Ballerina Universe Paradox

Question

Imagine a ballerina spinning with angular velocity $\omega$ on the ground, you will see her arms opening due to the centrifugal force.

Now think of the same ballerina but she is the only thing in the Universe. Technically, she wouldn't be able to tell that she is spinning. Will her arms open in the same way?

This question may be ill-posed, so you can think about it this way: she is first spinning on the ground and before her arms reach equilibrum, the rest of the universe vanishes. What would happen?

This is a variant of Mach's principle and no-one knows what would happen. None of the experimentally tested theories tell us what would happen, and sadly we can't make the universe disappear to test it. So I don't see how anyone can answer your question. — John Rennie, Aug 05 '22 at 11:07
Note also that there are many questions about Mach's principle on this SE. Have a look at them to see whether they address your question. — Yvan Velenik, Aug 05 '22 at 11:18
You should also take a look at Newton's Rotating Bucket argument https://en.wikipedia.org/wiki/Bucket_argument — foolishmuse, Aug 05 '22 at 16:02
"Technically, she wouldn't be able to tell that she is spinning..." How do you know that? — hft, Aug 05 '22 at 17:16
Related: Will a planet rotate if it is the only being in the universe? — Níckolas Alves, Aug 05 '22 at 23:58

Gerald · Answer 1 · 2022-08-05T16:15:34.833

Yes. She will still open her arms. The spacetime metric is one of a gravity induced by a rotating invisible mass around her. Like an accelerated observer in an empty space thinks there is an infinite massive plane giving rise to a uniform metric. He thinks he is at rest in this field. In the same way,the dancer thinks she is at rest in a frame in which an invisible shell of mass is rotating around her.

Ballerina Universe Paradox

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