I read on the Wikipedia page for proper length that proper distance is a quantity that is the same for all observers. I am not sure what it means exactly.
I learned from this answer that proper distance between two circumferential radius $r_1$ and $r_2$ in Schwarzschild spacetime is calculated using
$$\Delta R= \int_{r_2}^{r_1}\frac{1}{\gamma\sqrt{1-\frac{r_s}{r}}} dr.$$
(I would like to know how to derive this formula.)
Here the quantity $\gamma$ depends on how fast a distant Schwarzschild observer moves towards a black hole. For an ideal observer (Schwarzschild observer who is at rest in radial infinity), $\gamma=1$.
It seems to me that Schwarzschild observers travelling at different velocities will not agree on proper distance $\Delta R$?
Did I misunderstand anything?
