mathematical physics - Diffeomorphisms, Isometries And General Relativity

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.

Answer

If p is a point of the manifold then F at p is equal to ϕ∗F at ϕ(p), since they are related by the tensor transformation law, and tensors are independent of coordinate choice.

This is roughly true. Initially, there is no meaning when one says that tensors at different tangent spaces are equal. However, the diffeomorphism induces an isomorphism between $T_p M$ and $T_{\phi (p)} M$ (the isomorphism is nothing but the vector push forward). The two tensors are equal with respect to this isomorphism.

I have a feeling that I'm missing something crucial here, because this would seem to suggest that diffeomorphisms were isometries in general...

This is actually true in a sense that is relevant. If $(M,g)$ is a spacetime and $\phi \in \text{diff}(M)$, then while there is no reason to think that $\phi$ is an isometry between $(M,g)$ and itself, $\phi$ is always an isometry between $(M,g)$ and $(M,\phi_{\star}g)$.

This last point saves your concern about proper time. If $\gamma$ is a normalized timelike path between two events $a$ and $b$, we can always consider $\phi \circ \gamma$ as a timelike path in $(M,\phi_{\star}g)$. You can check that the new path is normalized with respect to the new metric $\phi_{\star}g$. The domains of the two paths are exactly the same so the proper time between $\phi(a)$ and $\phi(b)$ is the same as the original path's proper time.