High School Comoving distance in cosmology

[Apashanka]

For calculating the proper distance in cosmology why is the proper time between two points (galaxy) dΓ is taken 0??
e.g dΓ²=dt²-a(t)²dr²
Taking dΓ=0 and ∫dr=∫dt/a(t)

 

[Orodruin]

For calculating the proper distance in cosmology why is the proper time between two points (galaxy) dΓ is taken 0??
e.g dΓ²=dt²-a(t)²dr²
Taking dΓ=0 and ∫dr=∫dt/a(t)

It is not. What you have there is an expression for the comoving distance.

In the particular case you are referring to, it is part of a derivation of cosmological redshift based on expressing the comoving distance traveled by two consecutive signals in terms of the scale factor history and using that to find the time difference at emission to be compared to the time difference at reception.