Lorentz invariant relative velocity and relativistic binary collisions

TL;DR

This paper clarifies the concept of Lorentz invariant relative velocity, enabling accurate formulation of flux and cross section in relativistic collisions, with applications in collider luminosity and statistical collision theory.

Contribution

It provides a comprehensive review of Lorentz invariant relative velocity and its applications, clarifying misconceptions and emphasizing hyperbolic velocity space properties.

Findings

Clarification of Lorentz invariant relative velocity properties

Formulation of invariant flux and cross section without non-physical velocities

Application to collider luminosity and relativistic gas collisions

Abstract

This article reviews the concept of Lorentz invariant relative velocity that is often misunderstood or unknown in high energy physics literature. The properties of the relative velocity allow to formulate the invariant flux and cross section without recurring to non--physical velocities or any assumption about the reference frame. Applications such as the luminosity of a collider, the use as kinematic variable, and the statistical theory of collisions in a relativistic classical gas are reviewed. It is emphasized how the hyperbolic properties of the velocity space explain the peculiarities of relativistic scattering.