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Invariant spacetime - distance - Circular Motion

This is a question that I thought up a few years ago when I was taking mechanics. I asked the professor but didn't really get a straight answer.

Imagine a record spinning at relativistic speeds. In the lab frame, the circumference of the record should decrease according to the Lorentz contraction. However the radius of the record should remain fixed since it is orthogonal to the direction of motion. So does the shape of the record appear distorted since the circumference is smaller but the radius is the same? What would it look like?

I have a feeling the answer is obvious and/or well known since it would seem to be similar to the situation of relativistic particles traveling around a circular accelerator but I can't think of how it can be solved.

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BeauGeste
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As MBN pointed out, this is called the Ehrenfest paradox. Wikipedia summarizes

The modern resolution can be briefly summarized as follows: Small distances measured by disk-riding observers are described by the Langevin-Landau-Lifschitz metric, which is indeed well approximated (for small angular velocity) by the geometry of the hyperbolic plane, just as Kaluza had claimed. For physically reasonable materials, during the spin-up phase a real disk expands radially due to centrifugal forces; relativistic corrections partially counteract (but do not cancel) this Newtonian effect. After a steady-state rotation is achieved and the disk has been allowed to relax, the geometry "in the small" is approximately given by the Langevin-Landau-Lifschitz metric.

I don't have anything to add - I think Wikipedia is good for this one.

Mark Eichenlaub
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