Quaternionic Möbius Transformations: a Different Approach to Relativistic Kinematics (2026, Work in progress)
Year: 2026
Type: preprint
Abstract: **Monograph / Working Paper (2026 - Active)** This component hosts the ongoing research monograph introducing a non-linear geometric and algebraic framework for special relativity based on quaternionic Möbius transformations. * **Title:** *Quaternionic Möbius Transformations: a Different Approach to Relativistic Kinematics* * **Status:** Working Paper / Active Research Monograph * **Language:** English * **Canonical Archive (DOI):** https://doi.org/10.5281/zenodo.18344596 **Key Highlights:** * **Real Quaternionic Möbius Framework:** Re-formulation of 1+3D relativistic kinematics via fractional linear transformations $q \mapsto (aq-b)(bq+a)^{-1}$ over *Real Quaternions*, offering a distinct alternative to standard complexified biquaternion representations while establishing a rigorous isomorphism with it (both representations are unified via the Projectivized "LAST Notation"). * **Constructive Compass-and-Straightedge Geometry:** Novel ruler-and-compass constructions for relativistic velocity additions and Thomas-Wigner rotations, codified using a dedicated **Stroke Notation**, **Stroke Lemmata**, and culminating in the **Four-Stroke Theorem** (with its "Zorro Signature" geometric manifestation). * **The "Forgotten Fundamental Formula" (FFF):** Formalization of the non-linear tangent-like velocity addition law $v \star w := (v+w)(1-vw)^{-1}$ * **Classification** of Hyperbolic, Parabolic, Elliptic, and Loxodromic Transformations, and resolution of Fixed Points. * **Octonionic Generalization (Foreshadowing):** Demonstration that this entire non-linear geometric framework generalizes naturally to the non-associative setting of Octonions, leveraging Artin's theorem and the Moufang identities. *This active monograph represents the formal synthesis and maturation of the historical foundations established in the 2014 Master 1 project (DOI: 10.5281/zenodo.18322428) and the 2015 Master's Thesis (DOI: 10.5281/zenodo.18320378).*
Source: OSF Preprints (OSF Preprints)
Author: Gregory Hardt Lalinne
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171908785