What is the physics behind GR's diffeomorphism invariance?

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The discussion centers on the physical motivation for diffeomorphism invariance in General Relativity (GR), emphasizing that while special relativity is limited to the Poincaré group in flat spacetime, GR requires invariance under general diffeomorphisms due to the curvature of spacetime. This invariance reflects the principle that physical laws should be consistent across all observers, regardless of their coordinate systems. The conversation highlights that general covariance alone does not link the metric tensor to gravity; this connection arises through the equivalence principle, which posits that freely falling frames are inertial. The importance of making the metric tensor dynamical is underscored, as it adds substantive content to diffeomorphism invariance. Ultimately, the discussion illustrates the complexity of relating coordinate independence to the fundamental nature of gravity in GR.
  • #31
stevendaryl said:
The third is subtler, and I don't know whether it is equivalent to the second, or not. It's something like "the indistinguishabilty of points".

I don't understand this well either, but here are some thoughts. It's usually said here that the hole argument says that observables must be relational in gravity. I don't understand this because I think observables must also be relational in special relativity, ie. relative to an inertial frame.

However, what is known is that in pure gravity, there are no gauge invariant local observables. Classically, this problem can be solved if matter is introduced, so that relational aspects of the matter distribution create local observables. There's a discussion of this around Eq 1.1 of http://arxiv.org/abs/gr-qc/9404053.

The quantum situation seems trickier, eg. http://arxiv.org/abs/hep-th/0106109, http://arxiv.org/abs/hep-th/0512200.
 

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