# Ehrenfest paradox debate, how does a pinning disk look like?

1. Nov 23, 2013

### Trojan666ru

I was in a debate with a guy about "how does a spinning disk may look like in relativity."

His point was that the disk will be flat as always. But the radius will be curved, i couldn't actually imagine it.
He gave me a picture too

So i come up with a model, but he rejects it saying the disk will be always flat
This is my explanation with picture
It's my drawing and you can see that when the disk spinns at relativistic velocity, the circumference shrinks but the radius has to be kept same
In this way the radius of the sphere can be kept same and
the circumference will be length contacted, thus the disk
becomes a hollow sphere

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2. Nov 23, 2013

### yuiop

The picture your friend gave you is for a rolling disc that has linear motion relative to the observer on top of the rotation. That is why his disc is taller than it is wide.

The Ehrenfest paradox usually concerns a disc that is not allowed to curve out of the plane as suggested by your sketch. If the disc bent as much as your image suggests, the edge of the disc would be going slower than the mid parts of the spokes and so it would not length contract as much.

For a disc that remains flat, the circumference is 2*pi*r according to the external non spinning observer at rest with respect to the centre of the disc, but gamma*(2*pi*r) according to an observer attached to the perimeter of the spinning disc.

Length contraction means the proper length is greater than the coordinate length. For an unstressed straight rod, the proper length remains constant and the coordinate length appears to contract with increasing relative motion. In the case of the disc perimeter, the coordinate length is forced to remain constant, so the proper length of the perimeter increases with increasing rotation (and constant radius), but in this case there definitely will be stresses involved, as the perimeter is being stretched in its (non inertial) rest frame.

Last edited: Nov 23, 2013