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$$c^2 d\tau^2 = g_{00}(dx^{0})^2.$$

He then goes on to define the spatial proper distance ##dl^2## and notes that in SR ##dl^2## can be defined as the spatial distance between two events infinitesimally separated, but with the same coordinate time, i.e. by setting ##dx^0 = 0## in the line-element. He claims that this is not possible in GR because the proper time in a gravitational field may have a different dependence on ##x^0## at different points in space.

He then goes on to define ##dl^2## by sending and reflecting back a light ray from/to a stationary observer.

I remember naively doing exactly this when I started learning about GR, but getting the wrong answer. I never really understood why it was not allowed and I'm not sure if I understand what the author is saying here, so I wondered if someone could elaborate on why we can not define the proper distance in this way as one did in SR.

PS: I know it's become common to completely ignore the notion of a line-element and just work

with tensors, i.e. the metric tensor. But I would like to understand it in both ways.