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"Don't panic!"

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My question, given the above, is how it is possible to reconcile this when one considers applying these ideas to the surface of the Earth?

Maybe I have misunderstood things, regardless I'm feeling a bit confused. I realize that one can always approximate to Cartesian coordinates within a sufficiently small neighbourhood of a point, but shouldn't this be exact (rather than an approximation) in order for the equivalence principle to work?!