# Ehrenfest paradox

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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## Answers and Replies

PeterDonis
Mentor
2020 Award
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.

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:

http://en.wikipedia.org/wiki/Ehrenfest_paradox

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. ;)

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.

Ibix
Science Advisor
2020 Award
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 realise that the paradox is caused by your incomplete model of relativity, not relativity itself.

PeterDonis
Mentor
2020 Award
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.

Ibix
Science Advisor
2020 Award
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.

PeterDonis
Mentor
2020 Award
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.