# Resolving Ehrenfest Paradox: STR vs General Theory

• wellorderingp
In summary: No, that's not correct. SR can accommodate non-Euclidean spatial geometry perfectly well, and that's the only non-Euclidean (more precisely, non-flat) geometry involved here. The only requirement of flatness in SR is for flat spacetime. In summary, the Ehrenfest paradox can be resolved in Special Relativity using the principle of Special Relativity.
wellorderingp
Can the ehrenfest paradox be resolved using STR or does one require to go in general theory?
If it can be, please recommend a book or online source which explains it from the basics.
Also I'm somewhat unclear on what exactly the paradox is,does it state that-
Since the ratio of it's circumference and diameter is less than π it follows non euclidian geometry.
So what if it is a non euclidian geometry? How can that statement be a paradox?

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wellorderingp said:
Can the ehrenfest paradox be resolved using STR or does one require to go in general theory?

Since as it is usually formulated, the paradox is set in flat spacetime, that formulation can be resolved in SR. GR is only required if spacetime is curved, i.e., if gravity is present. In the usual formulation, the "rotating disk" in the scenario is assumed to have negligible mass and therefore negligible gravity. (Note, however, that if the disk is rotating, it will be under stress, and stress is a source of gravity; so we have to also assume that the stress is small enough, which requires assumptions about the disk's mechanical properties.)

There is a version of the paradox (mentioned in the Usenet Physics FAQ article I link to below) which attempts to take into account the mass and energy of the rotating disk and its effect on the spacetime geometry; but I don't know that anyone has ever done a full analysis of this (as the article notes, it would take a "full-blown, hairy GR calculation").

Also, sometimes the paradox is said to involve GR because considering it was one of the key lines of thought that helped Einstein in developing GR.

wellorderingp said:
please recommend a book or online source which explains it from the basics.

A decent discussion can be found in this Usenet Physics FAQ article:

http://math.ucr.edu/home/baez/physics/Relativity/SR/rigid_disk.html

If nothing else, this article makes it clear that there are a lot of complexities lurking in what seems like a simple scenario. Also, the Wikipedia page has useful information:

wellorderingp said:
I'm somewhat unclear on what exactly the paradox is,does it state that-
Since the ratio of it's circumference and diameter is less than π it follows non euclidian geometry.

It's no surprise that you're unclear on exactly what the paradox is, since many physicists have failed to agree on that. ;)

wellorderingp said:
So what if it is a non euclidian geometry? How can that statement be a paradox?

It isn't, literally speaking. It's just a very counterintuitive result; the word "paradox" can be used to mean that, not something literally self-contradictory.

PeterDonis said:
It isn't, literally speaking. It's just a very counterintuitive result; the word "paradox" can be used to mean that, not something literally self-contradictory.
Isn't the point that they are paradoxical from the point of view of an incomplete understanding of relativity? For example, the twin paradox is genuinely paradoxical if you understand that all motion is relative. The resolution is to realize that the paradox is caused by your incomplete model of relativity, not relativity itself.

Ibix said:
Isn't the point that they are paradoxical from the point of view of an incomplete understanding of relativity?

That's what I meant by "counterintuitive". There is no actual paradox, only an apparent one if you are relying on pre-relativistic intuitions.

PeterDonis said:
That's what I meant by "counterintuitive". There is no actual paradox, only an apparent one if you are relying on pre-relativistic intuitions.
If the paradoxes were only apparent, there'd be no need to abandon the pre-relativistic intuitions. I'd say that these are actual paradoxes, but only in misconceptions of SR that most of us have held (or still hold) at some point in our learning.

I'm arguing semantics, not physics, obviously.

My understanding of the paradox is that,it's a paradox in special relativity since STR only deals with flat spacetime and here our disc is showing non euclidian geometry,so the concept won't fit into STR.

wellorderingp said:
it's a paradox in special relativity since STR only deals with flat spacetime and here our disc is showing non euclidian geometry,so the concept won't fit into STR.

No, that's not correct. SR can accommodate non-Euclidean spatial geometry perfectly well, and that's the only non-Euclidean (more precisely, non-flat) geometry involved here. The only requirement of flatness in SR is for flat spacetime.

Oh,yes of course you are correct,I messed up geometry and spacetime. My bad.

## 1. What is the Ehrenfest Paradox?

The Ehrenfest Paradox is a thought experiment that highlights a discrepancy between the predictions of Special Relativity Theory (STR) and the General Theory of Relativity (GTR). It involves a rapidly rotating disk and the rotation of its radius, and how these motions affect the measurements of time and space.

## 2. How does Special Relativity Theory explain the Ehrenfest Paradox?

STR states that time and space are relative, and that the laws of physics remain the same in all inertial reference frames. In the Ehrenfest Paradox, an observer on the rotating disk would perceive the radius of the disk as shorter due to length contraction. This would result in the circumference of the disk being smaller, and the observer would measure time passing slower for an object moving along the circumference. This explains the paradox in the context of STR.

## 3. How does General Theory of Relativity explain the Ehrenfest Paradox?

GTR states that gravity is not a force, but rather a result of the curvature of spacetime caused by the presence of mass. In the Ehrenfest Paradox, the rotating disk would create a gravitational field due to its mass, resulting in a warping of spacetime. This warping would cause the radius of the disk to appear longer, and the circumference to be larger, resulting in a greater measurement of time passing for an object moving along the circumference. This explains the paradox in the context of GTR.

## 4. Which theory is correct in explaining the Ehrenfest Paradox?

Both theories, STR and GTR, have been experimentally proven and are considered to be correct in their respective domains. In the case of the Ehrenfest Paradox, both theories provide valid explanations, but they are based on different fundamental principles. Therefore, the paradox cannot be resolved by either theory alone, and it highlights the need for a unified theory that can reconcile the two.

## 5. How can the Ehrenfest Paradox be resolved?

The Ehrenfest Paradox can potentially be resolved with a unified theory of physics that combines the principles of STR and GTR. This theory, known as a Theory of Everything, is currently being sought after by physicists, but it has yet to be fully developed. It is also possible that the paradox may never be fully resolved and will remain a fundamental issue in our understanding of the universe.

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