In this simulation two points are connected by a black line. This line is straight path between the two points on the Schwarzschild map. The infinitesimal distance between points on the map is dl. But what is dl in terms of our coordinates? On the map dl2 = dx2 + dy2 = dr2 + r2 dφ2, since we are only looking at a 2-d slice of space. What is the proper length, dσ, between these same two points? The proper length can be determined from the metric, and since dt = 0, the proper length is just:
dσ2 = [1 − 2M/r]-1 dr2 + r2 dφ2.
So, in general, the proper length, dσ, will not equal the separation in the map, dl. When does it? The answer is when the factor in front of dr2 is 1. This occurs for two conditions: M = 0 and r >>1. In both cases, the space is said to be flat since there is no effect of gravity.
Explore this flat space with the simulation provided. The simulation allows you to drag the two points to anywhere in the space. Note that dσ = dl everywhere.