Reformulation of electromagnetic and gravito-electromagnetic equations for Lorentz system with octonion algebra

In this paper, the real, complex octonion algebra and their properties are defined. The electromagnetic and gravito-electromagnetic equations with monopoles in terms of S and \(\hbox {S}^{\prime }\) reference systems are presented in vector notations. Additionally, the duality transformations of gravito-electromagnetic situation for two reference systems are also represented. Besides, it is explained that Maxwell-like equations for gravito-electromagnetism are also invariant under Lorentz transformations. By introducing complex octonionic differential operator, a new generalized complex octonionic field term consisting of electromagnetic and gravito-electromagnetic components has been firstly suggested for Lorentz system. Afterwards, a complex octonionic source equation is obtained as in basic way, more compact and elegant notation. By defining a new complex octonionic general potential term, the field equation is attained once again. The components of complex octonionic field and wave equations are written in detailed for S and \(\hbox {S}^{\prime }\) reference systems.