Consider a coordinate-free metric tensor $g$ which can take different coordinate forms, say $g_{ij}(x)$ and $g_{i'j'}(x')$. Also consider a metric tensor $h$ which is related to $g$ by an active transformation.
Questions (which really boil down to one):
Are the possible coordinate forms of $h$ the same as those of $g$? My understanding of all GR textbooks is that there is a duality between passive and active transformations and so the answer must be 'yes'.
Should $g_{ij}(x)$ and $g_{i'j'}(x')$ be considered different representations of the same object? Universally I have seen the answer 'yes'.
Should we consider $g$ and $h$ as fundamentally different metric tensors? I have always seen that the answer is 'yes'.
Does this mean that the set of coordinate representations of $g$ also represent $h$ and so the answer to 2 or 3 above should be 'no'?
This post seems to focus on whether or not we consider the metric tensor as a dynamical field. That is a physics point whereas I am asking a differential geometry point. Misner, Thorne and Wheeler don't mention actives until near the end. Wald and Carroll focus on the duality. Norton makes a philosophical distinction referring to the hole argument. EDIT: I had misunderstood Gaul and Rovelli's argument in that they are making a point about metrics in relation to metric tensors, pointed out by user12262.
