The velocity addition rule is . Move at on a platform that is itself moving at , and the speed seen from the ground is .
Do it once more and you get , then . The same is being stacked, yet each step needs another pass through the denominator.
With rotation in a plane, degrees followed by degrees is degrees: the angles simply add. We look for a quantity that adds in this way, to use in place of speed.
Using , the with is called the rapidity. Solving the other way, .
For , . The point in the figure stops at , , and , where grows through , , and .
never exceeds , but has no upper limit. However large becomes, stays below .
obeys this addition formula:
The right side is exactly the addition rule for and . What needs a denominator when written in is plain addition when written in .
Stacking means adding over and over. Add it , and times and take , and you get , and , the values found at the start. Even after times it is , still below .
Using , . Further, .
The point in the figure stops at , , and . At , which corresponds to , it reads and .
The Lorentz transformation can be written and . The two coefficients and are set by the single .
A Lorentz transformation into a frame moving at speed is called a boost. Just as a rotation leaves unchanged, a boost leaves unchanged.
The and of a rotation become and in a boost, and angles add by the same rule. The one difference is the sign that turns into .
Under a rotation a point moves around a circle; under a boost it moves along a hyperbola. This is why, on a spacetime diagram, the two axes of a moving frame approach the degree line equally.