In this simulation two points are again connected by a black line. Again the proper length can be determined from the metric, which on the two-dimensional slice of space yields just:
dσ2 = [1 − 2M/r]-1 dr2 + r2 dφ2.
Explore this curved space around a black hole with the simulation provided. The simulation allows you to drag the two beacons to anywhere in the space. Note that dσ is not equal to dl everywhere and in fact only in special circumstances will they approximately be the same. In addition, in order to find the proper length between two points along the same radial direction, you must integrate dσ = dr/(1− 2M/r)1/2 from initial to final r-coordinate.