The Ehrenfest paradox exemplifies that a rigid body can't be defined in special relativity, but I can't convince myself that it implies non-Euclidean geometry as some sources say (https://en.wikipedia.org/wiki/Ehrenfest_paradox).

In this post question(Euclidean geometry in non-inertial frame), the answers vary from sayin that it only implies that rigid bodies can't exist in SR or that it implies, in addition to the former, that the geometry is non-Euclidian.

In my point of view, it only tell us that the theory is not consistent with this thought experiment.

Could someone elaborate on what the Ehrenfest paradox really tell us?

I'am an undergraduate in physics and I will be talking about rigid bodies in one of my classes. I want to clarify what would be non-sense to say to my professor. I appreciate the attention.

It is known that the length of a circle measured by non-inertial observers that are on a disk rotating around the center of this circle should exceed the length of the same circle measured by non-rotating inertial observers that are outside the disk. This circumstance, supplemented by the provision on the equality of the radius of the circle in the rotating and non-rotating frames of reference, formed the basis for recognizing the non-euclidean nature of geometry on rotating bodies.

In order that the disk in the inertial non-rotating frame of reference retain its size and shape that it possessed before it was rotated, all its concentric circles must be forcibly retained on the previous circumferences when we spin the disk faster and faster. Such a disk would be physically forcedly stretched in the peripheral areas.

The purely theoretical contraction of a disk or ring to disappearing dimensions is not at all more paradoxical than the effects of black holes and not more than other strange effects can be used to criticize the physical content of the theory of relativity. After all, after simple Lorentz contraction, the moving rod "having reached the speed of light" also "disappears".

Speaking about the difference of circumference of a disk in inertial and rotating frames of reference, the authors of geometrical metamorphoses on the disc as a rule do not consider the physical side of the difference in its length.

Nor did M. Born (see M. Born, Einstein's theory of relativity) explaining the reason for the non-Euclidean geometry on the Einstein disk and declaring the collapse of non-Euclidean geometry.

Was there any collapse at all? The following thought experiment shows, that it is hardly possible to "turn away" from the reduction of the rotating disk and reduce the physical processes to a simple geometrical portrayal.

Let us imagine a rotating ring, placed in the inertial reference frame $K$ on the circle $O$ of large diameter $R$.

Let the velocity $v$ of the points of the ring be equal to $(3/4)^{1/2} c$ , where $c$ is the velocity of light in a vacuum. Suppose that the observer who is at some distance from the plane in which the ring is located, sees the ring rotating counter-clockwise.

Let us now imagine that along the tangent to the upper point of the circle $O$ to the right of the observer at a speed of $(3/4)^{1/2} c$ approaches a rope. Rope’s length $L$ in the reference frame of the above observer is numerically equal to the length $L = 2\pi R$ of the circle on which the rotating ring is placed. Proper length of the rope is $2L$.

Suppose that at the moment when the front end of the “flying rope” comes to the upper point of the circle, it is grabbed by the ring and screwed onto it. Since the length $L$ of the moving rope is equal to the length $L_c$ of the circle $O$, and each section of the rope, continuing to move on the ring with the speed $v$, keeps its length unchanged, the rope also retains its length and completely covers the rotating ring in such a way that observers can tie end and beginning of the rope lying on the ring.

Observers on the ring, measuring in their rotating reference frame $K’$ length $L'$ of the circumference of circle $O$, on which the rope lies, will find that the length $L'$ of the circle $O$ and correspondingly, the length $L'$ of the rope is twice as long as the length $L$ of circle $O$ as measured in the reference system $K$ i.e. equal to $2L$.

In this case observers on the ring will argue that the length of the rope, the length of the ring and the same length of the circumference on which the ring and rope are located exceeds the value of $2\pi R$ due to the non-Euclidean nature of the geometry of the rotating reference frame $K'$.

Let's now slow the ring down until rotation completely stops.

What happens to the length of the circle $O$ on which the ring is located? The length of the circle should, according to the statements of observers on the ring, be equal to $L$. Such an expected decrease of the circumference of the ring the observers on the ring will explain by their transition from the non-Euclidean reference frame $K'$ of the rotating ring to the Euclidean space of the inertial reference frame $K$.

However, when the ring stops, the length of the rope becomes equal to its own length $2L$. What will be the length of the stopped ring? If we assume that the ring in the inertial reference frame $K$ after its stop has retained its dimensions, then the rope will be 2 times longer than the ring, which is completely incomprehensible - because the rope, like the ring, "passed" from non-Euclidean space to Euclidean space.

If we assume that the ring, will lengthen when stopped, and its radius $R_0$ for an inertial nonrotating observer becomes equal to $2R$, then we cannot speak at all about the change in the length of the ring for observers on the ring (and that line on which the ring lies in their reference frame and which they can call this circumference), since the length of the ring for them has not changed.

Of course, one can speak of a change in the radius of the ring for rotating observers and of the presence of transitions detected in the inertial reference system when it comes to deceleration from one circle to another, but then a lot of questions arise.

First, what is a circle in physics devoid of materiality (physicality)? Secondly, what in the reasonings of some authors determines the change in the metric on the disk when it is spinning - a change in radius or a change in the length of the circle? And thirdly, are rotating observers who are unable to detect a change in the length of the periphery of the disk can detect a change in the length of its radius?

This article shows that the disk does not bend when rotationally shortened and that the ratio of the circumference of the edge of the disk to its diameter on the rotating disk is exactly equal to the $\pi$.

http://aapt.scitation.org/doi/abs/10.1119/1.4942168?journalCode=ajp

The article gives two resolutions, the both are formally correct. But, if the rail and the wheel are toothed, only Lorentz – contracted wheel will run smoothly, because its teeth will match the ones on the rail.

The Ehrenfest paradox exemplifies that a rigid body can't be defined in special relativity

Well, I would say that rigid bodies can't exist simply because of the speed-of-light limit. Certain cases of rigid motion (of not-intrinsically-rigid bodies) make sense in SR, as you know, but there are only a few. I would count Ehrenfest's cylinder as one of those cases – it's a highly symmetric system. Almost anything else, that doesn't meet any definition of rigidity, would be a better example of the lack of rigidity in SR.

In my point of view, it only tell us that the theory is not consistent with this thought experiment.

If by "the theory" you mean SR then it absolutely is consistent with the thought experiment. SR is assumed to be correct in the thought experiment and all of the properties of the cylinder are predictions of SR.

It's called a paradox not because it's a logical inconsistency but because people are surprised by it. Length contraction also surprises people and has as much right to be called a paradox.

I would classify these paradoxes as "silly" in the following semi-rigorous sense: they have near exact analogues in Euclidean geometry, and no one would consider the Euclidean versions to be paradoxes. The Minkowskian versions only appear to merit peer-reviewed think pieces because people don't apply their evolved intuition about Euclidean geometry to spacetime geometry.

Here's the Euclidean version of the Ehrenfest setup. You have a right circular cylinder of radius $$r$$ and you want to cover it with strips of wallpaper of width $$w$$. If the strips run parallel to the length of the cylinder then you need $$2πr/w$$ strips. But if the strips are angled by $$\theta$$, so that they spiral around the cylinder, then you need only $$2πr/w\sqrt{1+v^2}$$ of them, where $$v = \tan\theta$$ is the slope. This means that the circumference of the circle, as measured by the strips of width $$w$$, is only $$w(2πr/w\sqrt{1+v^2}) = 2πr/\sqrt{1+v^2}$$, and the circumference divided by the radius is therefore $$2π/\sqrt{1+v^2}$$, so the cylinder is non-Euclidean when $$v\ne 0$$.

What are we to make of this? Well, it is what it is. It's obvious that there's something awry about the way we're measuring distances with these strips. The distances of length $$w$$ don't join together into the length of a single closed curve; for each "zig" of length $$w$$ there's a "zag" that we aren't counting. In Minkowski space it's the same story: no closed spacelike curve is actually measured out in increments of 1 meter by the metersticks. As is usual in the silly paradoxes, we're overlooking the time coordinate.

While I find this thought experiment somewhat interesting, there's nothing mysterious about it; all of the geometric rules are manifest, the setup is easy to understand, and it's just a matter of what words you think should be used to describe it. It is Euclidean or not? It's whatever you want it to be.