References: Griffiths, David J. (2007), Introduction to Electrodynamics, 3rd Edition; Pearson Education – Chapter 12, Post 23.
As an example of a spacetime diagram, suppose we have our usual two inertial frames with at rest relative to observer and moving at speed in the direction, with the origins of the two systems coinciding as usual.
We’d like to plot the lines of constant and on a spacetime diagram. Using Lorentz transformations we have
For various values of and , these two equations give two sets of parallel lines. The first equation gives lines with slope and -intercepts of , while the second equation gives lines with slope and -intercepts of . We can plot a few lines from each set as shown:
The red lines are lines of constant with the top line corresponding to and the bottom line to , in steps of 1. The green lines are lines of constant with the bottom line corresponding to and the top line to , again in steps of 1.
The thick blue line represents the world line of an object that starts at and moves to . We can find its velocity in by taking its slope on the graph. Finding the exact values of and is difficult by eyeballing a graph, so we can ‘cheat’ a bit and use the Lorentz transformations to find the corresponding values. We get for the starting point:
and for the end point:
The velocity is then
We can check this using the velocity addition formula. Its velocity in is
so it checks out.