3.4 The New Momentum-Energy Tensor of Field. 6-Formalism

3.4  The New Momentum-Energy Tensor of Field. 6-Formalism


…Each mechanical task concerning point mass motion using multidimensional space may be reduced to definition of light beam track in an appropriate environment.

Felix Klein

Improving Lorentz's electron model, Henri Poincaré (1905) has added momentum-energy tensor with component with scalar pressure. In Poincaré model negative pressure inside volume, which is limited by a spherical charged surface of final radius, counterbalances electrostatic repulsion forces. Hereinafter this procedure have become co-ordinated with the Max von Laue theorem.

In Gunnar Nordström's gravitation scalar theory the scalar function, equal to a trace of complemented momentum-energy tensor, thanking to conform multiplier in a linear element is transformed into a source of a scalar gravitation field. This theory can be classified as integration of the Poincaré electron theory, if we will forget about its initial purposes.

At transition to stationary ON-model of electron such counterbalance pressure is not required, but additional scalar repulsion field, acting on ON-currents of electron irrespective to a charge mark, is necessary. This role is executed by the field G.

G-repulsion, acting on positively charged ON-current, directed to electron centre, is reduced by E-attraction and the ON's speed is extinguished up to zero only upon achievement of a centre, then ON+ is again accelerated and leaves in infinity, restoring there individual speed.

G-repulsion and E-repulsion act on negatively charged ON-current, directed precisely to a electron centre. They cancel speed up to a zero in a distance of two units from the electron centre.

This universal scalar G-repulsion is interpreted as an ON's rest mass variability dynamic effect, which is determined by a scalar field of scale invariance.

The field G is entered into expressions of density and flow of field energy and expressions (accurate to within common multiplier, which is determined by choice of system of units) are obtained:

2u = EE + BB + GG, (1)
g = [EB] + (Gv)v. (2)

It is reasonable, but purely phenomenological approach. If we override stagnancy of thinking, we may write the density of field energy of an electron at rest in the following form:

2u = (–E+2 + l+–2G+2) + (E2 + l–2G2). (3)

Here we see a pair of vector fields with different signs and a pair of scalar fields. Summands of the first brackets describe the contribution of positively charged ON-currents filling the whole space into the energy of field. Summands in the seconds brackets describe the contribution of negatively charged ON-currents filling the space between spherical surface with two unit radius and infinity.

If k=1 we obtain:

l0 = r0 ± 1,    l± = r ± 1,    E±2 = G±2 = r–4 (4)

and energy of an electron at rest is equal to the integral:

òudV = –4π
(1 + r/2)dr
r(r + 1)2
 + 4π
(r2 + 1)dr
r2(r2 – 1)2

The first summand has a logarithmic divergence which is eliminated for vertex velocities meeting the condition:

l02 = r02 – 1 ± 2. (6)

In this case we obtain finite energy of an electron and degeneration of doublet of vector and scalar fields is eliminated. 6-formalism is well known where such fields appear in a natural way.

It is planar 6-dimensional variety of signature (+ – – – + –) with co-ordinates:

xα~= kxα,    x5~
(1 – x2);    x6~
(1 + x2),

where k is a new independent parameter. Linear transforms of Lorentz type for this variety have the form:

~xA ® LA~xB•B (A,B = α,5,6), (8)

where 21 conditions are imposed on 36 coefficients

0gABLA•MLBN =0gMN , (9)

i. e. 15 parameters are independent.

This 15-parameter rotation group of 6-dimensional planar variety (7) breaks down into four subgroups in Minkowski 4-variety:

1) 6-parameter Lorentz group:

xα ® Lα xββ; (10)

2) 4-parameter translation (or displacement) group:

xα ® xα + aα ; (11)

3) 1-parameter dilatation group:

xα ® μxα ; (12)

4) non-linear 4-parameter group of special conform transforms:

xα ® 
1 – 2cβxβ + c2x2

where cα – four parameters and

x2 =0gμνxμxν,    c2 =0gμνcμcν. (14)

(10)–(13) may be interpreted as subgroups of admissible co-ordinate transforms changing both co-ordinates and metric tensor components. In such interpretation, they are removed from planar Minkowski space-time.

The following facts comprise the basis for construction of 6-dimensional ON-theory in planar 6-variety (7):

1) ML-equations, as well as other basic physical equations for massless fields are invariant in respect to 15-parameter group of conform transforms;

2) correspondence 15-parameter group of conform transforms in Minkowski space to rotation group in 6-dimensional planar variety (7);

3) use of (1+1+4)-splitting of 6-dimensional variety and its equations enable to get all necessary material.

The variety (7) is a stage of 6-optics activity, since:

z2 =~xAxA~= 0,    dz2 =0gABdxAdxB~=~0. (15)

Light-like motion and equations are considered in the 6-world. The terminology of the description of motion on the tails of Minkowski world light cone is used. Formal difference is the result of dimensionality increase by space-like and one time-like dimension.

In the Minkowski world, all allowable motions (events) invariantly fall apart into three classes (fields): space-like, light-like and time-like depending of interval sign. Light-like motions describe massless fields governing motion of particle with finite mass at rest, and vice versa.

The particularity of the 6-world (7) is the existence of light-like motion only. There is nothing more than massless field and its equations in this world. The source of the field may be only the field itself. Equations for these fields do not have right parts, they are equations of pure field.

It is one of the mountains' peaks so sought by Albert Einstein.

It should be noted that the language of mathematics uses such words as «light-like» motion, «pure» field… Philosophically (spiritually) thinking brains (souls) will trace in it a connection with rich traditions of spiritual (esoteric) comprehension of the world, with the God's light, God's truth…

The uniqueness of our world lies in the uniqueness of the truth or its identity with countless number of «other» truths in the same way as the interval (15) between to arbitrary events of the world (7) is always a zero. The unique, initial truth shows us its various faces and is perceived by us in new forms (transcriptions) like motion in the 6-world reduced to its 6-rotation. Such world is awfully rich in its potential possibilities. The only thing we are lacking is a tolerance to accept this truth…

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