Frank Castle
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Geometry_dude said:If you want to stick to a coordinate point of view, isometries are those non-trivial 'coordinate transformations' that don't change the appearance of the gμνg_{\mu \nu}, while more general 'diffeomorphisms' might look like they change these functions, but actually it's just a reformulation of the theory. I used the parantheses to indicate that the statement is not really mathematically correct, but I though it might help you to get on the right track.
In what sense is the statement mathematically incorrect?
To be honest, I think the set of notes that I read (http://web.mit.edu/edbert/GR/gr5.pdf) really threw a spanner in the works for me. It has confused me about the difference between a diffeomorphism and a general coordinate transformation. Are they mathematically and physically equivalent (just philosophically different)? Is the Einstein-Hilbert automatically diffeomorphism invariant by virtue of the fact that it is invariant under general coordinate transformations?