This document contains comments about the document "Ehrenfest paradox" in Wikipedia
• The text in italics is copied from that url
• Immediate followed by some comments
In the last paragraph I explain my own opinion.

### Introduction

The article starts with the following sentence.
The Ehrenfest paradox concerns the rotation of a "rigid" disc in the theory of relativity.
The article should start exactly what rigid is and what it is not.
It discusses an ideally rigid cylinder that is made to rotate about its axis of symmetry. The radius R as seen in the laboratory frame is always perpendicular to its motion and should therefore be equal to its value R0 when stationary. However, the circumference (2pR) should appear Lorentz-contracted to a smaller value than at rest, by the usual factor?. This leads to the contradiction that R=R0 and R < R0.
In this case we consider the radius R at rest, the radius R0 when moving stationary, the circumference C = 2*pi*R when at rest and the C0=2*pi*R0 when the disc is statationary. The contradiction is that R = R0 and C < C0 which leads to R < R0
The solution is simple: by performing an experiment using fixed length rods In fact there is no paradox.

### 1. Essence of the paradox

 ``` R . R . R 1 | 2 . . |-> . . | . . . | . . ^ | 6- -|- - - R - - -|- -3 | V . . | . . . | . . <-| . . 5 | 4 R . R . R Figure 1 ``` Figure 1 at the left shows two circles. One large one at rest with a radius R. The points 1,2,3,4,5, and 6 are all on the large circle. The circumference is C = 2*pi*R One small one with a speed v and with radius R0. The circumference is C0 = 2*pi*R0 Accordingly to Erenfest paradox R should be equal to R0 and C > C0. The most logical situation is that both R0 and C0 are shorted by length contraction. What is correct can only be desided by means of a actual experiment, which is relatife simple.

### 2 Ehrenfest's argument

Ehrenfest considered an ideal Born-rigid cylinder that is made to rotate. Assuming that the cylinder does not expand or contract, its radius stays the same.
Of course you can assume that but if the result of a real experiment shows the opposite using a large metal solid disc than somewhere your argument is wrong.
When a Born-rigid cylinder does not contract than a Born-rigid rod should also not contract.
But measuring rods laid out along the circumference 2*pi*R should be Lorentz-contracted to a smaller value than at rest, by the usual factor gamma.
That means the rigid measuring rods do not touch each other.
This leads to the paradox that the rigid measuring rods would have to separate from one another due to Lorentz contraction; the discrepancy noted by Ehrenfest seems to suggest that a rotated Born rigid disk should shatter.
The only way out is to perform an actual experiment.
Thus Ehrenfest argued by reductio ad absurdum that Born rigidity is not generally compatible with special relativity.
To have an arguement does not solve any physical issue.

### 3. Einstein and general relativity

The rotating disc and its connection with rigidity was also an important thought experiment for Albert Einstein in developing general relativity.
Thought experiments are dangerous to discuss physical issues. You cannot solve the Ehrenfest paradox by means of a thought experiment.

### 5. Resolution of the paradox

Grřn states that the resolution of the paradox stems from the impossibility of synchronizing clocks in a rotating reference frame.
If you start from a frame at rest with synchronised clocks and you rotate them they stay synchronised.
The issue is if the time of each clock is identical and dilated with the same amount.

Following is a list with "Comments in Wikipedia" about related subjects

For a discussion in sci.physics.research see:
https://groups.google.com/forum/#!topic/sci.physics.research/_rQMwHe8I1o rigid rotating disc

### Reflection 1

The central issue of the Ehrenfest paradox is introduction of the concept rigid without a clear definition of what it is.
As such you can have rigid and non rigid rods. The same for cylinders, discs and balls.
The difference between rigid versus non rigid is stress. As such non rigid rods experience length contraction. For rigid rods this is not the case.
Our earth is considered non rigid and as such experiences length contraction.

### Reflection 2 - (Thought) experiment

The purpose of this experiment is to answer the question:
What happens with a disc made out of Fe3Al6Ni9 when it is brought into rotation
The material Fe3Al6Ni9 is an imaginary alloy
Figure 2A is at rest. In Figure 2B and 2C the disc is rotating.
 ```R . R . R 1----|----2 ./ \ | / \. / \ | / \ ./ \ | / \. / \|/ \ 6 - - - - R - - - - 3 \ /|\ / .\ / | \ /. \ / | \ / .\ / | \ /. 5----|----4 R . R . R Figure 2A ```
 ```R . R . R 1 ---|--- 2 . \ | / . / \ | / \ ./ \ | / \. \|/ 6 - - - - R - - - - 3 /|\ .\ / | \ /. \ / | \ / . / | \ . 5 ---|--- 4 R . R . R Figure 2B ```
 ```R . R . R | . 1---|---2 . / \ | / \ . / \ | / \ . / \|/ \ . 6 - - - R - - - 3 . \ /|\ / . \ / | \ / . \ / | \ / . 5---|---4 . | R . R . R Figure 2C ```
• Figure 2A shows the disc at rest made out of Fe3Al6Ni9.
• Figure 2B shows the situation that the size of the rotating disc stays the same and is not contracted in any way.
Calibration rods close to the circumference, attached to the rotating disc have contracted such that nearby rods don't touch each other any more.
• Figure 2C shows the situation that the size of the rotating disc, the radius and the circumference has decreased.
The question as:
• Which Figure is correct: 2B or 2C?
• Or are both correct and it depends about the alloy used.

One important remark is that the outcome is independent observed from an observer at rest or from an rotating observer. That means the observer stays on top of the rotating disc. For this observer the rotating disc is at rest and the surrouding rotates.

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Created: 15 February 2015

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