Rational Relativistic Collisions

TL;DR

This paper demonstrates that in one-dimensional relativistic collisions with rational initial parameters, the resulting velocities and gamma factors remain rational, revealing a structure related to Pythagorean triples and Lorentzian reflections.

Contribution

It establishes that relativistic collisions preserve rationality of velocities and gamma factors, linking the phenomenon to Pythagorean triples and Lorentzian geometry.

Findings

Outgoing velocities and gamma factors are rational if initial parameters are rational.

Collision results can be modeled as Lorentzian reflections of 2-momenta.

Numerous examples are constructed using Pythagorean triples.

Abstract

If two point particles collide relativistically in one dimension, and the masses, velocities and gamma factors of the incoming particles are rational numbers, then the velocities and gamma factors of the outgoing particles are rational. Numerous examples can be found using Pythagorean triples. At all velocities, the collision results in a Lorentzian reflection of the 2-momenta of the particles.