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I was reading "Natural Coordinate System in Curved Spacetime" by Ying-Qui Gu. The first sentence of the abstract struck me (bold emphasis added by me):

In this paper we establish a generally and globally valid coordinate system in curved space-time with the simultaneous hypersurface orthogonal to the time coordinate.

Is there a generally accepted set of criteria to determine if a coordinate system is valid? Are some coordinate systems invalid?

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    $\begingroup$ The "valid" should be read together with "globally". It's a coordinate system that is globally valid, in contrast to one that is only locally valid. But I think this question belong at physics.SE $\endgroup$
    – pela
    Dec 23, 2021 at 19:49

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Some desirable properties are that the coordinates of events are uniquely specified; that the coordinate system doesn't lead to singularities in the metric; and that the coordinates adequately span all the spacetime region of interest.

It is often not possible to satisfy all these conditions, which is why a variety of coordinate choices are often made.

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