The story of Ariel, Berta, and Clive. Ariel, Berta, and Clive all pop into existence at the same spacetime point, p0. Thereafter they are all inertial but in motion relative to one another. Relative to Berta: (i) Ariel is moving to the left at a certain speed, and (ii) Clive is moving to the right at that same speed. When Ariel begins to exist, she starts an accurate stopwatch which records the proper time that elapses along her worldline. Spacetime point p1 is the point on Arielâ€™s worldline at which her stopwatch reads â€˜10 hours.â€™ Hyperplane h1 passes through p1 and consists of spacetime points that are simultaneous relative to the inertial reference frame in which Ariel is at rest. Spacetime point p2 is the point at which h1 intersects Bertaâ€™s worldline, and point p3 is the point at which h1 intersects Cliveâ€™s worldline. Hyperplane h3 passes through p3 and consists of points that are all simultaneous relative to the inertial reference frame in which Berta is at rest. Hyperplane h3 intersects Cliveâ€™s worldline at p4 and Arielâ€™s worldline at p5. Relative to the inertial reference frame in which Clive is at rest, Ariel, Berta, and Clive all cease to exist simultaneously.
Assignment: Following our standard conventions for drawing spacetime diagrams, draw two diagrams set in Minkowski spacetime, with all the named points, hyperplanes, and individuals labeled on all four diagrams. Use graph paper.
The first diagram should be associated with the inertial reference frame in which Berta is at rest. The second diagram should be associated with the inertial reference frame in which Ariel is at rest.
Directions for drawing diagrams: When drawing Minkowski diagrams, be sure that lines representing the trajectories of all light rays intersect horizontal lines across the page at a 45 degree angle. Draw these lines in yellow, by hand [do as I say on this point, not as I do].
â€˜Scissors Ruleâ€™: When you are drawing hyperplanes in Minkowski spacetime, draw them as dashed lines, so that if s is a hyperplane associated with the rest frame of an inertial trajectory o, the trajectory of light rays emitted (or that would be emitted) at their intersection bisect the angle of their intersection. There is an example in the file attached.