Erlangen's Program for Space-Time through Space-Time Geometric Algebra Induced by the R Vector Characteristic of the Ring of Hybrid Numbers Z

TL;DR

This paper proposes a unified approach to characterizing flat space-times using hypercomplex rings and automorphisms, linking algebraic structures to geometric and physical properties of space-time.

Contribution

It introduces a novel automorphism in hybrid number rings that induces space-time metrics and develops hybrid trigonometric functions to analyze different geometries.

Findings

Unified algebraic framework for flat space-times

Derivation of space-time properties from hypercomplex automorphisms

Interpretation of anti-matter in Euclidean space-time

Abstract

This essay summarizes the efforts required to build a program of a unified, low-dimension topology that allows characterizing all these flat space-times. Since spatiotemporal manifolds are topological spaces equipped with metrics, their properties are characterized by Clifford algebras in hypercomplex rings associative with unity, so that Galileo's transformations are induced by a dual number; the Lorentz transformations, by a perplexed number and the Euclid transformations, by a complex number. This fact led us to establish an internal automorphism in the ring of hybrid numbers that acts as a map of the manifolds and induces the space-time metric based on the quality (characteristic) of the associated hypercomplex unit. From this automorphism, we built hybrid trigonometric functions, which we call Poincar\'e functions, which allowed us to deduce general properties of space-time, hyperbolic, parabolic and elliptical geometries and the groups SO (3), SO (4) and SO (1, 3). This approach allows us to highlight the global properties of space-time, suggests methods for geodynamic models and allows us to interpret anti-matter as matter in a Euclidean space-time where the nature of time is imaginary.