I How is SR applied to circular motion?

[Demystifier]

The author seems to believe Lorentz contraction is must. But sometimes a kind of "Lorentz extension" take place in rotation system.

At the end of page 9, the paper says:
"Now, as in Section 2, assume that the rotating ring is a series of independent short rods, uniformly distributed along the gutter. Each rod is relativistically contracted, but the ring is not. This means that the distances between the neighboring ends of the neighboring rods are larger than those for a nonrotating ring, so the proper length of the ring is also larger than that of a nonrotating ring. This is concluded also in [2]. This situation mimics a more realistic ring made of elastic material, where atoms play the role of short rigid rods. Owing to the rotation the distances between neighboring atoms increase, so there are tensile stresses in the material."
This quote shows that "Lorentz extension" is included.

 

[sweet springs]

Thanks for your suggestion. I agree him on this point.

This situation mimics a more realistic ring made of elastic material, where atoms play the role of short rigid rods. Owing to the rotation the distances between neighboring atoms increase, so there are tensile stresses in the material."

More ideal case is ring made of incompressible fluid rotating with speed u through the gutter with no friction. Depth or width of fluid from the gutter r outward side would become shallower or narrower as the speed u increases. Or metal ring melted by heat should congeal in new rotating environment. No $\phi$-stress inside the metal but get compressed when it stops. Best.

 

[PAllen]

At the end of page 9, the paper says:
"Now, as in Section 2, assume that the rotating ring is a series of independent short rods, uniformly distributed along the gutter. Each rod is relativistically contracted, but the ring is not. This means that the distances between the neighboring ends of the neighboring rods are larger than those for a nonrotating ring, so the proper length of the ring is also larger than that of a nonrotating ring. This is concluded also in [2]. This situation mimics a more realistic ring made of elastic material, where atoms play the role of short rigid rods. Owing to the rotation the distances between neighboring atoms increase, so there are tensile stresses in the material."
This quote shows that "Lorentz extension" is included.

You also noted in your paper that this is equivalent to linear cases, which I thought was a key observation. In my words, you have a rolled up version of Bell spaceship paradox. Because the circumference is constrained by the gutter to be the same as in the inertial frame, the local stretching increases without bound as rim speed approaches c. Just like a string constrained to maintain a fixed length in an inertial frame as it accelerates to c experiences unbounded local stretch.

 

[Demystifier]

You also noted in your paper that this is equivalent to linear cases, which I thought was a key observation. In my words, you have a rolled up version of Bell spaceship paradox. Because the circumference is constrained by the gutter to be the same as in the inertial frame, the local stretching increases without bound as rim speed approaches c. Just like a string constrained to maintain a fixed lenegth in an inertial frame as it accelerates to c experiences unbounded local stretch.

Yes, exactly!