Apologies if this question is too naive, but it strikes at the heart of something that's been bothering me for a while.
Under a diffeomorphism $\phi$ we can push forward an arbitrary tensor field $F$ to $\phi_{*}F$. Is the following statement correct?
If $p$ is a point of the manifold then $F$ at $p$ is equal to $\phi_* F$ at $\phi(p)$, since they are related by the tensor transformation law, and tensors are independent of coordinate choice. ()
I have a feeling that I'm missing something crucial here, because this would seem to suggest that diffeomorphisms were isometries in general (which I know is false). (*)
However if the statement isn't true then it menas that physical observables like the electromagnetic tensor $F^{\mu \nu}$ wouldn't be invariant under diffeomorphisms (which they must be because diffeomorphisms are a gauge symmetry of our theory). In fact the proper time $\tau$ won't even be invariant unless we have an isometry!
What am I missing here? Surely it's isometries and not diffeomorphisms that are the gauge symmetries?! Many thanks in advance.