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In general relativity, we assume that the connection is torsion-free. In some alternate theories, like Einstein-Cartan theory, we consider connections with torsion.

According to this answer, torsion can be measured through the precession of spins, but this requires understanding spinors in GR. Is there a way to directly measure torsion without using spin, ideally through a simple mechanical system? (I don't care if this is an impractical experimental test, I just want intuition for what the torsion tensor means physically.)

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You don't need quantum-mechanical spins to detect torsion. The diagram below shows a thought experiment that detects torsion. You set up a pair of orthogonal gyroscopes at A. You move a distance $\delta$ along the direction of one of the gyroscopes to B, then along the other to C. Now you go back to A and repeat the experiment, but interchanging the order, which takes you along ADE. In the limit of small $\delta$, The distance $\ell$ between C and E scales according to $\delta\sim\ell^3$ if there is no torsion, but $\delta\sim\ell^2$ if there is torsion.

diagram of experiment with gyroscopes

I have a longer presentation of this in my GR book, in section 5.8.

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The papers cited on this question may give you an answer or at least some insight. As far as I have read, the only paper that uses spinors is Flanagan and Rosenthal (arXiv:07041447).

(I'd have put that in a comment but I don't have enough reputation)

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