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at principle of equivalence I have restructured the Examples-section: added new subsections in “In physics” on gauge transformations and on general covariance (just pointers so far, no text), and then I moved the section that used to be called “In quantum mechanics” to “Examples-In category theory” and renamed it to “In the definition of $\dagger$-categories” (for that is really what these paragraphs discuss, not any notion of equivalence in quantum mechanics, the application of $\dagger$-categories in that context notwithstanding)
And I move the “How to break equivalence-invariance” to the end. At least it should go after the “General definition”, I’d say.
So general covariance is an example of the (mathematical) principle of equivalence. Remind me, how should I think of the (physical) principle of equivalence? I think somewhere you just said it’s a result of using Riemannian spaces. So the physical principle of equivalence is part of a precondition for the mathematical principle of equivalence to have as special case general covariance?
Remind me, how should I think of the (physical) principle of equivalence?
I am claiming that the mathematical formalization of the “principle ofequivalence” in the theory of gravity is (or was, in the time when people actually referred to it):
the infinitesimal neighbourhood of every point in spacetime is a vector space (the tangent space);
the Levi-Civita connection can be taken (up to its gauge freedom) to vanish on that neighbourhood (not it’s derivative, though, hence not the curvature). This is the existence of “Riemann normal coordinates”.
In conclusion: in first order approximation around any point, gravitational dynamics looks like dynamics in flat space.
So that “principle of equivalence” is not so very close to the mathematical “principle of equivalence” as we have on that page. What we have on the page principle of equivalence however directly subsumes what in gravity is the principle of general covariance.
But of course the relation is via those Riemann normal coordinates (or can be understood that way): the principle of general covariance says that if you see a field of gravity $g$ on spacetime $\Sigma$, you may regard it as being equivalent to the field of gravity $\tilde g \coloneqq f^* g$ for any diffeomorphism $f : \Sigma \to \Sigma$. The statement of Riemann normal coordinates is that you can always find such a diffeomorphism such that at a given point $f^* g$ induces vanishing Levi-Civita connection at that point, hence proving that up to gauge equivalence the dynamics in the infinitesimal neighbourhood of that point is that of flat space.
That’s how these things relate. Schematically we have implications
principle of equivalence in mathematics $\Rightarrow$ principle of general covariance $\Rightarrow$ principle of equivalence in physics
I turned this into a quick note here. But need to quit now.
One day we should probably extract some text from that discussion at In the concept of †-categories. But for now, I added a link to this old MO discussion – Are dagger categories truly evil?.
Added a reference
Touched the lead-in sentences, adding more hyperlinks (such as to higher structures).
Removed two more pointless appearances of the word “evil”.
I wonder if that long discussion on dagger categories could be condensed and extracted now with the benefit of hindsight.
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