Two-dimensional space-time symmetry in hyperbolic functions

TL;DR

This paper extends the Lie group properties of complex functions using hypercomplex systems to describe two-dimensional space-time symmetries, including invariance of the wave equation and the speed of light, generalizing Lorentz transformations.

Contribution

It introduces a hypercomplex framework that generalizes Lorentz transformations and describes accelerated frames in two-dimensional space-time.

Findings

Derived a set of PDEs defining an infinite Lie group for hypercomplex functions.

Established invariance of the wave equation and light speed under new transformations.

Obtained a relativistic hyperbolic motion as an application.

Abstract

An extension of the finite and infinite Lie groups properties of complex numbers and functions of complex variable is proposed. This extension is performed exploiting hypercomplex number systems that follow the elementary algebra rules. In particular the functions of such systems satisfy a set of partial differential equations that defines an infinite Lie group. Emphasis is put on the functional transformations of a particular two-dimensional hypercomplex number system, capable of maintaining the wave equation as invariant and then the speed of light invariant too. These functional transformations describe accelerated frames and can be considered as a generalization of two dimensional Lorentz group of special relativity. As a first application the relativistic hyperbolic motion is obtained.