A systematic search for a three-velocity gyrodistributive law in special relativity with the lorentz R package

Abstract

Here I present the lorentz package for working with relativistic physics. The package includes functionality for four-vector transformations, three-velocity addition, and other relativistic processes such as the behaviour of photons. It was designed to facilitate the search for a gyrodistributive law. In special relativity, three-velocities and scalars constitute a gyrovector space with addition $\oplus$ and scalar multiplication $\odot$. Standard vector spaces obey the distributive law $a(x + y) = ax + ay$ for scalar $a$ and vectors $x$, $y$; but no analogous gyrodistributive law for $r\odot (u \oplus v)$ is known. The package was designed to facilitate the search for a gyrodistributive law and includes functionality for four-vector transformations and three-velocity addition, which is noncommutative and nonassociative. I use the package to systematically sweep a large space of potential gyrodistributive laws, without success. The package is available on CRAN, at \url{https://CRAN.R-project.org/package=lorentz}.