smallphi
- 436
- 2
The so called 'physical' distance in cosmology doesn't have the status of invariance (independence of coordinate system) like the line element ds^2 because the 'physical' distance is a coordinate quantity.
It can be measured if you have a gazillion of comoving observers in straight line from you to the point where you want to measure. Those observers have clocks that all show the 'cosmic' time which is a coordinate time in FRW metric. You simply tell them at given fixed cosmic time to record the distance to the closest comoving observer. Assuming that distance is small it will be the line element ds. Then you tell them to report the distances to you and you add them up getting the coordinate distance to the object at that time. Basically this is integration of ds over a specified curve which as pervect pointed out is NOT a geodesic.
It can be measured if you have a gazillion of comoving observers in straight line from you to the point where you want to measure. Those observers have clocks that all show the 'cosmic' time which is a coordinate time in FRW metric. You simply tell them at given fixed cosmic time to record the distance to the closest comoving observer. Assuming that distance is small it will be the line element ds. Then you tell them to report the distances to you and you add them up getting the coordinate distance to the object at that time. Basically this is integration of ds over a specified curve which as pervect pointed out is NOT a geodesic.