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- How do we formally define "Proper time is the time measured by a clock travelling along the worldlike"?

In texts on General Relativity, the proper time ##d\tau^2 = -ds^2## (with an appropriate choice of metric signature) is commonly said that the time measured by a timelike observer traveling along a path is given by the integral of ##d\tau## along this path. Of course it's possible to construct a coordinate system (given some niceness assumptions about the path, presumably) where all the spatial coordinates are 0, and the timelike coordinate is the proper time. However, it seems to me that it would be equally easy to construct a coordinate system where the spatial coordinates are the same, and the time coordinate is halved, so in this system the coordinate time along the path would be half the proper time.

I'm wondering if there's a way to "single out" a coordinate system as corresponding to what an observer traveling along the path measures on their clock, or whether it's a postulate of GR that the proper time is what a clock traveling along the path would say.

I'm wondering if there's a way to "single out" a coordinate system as corresponding to what an observer traveling along the path measures on their clock, or whether it's a postulate of GR that the proper time is what a clock traveling along the path would say.

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