Some coordinate transformations relevant to refractive indices

Abstract

This paper focuses on applying the algebra of octonions to study some coordinate transformations in octonion spaces, exploring the contribution of partial field potential on the speed of light. 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 a 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 not only 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.