Thanks for finding that source Atyy. I've been busy with other stuff and haven't looked at it.There is a clear write up in Wuthrich's thesis, p49. http://philosophy.ucsd.edu/faculty/wuthrich/pub/WuthrichChristianPhD2006Final.pdf [Broken]
I did find confirmation in Sean Carroll's thing of what you see in Rovelli. Rovelli's certainly not presenting anything unusual.
In Carroll "Intro to GR" (which seems pitched to undergraduates, style-wise) on page 138 it says:
Let’s put some of these ideas into the context of general relativity. You will often hear it proclaimed that GR is a “diﬀeomorphism invariant” theory. What this means is that, if the universe is represented by a manifold M with metric gµν and matter ﬁelds ψ, and φ : M → M is a diﬀeomorphism, then the sets (M, gµν , ψ) and
(M, φ* gµν , φ* ψ) represent the same physical situation.
That is, you can totally moosh and morph and it is still the same physical reality. You aren't limited to an "isometric" diffeo.
In other words physical realities correspond to equivalence classes under diffeomorphism of metrics. They don't correspond to single metrics but to diffeomorphism classes of metrics. All those that you can make by morphing the one you started with.
So there is no rubber sheet. The equiv class carries the more abstract idea of a geometry-without-an-underlying-manifold.
Note that Carroll gives the mathematical truth, but then reassures his undergrads with some condescending pablum which basically says "don't worry your little heads about this". He gives the impression that diffeo is "just a highbrow change of coordinates" the basic message is "I gave you the equation, we aren't going to use it, so no need to think much about it."
Carroll is carrying on the "half century of confusion" that MTW complained about---diffeomorphism invariance going incognito, anonymous (as MTW put it) under the mask of "just like a change of coordinates".
Carroll is a master of comfortable communication---gift of the gab. The important thing is he gives the equation and says the two represent the same physical situation. His spin after that can be ignored.
Above just my humble view of course
“Dadurch verlieren Zeit & Raum den letzter Rest von physikalischer Realität.”
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