Instant Epstein diagrams

Max Born
In his book "Relativity Visualized", Lewis Carroll Epstein introduced a kind of spacetime diagram called a Space-Proper-time diagram. For a quick grasp of special relativity theory, there is none better. It actually makes it possible to use Pythagoras' theorem for triangles directly in many special relativity problems. The circle has a radius of cT, i.e. the speed of light multiplied by the time read on a clock that moves radially from the center. The black and red arrows from the origin are rotated relative to each other by an angle determined by the relative speed between two inertial observers. In one second of black coordinate time, the black observer has moved 1 light-second straight up, just like in any spacetime diagram. In the same time, the red observer has moved 0.6 light-seconds to the right and 0.8 light-seconds up, so the red unit vector points to 0.6,0.8 on the grid. Using Pythagoras, we know that 0.6^2 +0.8^2 = 1^2, so what we wanted to happen actually makes sense: add a space displacement vector to a time displacement vector and you get an invariant spacetime vector. What is more, it can be shown that if we take the origin and the two arrow-head positions each as a separate event in spacetime, the values conform to the Lorentz transformations (LTs) for converting space and time intervals observed in one inertial frame to another inertial frame. If the red frame's clock reads 1s when reaching the position (0.6, 0.8), a clock synchronized in the blue frame will observe the same event at 0.8 seconds. Just look at: