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


AnssiH
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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.

 

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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 

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