# I Rolling Rings in SR

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1. Jul 14, 2017

### Laurie K

This post is about the methodology required to produce the Fig. 9 Part C “optical appearance” plot as calculated by Øyvind Grøn in "Space geometry in rotating reference frames: A historical appraisal". http://areeweb.polito.it/ricerca/relgrav/solciclos/gron_d.pdf
In 2014 I contributed to a thread (on another forum, pm me for the details) that verified how the Fig. 9 Part C plot described above was created. This verification used t_Axle to represent the time at each event when photons would be emitted at a specific location on the circumference of the rolling ring x_SP, y_SP at t_SP where t_SP = t_Axle. The “optical appearance” chart produced in Part C is useful in that the emitted photons travel in a straight line from their emission point to the observer and can be measured as such. Also any point on the circumference of the ring can be regarded as a point on the tip of a wheel spoke that can be length contracted to identify the correct emission point at relativistic velocities. Section 7 of Grøn's conclusion refers to this transformation.
Grøn didn't really give much away on how he created his Part C plot so it's interesting to note that the plot shown below satisfies the rules given below even if the results are slightly different. The methodology to the solution is very simple as the oval shape shown in Grøn's Fig. 9 Part C is just the valid solutions for |t_Axle| - Sqrt[x_SP^2+y_SP^2] = 0. i.e. where the time the axle takes to get from the emission time to the camera point equals the length of a straight line from the emission point to the point where the camera is i.e. the photons direct paths to point 1.

In the image below the ring is rolling from left to right at 0.866025c
the Peak Value x, y, t is -3.464, 2, -4
which is also a valid solution for |t_Axle| - Sqrt[x_SP^2+y_SP^2] = 0.

Even the axle velocity between the emission events remains constant throughout.

Has anybody solved this problem in a different manner?

2. Jul 14, 2017

### Staff: Mentor

I would probably solve it in the axle frame and then just transform to the ground frame.