeit in more subtle manner. But the point is, the co-rotating observer cannot have a measurement rod in any single inertial frame if that rod is to be also attached to the rotating wheel. The approximation necessary to imagine that situation will always make the analysis invalid for the same reasons as described above.

 

The correct solution

First clue to understanding how this situation really gets plotted correctly is this; Rod B only shares inertial frame with an infinitesimally thin slice of the spinning wheel. This is true by the very definition of "spinning". Also from the perspective of the lab frame (where the hub of the wheel is stationary), each infinitesimal slice of the spinning wheel is sitting in a different inertial frame, and does not have any "length" assignable to any single inertial frame. This is a simple mathematical fact arising from the very definitions behind special relativity and "spinning wheel".
 
Second key to understanding this is also associated with properly understanding length contraction as coordinate transformation. Remember when I said "if we switch from one inertial frame representation to another, we must plot the world state "ahead" of us as pushing towards the future, and conversely the world state "behind" us as pushing towards the past.". Note what happens in-between; the world state in the infinitesimal slice exactly perpendicular to the motion does not transform at all! This is btw also why the spokes at the bottom and at the top of the spinning wheel were plotted as straight in the relativistic wheel visualization above. (And I can show why the spokes are plotted as curved with another thought experiment too if anyone is interested)
 
This leads into the simple fact that, in the above experiment, at the moment when the middle part of rod A and rod B meet, a non-rotating observer sitting at the hub of the wheel co-incides with this infinitesimal plane that is cutting through the wheel, and that observer will agree with simultaneity of all events that co-incide with that infinitesimal plane.

We could repeat the same experiment in any direction, and get the same result, because the wheel is symmetrical. Thus we can see how the observer in the middle will in fact agree with the notion of simultaneity of any infinitesimal slice of the wheel in any direction. Which is the same as saying, the inertial frame where the hub of the spinning wheel is stationary, will agree with the simultaneity notion of any infinitesimal slice of the spinning wheel. Thus, Lorentz length contraction never comes into play - the wheel circumference does not Lorentz contract at all.
 
So getting back to the original Ehrenfest Paradox, the correct solution is simply to realize that the definitions of Special Relativity do not in any shape or form even imply that the circumference of the wheel becomes Lorentz contracted for a non-rotating observer. Almost everyone instantly assume that it does, because - again - most everyone tend to think about length contraction as something happening to objects as oppose to being a transform to our coordinate system when we plot data down in self-consistent manner. This misconception is so pervasive that it's included in the very opening statement of the paradox without considering its validity. And this misconception makes people prone to analyze the situation by approximating it with measurement rods, always bringing in the exact error I just described, and coming up with all sorts of silly answers.
 
To summarize;
The non-rotating observer (at rest with the hub of the spinning wheel) actually shares simultaneity with every single infinitesimal part of the spinning wheel in exactly the same way the moving observer shares simultaneity with one infinitesimal slice of the wheel when passing by. Because each "inertial slice" of the wheel is infinitesimally thin (by definition), no part of the wheel actually occupies any length in any single inertial frame, and thus Lorentz contraction never comes into play.
 
If this still sounds like a strange claim to you, you are forgetting where length contraction comes from. It comes from dynamic notion of simultaneity, and only applies to how we plot spatial distances between two events.
 
So the TL;DR solution is, the spinning wheel circumference does not length contract at all. Nothing in Special Relativity implies that it does, other than a set of pervasive misconceptions and invalid simplifications, whose application just lead into erratic and inconsistent conclusions. Proper application of Special Relativity will show that the spinning disk actually has got exactly the same geometry as the non-spinning disk!
 
And as it turns out, all of the "commonly accepted" (or maybe there isn't one) solutions I can find are invalid. Contrary to Wikipedia, the geometry of the spinning disk is still exactly euclidean. As in, [math]\frac{circumference}{diameter} = \pi[/math] also for relativistic spinning disk. The only way to get the commonly cited result of [math]\frac{circumference}{diameter} = \pi \sqrt{1- (\omega R )^2 / c^2 }[/math] is to make the error in approximation I've explained above! And yes Einstein also made this same error.
 
Do note that this solution is all about how geometry gets plotted in terms of special relativity - it's not about how to set a wheel in rotation or about realistic materials. This solution simply arises from how objects get plotted into different inertial frames in self-consistent manner, following exactly the definitions of Special Relativity, and thus it is also exactly the correct solution to the original Ehrenfest Paradox.

Sorry about the length of this. I didn't want to just state what the correct solution is without explaining it in sufficient detail to give everyone a chance to convince themselves about this. Because it seems like the misconception here is so common that the actual end result probably just sound immediately wrong to most people until they think it through themselves.

I am going to revisit this because I feel your knowledge is worth reading in a clear consciousness... Been a long week so to speak 

[Fenmachsa]

On 8/1/2020 at 12:17 PM, LaurieAG said:

Hi Anssi,

 

The 2xpi discrepancy could merely be the difference between the standard Compton wavelength, the reduced Compton wavelength and the nature of universal matter calculations and observational data calculations as the Λ in ΛCDM is just 1 of 3 lambda's involved. Wikipedia describes the differences between the reduced/standard forms and the different types of equations they are used in. Gravitational mass and inertial mass are essentially the same while the CMBR is observed photons of a specific wavelength.

 

https://en.wikipedia.org/wiki/Compton_wavelength#Distinction_between_reduced_and_non-reduced

 

The following link is to my last post on an Against The Mainstream (ATM) thread on another forum. At the bottom of the post you will find a link to an image of the ΛCDM ratios as I can't post images here.

 

https://forum.cosmoquest.org/showthread.php?174969-Simple-Universal-DM-halo-calculations&p=2514394#post2514394

 

I was surprised that Δc, the virial overdensity constant used in the ΛCDM model is half of that as used for galaxies.

 

https://en.wikipedia.org/wiki/Virial_mass#Virial_radiusbackrooms game

 

Of course, there are no "rigid bodies" holding things together because there aren't infinite information speeds. However, the paradox's purpose is to raise the question of how relativistic transformations behave in the context of pure abstract geometry. Euclidean coordinates are used to express a single inertial frame by definition, so any inertial frame from which we take a snapshot of a disk must also be expressible in euclidean terms.

[Halc]

Hi. No idea if anybody is still reading replies to the OP, but I'd like to comment. Forgive if much of this is a repeat of posts of others. I didn't read it all.

On 7/21/2020 at 3:36 PM, AnssiH said:

I just recently stumbled upon a Quora question regarding Ehrenfest Paradox, and noticed that people were giving completely wrong answers to it.

I have no great opinion of Quora's ability to weed out incorrect replies, but there is usually not just one correct explanation to something like this. Really, they're all wrong? The twins 'paradox' has several different ways to correctly describe the reasons the twins are a different age, and the situation is similar here.

On 7/21/2020 at 3:36 PM, AnssiH said:

Basically the paradox is this; In terms of Special Relativity, how does spinning disk work in special relativity, if the circumference of the rotating disk undergoes length-contraction (since it's parallel to motion) while its radius does not (since it's perpendicular to motion), and this would imply that circumference/diameter != pi

Fist of all, I don't like this description of the paradox. A spinning disk has a circumference that is 2πr in the inertial frame of the center of mass. It has a circumference of greater than that as measured by say a tape turning with the disk.

A direct, and correct reply to what is written above is simply that relative to that inertial frame, the radius is not contracted because there is no radial motion. No matter is moving away from or towards the center.  The material is contracted along its direction of motion, which is tangential, so it is contracted in this direction. Hence the tape measure would be contracted and it would take more of it to wrap completely around the thing. None of this is paradoxical. Why is my simple explanation not valid? Your explanation seems to go the complicated route of tiny integrations, spacetime plots, etc, but is all that necessary?

Addressing some of the wrong answers found in Quora:

On 7/21/2020 at 3:36 PM, AnssiH said:

• Only the atoms length contract, but the space between them does not.
• The disk would tear into smaller pieces along the radius to give shorter total circumference.
• The disk would implode under pressure from the shrinking circumference.
• Centrifugal forces would counteract Lorentz contraction.
• There's no strong enough material to build such disk because of Born rigidity and elasticity, thus no paradox.
• You need to use General Relativity to solve the paradox.
• The geometry of the spinning wheel is non-euclidean. Just accept it.

First three answers imply a disk whose angular velocity has changed to rotating from not-rotating, which is not what the statement above states. Perhaps the original paradox says otherwise. These are all wrong if no angular acceleration was described. The disk is assumed to have been created in the rotating state.

The rest of the bullets are indeed blatantly wrong. It isn't about forces, not much at least.

The rotating disk can be built pre-stressed so there's zero deformation as it rotates. (*)

There's no need for strong material since any rotation at all will exhibit this, even if only in small degrees. No object can undergo angular acceleration at all and still exhibit Born Rigid motion.

Gravity is not involved, so GR is not involved.

The geometry is indeed non-Euclidean, but that's not an explanation. Minkowskian spacetime isn't Euclidean either, so the bullet is a non-answer. 'Just accept it' means the author doesn't understand.

I do like the train explanation, one very useful to illustrate that length contraction is real and not just coordinate magic. You have a circular roller coaster track with 314 cars on it bumper to bumper. You get them moving fast around the track (which never changes radius) and you can begin to fit additional cars in the gaps that form between them. This is not paradoxical, despite the incorrect rebuttal that from the PoV of one of the cars, it is the track that is contracting so fewer cars must fit, a sort of angular attempt at the barn-pole 'paradox'.

(*)

8 hours ago, Fenmachsa said:

Of course, there are no "rigid bodies" holding things together because there aren't infinite information speeds.

Speed of light does not make rigid motion impossible, per the comments just above. There indeed cannot be information travel through a rigid body, but there are ways to say accelerate an object without information travel, including force applied at all points (rail gun) and/or pre-stressed materials.

On 7/21/2020 at 3:36 PM, AnssiH said:

What really surprised me was when I went on to check how does Wikipedia see this, and it also doesn't explain the proper solution.

OK, that surprises me. I had to go to wiki to see what they said. This is the entire resolution section, not all at once.

"Grøn states that the resolution of the paradox stems from the impossibility of synchronizing clocks in a rotating reference frame.^[15] If observers on the rotating circumference try to synchronise their clocks around the circumference to establish disc time, there is a time difference between the two end points where they meet."

That sure seems wrong to me. It's pretty trivial to sync clocks on the edge of a spinning disk. You just put out a light pulse from the center and zero each clock as the detect it. They'll stay in sync as long as the angular rate stays the same, but the method cannot sync clocks at different radii, especially since such clocks objectively run at different rates.

Gron seems to be describing the Sagnac effect, attempting to sync adjacent clocks using a linear motion method. This seems similar to the work you describe below. It doesn't resolve the paradox. I'm more interested in the modern resolution:

"The modern resolution can be briefly summarized as follows:

1. Small distances measured by disk-riding observers are described by the Langevin-Landau-Lifschitz metric, which is indeed well approximated (for small angular velocity) by the geometry of the hyperbolic plane, just as Kaluza had claimed.
2. For physically reasonable materials, during the spin-up phase a real disk expands radially due to centrifugal forces; relativistic corrections partially counteract (but do not cancel) this Newtonian effect. After a steady-state rotation is achieved and the disk has been allowed to relax, the geometry "in the small" is approximately given by the Langevin–Landau–Lifschitz metric."

1) speaks of non-Euclidean geometry, which is one way to map distances between all points. This isn't wrong, but I don't find it to be much of an explanation.

2) Talks about real materials, which of course exhibit strain if you change their angular velocity, so while not false, it becomes off-topic. The issue is a mathematical one, not some kind of assertion that it is impossible to rotate any physical (non-rigid) object.

 

Born rigidity is a mathematical definition that cannot be met by any real material, which is why real objects can be rotated (from a halt), and Born-Rigid objects cannot. It is probably a mistake to mix the two concepts, but it can be done. For instance, a Born-Rigid rod can be accelerated linearly without violation of its rigidity. If the force is applied at the rear and the object is pre-stressed, the speed of light is not an issue since the front of the object will begin to accelerate immediately. This counters the argument in some posts about speed of light preventing such acceleration of extended objects. Problem is, it only works for continuous acceleration. To turn it on and off, every point of the object needs separate acceleration to prevent any strain at all and eliminate the speed-of-sound issue.
 

On 7/21/2020 at 3:36 PM, AnssiH said:

Third reason is that a mathematical analysis in the framework of special relativity is easiest to do by making certain approximations, which are exactly the approximations leading into wrong answers

Agree that approximations are a bad road to walk. Most of the 'proofs' that SR doesn't predict the Sagnac effect involve integration of small steps around the edge without doing the Lorentz-Transforms with each successive steps since the steps are so small that they're not needed.

On 7/21/2020 at 3:36 PM, AnssiH said:

it should be pretty easy to see how the "rigid rods along the circumference" analysis leads you down the wrong path

Isn't that what a tape measure is? Each mm is a little 'rigid' rod moving not quite in the same direction as the next mm on the tape. That we measure more than 2πr with it is not wrong.

On 7/21/2020 at 3:36 PM, AnssiH said:

• First, imagine a wheel-of-fortune, with pins sticking out from the outer circumference.
• Then we take a spoked wheel (a bicycle wheel), just proper size to snuggly fit inside the pins of the wheel-of-fortune.
• Last, let's enclose the whole two-disk setup inside a box with a snug fit.

Now let's take two straight rods - A and B - of exactly the same length, and tie rod A between two spokes of the inner wheel, so that both ends of the rod sit exactly on the circumference of the wheel.
Now let's set the inner wheel (and inner wheel only) spinning at a relativistic speed in a lab frame.

This seems to violate the initial description. You're accelerating that wheel, which cannot be done without violating rigidity. Something has to bend. When rotating, the rod A isn't going to reach between the spoke ends if it fit there when the wheel wasn't rotating.

I presume rods A and B have the same proper length, and thus that in the (momentary inertial) frame of either, the other is shorter, except A is tied to the spokes, so it actually gets stretched (strain) so that the spokes can retain their original length and even spacing. None of this is specified, so maybe I have your scenario wrong.

Another nit: Angular motion is absolute. No need to mention the inertial frame in which it is rotating, although it admittedly spins at different rates from one inertial frame to the next.

On 7/21/2020 at 3:36 PM, AnssiH said:

If we plot the external box of the whole setup, in terms of the inertial frame of rod B, it's easy to propose relativistic speeds where the entire box gets plotted as length contracted to shorter length than rod B.

You totally lost me. B is stationary in the inertial frame of the box, so the box isn't going to be contracted in that frame where both those objects are stationary. Relative to some other frame, sure the box gets contracted, but so does B with it. B always fits in the box, being just the distance between two pins on the non-rotating wheel.

On 7/21/2020 at 3:36 PM, AnssiH said:

If we investigate a moment where the exact middle points of the rods meet in the same inertial frame, and we choose to plot this in terms of rod B's simultaneity, then the "front" end of rod A (in terms of direction of rotation) has already passed the "front" end of rod B (in terms of direction of motion of rod B in lab frame) some time ago.

Similarly, in the lab frame, B is stationary and has no 'front end' since it isn't moving. Rod A is moving and contracted in this frame, and the front (north say) of A has not yet passed the north end of the uncontracted rod B. Your description makes no sense to me.

On 7/21/2020 at 3:36 PM, AnssiH said:

This is why, if you plot down the shape of the spokes of the wheel from the perspective of rod B, the end result looks like this;
 
https://en.wikipedia.org/wiki/File:Relativistic_wheels.gif

This is very wrong. Rod B is stationary relative to say the wheel  axis and the spokes are straight. That's a plot relative to the momentary inertial frame of rod A, effectively the frame of a hypothetical road along which the wheel is rolling.

 

I give up after this. Maybe I'm just reading it wrong.

Edited November 28, 2022 by Halc

[AnssiH]

On 11/28/2022 at 8:48 AM, Fenmachsa said:

Of course, there are no "rigid bodies" holding things together because there aren't infinite information speeds. However, the paradox's purpose is to raise the question of how relativistic transformations behave in the context of pure abstract geometry. Euclidean coordinates are used to express a single inertial frame by definition, so any inertial frame from which we take a snapshot of a disk must also be expressible in euclidean terms.

Yeah indeed. The point about handling relativistic transformations in purely geometrical terms is indeed the motivation for what I wrote - and I've come to realize the article doesn't really emphasize this thought strongly enough. There's a big difference in the idea of handling the shape of a disk as a "collection of points" (like collection of disconnected drones flying around a circle), or as collection of connected elements (whose connection is mediated by electromagnetic radiation, i.e. bindings that propagate at C). In the latter analysis you would be unable to avoid tension to the bindings when the object is rotating (due to the binding paths moving in straight line in terms of inertial frames -> in curved paths in terms of rotating frames)

[AnssiH]

On 11/28/2022 at 9:49 PM, Halc said:

Hi. No idea if anybody is still reading replies to the OP, but I'd like to comment. Forgive if much of this is a repeat of posts of others. I didn't read it all.

There are a lot of thought provoking comments in there. I'll probably return to this later as it's late, but giving some knee-jerk reactions at least first.

 

 

On 11/28/2022 at 9:49 PM, Halc said:

I have no great opinion of Quora's ability to weed out incorrect replies, but there is usually not just one correct explanation to something like this. Really, they're all wrong? The twins 'paradox' has several different ways to correctly describe the reasons the twins are a different age, and the situation is similar here.

Well admittedly I have not exhaustively examined all of the answers in Quora 😉

 

On 11/28/2022 at 9:49 PM, Halc said:

Fist of all, I don't like this description of the paradox. A spinning disk has a circumference that is 2πr in the inertial frame of the center of mass. It has a circumference of greater than that as measured by say a tape turning with the disk.

Now I think this already touches the subtlety that is not usually well defined in the phrasing of the problem (i.e. the same crime that I accused myself of in the previous post).

So this would completely depend now on what do we think would constitute a tape measure here. If we see it as a collection of elements (say, atoms) bound by some forces that mediate by C, and if this tape goes around the disk in completely symmetrical manner, then each element would become stressed by equal amount in both directions, by the binding force that cannot propagate in straight line in a rotating frame.

So I suppose in that case it would be fair to say that, even though each element must be experiencing an equal stress to both directions, together those forces would yield a tuck towards the center (in lesser degree with larger disks).

This is in direct contradiction with what I wrote earlier, because I was taking a different tack on this - purely concerned of geometrical effects. In this case our tape measure would not be made of connected elements (and I'd accept the argument that this is getting somewhat non-physical) but rather as milestone poles - say drones. In this case the space between the moving drones would be not affected by their rotating motion, and we would count the equal number of drones around the circumference whether they are flying around the perimeter or not.

 

On 11/28/2022 at 9:49 PM, Halc said:

I do like the train explanation, one very useful to illustrate that length contraction is real and not just coordinate magic.

I think this sentiment perhaps reflects one of the reasons why I liked to write the solution I did, to point out a subtlety in the argument of saying length contraction is "real" without disclaimers. I know people usually use the phrase to differentiate from the idea of "optical illusion". But then there's also still two senses of "real", and the length contraction that follows from relativistic simultaneity is of course only "real" in relativistic sense. I.e. nothing "really" happens to a spaceship when we choose to view it from different inertial frame.

Of course Lorentz transformation originally arose in a context in which length contraction would be considered "real"; but in that case also it was never measurable by natural observers in any non-relativistic sense.

And yeah this does touch Bell spaceship paradox in ways, because that's specifically a problem stated as what happens when we consider elements that are not considered as simultaneously accelerating...

I'm thinking majority of the rest of your points have to do with the choice of whether or not we are looking at the  proposed shapes as purely geometrical shapes ("as a collection of drones") or whether we take object bindings into account.

Regards,

Edited December 12, 2022 by OceanBreeze

[LaurieAG]

3 hours ago, AnssiH said:

There are a lot of thought provoking comments in there. I'll probably return to this later as it's late, but giving some knee-jerk reactions at least first.

Hi AnssiH, good to see you back again.

Here are the links to the relevant pages of a forum thread I referred to before that gives a solution for determining Gron's Fig 9 part C. The site is no longer considered dangerous by Norton and the 2 posts themselves provide further information about how it was done.

https://www.thephysicsforum.com/special-general-relativity/5577-relativistic-rolling-wheel-ii-3.html#post12639

https://www.thephysicsforum.com/special-general-relativity/5577-relativistic-rolling-wheel-ii-3.html#post12704
 

[Halc]

5 hours ago, AnssiH said:

On 11/28/2022 at 9:49 AM, Halc said:

A spinning disk ... has a circumference of greater than [2πr] as measured by say a tape turning with the disk.

So this would completely depend now on what do we think would constitute a tape measure here.

Well I see it as a set of marking printed on the edge of the spinning disk, exactly as if it had rolled along linear wet-paint markings on the ground. Remember, I don't see the disk as ever having been accelerated. So I have a disk of say radius 1 with the edge moving at 0.19c and the markings on the edge indicate the circumference is 6.4, not a lot more than 2π, but more. There's no stress other than whatever radial tension is needed to effect the centripetal acceleration.

5 hours ago, AnssiH said:

So I suppose in that case it would be fair to say that, even though each element must be experiencing an equal stress to both directions, together those forces would yield a tuck towards the center (in lesser degree with larger disks).

You seem to envision a 2π rotating tape that shrinks to a smaller radius as it contracts, which implies angular acceleration.

As for how much it does this, it isn't a function of the disk size. If the edge is moving at say 0.19c, the contraction will be about 1.5% regardless of the disk size. Interesting thing happens with faster speeds. Say the edge of the disk moves at .866c relative to the center, which contracts the tape by a factor of 2, but the tape moves far less than half way to the center since as it moves in, its speed slows and so does its contraction. So exercise for those curious: What would be the radius of a 2π tape after it shrinks from the edge of a disk that is moving at .866c? The tape still has to be rotating at the same RPM as the edge, with RPM measured by a clock at the axis.

5 hours ago, AnssiH said:

I.e. nothing "really" happens to a spaceship when we choose to view it from different inertial frame.

and yet more train cars fit on the circular track when the train moves fast, despite the fact that relative to any given train car, more track fits under it at a given moment than if both were mutually stationary. The larger number of cars fitting on the track is an objective fact, true regardless of choice of frame. That's why I like the example.

5 hours ago, AnssiH said:

I'm thinking majority of the rest of your points have to do with the choice of whether or not we are looking at the  proposed shapes as purely geometrical shapes ("as a collection of drones") or whether we take object bindings into account.

I don't think I ever got down to the molecular level with bindings and such. It seems unnecessary. All the geometry still works with objects not composed of particles / drones.

Edited December 13, 2022 by Halc

[sluggo]

forum;

 

Length contraction occurs for an object as it accelerates in the direction of motion.

Rotation of the circumference of a disk does not qualify as translational motion. There are radial or transverse forces acting on an arc as shown. These centrifugal forces f are directed radially outward for each element, increasing the arc length