- #36

PeterDonis

Mentor

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because it has the same functional form as ##g_{ab} (x)## but belongs to a different coordinate system, it imposes a different spacetime geometry!

I don't see how this follows, because you have not said how the ##x## and ##y## coordinates are actually different. Changing from ##x## to ##y## is just changing a label. But that means that you cannot assume that the two metrics ##g_{ab}(x)## and ##g_{ab}(y)## are actually different geometries. It might turn out that the ##x## and ##y## labels actually label exactly the same points in exactly the same geometry--you just didn't realize it because you started out using two different labels.