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8. RELATIONS BETWEEN ELECTROMAGNETIC FIELD PARAMETERS IN DIFFERENT REFERENCE FRAMES DERIVED FROM NEW TRANSFORMATIONS

8.1. Invariance of Maxwell's equations with respect to new transformations and dependence of charge upon speed

Let us consider the same two inertial reference frames A and B, which we have considered in section 3. All variables pertaining to the inertial reference frame B we shall as before designate as primed values, and all variables pertaining to the inertial reference frame A we shall designate as unprimed values.

Let a source of electromagnetic field be at rest in the inertial reference frame B in a point, which is very far from the reference frame B origin. Then in the vicinity of the reference frame B origin the following Maxwell's equations can be used to describe this electromagnetic field (see [55]. Bredov M. M. and others. Classical electrodynamics/ Bredov M. M.,  Rumyantsev V. V. , Toptygin I. N., Moscow, Nauka, 1985, p. 76.)

.(8.1)

where are  the vectors of electric and magnetic inductions, respectively, in the inertial reference frame B;  are the vectors of electric field strength and magnetic field strength, respectively, in the inertial reference frame B; r' is electric charge density in the inertial reference frame B; is the vector of current density in the inertial reference frame B;

;      

  are unit vectors for the axes x', y', z' of the inertial reference frame B;

(8.2)

From the system (8.1) of Maxwell's equations at r' = = 0  we can derive Dalember's wave equation (see [[55]. Bredov M. M. and others. Classical electrodynamics/ Bredov M. M.,  Rumyantsev V. V. , Toptygin I. N., Moscow, Nauka, 1985, p. 117)

(8.3)

Taking into account that the source of electromagnetic field is at rest in the inertial reference frame B, let us apply transformations (6.9) to Maxwell's equations (8.1) and to Dalember's equation (8.3). Then we shall obtain Maxwell's equations for the same electromagnetic field in the inertial reference frame A (derivation see in Appendix 3)

.(8.4)

where are the vectors of induction and strength of electric and magnetic fields in the inertial reference frame A, the components of which depend upon the components of electromagnetic field vectors in the inertial reference frame B according to the equations

.(8.5)

.(8.6)

.(8.7)

and, in addition,

.(8.8)

(8.9)

(8.10)

..(8.11)

And Dalember's equation in the inertial reference frame A for the field, the source of which is at rest in the inertial reference frame B, takes the form

(8.12)

The wave equation (8.12) is the differential equation for propagation of the electromagnetic field at a speed, which is greater than the constant c_o = 299792458 m/s   and which depends upon the source motion speed under the square-law formula (2.1) c_u = c_o (1 + u²/c_o²)^1/2.

If we shall find from equations (8.5), (8.6) and (8.7) the values

.(8.13)

corresponding to propagation of electromagnetic wave (emitted by a source, which is at rest in the reference frame B) in vacuum of the inertial reference frame A, then we shall have

.(8.14)

.(8.15)

.(8.16)

where e , m are permittivity and permeability of vacuum for an electromagnetic wave emitted by a moving source.

We must make a point of the fact, that if in the inertial reference frame A there are two sources of elecctromagnetic field, one of which is at rest in the reference frame A and the other one is moving with respect to it at a speed u, then in vacuum of the inertial reference frame A the electromagnetic wave emitted by a moving source propagates at a speed c_u, which is determined by the formula c_u = c_o (1 + u²/c_o²)^1/2, and the wave emitted by an immovable source propagates at a speed c_o.

The invariance of Maxwell's equations with respect to transformations of the new space-time theory follows from the well-known fact of invariance of Maxwell's equations with respect to arbitrary nondegenerate transformation of space and time variables (see [56]. Miller M. A. and others. Covariance of Maxwell's equations and collation of electrodynamical systems/ Miller M. A., Sorokin Ju. M., Stepanov N. S. // Uspekhi physicheskikh nauk [Achievements of physical sciences],  1977, v. 121, issue 3, pp. 525 - 538).

Dissimilarity of coordinates and time transformation of the new space-time theory from all other linear transformations of coordinates and time consists in the fact, that material equations corresponding to coordinate and time transformation of the new space-time theory have the same simple form like material equations (8.2). Indeed, from expressions (8.5), (8.6) and (8.7) it follows, that in the inertial reference frame A the following equations are valid

.(8.17)

where e , m are determined by expressions (8.14) and (8.15).  

From the expression (8.8) at r' = 0  we have

. (8.18)

From the formula (8.18) it follows, that if in the inertial reference frame B the charge density is equal to zero, but the current density is not equal to zero, then in the inertial reference frame A the non-zero charge density  appears. This result qualitatively coincides with the assertion of the special relativity theory    (see [53]. Ugarov V. A. Special theory of relativity. Moscow: Nauka, 1977, p.  178-180).

From the expression (8.8) at we have

r = r'.  (8. 19)

This means, that according to the new space-time theory if  in the inertial reference frame B a longitudinal current is absent, then the charge density is an invariant value. But  in case of absence  of longitudinal current in the reference frame B charge densities in the inertial reference frames B and A are determined by the expressions

.(8.20)

where are volumes occupied by charges q ' and q  in the reference frames B and А, and these volumes depend upon each other according to the formula

.(8.21)

Then from formulas (8.19), (8.20) and  (8.21) it follows, that in the new space-time theory the following formula is valid if longitudinal current is absent in the reference frame B

.(8.22)

The expression (8.22) means, that in the new space-time theory the value of a moving charge depends upon the charge motion speed, the more the speed of a charge is, the less is the charge value.

At the first glance the expression (8.22) may seem absurd. Really, at present time we are convinced, that the full charge in the given volume remains unchanged in any reference frame (for example, see p. 178 in  [53]. Ugarov V. A. Special theory of relativity. Moscow: Nauka, 1977). But this persuasion is based upon transformation formula for 4-dimensional vector of current from the special relativity theory, but not upon the results of experiments. That is why the expression (8.22) may seem as an absurd one only from the point of view of the special relativity theory. But from the point of view of the special relativity theory even an assumption about square-law dependence of light speed upon the light source speed seems as absurd assumption. Nevertheless, as we have shown in section 4, all ever made experiments on checking the validity of the second Einstein's postulate do not contradict the existence in nature of the square-law dependence c_u = c_o (1 + u²/c_o²)^1/2 of light speed upon the light source speed. That is why future experiments can confirm the existence of such dependence in nature.

So, if the square-law dependence of light speed upon the light source speed in the form c_u = c_o (1 + u²/c_o²)^1/2 exists in nature, then the value of a moving charge must depend upon the charge speed in accordance with the formula (8.22).