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8.3. Maxwell's theory in vacuum in a new relativistic form

The system (8.1) of Maxwell's equations can be written using not only electromagnetic field vectors , but also using scalar and vector potentials and . All variables pertaining to the inertial reference frame B we shall  continue designating as primed values, and all variables pertaining to the inertial reference frame A we shall continue designating as unprimed values. In such homogeneous isotropic envvironment as vaccuum for such potentials of the field, the source of which is at rest in the inertial reference frame B, the following equations are valid (see page 174 in [53]. Ugarov V. A. Special theory of relativity. Moscow: Nauka, 1977)

.(8.54)
.(8.55)
.(8.56)
.(8.57)
.(8.58)
.(8.59)

where

.(8.60)
.(8.61)
. (8.62)

Let us introduce into consideration  a vector of 4-dimensional potential and a vector of four-dimensional current density

.(8.63)

.(8.64)

Taking into account designations (8.63) and (8.64) the equations (8.56) and (8.57) can be written as one formula

(8.65)

Indeed, in case of k = 1, 2, 3 the equation (8.65) coincides with equations for three vector components in the equation (8.56), and in case if k = 4 we have

.(8.66)

but and we have the equation (8.57).

Taking into account designations   (8.63) and (8.64) the Lorentz's condition (8.58) and the continuity equation (8.59) we can write in the form

..(8.58а)

...(8.59а)

And now let us write down in the sane inertial reference frame B expressions (8.54) and (8.55) using designations (8.63) and (8.64) [67]. R. P. Feynman, R. B. Leighton,  M. Sands The Feynman lectures on physics, v. 6. Electrodynamics, London, 1964, (in Russian, Moscow, "Mir", 1977,  p.p. 271 - 272)

..(8.67)
.(8.68)
.(8.69)
.(8.70)
.(8.71)
.(8.72)

Each equation (8.67) - (8.72) can be represented in the form (see p.p. 180-181 in [53]. Ugarov V. A. Special theory of relativity. Moscow: Nauka, 1977)

.(8.73)

where

..(8.74)

In section 8 we till here did not make any changes as compared with 4-dimensional formulation of electrodynamics in the special relativity theory (exept  change of designation for light speed constant С for c_o).  This is conditioned by the fact that till now we considered electromagnetic field in the inertial reference frame B, which was generated by a source, which is at rest in the same inertial reference frame B. But as soon as we have written the 4-dimensional tensor of electromagnetic field in the inertial reference frame B, we can use the general formulas for transformation of 4-dimensional tensor components (see  [53]. Ugarov V. A. Special theory of relativity. Moscow: Nauka, 1977, p. 359)

..(8.75)

where summation is performed for repetitious indices, and

..(8.76)

are components of matrix of new coordinates and time transformation (for an event  happening to a body, which is at rest in the inertial reference frame B (see the formula (9.54)))

.(8.77)

.(8.78)

Subsstituting the values (8.76) into expression (8.75) we have

.(8.79)
.(8.80)
.(8.81)
.(8.82)
..(8.83)
.(8.84)
.(8.85)
.(8.86)
..(8.87)
.(8.88)
.(8.89)
.(8.90)
.(8.91)
.(8.92)

Substituting the values (8.74) and (8.78)   into expressions (8.79)...(8.92) we shall have exxpressions (8.6) and (8.7).

The tensors (8.72) and (8.76) are expressed using vectorsand of electromagnetic field. For description of electromagnetic field in the substance we also use vectors , which are dependent upon each other in the inertial reference frame by equations (see p. 183 in [53]. Ugarov V. A. Special theory of relativity. Moscow: Nauka, 1977)

...(8.93)

..(8.94)

where is the vector of electric induction in the reference frame B; is the vector of magnetic field strength in the reference frame B; is the vector of electric polarisation in the reference frame B;  is the vector of magnetization in the reference frame B, at that for vacuum   we have

For electromagnetic field 4-dimensional tensor expressed by means of vectors we shall use designations:

а) in the inertial reference frame B

.(8.95)

б) in the inertial reference frame  А

.(8.96)

Transformation of 4-dimensional tensor  (8.95) components from the reference frame B to the reference frame A is perfomed according to the formulas

.(8.97)

i. e. according to formulas similar to formulas (8.79) ... (8.92).

When performing transformation (8.97) using tensors (8.95) and (8.96) we shall have expressions (8.5) и (8.8).

Now let us use general formulas for transformation of 4-dimensional vectors (see  the formula (9.55))

..(8.98)

In accordance with these transformations using known values of 4-dimensional current density vector (8.64) in the reference frame B we shall obtain values of 4-dimensional ccurrent density vector in the reference frame A

,.(8.99)

where

.(8.100)

Substituting vaalues (8.100) into the formulas (8.99) we shall   obtain expressions (8.9), (8.10), (8.11) and (8.12).

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