[Paper Review] Hypercomplex Numbers, Associated Metric Spaces, and Extension of Relativistic Hyperboloid

D. G. Pavlov|ArXiv.org|Jun 3, 2002

Advanced Differential Geometry Research15 references3 citations

TL;DR

This paper introduces a framework for commutative-associative hypercomplex numbers—specifically quadrahyperbolic numbers in four dimensions—by defining a multilinear scalar polyproduct that generalizes the Euclidean and pseudo-Euclidean scalar product. It establishes a correspondence between these numbers and Finsler-type metric spaces, enabling a geometric and relativistic extension of the hyperboloid model to anisotropic spacetime structures through isometric transformations and generalized Pythagorean theorems.

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.

Motivation & Objective

• To develop a consistent framework for commutative-associative hypercomplex numbers with multilinear metric forms.
• To generalize the concept of scalar product to a scalar polyproduct for use in geometric and relativistic applications.
• To establish a one-to-one correspondence between quadrahyperbolic numbers and Finsler-type vector spaces via homogeneity of degree one in the 4th radical of the polyform.
• To extend the relativistic hyperboloid model to anisotropic spacetime by introducing distinguished directions corresponding to hyperunits.
• To propose new geometric concepts such as multilinear isometry, transversality, orthogonality, and generalized Pythagorean theorems in this algebraic-geometric setting.

Proposed method

• Defining quadrahyperbolic numbers as a 4-dimensional algebra with commutative, associative, and distributive operations, including zero and unity.
• Introducing the scalar polyproduct as a symmetric multilinear form generalizing the standard scalar product.
• Using the 4th radical of the 4th-degree polyform to define vector lengths that satisfy homogeneity of degree one, a key property of Finsler geometry.
• Establishing isomorphism between the vector space of quadrahyperbolic numbers and a Finsler space via the indicatrix equation $F(\mathbf{A}) = 1$.
• Deriving generalized Pythagorean theorems and defining geometric objects such as arcs, triangles, and cones in the multilinear space.
• Proposing the concept of multilinear isometry to generalize transformations preserving vector lengths and geometric structures.

Experimental results

Research questions

• RQ1 Can a multilinear scalar polyproduct be defined such that it generalizes the scalar product and supports a consistent geometric metric in hypercomplex algebras?
• RQ2 How can the Finsler geometry framework be naturally embedded in the structure of quadrahyperbolic numbers through the homogeneity of the 4th radical of the polyform?
• RQ3 To what extent can the relativistic hyperboloid be extended to model anisotropic spacetime by assigning distinguished directions to hyperunits $1, I, J, K$?
• RQ4 Can the concepts of angle, orthogonality, and transversality be consistently generalized in this multilinear space, and do they satisfy a generalized Pythagorean theorem?
• RQ5 What are the implications of the algebraic independence of the five canonical metric forms (8.1)–(8.5) for classifying four-dimensional multilinear spaces?

Key findings

• The scalar polyproduct generalizes the bilinear scalar product and enables the definition of multilinear isometry, a fundamental geometric structure in the new algebraic framework.
• The 4th radical of the 4th-degree polyform yields a length function that is homogeneous of degree one, satisfying the defining property of Finsler geometry.
• The Finslerian indicatrix $F(\mathbf{A}) = 1$ is equivalent to the ${\cal H}_4$-hyperboloid defined by the modulus of a unit quadrahyperbolic number.
• The metric forms (8.1)–(8.5) are algebraically independent, meaning no linear transformation maps one to another, establishing them as canonical elements for classifying quadralinear spaces.
• The generalized Pythagorean theorem is derived in the multilinear space, extending the classical relation to higher-order geometric configurations.
• The paper identifies 12 open research problems, including extending holomorphic functions to polynumbers, classifying spaces with symmetric forms, and investigating Riemannian-type submanifolds within the multilinear framework.