Biquaternionic Model of Electro-Gravimagnetic Field, Charges and Currents. Law of Inertia

One the base of Maxwell and Dirac equations the one biquaternionic model of electro-gravimagnetic (EGM) fields is considered. The closed system of biquaternionic wave equations is constructed for determination of free system of electric and gravimagnetic charges and currents and generated by them EGM-field. By using generalized functions theory the fundamental and regular solutions of this system are determined and some of them are considered (spinors, plane waves, shock EGMwaves and others). The properties of these solutions are investigated.


Introduction
The one biquaternionic model of electro-gravimagnetic (EGM) fields and their interaction was elaborated by author in [1] [2]. There the fields analogues of three Newton laws for densities of mass and electric charge and current, acting forces and their powers have been built.
Here we consider the EGM-field created by free system of mass, charges and currents and their motion under action only internal electric and gravimagnetic tensions. In this model gravitational field (which is potential) is united with magnetic field (which is torsional) what gives possibility to enter gravimagnetic tension, charge and current. Lasts contain gravitational mass and their motion but not only them. Also here the new scalar α-field of attraction-resistance is entered and their existence is justified. This phenomenon explains existence of longitudinal EM-wave which is observed in practice. We use here differential algebra of biquaternions in hamiltonian form which more full were described in [3]. The scalar-vector form of biquaternion representation (hamiltonian biform) is very demonstrative and strangely adapted for writing the physical values and equations.
The base of this model is generalized biquaternionic form of Maxwell equations which includes differential part of Dirac operator [4]. From this form follow bigradiental representation of electric and gravimagnetic charges and currents. Differential operator bigradient is the generalization of gradient operator on the space of biquaternions which characterizes a direction of more extensive change of biquaternionic functions.

Biwave Equation and Its Solutions
To use biquaternions algebra we give some definitions. We enter on (Minkowski space) the functional space of biquaternions in hamiltonian form: e e e = are basic elements [3]. We assume are locally integrable and differentiable on  or, in general case, they are generalized functions [5].
Summation and quaternionic multiplication are defined as are usual scalar and vector productions in 3 R (here over repeated indexes there are summation from 1 to 3), jkl  is Levi-Civita symbol. The norm and pseudonorm of Bq. are denoted ( ) We'll use convolution of biquaternions: For regular components a convolution has the form: to take a convolution for singular generalized function and conditions of convolution existence see [5]. Mutual bigradients , Composition of mutual bigradients gives classic wave operator: It gives possibility easy to construct the solutions of biquaternionic wave equation (biwave Equation) (1) which are presented in the form of the convolution: is the fundamental solution of D'Alember equation (a simple layer on light cone): is arbitrary solution of homogeneous D'Alember equation: In formulae (2) the second equality is written for regular G . We name

Characteristics of Electro-Gravimagnetic Field
Let introduce known and new physical values which characterize EGM-field, charges and currents: • real vectors E and H are the tensions of electric and gravimagnetic fields; • real scalars , E H ρ ρ are the densities of electric and gravimagnetic charges; • real vectors , E H j j are the densities of electric and gravimagnetic current. Here we united the gravitational field (which is potential) with magnetic field (which is torsional) in one gravimagnetic field H. Also we united mass current with magnetic currents. As well known classic electrodynamics refuse the existence of magnetic charges and currents. But here we'll show that H ρ and H j can the rights on the existence.
By using these values we introduce the complex characteristics of EGM-field: Here values , ε µ are constants of electric conductivity and magnetic permeability of corresponding EMmedium.

Biquaternions of Electro-Gravimagnetic Field
We construct the next Bqs. of EGM-field and charge-currents field (CC-field): In case 0 α = here you see the energy density W and Pointing vector P of EM-field: c is light speed. By analogue we enter biquaternion of energy-pulse of charge-current field (CC-field) ϖ contains ρ and energy density of currents: where the first summand includes Joule heat of electric current; second one includes energy density of gravimagnetic current, which contains kinetic energy of mass current. Here vector J P is analogue of Pointing vector, but for the current:

Connection between EGM-Field, Charges and Currents
Postulate 1. Connection between EGM-intensity and charge-current is bigradiental: This assumption follows from Maxwell equations. In particulary by 0 α = from here follow the known Hamiltonian form of Maxwell equations [6]: When EGM-field and charge-currents are independent on time, we get from ((8) equations for stationary charges and currents: Remark. We must note that the first scalar equation of classic Maxwell Equation (9) (where electric charges can depend on time) contradict to wave nature of EM-field. But Equation (9) is true only if charge and currents are independent on time. The same one relates to Eq. for gravitational field which is true only for static mass.
All this confirm postulate 1, which shows, that charges and currents of EGM-field are physical appearance of bigradient of EGM-intensity! From here follow, if bigradient of EGM-intensity is equal to zero then charges and currents are absent! Equation (5) is generalization of Maxwell equation in biquaternions algebra. The differential operator corresponding to it coincides with the differential part of matrix operator of Dirac [4]. By this course Equation (5)

we name Maxwell-Dirac equation of EGM-field or simply the EGM-equation.
EGM-equation is hyperbolic, and corresponding to it system of differential Equation (8) is hyperbolic and connected. It's known that classic system of Maxwell Equation (9) doesn't possess such properties.

Generalized Solutions of EGM-Equation
As Equation (5) is biwave equation, to construct its solution it's need to use formulae (2): According to (2) the scalar and vector parts of EGM-intensity have the form:

Spinors of EGM-Field
At absence of charges and currants EGM-field satisfies to homogeneous biwave equation 0, which solutions can be constructed by use spinors in (2). We consider here some unconventional spinor which can explain longitudinal electromagnetic waves, which are observed in practice [7] [8].
Plane spinors. Let construct some plane waves generated by scalar potentials: ...