Ehrenfest Paradox

???

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Ehrenfest Paradox

#1  Postby hackenslash » Jan 16, 2022 7:10 pm

OK, so two odd things. The first is that, for some reason, I've never encountered this before. The second is the paradox itself. been scratching my head all afternoon on it, and I've made so much headway the spot I'm standing on is substantially rutted (though no females involved this time).

For those who haven't encountered it, it's a paradox that arises when special relativity meets rudimentary geometry in a very specific but extremely simple circumstance. We're to picture a rotating disc. It's probably helpful to think of this disc as being sufficiently large and rotating sufficiently quickly for relativistic effects to be significant. Fairly straightforward, right?

The paradox arises because, in order to maintain the constancy of c for all observers, Lorentz contraction applies in the direction of rotation, but only in that direction. The result is, of course, that this results in contraction of the circumference and, because C=2πr, the radius needs to contract, but the radius is orthogonal to the direction of travel, which basically means that you have to shrink a circle without shrinking a circle.

Anybody got any ideas?

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Re: Ehrenfest Paradox

#2  Postby Cito di Pense » Jan 16, 2022 7:15 pm

Is it a circular argument? Asking for a friend. I mean, on behalf of a friend. I have all the friends I can stand.
Хлопнут без некролога. -- Серге́й Па́влович Королёв

Translation by Elbert Hubbard: Do not take life too seriously. You're not going to get out of it alive.
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Re: Ehrenfest Paradox

#3  Postby hackenslash » Jan 16, 2022 7:25 pm

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Re: Ehrenfest Paradox

#4  Postby newolder » Jan 16, 2022 7:33 pm

Wolfram alpha includes (image attached):
Ehrenfest.png
Ehrenfest.png (100.86 KiB) Viewed 729 times
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Re: Ehrenfest Paradox

#5  Postby hackenslash » Jan 16, 2022 7:44 pm

I was just pondering synchronising clocks, and that all frames on the disc were accelerating. That makes sense.

Off to look up that metric.

Edit: Thanks for that, dude. I feel suitably unscratched. That was exactly what I needed.
Last edited by hackenslash on Jan 16, 2022 7:51 pm, edited 1 time in total.
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Re: Ehrenfest Paradox

#6  Postby newolder » Jan 16, 2022 7:50 pm

At high enough accelerations we enter the realm of Unruh radiation, iirc.
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Re: Ehrenfest Paradox

#7  Postby hackenslash » Jan 16, 2022 7:55 pm

Looking for the LLL metric brought me to the wiki for Born Coordinates, which has a good section on it.

https://en.wikipedia.org/wiki/Born_coordinates
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Re: Ehrenfest Paradox

#8  Postby Evolving » Jan 17, 2022 8:49 am

That was very interesting, thank you for bringing that to our attention!

Definitely more unscratching than Einstein's deeply unsatisfying initial response (if https://www.quora.com/How-is-the-Ehrenfest-paradox-resolved is to be believed) that the disc (or cylinder) would break if you tried to accelerate it to relativistic speeds.
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Re: Ehrenfest Paradox

#9  Postby hackenslash » Jan 17, 2022 10:32 am

Indeed, not least because there's a way to circumvent that objection, namely to construct a disc already rotating from small rigid bodies.
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Re: Ehrenfest Paradox

#10  Postby newolder » Jan 17, 2022 10:54 am

I wonder if someone at quora is thinking about Bell's spaceship paradox where 2 spacecraft connected by a string and maintaining equal acceleration as measured in the lab. frame and noting that the string eventually breaks? :dunno:
...
Bell considered two rocket ships connected by a string, with both having the same acceleration in the inertial "lab frame", with one ship trailing the other and both moving along one line. The ships start out at rest in the lab. Their accelerations in the lab frame are required always to be equal, but these accelerations can vary with time. Because the ships always maintain equal accelerations in the lab frame, their speeds will be equal at all times in the lab, and so they'll remain a constant distance apart in the lab. But after a time they will acquire a high speed, at which point we ask what has become of the idea of the distance between them being Lorentz contracted. The actual paradox posed by Dewan & Beran and by Bell was that the string should eventually snap. But why should it snap, if the ships are maintaining a constant separation?
...

ucr.edu link

ETA
Having read the quora link, I apologise for introducing this red herring. :oops:
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Re: Ehrenfest Paradox

#11  Postby hackenslash » Jan 17, 2022 11:15 am

That popped up in te Talkrat thread where I came across this. To my mind, the string shouldn't snap because the whole caboodle is in a single Lorentz contracted frame, thus the same Lorentz contraction applies to the entire assemblage, maintaining their relative displacement in the frame during contraction.
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Re: Ehrenfest Paradox

#12  Postby Evolving » Jan 17, 2022 11:16 am

The herring wouldn't snap either.
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Re: Ehrenfest Paradox

#13  Postby hackenslash » Jan 17, 2022 11:26 am

Indeed not, because they have to make it all the way to purple before they snap. You definitely don't want to be around when that happens.

Of course, the herring making it to purple would necessitate it coming directly toward you at precisely the right velocity...
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Re: Ehrenfest Paradox

#14  Postby Evolving » Jan 17, 2022 11:28 am

That is the core of the Herringfest Paradox.
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Re: Ehrenfest Paradox

#15  Postby newolder » Jan 17, 2022 11:40 am

Ralph Berger has a treatment:
Abstract

The Bell Spaceship Paradox has promoted confusion and numerous resolutions since its first statement in 1959, including resolutions based on relativistic stress due to Lorentz contractions. The paradox is that two ships, starting from the same reference frame and subject to the same acceleration, would snap a string that connected them, even as their separation distance would not change as measured from the original reference frame. This paper uses a Simple Relativity approach to resolve the paradox and explain both why the string snaps, and how to adjust accelerations to avoid snapping the string. In doing so, an interesting parallel understanding of the Lorentz contraction is generated. The solution is applied to rotation to address the Ehrenfest paradox and orbital precession as well.

...

This measured change in tilt is proof of a measured change in orbit length. If no acceleration is involved, a second GP-B satellite can be tethered by string with the first and the string will not break no matter how long in orbit. If, however, through some machination, we accelerated the two GP-B satellites up to a much higher velocity while at the same radius, the overshoot in orbits will be proportionally larger, and the distance between satellites will grow, and the string will snap. Just like the edge of an Ehrenfest disk, the points separate, with motive power being whatever caused the satellite velocities to increase.

...


Is this stuff the origin of the spaghettification experienced when falling towards a gravitational black hole or have I got my phenomena mixed up again?
Last edited by newolder on Jan 17, 2022 11:49 am, edited 2 times in total.
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Re: Ehrenfest Paradox

#16  Postby hackenslash » Jan 17, 2022 11:42 am

Evolving wrote:That is the core of the Herringfest Paradox.


I think I attended one of those in Sweden a while back, though it's hard to tell, as I'd been punsched pretty hard...
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Re: Ehrenfest Paradox

#17  Postby BlackBart » Jan 17, 2022 11:45 am

Evolving wrote:The herring wouldn't snap either.


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Re: Ehrenfest Paradox

#18  Postby hackenslash » Jan 17, 2022 11:56 am

newolder wrote:Ralph Berger has a treatment:
Abstract

The Bell Spaceship Paradox has promoted confusion and numerous resolutions since its first statement in 1959, including resolutions based on relativistic stress due to Lorentz contractions. The paradox is that two ships, starting from the same reference frame and subject to the same acceleration, would snap a string that connected them, even as their separation distance would not change as measured from the original reference frame. This paper uses a Simple Relativity approach to resolve the paradox and explain both why the string snaps, and how to adjust accelerations to avoid snapping the string. In doing so, an interesting parallel understanding of the Lorentz contraction is generated. The solution is applied to rotation to address the Ehrenfest paradox and orbital precession as well.

...

This measured change in tilt is proof of a measured change in orbit length. If no acceleration is involved, a second GP-B satellite can be tethered by string with the first and the string will not break no matter how long in orbit. If, however, through some machination, we accelerated the two GP-B satellites up to a much higher velocity while at the same radius, the overshoot in orbits will be proportionally larger, and the distance between satellites will grow, and the string will snap. Just like the edge of an Ehrenfest disk, the points separate, with motive power being whatever caused the satellite velocities to increase.

...


Is this stuff the origin of the spaghettification experienced when falling towards a gravitational black hole or have I got my phenomena mixed up again?


That's awesome, and gels completely.

So yes, it is the spaghettification principle, but massively (geddit) reduced. In the planetary scenario, the difference in elevation means that they're experiencing fundamentally different degrees of acceleration due to gravity, and that's why acceleration corrections are required to maintain equilibrium of acceleration between front and back so that their acceleration is equivalent in all the accelerated frames along the gradient. Free of a gravity well, these corrections are unnecessary and the string won't break.
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Re: Ehrenfest Paradox

#19  Postby hackenslash » Jan 17, 2022 1:05 pm

Incidentally, the wiki on proper motion is excellent, and contains a section treating Ehrenfest problem in a cartesian setting.

https://en.wikipedia.org/wiki/Proper_reference_frame_(flat_spacetime)#Uniform_circular_motion
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Re: Ehrenfest Paradox

#20  Postby newolder » Jan 17, 2022 1:38 pm

The Sagnac effect and it's applications to gyroscopes, interferometers and 3rd-generation LIGO detections sounds cool too.
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