I have read through the numerous questions on 'degrees of freedom' in the metric tensor, and won't list them all here. However none of them address my question on 'functional' degrees of freedom in the metric tensor, as referred to in "General Relativity, An introduction for physicists" by Hobson et al.
Specifically, I get that requirements on the metric tensor to be symmetric means that there are $N(N+1)/2$ independent metric functions, where $N$ is the manifold dimension.
What I am not sure about is the following statement:
Since there are $N$ arbitrary coordinate transformations, there are really only $(N+1)/2-N=N(N-1)/2$ independent degrees of freedom associated with $g_{ab}(x)$.
I am trying to think about it in the following way:
Given a manifold with a certain geometry, the intrinsic distances between points exist there without reference to a coordinate system. If we then specify some coordinate system, then the metric functions are not all specified. Because if they were, then we could use our $N$ degrees of freedom to transform to another coordinate system, in which all the metric finctions would also need to be specified. But the $N$ degrees of freedom associated with a coordinate change mean we can't access all of the $N(N+1)/2$ degrees of freedom in the metric functions - a contradiction.
For some reason, this argument isn't seeming convincing to me (maybe thinking about it in terms of mappings would clarify). But I think the 'independent degrees of freedom' means that we necessarily (for $N>1$) have degeneracy in the possible metric functions? And this degeneracy is in $N(N-1)/2$ degrees of freedom?
Is this redundancy in the metric function the connection with gauge freedom in electromagnetism that I have been reading about like here? Note that I am just starting to learn about field theory, so the concepts aren't entirely clear yet.
