About the definition of Born rigidity

[A.T.]

Regarding "But here the static rulers in O_r should be twice as long": This is only true, if you apply the standard Lorentz transformation. You can do this only if you define, that the one-way speed of light with reference to O_r is the same clockwise and counterclockwise.

Yes, this is the flaw in the reasoning. The usual length contraction factor is isotropic: It depends only on the speed, not on the direction. The moving ruler (aligned with direction of motion) is always shorter than the resting ruler of the same proper length.

But this isotropic length contraction factor is derived using the assumption that light propagation is isotropic. Which is not valid in the rotating frame along the circumference. In the example I gave, the rulers forming O_i are not at half their proper length in the rotating rest frame of O_r, but rather must be at twice their proper length, in order to span the circumference measured in the rotating frame by the resting rulers forming O_r. So in rotating frames, you can have not only kinematic length contraction, but also kinematic length elongation, depending on the direction of motion.

 

[Sagittarius A-Star]

assumption that light propagation is isotropic. Which is not valid in the rotating frame along the circumference.

I think for clarificaion one should distinguish between one-way- and two-way speeds.

For applying the LT for a segment of the rim of the disk, one defines (not assumes) that the one-way speed of light is isotropic for synchronizing clocks, fixed to the rim of the disk. For "measuring" a one-way-speed, one needs always 2 synchonized clocks.

If you use only 1 clock, fixed to the rim of the disk, you will measure, that a two-way-speed is anisotropic around the complete rim of a rotating disk: The clockwise two-way-speed is different from the counterclockwise two-way speed. Two-way-speeds have a physical effect (i.e. Sagnac-effect). What one measures with only 1 clock should not be called a one-way-speed.

 

[cianfa72]

Yes, this is the flaw in the reasoning. The usual length contraction factor is isotropic: It depends only on the speed, not on the direction. The moving ruler (aligned with direction of motion) is always shorter than the resting ruler of the same proper length.

Yes, this turns out by looking at the worldsheet of the moving ruler described in the inertial rest frame of the center's disc (cut the cylinder in #19 along a vertical side and unroll it on the plane using the induced Minkowski metric).

My confusion is about which circumference Langevin observers actually measure. Do they measure (by using their standard rulers) the proper lenght $2\pi R$ where R is the proper lenght of the radius using rulers at rest in inertial rest frame of the rotating circumference?

 

[Sagittarius A-Star]

My confusion is about what circumference Langevin observers actually measure. Do they measure (by using their standard rulers) the proper lenght $2\pi R$ where R is the proper lenght of the radius using rulers at rest in inertial rest frame of the rotating circumference?

The circumference will be measured in the non-Euclidean geometry of the rotating disk as
$U=\gamma 2\pi R$ with $\gamma=1/\sqrt{1-\omega^2R^2/c^2}$.

Source:
https://en.wikipedia.org/wiki/Ehrenfest_paradox#Einstein_and_general_relativity

 

[Ibix]

My confusion is about which circumference Langevin observers actually measure. Do they measure (by using their standard rulers) the proper lenght $2\pi R$ where R is the proper lenght of the radius using rulers at rest in inertial rest frame of the rotating circumference?

A (small) ruler they carry with them will measure the same distances that an instantaneously co-moving inertial ruler would measure. So it'll measure (small) distances along the blue line because its tangent is one of the spatial basis vectors of the instantaneously co-moving frame.

One way to see it is to imagine starting with the disc at rest and laying a chain of small rulers around a circle, touching end-to-end. Hold each one in place with a single nail in the middle (and maybe supported against a rim so they won't bend due to centrifugal force when we start the disc rotating). Then start it rotating. Seen from the initial inertial frame, each ruler will length contract, so will open a small space between the ends. Thus (after spin up, at steady speed) a disc-riding Langevin observer must measure the circumference lengthened compared to before the spin up.