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We can't use Euclidean coordinates (Cartesian, spherical or polar) to label the points in curved spacetime except for the local points in local inertial frame. So for a general solution of Einstein's equation, how can we construct a coordinate system to label the points?

Is the labeling is important? (i.e.) Can we extract all the physics from the line element only?

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Not quite sure I understand your point: To take an example: suppose you have a solution which is non flat, but nevertheless spherically symmetric, then you can use spherical polar style coordinates $r, \theta, \phi$, but the metric components, expressed in these coordinates, won't be the usual flat ones. –  twistor59 Jul 13 '13 at 11:01
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