Some coordinate transformations relevant to refractive indices

TL;DR

This paper uses octonion algebra to analyze coordinate transformations in electromagnetic and gravitational fields, revealing how partial electromagnetic potential influences the speed of light and refractive indices in optical waveguides.

Contribution

It introduces a novel application of octonion composite space to derive transformations affecting light speed and refractive indices, incorporating electromagnetic potentials.

Findings

Partial electromagnetic potential affects light speed in optical waveguides

Derived Galilean-like and Lorentz-like transformations in octonion space

Explored influence of relative motion on the speed of light

Abstract

The paper focuses on applying the algebra of octonions to study some coordinate transformations in the octonion spaces, exploring the contribution of partial field potential on the speed of light. J. C. Maxwell was the first to introduce the quaternions to describe the physical properties of electromagnetic fields. Nowadays the octonions can be applied to study simultaneously the physical quantities of electromagnetic and gravitational fields, including the transformation between two coordinate systems. In the octonion space, the radius vector can be combined with the integrating function of field potential to become one composite radius vector. The latter is considered as the radius vector in an octonion composite space, which belongs to the function spaces. In the octonion composite space, when there is the relative motion between two coordinate systems, it is capable of deducing the Galilean-like transformation and Lorentz-like transformation. From the two transformations, one can achieve the influence of relative speed on the speed of light (or Sagnac effect), but also the impact of partial electromagnetic potential on the speed of light. The study states that the partial electromagnetic potential has a direct influence on the speed of light in the optical waveguides, revealing several influencing factors of refractive indices in the optical waveguides.