It is also clear that GR is special because the metric is a dynamical field for it. No theory in which the metric is not dynamical can have this symmetry because the $\sqrt{-g}$ factor from the volume form is always in the action, dynamical or not

@Slereah I suppose you could call it a tetrad rotation, yes.

Would you prefer I call it a linear frame bundle gauge

No, I mean the tetrad is an additional thing, it encodes the reduction of the structure group to the Lorentz group.

Locally it is always true though :p

ncatlab.org/nlab/show/general+covariance

nlab will make everything clearer

"The configuration space is not the naive quotient of fields by diffeomorphisms as above, but is the homotopy quotient, or action groupoid, denoted $[\Sigma,\mathbf{Fields}] // \mathbf{Diff}(\Sigma)$. "

Does this help

They do call it a gauge transformation, though

this should resolve the matter

Ah but of course I was missing the $\mathbf{BAut}$ intuition...

Missed the boat indeed

in some cases the examples of nlab are worse than the definitions

I blame him

Are moduli spaces generalizations of fiber bundles?

Is $P = k \rho ^ \gamma $ valid for non-adiabatic fluids?

@Slereah Not really? "Moduli space" - like "classifying space" - is a malleable concept that is applied to spaces where in some sense the points correspond each to a space or other geometric structure. E.g. in string theory you'll often hear people talk about something like the "moduli space of complex structures", where each point is a different complex structure (or Kähler metric or whatever) on some fixed base space

Or, well, yes, maybe you might call it a generalization of the fiber bundle. But the fibers

The fibers don't consist of points, though, but of spaces/structures

this is also a bit confusing because pure mathematicians tend to use the term differently from physicists

Discussions have been going on in this question way longer than I thought they would haha

Is the set of all circles in the plane about the origin, parameterized by the radius (a moduli space with modulus given by the radius) a fiber bundle

I mean that's just the cylinder?

Which is famously a bundle

Where do the $u_m$'s come from in (8) and (9)

It only seems to makes sense if the $c_m$'s are the $u_m$'s and you use $\phi_m = 0$ but you're not supposed to invoke $\phi_m = 0$ as you need to be able to use the P.B.'s which assumes the p's and q's are independent...

I think those constraint videos used $\phi_m = 0$, hmm

Yep, yikes, that can't be right