Do you have a reference? I'm not familiar with your argument about six arbitrary functions. I can, however, follow that there are 10 independent numbers in a metric. This excludes any considerations of diffeomorphisms, so it's not necessarily in conflict with what you said (though I'm not qutie following what you said). And the tensor transformation rules give us 16 possible linear transformations (a 4x4 matrix) , which gives us more than enough degrees of freedom to transform away all the components of the metric tensor at a single point. Which means that specifying the metric tensor alone at a single point can't tell us anything physical, as we can always find a coordinate system in which the metric tensor is diag(-1,1,1,1).
The argument "with a little imagination" made by Melgrin isn't quite rigorous enough for me to want to defend, though I thought it was interesting.
The conclusion that we can transform the metric to diag(-1,1,1,1) and also make all it's first-order derivatives vanish at a single point also seems reasonably obvious on physical grounds from the existence of Riemann normal coordinates.
Sorry, I'm not following this at all. Do you have any references or can you explain further?
