In this set of simulations points will be added to the Schwarzschild map and the connection between said points will be either a straight line on the map or 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.
1. Load the Simulation Ex. 1. Note that one point in the line segment is fixed in length on the map at 5. Initially the segment is from r = 5 to r = 10, but it can be moved. 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 this result, calculate the proper distance between the points on the map r = 5 and r = 10 when φ = 0. The simulation does this for you, but in order to do the next exercise you should calculate (approximately) the proper distance yourself and compare to the result given. NOTE: the infinitesimal proper distance changes as a function of r so you must made a reasonable estimate for r in the above equation. Once you have calculated the proper distance, check to make sure your result agrees with the simulation. Now drag the near point to as close to the black hole as possible (here 2.001) when again φ = 0. What is the proper length now?
2. Load the Simulation Ex. 2. Compare the simulation's proper length between the points on the map r = 5 and r = 10 when φ = 0 to that of the known proper length for M = 1. Use this information to determine whether M > 1 or M < 1. From the proper length, determine the mass of the central object.
3. Load the Simulation Ex. 3. This simulation connects the two points by a null geodesic: the curve light would follow between the points. What is the proper distance between the points on the map r = 5 and r = 10 when φ = 0? Now drag the far point to 15 when φ = 0 and determine the proper distance. Now drag the near point to as close to the black hole as possible (here 2.001) when again φ = 0. What is the proper length this time? How do your results for the null geodesic agree/disagree with the result from Exercise 1?
4. Load the Simulation Ex. 4. In this simulation two points are connected by a black and a blue curve. This line represents the straight path between the two points as measured on the map, while the blue curve represents the straight path in space (a straight line path that light takes, a null geodesic) between the two points as measured on the map. Explore this curved space with the simulation provided. Describe the differences between the straight line on the map and the null geodesic as shown on the map as well as their proper lengths. How do your results for the null geodesic agree/disagree with the result from Exercise 3?
5. Why is the path light takes called a null geodesic?