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Can someone explain what is and why there is diffeomorphism invariance (I knoe that it's a GR question but it is so central to the whole loop quantum gravity approach that I thought this would be a good place to ask)?

Diffeomorphism invariance is an invariance under a mapping (obeying certain conditions) of the points of the manifold into different points on the manifold (as ooposed to a simple relabelling of the points in which case there is no real change of the manifold). This sounds crazy at first. I mean, if there an observer falling into a black hole, let's say, clearly the physics is changed is the spcetime point of the observer is suddenly mapped to another point very far away from the black hole. The curvature near the observer can't stay the same (since the metric would no longer be a Schwarschild materic). So what is going on?

Thanks in advance