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This is a question about Mach's principle. Why is Mach's principle still an open question? In particular, if inertia arises from distant matter, Mach's principle raises the question would a particle have inertia in a universe with only that one particle. Why is this still an open question? Do the thoughts below help settle such questions?

This question (and the title) come from How does an isolated body in deep space 'know' it's rotating? The most highly upvoted answer says

This is a longstanding problem in physics and has not been wholly solved to anyone's satisfaction. It's not just rotational motion, any motion is subject to this concern. Very basically, what is "motion" for a singular object in its own universe?

The question talks about Mach's principle, General Relativity, and Brans-Dicke theory.

Asking a similar question about electron spin seems obvious, but I didn't see one anywhere.

As Why does spin have a discrete spectrum? shows, an electron has an intrinsic spin with half integer values. The reason is that the wave function has boundary conditions. The value at $\theta=0$ must be the same as at $\theta = 2\pi$. Other sources say it is just an experimental fact. Either way, and electron has spin and angular momentum.

It seems this must apply to an electron in otherwise empty space. It doesn't seem right to argue that it may not have an angular momentum. Is there something wrong with this thought?

Qmechanic
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mmesser314
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First off, you are assuming that Mach's principle holds. Mach's principle is not something we can test empirically if it does or doesn't hold because we cannot take out all matter from the Universe and leave an isolated spinning object as a point of comparison.

That said, properly tackling the question of an electron's spin's effects on spacetime is something that fundamentally requires a quantum theory of spacetime. Spin is inherently quantum - an intrinsic, so in effect quantized in the most extreme way, to exactly one quantum level, angular momentum and hence any coupling between it and spacetime will mean that spacetime will also have to be treated quantum mechanically as inheriting the influence of that quantization effect. We don't have such a theory yet, so we can't really answer this question. For what it's worth, this doesn't just apply to elementary spin only, but also, say, the analogous question for a rotating molecule's relationship to its surrounding spacetime: such a thing is also quantum, even though its angular momentum is less constrained than that of a rotating elementary particle.

Nonetheless, I'd propose one might be able to draw two broad conclusions about what it might imply for spacetime:

  1. This is actually a counterargument to Mach's principle precisely because electron (and other elementary particles') spin is so heavily constrained. It would need to mean that some large-scale element of the Universe would somehow have to mirror that heavy constraint in some fashion given you can always set up a rotating reference frame at any rotation speed, and it's hard to see how that could work. And so rotation really is a self-contained property, not a relational one.

  2. Mach's principle does hold, and a quantum theory reconciling it and electron spin might of necessity produce potentially empirically examinable predictions of deviations from normal behavior under certain circumstances which depend on whatever imperfections the reconciling mechanism must permit - if not for the electron, then for the Universe as a whole and hence potentially detectable either subatomically or astronomically. This would perhaps be the more interesting option because then it might help get us that quantum theory of spacetime we'd like to have by suggesting doors for future experiments which, right now, are one of THE biggest problems with any further advances in fundamental physics: we have none that have let us on to any real answers.

Nice Q!

The_Sympathizer
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