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

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?

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