5.2 Maxwell Equations single out Hyperbolic Motion of Field Sources
Paul Dirac 
For the description of a single isolated electron at a subquantum level it is necessary to have at least a centrally (spherically) symmetrical, stationary field totality of charged subcurrents moving with acceleration in the own integral field (of electron). Each of these charged subcurrents separately or its part, being a source of a field, in full conformity with the MLequations, should generate strictly conservative partial field.
Hyperbolic motion of field sources, which met all these requirements, was revealed and described already in the first decade of the XX century. But both then and later on, prepotent researches appeared to be limited by hyperbolic motion of corpuscular electrons (or their «parts»), identified with field sources of electronic theory of Lorentz (or its relativistic version). This important identification in particular has predetermined all further development of the quantum theory, yet not having received the necessary substantiation either within the framework of the classical theory, or – of the quantum one. It was and still remains a source of the latent and obvious contradictions (paradox of Born). Today as well it regularly blocks the way to the construction of subquantum physics. Owing to this identification, fundamental character of hyperbolic motion of field sources has not received its due estimation and application in physics up to now. Let us give the floor to the dramatis personae of that time.
Wolfgang Pauli: – «Hyperbolic motion was for the first time investigated by Minkowski [2] as especially simple motion; then it was considered in detail by Born [3] and Sommerfeld [4]».[1] «Abraham [5] proved also that the integral taken over time of K, propagated over duration of radiation, is equal to the pulse of radiated light as well as that the integral taken over time of vK is equal to radiated energy. In hyperbolic motion the K disappears as it should be because in this case there is no any radiation». [1]
Max Born: – «It should be noted that an electron in hyperbolic motion does not have any selfradiation, no matter how great its acceleration is, but hauls its field along. Up to now this circumstance has been known for electrons in uniform motion only. The radiation… shows up only in cases of deviation from hyperbolic motion». [3]
Thus, still at the dawn of development of a relativistic paradigm it was established, that: – In strict conformity with MLequations of the field, the charged field sources being in hyperbolic motion, generate (a conservative) field without radiation (the source hauls the field along; the field does not break away from the source; there is no formation of wave zone; the item of field, decreasing under the law 1/R, is absent).
This fundamental result of the classical field theory, basing on already discovered natural kinematics in 4vector World of Minkowski – Mkinematics can be unambiguously defined (singled out) solely by the symmetries of the MLequations. It does not depend on a concrete type of the equations of motion of field sources in operating integral field, does not require the equations of motion for its definition and singularity. It is a reflection of only one additional kinematic singularity of a class of world lines of hyperbolically moving field sources in Mkinematics. The symmetry of a field, expressed in full absence of radiation at hyperbolic motion of sources – is a purely kinematic effect in the field theory based on MLequations.
At calculation of a field of sources moving with acceleration or resistance to radiation (radiating friction) the following characteristic values are being received, responsible for the absence of radiation or resistance to radiation:
G(U,W,Y) := DW + (WW)U, D := d/ds,  (1.G) 
g_{3}(v,a,b) := b + 3l^{2}(va)a,  (1.g_{3}) 
g_{1}(v,a) := a^{2} – [va]^{2} = a^{2}(1 – v^{2}) + (va)^{2}.  (1.g_{1}) 
These values are modelled of 4vectors and their components obtained by successive differentiating world 4radiusvector R on the proper invariant time s according to Mkinematics scheme, – or of the corresponding values of Newton's kinematics – Nkinematics:
x^{i} := R := (t, r), ds^{2} := dt^{2} – dr^{2},  (2.0) 
DR := u^{i} := U := (l, u), Uds := dR, vdt := dr,  (2.1) 
DU := w^{i} :=W := (p, w), Wds := dU, adt := dv,  (2.2) 
DW := y^{i} := Y := (q, y), Yds := dW, bdt := da…  (2.3) 
Corresponding t and rcomponents of 4vectors of Mkinematics are in standard way connected with 3vectors r,v,a,b,… and time t of Nkinematics:

(3.t) 
u = lv, w = pv + l^{2}a, y = qv + 3pla + l^{3}b, … .  (3.r) 
The abovewritten values assume such a form in a metric with signature of scalar product (+ – – –) and in such a system of units of length and time measuring, in which velocity of light is equal to unit. Total differentials of proper (invariant) time ds of Mkinematics and time dt of Nkinematics, coincident with differential of tcomponent of 4vector R, are connected (lds=dt) by Lorentzfactor l with the help of the fundamental equation of Mkinematics
UU = l^{2}(1 – v^{2}) = 1.  (4) 
Here indexless designations of 4vectors are accepted which are typed by UPPERCASE italic bold letters: – R,U,W,G… In order to obtain the necessary signature of scalar product of 4vectors A:=(a^{0},a) and B:=(b^{0},b), singled out by their tcomponents a^{0} and b^{0}, and by rcomponents a and b, it is necessary to follow the rule:
AB := (a^{0}, a)(b^{0}, –b) = a^{0}b^{0} – ab,  (4.def) 
where ab – a usual scalar product of 3vectors with the signature (+ + +). Usually, for this purpose, 4vector A is written down in countervariant components a^{i}:=(a^{0},a), and B in covariant b_{i}:=(b_{0},–b), or they multiply rcomponents of 4vectors by imaginary unit. 3vectors and rcomponents of 4vectors are typed as normal bold letters (nonitalic) and can be both lowercase, and UPPERCASE.
Used by Abraham and Pauli 3vector K (formula (265a) [1]) and 4vector K (formula (265) [1], written down there in a metric with the signature (+ + + –)) assume in our designations the following form:
K = cl^{2}(g_{3} + l^{2}(vg_{3})v), K = cG,  (5) 
where c – the dimension matching factor, dependent on the system of units used.
Equality of kinematic values g_{3} and G to zero involves meeting the conditions of dynamic character K=0 and K=0, agreeable to the absence of resistance to radiation (by Abraham and Pauli) at such accelerated motion of sources. Motions within the framework of Mkinematics – the Mmotions satisfying an additional condition G=0, we shall for distinctness call hyperbolic motions.
Minkowski has singled out in Mkinematics revealed and constructed by him uniformly accelerated motion, as especially simple motion. Uniformly accelerated motion in relativistic kinematics is believed to be such motion, for which acceleration constantly has the same value a in a frame of reference K' corresponding at the present moment to a body or a material point. The frame of reference K' for each moment is different; in one certain Galilean frame of reference K acceleration of such motion is not constant in time. [1] Uniformly accelerated motion in the Mkinematics, determined by a constancy (preservation) of a 3vector of acceleration a along all world way in family of (instantaneously) accompanying inertial (Galilean) frames of reference, – is singled out in Mkinematics only by considerations of symmetry in the family of vectors of acceleration a_{K'} , which have purely kinematic nature.
Definition 1 Mmotion in conformity with kinematic conditions:
DW + kU = 0 & Dk = 0,  (6.G) 
will be called hyperbolic or uniformly accelerated motion, or – Gmotion, and kinematics of Gmotions will be called Gkinematics.
From the equation (6.GII) follows, that k is a constant preserving along all Gmotion. At such k the equation (6.GI) coincides with the 4vector covariant condition, singling out uniformly accelerated motion, and conformable to its 3vector condition b=0 in family of the own (instantaneouslyaccompanying) frames of reference, correlated to the flock of all points of the world way. Thus, any Gmotion is uniformly accelerated motion.
Multiplying (6.GI) by U, we receive (DW)U=–k. Twice differentiating the equation UU=1 by s, we receive (DW)U=–WW. From the received results the equation follows
WW = k = –W^{2},  (6.k) 
that is fair for any Gmotion. We shall name such motions Mmotions, preserving square of 4acceleration. Substituting WW into the equation (6.GI) instead of k, we receive the equation of hyperbolic motion G=0. Thus, any Gmotion is hyperbolic motion.
The system of the equations (6.G) is logically «closer» to the definition of uniformly accelerated motion in Mkinematics and it is easier than the equation G=0, can easier be solved with regard to U. Multiplying equation G=0 by W, it is possible to receive the equation D(WW)=0 and further, – the equations (6.k) and (6.GII), allowing to substitute WW for k in the equation G=0 and to receive the equation (6.GI). Since, as the previous analysis has shown, equation G=0 and the system of equations (6.G) are mathematically equivalent for Mmotions, in definition (1) for singling out Gmotions from Mmotions the more simple system of equations (6.G) was chosen. This very system of equations meets the logic of singling out the uniformly accelerated motion according to Minkowski.
The set of all Gmotions forms the singleparametric family of Gkmotions with various values of the constant k=WW. If we limit ourselves to the studying of Gk_{0}motions with the fixed value k_{0}=–W_{0}^{2}, it is expedient to redefine the system of units of length and time measuring according to the conditions:
U := W_{0} := 1.  (7.UW_{0}) 
In such natural and complete, for Gk_{0}motions with the constant k_{0}, a system of units UW_{0} 4vectors of Gk_{0}motions demonstrate quite clearly the remarkable symmetries peculiar to them. In order to be convinced in it, we shall write down all values of Gk_{0}kinematics in their complete system of units UW_{0} and give that kinematics a new name.
Definition 2 Mmotion, meeting the kinematic conditions:
DW = U & WW = –1;  (7.H) 
DU = W & UU = +1,  (7.M) 
we will name the hyperbolic motion or Hmotion, and the kinematics of Hmotions – Hkinematics.
Under the system of equations (7.H), which completely determines (singling out) Hmotions among Mmotions, the it system of equations (7.M), made of definition of 4vector W in Mkinematics and the basic equation of Mkinematics, is written down clearly. This «liberty» is allowed for giving better presentation to the presence of symmetry in the pair of 4vectors U and W, that describe Hmotion.
If not to redefine the system of units according to (7.UW_{0}), but at once to define Hmotion as Gmotion with a constant k_{0}=–1, it is possible to receive the definition of Hmotion formally coincident with the definition (2). But then the fact remains unseen that any Gk_{0}motions, with the constant k_{0} chosen at random, automatically turn into Hmotions by simple conversion into their own kinematically complete (absolute) system of units determined by the correlations (7.UW_{0}).
Structure and properties of values of Hkinematics
It is expedient to write down the common solution of the system of equations (7.H)&(7.M) at once for the pair of velocity 4vectors U and the acceleration W, corresponding to the moment of the proper time s:
U = U_{0}chs + W_{0}shs,  (8.U) 
W = U_{0}shs + W_{0}chs,  (8.W) 
where: U_{0} and W_{0} – vertex values of 4vectors U and W, corresponding to the zero value of world parameter of evolution (invariant proper time) s=0.
The possibility of such a notation of solutions is evidence of the fact that the pair of 4vectors (U,W) of Hkinematics, corresponding to arbitrary value s, is the result of:
• linear homogeneous transformation of the vertex pair of 4vectors (U_{0},W_{0}), corresponding to the value s=0;
• hyperbolic turn of the vertex pair (U_{0},W_{0}) at the «angle» s.
It is characteristic for Hkinematics that combinations of 4vectors U±W are isotropic: – (U±W)^{2}=0, while the 4vectors U and W are equal to the halfsum and the halfdifference of the isotropic 4vectors U+W and U–W. The rcomponents u and w of 4vectors U and W which are laid off from a common point O draw at their motion hyperbolas in a commom plane uOw, whereas the rcomponents u±w of isotropic 4vectors U±W draw the asymptotes of these hyperbolas, intersecting in a point O. Such a hyperbolic behaviour of the rcomponents of 4vectors U and W at their Hmotion gave cause for naming such a motion as hyperbolic one.
The characteristic property of Hmotions reflecting symmetry peculiar to them, is the following sets of equations for 4vectors describing Hmotion:
U = D^{2}U = … = DW = D^{3}W = … ,  (8.1) 
W = D^{2}W = … = DU = D^{3}U = … .  (8.2) 
Solutions (8.U)&(8.W) may be obtained by way that clearly illustrates the structure of Hkinematics. Let's expand the velocity 4vector U(s) in Taylor series in powers of proper time s:
U(s) = U(0) + DU(0)s + D^{2}U(0)s^{2}/2! + … .  (8.3) 
According to the series of equations (8.1) and (8.2) all the odd derivatives, when expanded, are equal to W(0) and the even derivatives – to U(0). Using these identifications and rearranging components we obtain – with regard to determination of hyperbolic functions – the required solution:
U(s) = U(0)(1 + s^{2}/2! + …) + W(0)(s + s^{3}/3! + …).  (8.4) 
Acting similarly with W(s), – we obtain the resolution (8.W).
The geometry of Hmotions is also singled out by the fact that the values of 4vector total differentials dR and dW coincide and 4vector R turns out to be a shift (translation) of 4vector W on a (common for all the way) constant 4vector, equal to the difference R_{0}–W_{0} of values of these 4vectors in the vertex (a turning or return point) of Hmotion:
dR = dW, R – R_{0} = W – W_{0},  (8.R) 
where: R_{0} and W_{0} – vertex values of 4vectors R and W, corresponding to the zero value of world parameter of evolution (invariant proper time) s=0. The vertex 4vectors and their vertex components resemble «initial» values, still which they are not, as any Hmotion «begins» with value of world parameter s=–¥, passes through the vertex value at s=0 and «comes to an end» at s=+¥.
In Mkinematics Minkowski's world metrics is set with the help of the square of differential of 4radiusvector, which is equated to the square of differential of its proper invariant time:
ds^{2} := dR^{2} := dt^{2} – dr^{2}.  (8.M) 
In Hkinematics there exist its proper, not depending on Mkinematics, ways of definition of its proper invariant time s, the time which also stands here for the angle of hyperbolic turn of the pair of 4vectors U and W:
ds^{2} = – dU^{2} = dW^{2} = … .  (8.ds^{2}) 
In Hkinematics there exist following remarkable equations for the differentiation operator D including tcomponents t, l and p 4vectors R, U, W and the differentials of these components:
D := d/ds = l d/dt = p d/dl = l d/dp.  (8.D) 
dp = dt, p – p_{0} = t – t_{0}.  (8.p) 
Comparison of Hmotions with harmonic oscillations
It is useful to trace similarity and difference in the description of Hmotions and harmonic oscillations with the frequency equal to one. For that let us write out the corresponding systems of equations of first order and their common solutions in left and right columns:
dW/ds = U, dU/ds = W; dv/dt = – x, dx/dt = + v;  (9.1) 
U = U_{0}chs + W_{0}shs, x = x_{0}cost + v_{0}sint,  (9.2) 
W = U_{0}shs + W_{0}chs. v = – x_{0}sint + v_{0}cost.  (9.3) 
Let us define the universal operator of differentiation by angular parameter of evolution (of motion) or by parameter of duality, designating it by upper star: *:=d/ds and *:=d/dt. With its help our equations at once take more compact and heuristically useful form:
*(W,U) = (U,W). *(v,x) = (–x,v).  (9.*) 
Input equations are written down in the form of dual equations. Applying twice the operator of differentiation by the parameter of duality «*» to the value pairs (W,U) and (v,x), we obtain characterestic eqautions
**(W,U) = (W,U). **(v,x) = – (v,x).  (9.**) 
Let us give the following definitions to these two qualitatively different types of «duality»
Definition 3 The symmetry which is inherent to the equations *(W,U)=(U,W) and coresponding values we shall call hyperbolically dual symmetry or Hdual symmetry.
Definition 4 The symmetry which is inherent to the equations *(v,x)=(–x,v) and coresponding values we shall call Euclidean dual symmetry or Edual symmetry.
Hduality is called hyperbolic in view of possibility of representation of Hmotions, for which are true the equations *(W,U)=(U,W), as hyperbolic rotation of the pair of descibing it values (U,W). In its turn – the description of hyperbolic rotations is in need of hyperbolic functions of the angular parameter of evolution s, by which the differentiation is made.
Eduality is called Euclidean duality in view of possibility of representation of harmonic oscillations, for which are true the equations *(v,x)=(–x,v), as Euclidean rotation of the pair of descibing it values (x,v). In its turn – the description of Euclidean rotations is in need of circular trigonometrical functions of the angular parameter of evolution t, by which the differentiation is made.
Eduality of bivectors of free electromagnetic radiation field (E,H) and *(E,H)=(–H,E), another notation of which appear to be the antisymmetric tensors of the 2nd rank F and *F, is of particular interest.
Comparison of Hmotions with Lorentz transformations
In Hdual transformations (8.U)&(8.W) it is possible to see the formal features of «Lorentz's transformations» of U and a Wcomponents of a vertex (1+3)×2state vector (U_{0},W_{0}), which describes Hmotion, towards a new «frame reference» with a parameter of evolution (of duality) s. Indeed, – transformations of Lorentz (Ltransformations) of components of the 1×2vector (t_{0},x_{0}) into t and xcomponents in a new frame of reference look the following way:
t = t_{0}chĴ + x_{0}shĴ,  (10.t) 
x = t_{0}shĴ + x_{0}chĴ,  (10.x) 
where: Ĵ – a parameter of velocity (speed) of the Ltransformation, connected (chĴ=l) with the Lorentzfactor l (of Ltransformation); relative speed V of the new frame of reference is connected with l ratio l^{2}(1–V^{2})=1. As an occasion for analogy the fact serves, that both Hdual transformations (8.U)&(8.W), and Ltransformations (10.t)&(10.x) are set by equally arranged hyperbolic 2×2 matrixes H(0,s) and L(0,Ĵ).
When comparing Hmotions with harmonic oscillations it was noticed that the structure of hyperbolic matrix H(0,s), describing the transformations of the pair of 4vectors (U,W) at Hmotion, is the reflection of Hdual symmetry of such transformations. Just as that, – the hyperbolic structure of matrix of Ltransformation L(0,Ĵ) is determined by the Hdual equation of Ltransformation:
*(x,t) = (t,x), *: = d/dĴ,  (10.*) 
where: Ĵ – the angular parameter of duality of Ltransformation, by which differentiation is made.
All that proves that the Ltransformations as well as the Htransformations are the elements of the same, common for them, group – the Lgroup (group of Lorentz).
Heuristic considerations at parade of revealed SYMMETRIES
in the structures of field and of motion of its sources
Albert Einstein 
For the elaboration of of ideas of geometrization of electrodynamics in works of Poincare and Minkowski, by all previous development of mathematics and theoretical physics both formal and experimental bases were prepared. Symmetries of MLequations (MaxwellLorentz equations), concealed for the time being, more and more distinctly came out in the forefront. To the determination of these symmetries contributed the rationalization of system of units as well as the development of vector and tensor designations. Crucial experiments strengthened hte belief in the actual presence of these symmetries in the very nature of the physical processes being described. All that cleared the way for novel steps in the direction of dimension enlargement of the physical space and giving to it hyperbolic metrics.
• In the very form of record of differential D'Alambertian operator:
¶^{2}/¶t^{2} – Ñ^{2} = (¶/¶t, –Ñ)(¶/¶t, Ñ) = (¶/¶t, –Ñ)^{2}, Ñ := (¶/¶x, ¶/¶y, ¶/¶z),  (11.1) 
it was possible to guess the indication of 4vector character of Hamilton's 4operator (¶/¶t,–Ñ) and the hyperbolic signature (+ – – –) of metrics and scalar product of 4vectors.
• The same D'Alambertian operator affected both the scalar potential φ, and the 3vector potential A in the left parts of the MLequations.
(¶^{2}/¶t^{2} – Ñ^{2})φ = ρ, (¶^{2}/¶t^{2} – Ñ^{2})A = ρv.  (11.2) 
• It indicated the naturalness of their association in 4vector potential (φ,A) and formation of uniform 4vector density of a current of charges (ρ,ρv) entering the MLequations:
(¶^{2}/¶t^{2} – Ñ^{2})(φ, A) = (ρ, ρv).  (11.3) 
Electromagnetic potentials of Lorentz's theory: the scalar potential φ and the vector one A, they also have simple fourdimensional interpretation. As it was mentioned by Minkowski first (look [2], Minkowski I), they can be joined in one vector of fourdimensional world – fourdimensional potential… [1,§28]
• The condition (of potential calibration) of Lorentz as well as the continuity equation, – supposed the possibility of their record in the form of 4orthogonality condition of corresponding pairs of 4vectors:
(¶/¶t,–Ñ)(φ,A) = 0, (¶/¶t,–Ñ)(ρ, ρv) = 0.  (11.4) 
This natural scheme of construction of 4vector Minkowski's World, no doubt, – came as the result of his meditations on the essence of Lorentz transformations, which have an affect, above all, on the values, entering the MLequations.
Interestingly, it was namely Maxwell's wave equation which first made Lorentz transformation known to the public… Woldemar Voigt showed in 1887 [52], that the equations of the type (¶^{2}/¶t^{2}–Ñ^{2})φ=0 preserve the form at the transition to new spacetime variables… , congruous, to scaling multiplier, to Lorentz transformations. [11,]
Both then and today, many people still fail to notice or deliberately ignore the next obvious fact – in the structure of Minkowski's World there is nothing that goes beyond:
• the demands of the mathematical notation of MLequations for the values they contain;
• anything that was conditioned by the character of Lorentz transformations, which force on these values.
It should be emphasized that, staying within the framework of electrodynamics, the postulates (principles) of Einstein, – which form the basis of his Special Theory of Relativity, – are just theorems in the World of Minkowski. The Scheme of construction of electrodynamics in the World of Minkowski, which is specially built up to anwer its needs, doesn't need the postulates of Einstein as the formal basis. Instead of those postulates, the leading hand was that of the already revealed symmetries of MLequations together with the symmetries, which were set by Lorentz transformations.
Hyperbolic mixing of t and rcomponents of 4vectors during Ltransformations, reminding Euclidean mixing of spatial coordinates of 3vector (rcomponent) at spatial turn of a coordinate reference point, – was for Minkowski a final stroke that made him accept the physical reality of 4vectors in electrodynamics and its physical geometry.
Hyperbolic mixing of time and spatial coordinates during Ltransformations, reminding the mixing of coordinates of Euclidean 3vector at spatial turn of a coordinate reference point, has served Minkowski as the convincing basis for their unification in uniform into spacetime – 4vector World of Minkowski.
It is authentically known, that the Program of researches of Minkowski on geometrization of electrodynamics was not limited to the restoration of symmetry of Maxwell equations concerning Ltransformations by transition from Nkinematics to Mkinematics in the fourdimensional World. He was the first to write down the first pair of equations of Maxwell with the help of both bivector (E,H) (Maxwell tensor F), and the one dual to it (–H,E) (*F, dual to Maxwell tensor F) with the purpose of restoration of dual symmetry of Maxwell equations [1,§28].
This Minkowski's achievemnet actually led to the restoration of Edual symmetry in the part of Maxwell equations that described of free radiation field. In the solutions themselves, reperesenting electric waves, this Edual symmetry was already present. This is quite natural, for in (plane) electric wave harmonic oscillations of respective field components take place. The doubling of number of field equations, which at first sight seems unnecessary, is justified by the fact that it is always expedient to deal with such a notation of equations, to which the symmetry, observed in the solutions of these equations, is intrinsical in an explicit form.
At the attempt of immediate record of the second pair of Maxwell's equations with sources in dual form, inevitable problems appear. Mmotions of sources of a field are still too wide for the maintenance of Hdual symmetry in the second pair of Maxwell equations.
Minkowski turns to hyperbolic motion of field sources (additional symmetry) as to the necessary to him additional narrowing of Mmotion of sources. We do not know for certain how long he advanced in this direction. The work of Minkowski on hyperbolic motion remained unpublished after his sudden death. Another work on hyperbolic motion of Max Born [3], who under David Hilbert's patronage was engaged directly in an estimation of the importance and expediency of preparation for the edition of the unpublished physical works of his teacher has appeared. (See: – 6.2 Goettingen's Tragedy as Choice of History)
Heroic labour of great physicists and mathematicians on the boundary of the XIX and XX centuries, directed at «removal» of the equations of Maxwell into a new 4vector World of Minkowski essentially narrowed both allowable Mmotions of sources in comparison with Nmotions in the Absolute World of Newton, and allowable structure of the field generated by charged sources. All these performances appeared as a result of the work done on restoration of the symmetry (covariance property) of the equations of Maxwell concerning the Ltransformations from Lgroup.
In SubQFT there is a need of additional symmetrization of the equations of Maxwell concerning the Htransformations from Lgroup and the subsequent development of these remarkable equations, – a fascinating acquaintance with drawings of the subquantum base by which both the World of Minkowski, and the World of Quantum together with all their already known or for the time being latent faces are supported.
The theme of additional symmetrization of Maxwell's equations for the subquantum field sources moving hyperbolically is enlightened in a separate article – 5.4 Symmetrization of the Maxwell Equations. Below are given two alternative variants of «symmetrized» equations of Maxwell in widely known designations of the Problem 4.21. from «Problem book in Relativity and Gravitation» [32] and in designations of Pauli in his «Encyclopedic article» [1,§28]:
F^{μν}_{,ν} = 4πJ^{μ} « div F = ρ_{0}U,  (12.U) 
*F^{μν}_{,ν} = 4πK^{μ} « div*F = μ_{0}U.  (13.U) 
rot*F = ρ_{0}W.  (12.W) 
A really new thing here is the replacement of the 4operator div by 4operator rot, that is the transition in the left part of the «dual» equations from [32] from the 4vector to the tensor of the 3rd rank and the assumption of the coincidence of its corresponding components with the components of the 4vector ρ_{0}W. It is clear, that it cardinally changes all the physical interpretation of the *MLequations. To the substantiation and analysis of thus symmetrized system of the equations of Maxwell:
div*F = 0 & div F = ρ_{0}U,  (12.ML) 
rot F = 0 & rot*F = ρ_{0}W,  (12.*ML) 
the article 5.4 Symmetrization of the Maxwell Equations will be given up.
Originally, hyperbolic motion of field sources has attracted attention due to the absence of radiation. At the same time it was found out, that Hmotion is naturally singled out from set of Hmotions only by the additional symmetries of its kinematics. Hmotion in the family of its own (instantaneously accompanying) frames is the most symmetrical among arbitrarily arranged accelerated motions – it has there a constant, permanent (in its value and direction) along all the way, unit 3vector of acceleration a_{0}. General 4motion in set of proper Lorentz frames, being realized by sequence of infinitesimal Lorentz transformations, comes to hyperbolic rotation of taxis according raxes, whereas – at such a transformation no Euclidean rotations of the very raxes take place. Spatial raxes undergo strictly parallel shift at their transformation. At this very motion of raxes, which is realized by Lorentz transformations, parallel shift of both spatial raxes and permanent unit 3vector of Newton acceleration a_{0}, resolved along these axes, take place.
Let us absolutely arbitrarily fix one of these instantaneously concomitant Lorentz frames K_{0} and describe in it Hmotion along all Hway. In this frame K_{0} at the moment of its fixation s_{0}=0 (freezing of its velocity of motion) 4vectors U and W took on values U_{0}=(1,0) and W_{0}=(0,a_{0}). When reevaluating in K_{0} the values of U and W from all other instantaneously concomitant Lorentz frames (performing corresponding Lorentz tranformations), we will get the values of these 4vectors (chs,a_{0}shs) and (shs,a_{0}chs) in K_{0}, fitting with the moment of proper time s (ar the angle of hyperbolic turn). Spatial rcomponent of these 4vectors is in strictly onedimensional motion along permanent direction a_{0}. It is characteristic for Hmotion that the just obtained picture of hyperbolic motion in K_{0} doesn't depend in the least on its concrete choice from corresponding set. Description of Hmotion is invariant relatively to the choice of concrete frame K_{0} from set of all instantaneously concomitant (to arbitrary point of Hway) Lorentz frames.
It is remarkable that this very especially simple motion became that very accelerated motion of field sources which does not generate radiation. As radiation is a constituent of the field with qualitatively singled out structure, it is natural to make the assumption of the existence of an unknown to us (conservative) symmetry of equations of Maxwell, at breaking which there takes place a structural reorganization of a field with the formation of qualitatively new component – radiation.
Spherically symmetrical set of Hmouvements
Let us write down in more detail the common solution of the system of equations (7.H)&(7.M) at once for the pair of 4vectors U and W, corresponding to the moment of the proper time s, in the form (8.U)&(8.W) chosen before:
U = U_{0}l/l_{0} + W_{0}t/l_{0} = (l, lv),  (14.U) 
W = U_{0}t/l_{0} + W_{0}l/l_{0} = (t, tv + l^{2}a), p = t,  (14.W) 
U_{0} = (l_{0}, l_{0}v_{0}), W_{0} = (0, l_{0}^{2}a_{0}), v_{0}a_{0} = 0, l_{0}^{4}a_{0}^{2} = 1,  (14.0) 
l = l_{0}chs, t = l_{0}shs, l^{2} = l_{0}^{2} + t^{2}, t_{0} = p_{0} = 0,  (14.l) 
where: U_{0} and W_{0} – vertex values of 4vectors U и W; l and t – tcomponents of 4vectors U and W; v_{0} and a_{0} – vertex values of 3vectors of Newton's velocity v and acceleration a; l_{0} – vertex Lorentz factor or the vertex value of tcomponent of 4vector U_{0}; the vertex values of tcomponents of 4vectors R_{0} and W_{0} are chosen equal to zero (t_{0}=p_{0}=0).
The choice of zero values for vertex tcomponents (t_{0}=p_{0}=0) of vertex 4vectors R_{0} and W_{0} right away leads both to the equation t=p (to complete identification of values of tcomponents of 4vectors R and W) and orthogonality of vertex 3vector Newton velocity v_{0} and acceleration a_{0}. The latter is connected to the presence of the equation p_{0}=l_{0}^{4}v_{0}a_{0}. The necessity of such a choice is caused by the attempt to obtain strictly spherically symetric Hmotions for subcurrents of electron. In order to ensure such a symmetry it is strictly necessary to fulfil the condition p_{0}=0, bringing with it v_{0}a_{0}=0 (and vice versa). The choice t_{0}=0 is not of of fundamental importance but leads to the simplification of notation of formulas.
The rcomponents u and w of 4vectors U and W which are laid off from a common point O draw at their motion hyperbolas in a commom plane uOw, whereas the rcomponents u±w of isotropic 4vectors U±W draw the asymptotes of these hyperbolas, intersecting in a point O. The angle of inclination of asymptotes ±α to waxis of symmetry of whyperbolas, described at the motion w, is connected to vertex Lorent factor l_{0} by the ratio l_{0}cosα=1. On waxis lies a_{0}. Accordingly, — on uaxis symmetry of uhyperbolas, described at the motion u, – lies v_{0}. Both vertex 3vectors v_{0} and a_{0}, and u and waxes themselves are (Euclidean) orthogonal. Expansion of rcomponents U and W by Newton 3vectors of vertex velocity v_{0} and acceleration a_{0} (by u and waxes) looks like:
u = lv_{0} + l_{0}ta_{0}, v_{0} = v_{0}i,  (15.u) 
w = tv_{0} + l_{0}la_{0}, a_{0} = m j/l_{0}^{2},  (15.w) 
where: i and j – unit vector (orts) on u and waxes. The expansions of Newton velocity 3vectors v and of acceleration a look like:
v = v_{0} + l_{0}ta_{0}/l = v_{0}i m tj/l_{0}l,  (16.v) 
a = l_{0}^{3}a_{0}/l^{3} = m l_{0}j/l^{3}.  (16.a) 
In Hkinematics a point of view is preferable, in accordance with which – fundamental (basic, determinant, initial, primary, base) value appears to be the very pair of 4vectors (U,W), whereas the 4radius vector R is secondary and can be reconstructed on their basis, taking into consideration corresponding conditions. Formal status of R in Hkinematics is similar to role of 4vector potential (φ,A) in electrodynamics.
In view of the identity of Hkinematics r–r_{0}=w–w_{0} for reconstruction of rcomponent R it is enough to choose its 3spatial vertex direction. With the condition of spherical symmetry only its vertext waxis direction is compatible r_{0}=r_{0}j. The reconstruction in this way of expansion of rcomponent R at u and waxes finally looks like:
r = tv_{0}i + (r_{0} ± 1 m l/l_{0})j.  (17.r) 
The very procedure of 3vector r reconstruction, as well as its expansion (17.r) evidence that r at its motion draws a hyperbola. Both this rhiperbola and its asymptotes result from the spatial shift of whyperbola and its asymptotes by constant 3vector r_{0}–w_{0}.
For the determination of position of asymptotes of rhyperbolas it is useful, addiotionally to their angle of inclination to waxis ±α, to determine aiming parameter o equal to (the shortest) distance of asymptotes from point O. Aiming parameter o is connected with vertex parameters v_{0} and r_{0} by the correlation:
o = v_{0}(r_{0} ± 1).  (18.o) 
With the help of symbols «±» and «m» there are described values of at once two qualitatively different spherically symmetrical subfamilies of Hmotions, having oppositely directed Newton accelerations a. Strictly speaking, it is necessary either to write two groups of formulae or to add an index «±» to all values. It is supposed that the reader, when necessary, will always be able to do it. Full spherically symmetrical set of solutions of system (7.H)&(7.M) may be obtained from described above by means of threedimensional rotations around u, w and zaxes. The third spatial zaxis is laid from point O at right angle to the plane uOw in the corresponding direction.
The translation from Russian was made by Masha and Natasha Zazerska Last modifications: July 16 2005  RU  Back to Contens 
Spherically symmetrical set of Zmouvements*
For trajectories of motion (17.r) there takes place a relationship between Lorentzfactor l(t) and the distance to the center of symmetry r(t) of an arbitrarily selected point
r^{2} = (l m k)^{2} + (l_{0}^{2} – 1)(k^{2} – 1), k := (r_{0} ± 1)/l_{0},  (19*) 
that shows that the Lorentzfactor l and the value of Newton velocity v, connected with it, are nonizotropic and depend on an individual parameter of the trajectory. If to set up, in volitional way, a connection between the vertex parameters
l_{0} = r_{0} ± 1, k = 1,  (20*) 
this additional symmetrization of spherically symmetrical Hkinematics will result in substantial simplification of its structure and, in particular:
o = v_{0}l_{0}, l = r ± 1.  (21*) 
The set of solutions (17.r) after degeneration by one of the parameters (selection in conformity with the symmetry (20*)) becomes isotropic by vv (consequently: – by l and uu) and singleparametric (depending, for instance, only on the aiming parameter o).
Definition 3* Spherically symmetrical Hmotions isotropic everywhere by vv (Euclid norm of 3vector v) we call Zmotions and kinematics of Zmotions – Zkinematics.
In Zmotion the magnitude of W displacement for obtaining Z 4radiusvector coincides with the vertex Lorentzfactor (20*) along the axis OY
Z = W + (0, 0, l_{0}, 0),  (22*) 
which is a result of Zsymmetrization of vector R(n). The use of a new letter Z for description of 4radiusvector is justified by change of its nature. «SpaceTime» of Z, H, G, Mkinematics inserted into each other requires a definition independent of relativistic kinematics, which will become possible in the process of determination of the law of motion. It should not be just well selected stage for the presentation (description) of physical collisions (events), but it is to be organically woven into the closed structure of subquantum field theory. Moreover, it is not possible to apply measuring rulers and clocks to subquantum level.
The translation from Russian was made by Yuri Nezhentsev Last modifications: March 30 2005  RU  Back to Contens 
The Literature Quoted:
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