A diagram with position across and up the page is called a spacetime diagram. Plot where a body is at each moment and the points join up into a line.
This line is called a world line. A body at rest keeps the same position, so its world line runs straight up.
The horizontal line in the figure is the present moment, moving up from the bottom. Carry the points where it crosses the world lines down to the strip below, and you get the motion as we ordinarily see it.
A body moving to the right at speed covers in time . For each unit increase of on the vertical axis, increases by , so the world line tilts to the right away from the vertical by the ratio .
Light travels along , so its world line is a line at exactly degrees. Using rather than for the vertical axis is what gives light this direction.
Since never exceeds , no body's world line leans past the degree line. The world line in the figure tilts through , , , and in turn, approaching the degree line without reaching it.
The axes of a frame moving at speed can be drawn on the same diagram. Its time axis is the world line of a body at rest in that frame, tilted to the right away from the vertical by the ratio .
Its position axis is the line of simultaneity along which that frame's simultaneous events lie. By the relativity of simultaneity, it is a line of slope against the horizontal axis.
The two axes do not rotate rigidly at right angles; both move toward the degree line by the same amount. The closer is to , the closer the two come to the degree line from either side.
The ticks on a tilted axis are not set by lengths on the paper. Take one light second as the distance light travels in one second. The event at which a moving clock has ticked one second is, in the ground frame, at time seconds and position light seconds.
There , the same value as at the point where a clock at rest has ticked one second.
For a frame of any speed, the one second tick lies on the hyperbola . On the paper, one tick on a tilted axis looks longer.
In an ordinary rotation the ticks would lie on the circle . A hyperbola takes the place of the circle on a spacetime diagram because and are subtracted rather than added.
To read the event in the figure in the ground frame, draw lines from it parallel to the two axes. The axes are at right angles, so the lines go straight down and straight to the left, and on the figure's scale the event is at .
Reading it in the moving frame, draw lines parallel to each of the two tilted axes. On the scale of the axes, it is at .
Written as formulas, this change of reading is and , called the Lorentz transformation. With , .
In both sets, is the same: and . Changing the axes you read along does not change this value.