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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 EGM-waves and others). The properties of these solutions are investigated.

The one biquaternionic model of electro-gravimagnetic (EGM) fields and their interaction was elaborated by author in [

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 a-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 [

The base of this model is generalized biquaternionic form of Maxwell equations which includes differential part of Dirac operator [

To use biquaternions algebra we give some definitions. We enter on

f is a complex function,

Summation and quaternionic multiplication are defined as

where

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 [

Mutual bigradients

Composition of mutual bigradients gives classic wave operator:

It gives possibility easy to construct the solutions of biquaternionic wave equation (biwave Equation)

which are presented in the form of the convolution:

where

is the fundamental solution of D’Alember equation (a simple layer on light cone):

In formulae (2) the second equality is written for regular

We name

If

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

・ real vectors

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

By using these values we introduce the complex characteristics of EGM-field:

・ complex vector of EGM-intensity

・ complex charges field :

・ complex currents field:

・ complex scalar field of attraction-resistance

Here values

We construct the next Bqs. of EGM-field and charge-currents field (CC-field):

・ EGM-potential

・ EGM-intensity

・ charge-current

・ energy-pulse of EGM-field

In case

c is light speed.

By analogue we enter biquaternion of energy-pulse of charge-current field (CC-field)

If to calculate

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

Postulate 1. Connection between EGM-intensity and charge-current is bigradiental:

This assumption follows from Maxwell equations.

In particulary by

As

we have from here the known formulas for electric charge and currents:

From (5) for real and imaginary part we get

generalized Maxwell equations:

which, when

classic Maxwell equations:

When EGM-field and charge-currents are independent on time, we get from ((8)

equations for stationary charges and currents:

From last two scalar equation easy to get the known Coulomb’s equation for potential of electrostatic field. The second one gives the Poisson equation for potential of Newton gravitational fields if to put

mass density and

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 [

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.

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:

For classic Maxwell equation

In particulary it’s performed if

But if

As EGM-equation (5) is hyperbolic, it has characteristic surface (F). Its equation has the form [

where

In the space of distributions the classical solution of (5) (considered as generalized biquaternions

Here singular generalized function

conditions on EGM-wave front:

From here follow the next conditions for real and imaginary parts.

On fronts of shock EGM-waves the gaps of intensities satisfy to the next conditions:

Here the sigh “×” notes vector production.

You see that for generalized Maxwell-Dirac equations shock waves are not transversal. Only if

It’s well known for EM-waves as generalized solutions of classic Maxwell Equation (9).

You see that the longitudinal components at the front of EGM-wave are connected with a jump of scalar a-field, which describes property of attraction-resistance of EGM-field to the movement of external charges and currents. It was shown in [

At absence of charges and currants EGM-field satisfies to homogeneous biwave equation

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 [

Plane spinors. Let construct some plane waves generated by scalar potentials:

where

1) Longitudinal magnetic wave in direction H:

2) Longitudinal electric wave in direction E:

3) Tesla’s wave―EM-wave in direction with torsion component H:

as

4) Torsion wave―EM-wave in direction H with torsion component E.

Here waves names correspond to ones in paper of V.A. Etkin [

EGM-equation (5) gives possibility to construct EGM-intensity if charge-current are known. And vice versa if EGM-field is known its bigradient determine charges-currents.

Hence this equation and corresponding system are unclosed. To close this equation let assume that the next proposition in true.

Postulate 2. If the charge-current are free (there are absence the action of external EGM-fields) then

which is equivalent to equations:

or for real and complex parts

Here the scalar equations are the known conservation laws of the electric and gravimagnetic charges which must be executed at absence of external actions (influence).

The Equation (13) is field’s analogue of the first Newton law (inertia law). The systems (14) show that gradients of charges and rotors of currents stipulate their motion and changing.

Free

where

Solutions of wave equation have been studied very well. We give some examples of solutions of this equation which may be used for construction of elementary particles and more complex matter.

1) Harmonic spherical w-spinor. Their potentials are presented in the form:

2) Spinors field

3) Fibers

4) Tissue

5) Body

Here

Equations ((5) and (13)) give full and closed system of hyperbolic type for determination field of free charge and currents and generated EGM-field, which we formulate as postulate 3.

Postulate 3. The EGM-field of free charge-current are described by the next biquaternionic system:

Its hamiltonian form

Its differential equations system

Their general solutions are presented in the form (15), (11).

Here we consider the fields analogue of first Newton law which has been postulated not for material point but for distributed electric and gravimagnetic charges and currents. We show, that charges and currents of EGM-field are physical appearance of bigradient of EGM-intensity. If bigradient of EGM-intensity is equal to zero then charges and currents are absent.

From this model follow that charges

We introduced postulates for EGM-field on the base of generalization of biquaternionic form of Maxwell equations and obtained closed hyperbolic system which connects EGM-field, charges and currents in united system of equations. For this we enter new scalar a-field of attraction-resistance which gives possibility to explain some physical phenomena which are observed in practice.

In particular, the solutions of EGM-field describe electric and gravimagnetic waves which, in general case, are not transversal and have longitudinal component. Longitudinal EM-waves are observed in practice but classic electrodynamics doesn’t explain their existence.

Many interesting physical properties of this model appear by interaction of different system of charges and currents and their EGM-fields. Some of them were described in papers [

L. A.Alexeyeva, (2016) Biquaternionic Model of Electro-Gravimagnetic Field, Charges and Currents. Law of Inertia. Journal of Modern Physics,07,435-444. doi: 10.4236/jmp.2016.75045