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

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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 $\frac{circumference}{diameter} \neq \pi$.

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, $\frac{circumference}{diameter} = \pi$ also for relativistic spinning disk. The only way to get the commonly cited result of $\frac{circumference}{diameter} = \pi \sqrt{1- (\omega R )^2 / c^2 }$ 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.

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I don't get it.  If paths and speeds of rods A and B coincide, what is relative about it?!  The frames?!

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The point is, they cannot possibly co-incide except by an infinitesimally small slice of the rotating rod.

When you imagine the rods as fully co-inciding, you are committing exactly the logical error I try to explain - and map the situation in inconsistent manner. As I said, this error is incredibly pervasive and everyone just does it without giving it a second thought.

A rod cannot be tied to the rotating wheel and and still also occupy a single inertial frame through its length. Taking this simple fact into account correctly solves the paradox entirely.

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Posted (edited)

In relativity it is said there are no rigid bodies. This has an extensive history.

Edited by Dubbelosix
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Yeah, everything's a collection of microscopic elements that can only communicate at maximum speed of C. That puts an upper bound on "rigidity".

But I have no idea why you are bringing that up?

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The point is, they cannot possibly co-incide except by an infinitesimally small slice of the rotating rod.

When you imagine the rods as fully co-inciding, you are committing exactly the logical error I try to explain - and map the situation in inconsistent manner. As I said, this error is incredibly pervasive and everyone just does it without giving it a second thought.

A rod cannot be tied to the rotating wheel and and still also occupy a single inertial frame through its length. Taking this simple fact into account correctly solves the paradox entirely.

AnssiH, Oyvind Gron covers much of the historic ground in his paper "Space Geometry in Rotating Reference Frames: A Historical Appraisal".

https://www.researchgate.net/publication/252135276_Space_Geometry_in_Rotating_Reference_Frames_A_Historical_Appraisal

The result is shown in Fig. 9. Part C of the figure shows the “optical appearance” of a rolling ring, i.e. the positions of emission events where the

emitted light from all the points arrives at a fixed point of time at the point of contact of the ring with the ground. In other words it is the position of the points when they emitted light that arrives at a camera on the ground just as the ring passes the camera.

The 2 images below are from another physics forum (it doesn't allow new users or logins, luckily these images were stored on another website) where several members worked out a solution (the only one I have seen) as Gron doesn't give his working in his paper. I suggested they order their emission events to cross check that the actual axle velocity remained constant between events, as it does as shown in the second image.

Note:

The optical appearance plot in part C is not a frame, if anything it is a photon travel 'time space' as opposed to a space time as all dimensions are in c. While a Wheel frame, Axle/Carriage frame and a Road frame are used in the solution the plot shown is the plane of the rotation of the wheel with the camera being at a fixed point on that plane.

Per Gron's paper the z axis is zero to ignore Born rigidity issues.

Basically the solution uses the length contracted position of any spoke, wrt the axle and therefore the road, to determine the emission time from when the axle can travel at the consistent velocity to be above the camera location in the same time that a photon will travel, in a straight line at c, from the emission point to the camera.

Hope this helps.

The link to the thread in the original forum is below, scroll down to post 253 for the summary. While the site seems to be flagged as dangerous it doesn't load anything nasty according to Norton, you can bypass the Norton security message, but no information is available on why the site is classified as dangerous.

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

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I think the correct answer is, is that 'contraction' is the anomaly, and is not a 'real thing', contraction has never been observed or experimentally confirmed, it also does not fit with what we do observe.

So we don't need a mechanism to explain a paradox because the assumptions that make up that paradox are incorrect. That is until (and IF) we confirm that contraction is actually a thing, and I do not expect that to happen.

Also it is trying to describe it in Newtonian dynamics, when the process if relativistic.

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An object does not contract as such, it in fact rotates in space. This is why a moving sphere does not "contract" though its apparent size (might) appear slightly smaller. For more on Lorentz rotations,see Penrose.

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An object does not contract as such, it in fact rotates in space. This is why a moving sphere does not "contract" though its apparent size (might) appear slightly smaller. For more on Lorentz rotations,see Penrose.

Apart from a mathematical conclusion, is there any observational or experimental evidence that supports rotation or contraction?

I am wary of purely mathematical based conclusions, I like to work from first principles (observations) up.

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Yes. As far as I am aware, these rotations have been measured.

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That is still only a mathematical transform, and it really just describes or shows that you get differences in the measurement of space length and time length when observed from a different relative position (in SR).

But we do observe a one way (one sided) time length variance in things like GPS satellite atomic clocks, where we see the clock ticking slower due to SR time dilation. But it's a very interesting subject..

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Hey Laurie, I remember you from this forum back in the day. I can hardly believe it's been over 10 years... :I

AnssiH, Oyvind Gron covers much of the historic ground in his paper "Space Geometry in Rotating Reference Frames: A Historical Appraisal".
https://www.researchgate.net/publication/252135276_Space_Geometry_in_Rotating_Reference_Frames_A_Historical_Appraisal

Thanks, that was an interesting read. It reinforces my views on the reasons why people tend to get this so wrong. Everyone make a wrong assumption about infinitesimal Lorentz contraction right at the get-go without thinking about it critically. The correct solution does not seem to appear anywhere on that paper (they get the rolling disk case right though. But that's not the original paradox).

Truthfully, I'm quite dumbfounded about all of this. The commonly presented solutions seem to be genuinely terrible across the board. I mean analyzing Born rigidity is all fine and well but it is also a different topic, not relevant to the paradox. I think people who understand relativity well enough should be able to quite easily convince themselves that the case I present in the OP must be exactly valid.

I want to comment on few things in that paper, but first;

The 2 images below are from another physics forum (it doesn't allow new users or logins, luckily these images were stored on another website) where several members worked out a solution (the only one I have seen) as Gron doesn't give his working in his paper. I suggested they order their emission events to cross check that the actual axle velocity remained constant between events, as it does as shown in the second image.

Yeah these look absolutely correct to me. I mean I didn't check the exact numbers but the form of the distortions is correct, and incidentally the method of figuring this out appears to be exactly the same idea that I'd do when just thinking about this in my head. As in, when boosting to a new frame, you push the events "ahead" forward in time, and you push the events "behind" backward in time, "events" being a spatial/temporal representation of the wheel & it's motion. (And importantly, the events perpendicular to yourself do not transform at all).

So what that gives you for a rolling wheel case is first of all spatially squashed wheel in direction of motion since the events behind and ahead all effectively push closer to you, but also since the wheel is rotating, the spokes get plotted as bending away from that infinitesimal slice on the wheel that is moving at 0 velocity in our reference frame. And importantly, the part with 0 velocity is indeed an infinitesimal slice, not a "rod" or "drawn rod" or "area between two dots" or any sort of concept with spatial length.

So basically that result is the same as what I cited as correct in the OP for rolling wheel.

The big benefit of analyzing this in the form of spatial/temporal events is that you avoid the semantical pitfall of handling macroscopic objects as naive-realistic entities by themselves. It is clearer to remember that length contraction comes out from the fact that when you do a frame transform, you are transforming the apparent spatial location of two separate events, which also just so happen to represent the instantaneous locations a some part of a macroscopic object.

The single most important point to be made here is, an infinitesimal point does not constitute two events, and Lorentz transformation does not impact the "length" of an infinitesimal point - there are no two events to move around, since you are by definition looking at a single event. A dynamic notion of simultaneity can only impact spatial/temporal lenghts if you are differentiating between two separate events. A single event is always simultaneous with itself no matter how much you try to transform it.

This is the point that everyone are getting wrong when analyzing Ehrenfest Paradox, without explicitly realizing their error. Because they are thinking of length contract as something fundamental to matter or space "just because it does", as oppose to something that happens between spatial/temporal because we choose to plot the same system from different basis of simultaneity.

Keeping those facts in mind, I think it's easy to see where everyone are going wrong in the historical attempts, and the correct solution becomes clear as a day.

The following are quotes from Gron's research and my reactions to them;

To say that a body remains relativistically rigid means: It deforms continuously by arbitrary
motion so that each of its infinitesimal elements Lorentz contracts (relative to its rest length) all
the time in accordance with the instantaneous velocity of each of its elements, as observed by
an observer at rest.

So here Ehrenfest automatically assumes that a new notion of simultaneity is somehow meaningful to the length of an infinitesimal point. That is a critical error, and leads him to assert invalid conclusion;

The circumference of the cylinder must obtain a contraction $2\pi R' < 2\pi R$ relative to its rest
length, since each of its elements move with an instantaneous velocity $R' \omega$.

That result is actually not consistent with Lorentz transformation, as it turns out in a proper analysis.

Planck then argued that the task of specifying the final state of a body set into rotation is a
dynamical problem involving the theory of elastic media.

I guess here is where - historically - things shot off on a tangent onto a different topic entirely. A topic which is much less interesting to me. Of course the real question is about the geometry of special relativity, and the fact that Euclidean 3D space still must be able to represent also the instantaneous events making up "a spinning disk" consistently in any and every inertial frame. And indeed the end result must be Euclidean, since we are by definition drawing up that situation in an Euclidean coordinate system...

I mean the question is "what is the fundamental nature of the spinning disk". The question is simply "what is the correct way to plot the spinning disk in a non-spinning lab-frame", and rest assured, it is not possible to mark down events in a coordinate system and by that procedure change the definition of the coordinate system itself... Obviously.

In response to an article by W. von Ignatowski [10] on relativistic kinematics Ehrenfest [11]
offered a gedanken experiment: Consider a disk at rest with equally spaced circles about the
origin of the rotational axis engraved on its surface. Let these circles be recorded on a piece of
tracing paper by a stationary observer. Assume now that the disk could be put into rotation
while remaining Born rigid and then rotate at constant angular velocity about an axis through
its centre, while the observer remains at rest. If the observer instantaneously registered the
rotating disk’s markings on another piece of tracing paper, he would find upon comparison
with the other piece of paper that the radial lengths are the same, but the circumference
measured during rotation is less than before.

So basically he's made the same error here as discussed above - approximating Lorentz contraction to apply to "areas between the holes" as if they represent an inertial frame. Interestingly he doesn't realize it even though it causes him to make a rather silly claim - that the observer could actually somehow draw a non-euclidean picture inside an euclidean coordinate system, using just a pen and paper :facepalm:

We suppose that the circumference and diameter of a circle have been measured with a
standard measuring rod infinitely small compared with the radius, and that we have the
quotient of the two results. If this experiment were performed with measuring rods at rest
relatively to the Galilean system K', the quotient would be $\pi$. With measuring rods at rest
relatively to K, the quotient would be greater than $\pi$. This is readily understood if we
envisage the whole process of measuring from the “stationary” system K', and take into
consideration that the measuring rods applied to the periphery undergoes a Lorentz
contraction, while the ones applied along the radius do not. Hence Euclidean geometry does
not apply to K.

Einstein has here for the first time made it clear that the length of the periphery of a rotating
disk in longer than not shorter as stated in Ehrenfest’s paradox.

Now this is very interesting statement because it contains multiple logical errors stacked together.

First, he makes the same error of assuming that "infinitely small measuring rods" still have length dependency to Lorentz transformation - forgetting that a single event cannot transform any closer to itself.

Second, he makes a very interesting error of defining length as the count of "moving measuring rods" against the count of "non-moving measuring rods". It is rather incredible for Einstein to make this mistake, because by the same token we would also measure the length of a moving train by first accelerating the measuring rods into speed, measuring their length, and the counting how many of those rods we can fit within a train. That analysis would indeed show that the train has become "longer" because more rods fit inside. But it is also completely self-contradictory definition of length... Ugh, not his finest hour, definitely...

To be fair, the reason he makes this second mistake may be little more subtle, and has more to do with him approximating the entire circumference as behaving like a straight line instead of something that does a full spatial circle. But that is also completely invalid procedure. When you do that you end up attributing a temporal discontinuity somewhere along the circumference (as in when you return to where you started), which is obviously an error. Well, I'll explain later in this post as other people made the same mistake.

A rigid circular disk must break up if it is set into rotation, on account of the Lorentz contraction
of the tangential fibres and the non-contraction of the radial ones. Similarly, a rigid disk in
rotation must explode as a consequence of the inverse changes in length, if one attempts to
bring it to the rest state.

I assume he is thinking about the accelerations not being simultaneous between the infinitesimal frames. Basically Bell spaceship paradox applied to the circumference of the disk. In which case basically this represents the same underlying error of assuming spatial dependency onto the infinitesimal slices of the disk, and additionally ignoring the critical fact that the circumference actually loops back to the beginning so the frame boosting during acceleration cannot be taken to occur to the same direction constantly as you traverse around the circumference. In fact the frame boosting direction does also perform a full circle.

Another way of obtaining an identical synchronisation to that which results from the
procedure of Goy and Selleri, is to use a time signal emitted from the axis. This will reach all
clocks at a circle with center at the axis simultaneously both as measured in K' and as
measured in the rotating rest frame K of the disk. The clocks synchronised in this way are just
the coordinate clocks on the rotating disk. They show the same time as the clock at the axis,
as the clocks in K'. However, these clocks are not Einstein synchronised.

Oh, but they are! Ok, ok, there's no such thing as "rotating rest frame", but there's an interesting point to be made here nevertheless - if you synchronize the rotating clocks by a pulse from the center axis, they most certainly can be synchronized together as per Einstein synchronization convention. That's not paradoxical whatsoever though - it just is another way to remind everyone that the spinning disk indeed must be taken as symmetrical in the lab frame no matter how fast it spins - and furthermore acceleration originating from the axis will in fact also reach all the points at the circumference symmetrically - there's no Lorentz "tearing of fibres" expected here because this situation is critically different from co-accelerating Bell spaceships.

Also he gave a nice illustration of the difference
between the two geometries. Imagine an arbitrary
point P on the disk. Draw a circle around P with
radius 1 dσ = (see Fig.1).
Fig.1. Rotating disk with measuring rod. The dashed
ellipse is the curve followed by the end of the rod when it is rotated about P.

Okay now this example is great because it makes it explicitly clear that this person is making exactly the critical error I am warning about in the OP, and what my thought experiment with two rods shows as unequivocally invalid.

Check out the illustration in this part of the paper. It shows a sphere drawn on a disk, and shows it contracted in a way that a sphere would contract in an inertial frame.

The sphere drawn on a spinning disk is not in a single inertial frame; each infinitesimal slice of it is in a different inertial frame. It would not transform like that.

This is the reason I described the rolling disk example with sufficient detail - the fact is, if that drawn sphere is approximated to behave just like a sphere in an inertial frame behaves, then that leads into an inconsistency where the drawn sphere would have to match the size of the sphere in an inertial frame. Obviously it cannot; it is trivial to describe a circumstance where the drawn sphere would become plotted as wider than the entire disk it is drawn on... :facepalm:

That just goes to show the mistake of trying to approximate this situation with straight rods - it is critically invalid approach.

Hence, going around a circle about the axis of
rotation one arrives at a different point of time than at the start. This means that a certain
event corresponds to different points of time in the rotating coordinate system. In other words
there exists a time discontinuity along a radial line in this coordinate system.

Yeah, nope... This is an explicit statement of the same error that Einstein seemed to make tacitly.

So basically they imagine frame jumps from one "rod" to another, and just bluntly think of the idea that each jump skips simultaneity further away from where we started.

That would be true (and unproblematic) if you were analyzing acceleration in spatially straight line. But we are traversing around a circle, so actually we are boosting simultaneity one way for half of the time, and the other way half of the time, ending up right where we started.

I'm really surprised this has been so difficult to everyone... :I

Lorentz adds a comment which seems to indicate that the principle of relativity is not valid
for rotating motion:

Absolutely ridiculous, but also indicative of him imagining this situation in terms of naive realistic objects instead of spatial/temporal events. In the latter view, it's quite clear that "rotating disk" is just semantical definition applied on some specific type of pattern of events, and quite obviously any pattern of events transforms from one frame to another. Just because we call something "rotating" doesn't suddenly change the nature of reality.

Basically this just means Lorentz is thinking of this whole thing completely upside down.

Einstein’s point is that because
each measuring rod along the periphery is Lorentz contracted there is place for more of them
around the circumference the faster the disk rotates, and the length of the circumference is just
the number of measuring rods around it.

So basically he is thinking the same exact concept as measuring the length of a train by boosting the measuring rods and then counting how many you can fit inside the train... :facepalm:

Well, we are all only humans.

However, Eddingtons’s result is correct because the angle around a circle with an arbitrary
centre on a curved surface is defined on the tangent plane of the centre of the circle, i.e. it is
defined locally. Hence, in the case of a circle around the axis of a rotating disk one has to take
the limit 0 r → to find the angle. This implies that the angle around a circle is 2π even on a
surface with non Euclidean geometry.

Just wow... So basically someone had an analysis that started to show signs of the correct solution, but even then the assumption is some kind of insane pseudo-euclidean-but-non-euclidean insanity, simply because they were by default so convinced that the circumference must be dependent on Lorentz contraction even when some analysis manages to show it actually is not.

Clark’s analysis was followed up by Cavalleri [65] in 1968 in an interesting, although
somewhat controversial paper. He first gave a thorough review of earlier work on this topic.
Then he concluded that “Ehrenfest’s paradox cannot be resolved from a purely kinematical point of
view”, and inferred that the relativistic kinematics for extended bodies is not generally selfconsistent.

Which means he had very poor idea of how Lorentz transformation works, and he was thinking of naive realistic kinematic objects instead of spatial/temporal events. Otherwise he would have realized that relativistic frame transformation really can be seen as just an act of moving events around on a piece of paper and it's not possible to arrive at paradoxical situations other than holding misconceptions over behavior of naive-realistic macroscopic objects.

The following discussion is believed to show that a careful definition of
all quantities involved eliminates the paradox and, with it, the alleged
inconsistency of relativistic kinematics.
At any rate, should relativistic kinematics not be self-consistent, it
would seem hard, on logical grounds, to accept the view that the addition
of dynamical arguments might improve the situation.”

On the latter point - indeed!

On the former, unfortunately he then went on to present a ridiculous attempt with discontinuous time jump in arbitrary slice (of a symmetrical disk). So basically the same mistake as already discussed above, and also;

We now consider this acceleration program from the point of view of the inertial rest frame
of the axis. The marks are enumerated from 1 to n in the direction of the rotation. Performing
Lorentz transformations from the instantaneous inertial rest frames of the marks to the
laboratory frame, one finds that 2 gets a blow later than 1, 3 later than 2, and so forth. Going
around the disk one finds that n happens later than n-1. Hence, n happens later than 1. But
passing on from n to 1, the blow 1 should happen later than n since the events should be
simultaneous as measured in the instantaneous rest frame of the element between the marks n
and 1.

Yeah, no... The circumference of the disk is not laid down on a straight line, it actually goes around spatially and comes back to itself. Which means the frame jumps come back to the beginning with no discontinuity anywhere. Indeed, symmetrical disk is symmetrical; it cannot have a discontinuity on the slice that we arbitrarily chose as our starting point...

That research paper just seriously makes me question the intelligence of many of those contributors...

If the disk is regarded as a 2-dimensional surface it can be put into rotation in a Born
rigid way, that is without any displacements in the tangential plane of the disk, by
bending for example upwards so that it obtains the shape of a cup.

Oh sweet baby jesus... :facepalm:

In addition to all of the above, Fig.7. of the paper shows an illustration of measuring rods shrinking but the space between measuring rods not shrinking. That is pretty obviously also invalid idea, I trust it's not even necessary to explain to anyone why...

Interestingly, the rolling disk case is completely correct in the paper. The reason seems clear, there's no involvement of infinitesimally small frames in here, and thus it's impossible to make the mistake of applying length contraction on "lengthless" elements.

Anyway, just to re-iterate the OP, the correct solution is to realize that each and any "inertial slice" of the spinning disk is infinitesimally thin, and thus jumping from lab frame to any of the "spinning" frames yields no spatial/temporal changes to any events that make up that infinitesimal slice. There is no Lorentz contraction expected because there are no spatially separated events in the direction of motion, on any such slice.

Obviously this applies to any and all infinitesimal slices that make up the disk. Thus, there are no changes to any events that make up the spinning disk That is why the shape of the spinning disk does not change in the lab frame, and it remains fully euclidean through and through. And this is true btw regardless of the shape of the spinning object - the fact that it is spinning means it does not represent any inertial frames with length in the direction of motion.

Another way to realize this fact - even if you expect that some events that make up the spinning disk ought to move somewhere temporally during the transformation, note that the shape of the disk in the lab frame is circular at all times. Meaning, no matter how much you stretch simultaneity around, you would still get a circular shape. Then combine that with the fact that the disk is symmetrical - and you only really have one possibility left. (But yeah this is somewhat misleading because actually the spinning disk doesn't have to be symmetrical - just pointing out the absurdity of non-symmetrical solutions).

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I think the correct answer is, is that 'contraction' is the anomaly, and is not a 'real thing', contraction has never been observed or experimentally confirmed, it also does not fit with what we do observe.

So we don't need a mechanism to explain a paradox because the assumptions that make up that paradox are incorrect. That is until (and IF) we confirm that contraction is actually a thing, and I do not expect that to happen.

Yes but by "paradox" we mean an apparent inconsistency within the framework of a theory. It has got nothing to do with observations. And if the theory is actually self-consistent, the solution is also purely logical work. It's just about identifying the error in the assumptions leading to paradox, and how the situation actually must be represented in the framework of that theory.

Generally paradoxes are cases of applying the rules of a theory in inconsistent manner. That is exactly what has happened with Ehrenfest Paradox. That is why I'm so surprised that the actual correct, purely logical and purely geometrical solution, is not generally known. I guess I've never heard anyone voice out this solution, I've just always assumed this is common knowledge after figuring it out myself.

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Yeah I would consider that one of the better representations of the issue, because it draws out the fact that the whole thing arises from the notion of dynamic simultaneity. And note how he represents objects as a set of spatial/temporal events to make the actual logical mechanism clear - exactly what I'm talking about in my previous long post.

Another thing that his apparatus also makes very clear is that the frame transformation really does just scale events around - there is no way to end up into a self-contradiction even if you are transforming events that represent a spinning disk. Those events will not move around in random fashion, getting all mixed and jumbled up. But looking at this just from the perspective of math equations has made everyone assume that the transformation changes the length of a single event too along the circumference of the disk.

But actually it does not - length contraction only appears if we are talking about multiple events; they morph because we change our notion of simultaneity.

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Hey Laurie, I remember you from this forum back in the day. I can hardly believe it's been over 10 years... :I

...

Interestingly, the rolling disk case is completely correct in the paper. The reason seems clear, there's no involvement of infinitesimally small frames in here, and thus it's impossible to make the mistake of applying length contraction on "lengthless" elements.

Anyway, just to re-iterate the OP, the correct solution is to realize that each and any "inertial slice" of the spinning disk is infinitesimally thin, and thus jumping from lab frame to any of the "spinning" frames yields no spatial/temporal changes to any events that make up that infinitesimal slice. There is no Lorentz contraction expected because there are no spatially separated events in the direction of motion, on any such slice.

Obviously this applies to any and all infinitesimal slices that make up the disk. Thus, there are no changes to any events that make up the spinning disk That is why the shape of the spinning disk does not change in the lab frame, and it remains fully euclidean through and through. And this is true btw regardless of the shape of the spinning object - the fact that it is spinning means it does not represent any inertial frames with length in the direction of motion.

Another way to realize this fact - even if you expect that some events that make up the spinning disk ought to move somewhere temporally during the transformation, note that the shape of the disk in the lab frame is circular at all times. Meaning, no matter how much you stretch simultaneity around, you would still get a circular shape. Then combine that with the fact that the disk is symmetrical - and you only really have one possibility left. (But yeah this is somewhat misleading because actually the spinning disk doesn't have to be symmetrical - just pointing out the absurdity of non-symmetrical solutions).

Hi AnssiH,

Yes, that's correct, they found out that the long way in the other website as only the single length contraction to the spoke was necessary (no correct answer with more than 1) to identify the emission point at its tip and the rest was the photon traveling at c in a direct straight line. As Gron put it 'optical appearance' 'at retarded points in time'. One interesting thing, that was not identified in the paper, is that the lengths of the lines from each emission point to the camera in the part C plot are the photon travel times scaled to the radius length in c (the plot is 2D + t, effectively using 'radians' in 'plain sight' with no distortion of pi).

It just seems that while the speed of light may be the same in all frames, more than just a frame is required to keep the basic geometry intact. (Special Special Relativity ;) ).

These rolling wheel/ring plots are just the relativistic extension of a Cycloid arc over one complete cycle so it's interesting that the plot for both (part C and Cycloid arc) is a rectangle with the diameter as the height and the circumference as the length. The area under a cycloid line equals the area of the circle that made the line and the area above the cycloid line equals 3x the area of the circle while the total area is the surface area of a sphere with the same radius as the circle.

Harking back to the old days on this forum, I questioned Dr D for masking an axis in his proofs with something else, mainly because I thought the masked axis may take up the natural properties of that which was being masked.

In a way this may be similar to what is happening on a higher scale in physics, mathematically wise (especially with radians 'not in plain sight'), to give us other paradoxes like dark matter and dark energy. After all, in the ΛCDM model, the ratio of total universal mass to baryonic mass is 2*pi +/-1.1 % in both the PLANCK and WMAP data, and the ratio of total universal energy and mass to total mass is pi +/-3 %.

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In a way this may be similar to what is happening on a higher scale in physics, mathematically wise (especially with radians 'not in plain sight'), to give us other paradoxes like dark matter and dark energy. After all, in the ΛCDM model, the ratio of total universal mass to baryonic mass is 2*pi +/-1.1 % in both the PLANCK and WMAP data, and the ratio of total universal energy and mass to total mass is pi +/-3 %.

Hmm, that's an interesting idea. Can you elaborate on it a bit?

-Anssi

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