What's the definition of distance in curved space-time in general relativity?

While I'm learning general relativity, the definition of the distance really confuses me. For example, we observe the distance between the Earth and the Sun (usually by a transit of Venus), what does the distance mean? When we say the PSR1913-16's semi-major axis is 1,950,100 km, what does it mean?

In Schwarzschild space-time, we can be a stationary observer because the space-time is stationary. Then we have a clear definition of the simultaneity surface, so we can define the distance as the proper length in the simultaneity surface between two objects. (However,I'm not sure whether we can do so if we are not a stationary observer). It is the same as we do in flat space-time.

However, in general, the space-time isn't always stationary. We may even NOT have a time-orthogonal coordinate system and NOT have any Killing vector fields. So we may have NEITHER special coordinate systems NOR special vector field. We cannot give a suitable and unique definition of the simultaneity surface and the distance.

We have 3+1 formalism, but when the certain observer is given, it also gives arbitrary simultaneity surfaces so we have arbitrary definitions of the distance. I know little about this, so I'm not sure.

In short, I just want to know:
1. Given a certain observer(such as human in the earth),is there a suitable and unique definition of the distance in curved space-time in general relativity? If the answer is yes,what is it?
2. If the answer of the first question is no, what does the PSR1913-16's semi-major axis mean? What does the distance between the Earth and the Sun mean?

Thanks for your help.

Answer

As far as I know, the answer to the first question is: No, there is no unique definition of a spacial distance in a curved space-time. Already in simple cases like the Friedmann metric there are several definitions of distance: https://en.wikipedia.org/wiki/Distance_measures_(cosmology) However, for small distances and small curvatures (which applies in the case of the Earth and Sun example) all the definitions give approximately the same result (the Minkowski one). As soon as the curvature or the distance (e.g. points next to a black hole or points far apart in a Friedmann universe respectively) the deviations of the different definitions become large.