In this study, octonions with eight dimensions and their algebra, which are both noncommutative and nonassociative, are presented. Moreover, the general properties of complex octonions with 16 dimensions and the products of basis are defined by using Cayley–Dickson multiplication rules. Maxwell's equations are taken into consideration when it comes to bi-isotropic media in which the electric and magnetic fields are coupled by means of bi-isotropic constitutive relations. The Drude–Born–Fedorov constitutive relations defined with the complex representations of electric and magnetic fields are used for all calculations. In the next stage, the complex octonionic differential operator is introduced and the octonionic field and source equations are defined for bi-isotropic media. As a result, Maxwell's equations for bi-isotropic media, which are fundamental features that serve as the groundwork for electromagnetism, all of them as being unique, more compact, and a convenient form with magnetic monopole.
In this paper, we propose the generalized description of electromagnetism and linear gravity based on the combined dual numbers and complex quaternion algebra. In this approach, the electromagnetic and gravitational fields can be considered as the components of one combined dual-complex quaternionic field. It is shown that all relations between potentials, field strengths and sources can be formulated in the form of compact quaternionic differential equations. The alternative reformulation of equations of gravitoelectromagnetism based on formalism of [Formula: see text] matrices is also discussed. The results reveal the similarity and isomorphism of distinctive algebraic structures.
Abstract:This study investigates whether the electromagnetic and gravitoelectromagnetic energy conservation equations are obtained together by using octonion algebra or not. Maxwell and Maxwell-like equations for linear gravity with magnetic monopole terms are used in the SI unit system. A new complex octonionic field term is suggested for the first time. The complex octonionic source equation is then obtained. Finally, Poynting theorems for both electromagnetism and gravitoelectromagnetism are defined for the first time by using higher dimensional algebra.
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