gyrogroup: The Lorentz Transform in Relativistic Physics

title: 'Special relativity in R: the lorentz package ' authors: - affiliation: 1 name: Robin K. S. Hankin orcid: 0000-0001-5982-0415 date: "September 05, 2019" output: pdf_document bibliography: ref.bib tags: - special relativity - Lorentz transform - Wigner rotation - Nonassociative operation - gyrogroup - Thomas precession affiliations: - index: 1 name: Auckland University of Technology

Introduction: the Lorentz transform in special relativity

In special relativity, the Lorentz transforms supercede their classical equivalent, the Galilean transforms [@goldstein1980]. Lorentz transforms operate on four-vectors such as the four-velocity or four-potential and are usually operationalised as multiplication by a $4\times 4$ matrix. A Lorentz transform takes the components of an arbitrary four-vector as observed in one coordinate system and returns the components observed in another system which is moving at constant velocity with respect to the first.

Previous related work

There are a few existing software tools for working with Lorentz transforms, mostly developed in an educational context. Early work would include that of Horwitz et al. [-@horwitz1992], who describe relLab, a system for building a range of gendanken experiments in an interactive graphical environment. The author asserts that it runs on "any Macintosh computer with one megabyte of RAM or more" but it is not clear whether the software is still available. More modern contributions would include the OpenRelativity toolkit [@sherin2016] which simulates the effects of special relativity in the Unity game engine. There are also many excellent github repos that provide functionality to create simple visual displays of Lorentz transforms of events (eg https://github.com/nick1627/RelativityVisualisation) although these are generally restricted to a single spatial dimension.

However, there does not appear to be an R package designed for systematic numerical investigation of the Lorentz group. Here, I present lorentz, a consistent and documented suite of software that allows the user to manipulate Lorentz boosts---considered as members of $O(3,1)$ or $SO(3,1)$---and also facilitates investigation into the nonassociative and noncommutative "gyrogroup" structure of three-velocities [@ungar1997]. The package allows the user to employ efficient and natural vectorised R idiom in the context of relativistic kinematics.

The `lorentz` package: summary of high-level functionality

The lorentz package provides R-centric functionality for Lorentz transforms. It deals with formal Lorentz boosts and converts between three-velocities and four-velocities. Computational support for the nonassociative and noncommutative gyrogroup structure of relativistic three-velocity addition is included. Some functionality for relativistic transformation of tensors of order two such as the stress-energy tensor is given, with examples. In the package, the speed of light is one by default, but is user-settable and the classical limit is recovered by setting $c=\infty$. Both passive and active transforms are implemented. An extensive heuristic vignette detailing package idiom is included.

Research application

There does not seem to be a known relativistic generalization of the classical distributive law $r\left({\bf u} + {\bf v}\right)=r{\bf u} + r{\bf v}$. Ungar [-@ungar1997] states that ``It is hoped that one day a gyrodistributive law \ldots will be discovered. If exists, it is expected to be the standard distributive law relaxed by Thomas gyration in some unexpected way''. The package is used to execute a systematic sweep through a total of $2^{15}\cdot 3\cdot 7=688128$ possible distributive laws consistent with Ungar's suggestion, unfortunately without success.