Rapidity

Definition

Rapidity $\eta$ is defined by

$$\tanh\eta=\frac{v}{c}.$$

Boosts

Lorentz boosts act like hyperbolic rotations:

$$\begin{pmatrix}ct'\\x'\end{pmatrix} = \begin{pmatrix}\cosh\eta&-\sinh\eta\\-\sinh\eta&\cosh\eta\end{pmatrix} \begin{pmatrix}ct\\x\end{pmatrix}.$$

Rapidities add under successive boosts in the same direction.