Schwarzschild proper distance
This video looks at the Schwarzschild space-time interval and goes through the process for determining various proper distances between two given events. It shows how proper time and coordinate time are related in a Schwarzschild space-time. It also shows that coordinates are not the same as physical measurements and so coordinates do not have direct metrical significance.
NB. Where differentials appear they have to be integrated to produce actual physical values. For example, the proper length is s = Integral[g_11 dr] between r_1 and r_2.