# Can we extract all the physics from the line element only? (i.e.) without any knowledge of coordinate system?

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 at 11:01