Hyperbolic trigonometry in two-dimensional space-time geometry

TL;DR

This paper introduces hyperbolic numbers linked to space-time geometry, enabling a Cartesian-based formalization of hyperbolic functions and pseudo-Euclidean trigonometry aligned with special relativity principles.

Contribution

It develops a hyperbolic number system and demonstrates how to formalize pseudo-Euclidean trigonometry using elementary Cartesian methods.

Findings

Hyperbolic functions can be defined via hyperbolic numbers respecting Lorentz invariance.

Pseudo-Euclidean trigonometry can be formalized similarly to Euclidean trigonometry.

The approach simplifies the mathematical framework of space-time geometry.

Abstract

By analogy with complex numbers, a system of hyperbolic numbers can be introduced in the same way: z=x+h*y with h*h=1 and x,y real numbers. As complex numbers are linked to the Euclidean geometry, so this system of numbers is linked to the pseudo-Euclidean plane geometry (space-time geometry). In this paper we will show how this system of numbers allows, by means of a Cartesian representation, an operative definition of hyperbolic functions using the invariance respect to special relativity Lorentz group. From this definition, by using elementary mathematics and an Euclidean approach, it is straightforward to formalize the pseudo-Euclidean trigonometry in the Cartesian plane with the same coherence as the Euclidean trigonometry.