In this set of simulations points will be added to the Schwarzschild map and two different connections between the points will be given: a straight line on the map and a straight line in space (which is a null geodesic: the path light would travel). In addition, the proper length, dσ, between the points will be shown so that you can compare the two quantities. Points partway on the straight path on the map are moveable.
Load the Simulation Ex. 1a. Note that the straight line on the map and the straight line in space are fixed on the map at r = 5 and r = 15, and in this case are on top of each other. We know that for infinitesimal distances, dσ2 = [1 − 2M/r]-1 dr2 + r2 dφ2, which reduces to dσ2 = [1 − 2M/r]-1 dr2 when dφ = 0. For this simulation M = 1. From the simulation, the proper distance between the points on the map r = 5 and r = 15 when φ = 0 is 11.36.
1. Initially the second point that forms the straight line is at r = 10, but it can be moved. Move the draggable point left and right. Does this change the proper length? How? Does it increase, decrease, or stay the same.
2. Now, reset the simulation. Drag the mid point of the line up and down. Does this change the proper length? How? Does it increase, decrease, or stay the same. Which length is smaller, the one from the null geodesic or the new one with the displaced midpoint?
3. Load the Simulation Ex. 1b. Drag all of the way points along the path up and down. Does this change the proper length? How? Does it increase, decrease, or stay the same. Which length is smaller, the one from the null geodesic or the new one with the displaced waypoints?
Load the Simulation Ex. 2a. Note that the straight line on the map and the straight line in space are fixed on the map at (x,y) = (+/-10, 5) and in this case are not on top of each other.
4. Which proper length is shorter?
5. Load the Simulation Ex. 2b. Initially the second point that forms the straight line is at x = 0, but it can be moved. Move the draggable point left and right. Does this change the proper length? How? Does it increase, decrease, or stay the same.
6. Now, reset the simulation. Drag the mid point of the line up and down. Does this change the proper length? How? Does it increase, decrease, or stay the same. Which length is smaller, the one from the null geodesic or the new one with the displaced midpoint?
7. Load the Simulation Ex. 2b. Drag all of the way points along the path up and down. Does this change the proper length? How? Does it increase, decrease, or stay the same. Which length is smaller, the one from the null geodesic or the new one with the displaced waypoints? Can you minimize the draggable straight line paths? If so, for what orientation is the new path minimized?