Why the parsec (pc) is still a good cosmological definition in the curved spacetime in a large scale of universe?

Question

The parsec (pc) is a unit of length used to measure the large distances to astronomical objects outside the Solar System, approximately equal to 3.26 light-years or 206,000 astronomical units (au), i.e. 30.9 trillion kilometres (19.2 trillion miles).

The definition can be understood from this figure below.

question

However, because the universe should be a curved spacetime instead of a flat spacetime, why the use of the 1 arc second measurement: $$\text{1 arc second = 1 AU / 1 pc}$$ is still a good cosmological definition? Should we include the possible global property of the curvature in the universe into account?

I mean why not taking into account of hyperbolic or differential geometry in the definition to make the "distance unit" has a curvature dependence?

Answer 1

Space is not curved. Space is flat, within observational measurements. Spacetime is curved. Big difference. There's some subtlety here, and I'm not the best expert to tell you about it in depth, but one example of the difference: If you measure the distance between two points far apart in space:

$r^2=x^2+y^2+z^2$ will hold true, for a given Cartesian Coordinate system you impose, and the coordinate distances are $(x,y,z)$

Now, for two events far apart in spacetime (say, a distant supernova emitting light, and that light reaching Earth)

$s^2=-(ct)^2+x^2+y^2+z^2$

will NOT hold true (as it would on a flat spacetime). $(x,y,z)$ are the coordinate distances as before, and $t$ is the time interval between the events. [all these measurements according to some single observer].

So for measuring distances in space, at least globally in general, the Cartesian rules of geometry are the correct tools