% miko:type: atomic
% miko:title: Rapidity
% miko:summary: A hyperbolic angle parameter that makes relativistic velocity addition simple.
% miko:tags: relativity
% miko:difficulty: undergrad
% miko:related: lorentz-transformation, minkowski-interval, proper-time
% miko:keywords: boost, hyperbolic rotation, velocity addition

\documentclass{article}
\usepackage{amsmath}
\title{Rapidity}
\begin{document}
\maketitle

\section{Definition}

Rapidity $\eta$ is defined by

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

\section{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.

\end{document}
