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So I'm trying to understand spinor geometry (and not getting anywhere).

Is it possible to define relative velocity for spinors? (at a point in the manifold similar to How to calculate relative velocity in curved spacetime?) If so, how?

This question was inspired by this: https://math.stackexchange.com/a/2281305/430082

More Anonymous
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    I don't know what you mean by a "relative velocity for spinors". What about relative velocity (as treated in the question you link) is "for non-spinors" so that you need a different notion "for spinors"? Relative velocity is just about a way to compare two velocities, which are vectors, at different points. Do you perhaps mean that you're looking for ways to compare two different spinors at different points? (I don't see why one would call that relative velocity for spinors - what do spinors have to do with velocities?) Why do you need such a way? – ACuriousMind Apr 01 '22 at 12:17
  • Hi More Anonymous. Echoing @ACuriousMind: In which context? Is this question inspired by some reference? Which page? – Qmechanic Apr 03 '22 at 22:44
  • @Qmechanic I've linked the square root of the line element - link which inspired this question. Even an answer explaining why this can't be done will suffice. – More Anonymous Apr 04 '22 at 04:49

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If you have a spinor field, there may be ways to obtain a natural velocity vector. For a Dirac field for example, there is the conserved Dirac current vector field $\bar\psi\gamma^\mu\psi$, which is timelike and future-pointing. You can then normalise it to obtain a 4-velocity vector field. You can compare this velocity (field) to any other velocity, in a standard way. (This answer should be treated as preliminary thoughts, and feedback is welcomed.)

Colin MacLaurin
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