K12 STR and Penrose Diagrams
There are many other possibilities to graphically represent relativistic relationships. The two British mathematicians B. Carter and R. Penrose have a model in which the infinitely extended Minkowski plane is projected onto a square in which light beams continue to be straight when they run parallel to one of the two angle bisectors ?. Lines of constant time or constant location become hyperbolas. In the center of the picture one sees the undistorted current local event, while distant events are compressed together:
This figure is taken from the free encyclopedia Wikipedia. On the website of Franz Embacher http://homepage.univie.ac.at/Franz.Embacher/Rel/ you can find a Java applet, which allows you to play with this coordinate transformation. If you play with it seriously and use it to solve the problems that are suggested, then you gain a good understanding of what this model offers.
The arc tangent is used since it is a monotonically increasing function; approximates the identity function in the vicinity of the origin (i.e., f (x) ≈ x); and has a finite limit as x approaches ∞. Thus the corners of the square lie ± π / 2 from the origin. The mapping of the Minkowski plane into the Penrose-world is defined by the following equations:
x’ + t’ = ArcTan (x + t) and x’ – t’ = ArcTan (x – t)
Addition (and subtraction) of these equations immediately gives the transformation equations for x’ and t’ respectively. It is easy to show that light beams that go out from the origin in the Penrose diagram are angle bisectors. Show that all the light beams run parallel to these!
Also: Find a model yourself, which has similar properties! Is the arc tangent really better than your model?