Hypercomplex Numbers, Associated Metric Spaces, and Extension of Relativistic Hyperboloid

TL;DR

This paper develops a framework for commutative-associative hypercomplex numbers and their associated metric spaces, extending traditional geometric and relativistic concepts with multilinear metric forms for potential physical applications.

Contribution

It introduces a generalized scalar polyproduct and multilinear isometry, expanding the mathematical tools for hypercomplex numbers in relativistic geometry.

Findings

Proposes a scalar polyproduct extending the scalar product.

Introduces the concept of multilinear isometry.

Suggests applications in anisotropic relativistic physics.

Abstract

We undertake to develop a successful framework for commutative-associative hypercomplex numbers with the view to explicate and study associated geometric and generalized-relativistic concepts, basing on an interesting possibility to introduce appropriate multilinear metric forms in the treatment. The scalar polyproduct, which extends the ordinary scalar product used in bilinear (Euclidean and pseudo-Euclidean) theories, has been proposed and applied to be a generalized metric base for the approach. A fundamental concept of multilinear isometry is proposed. This renders possible to muse upon various relativistic physical applications based on anisotropic {\it versus} ordinary spatially-rotational case.