5.2  Maxwell Equations single out Hyperbolic Motion of Field Sources

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 ML-equations, 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 self-radiation, 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 ML-equations 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 4-vector World of Minkowski – M-kinematics can be unambiguously defined (singled out) solely by the symmetries of the ML-equations. 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 M-kinematics. 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 ML-equations.

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:

These values are modelled of 4-vectors and their components obtained by successive differentiating world 4-radius-vector R on the proper invariant time s according to M-kinematics scheme, – or of the corresponding values of Newton's kinematics – N-kinematics:

Corresponding t- and r-components of 4-vectors of M-kinematics are in standard way connected with 3-vectors r,v,a,b,… and time t of N-kinematics:

The above-written 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 M-kinematics and time dt of N-kinematics, coincident with differential of t-component of 4-vector R, are connected (lds=dt) by Lorentz-factor  l with the help of the fundamental equation of M-kinematics

Here indexless designations of 4-vectors 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 4-vectors A:=(a⁰,a) and B:=(b⁰,b), singled out by their t-components a⁰ and b⁰, and by r-components a and b, it is necessary to follow the rule:

where ab – a usual scalar product of 3-vectors with the signature (+ + +). Usually, for this purpose, 4-vector A is written down in counter-variant components aⁱ:=(a⁰,a), and B in covariant b_i:=(b₀,–b), or they multiply r-components of 4-vectors by imaginary unit. 3-vectors and r-components of 4-vectors are typed as normal bold letters (non-italic) and can be both lowercase, and UPPERCASE.

Used by Abraham and Pauli 3-vector K (formula (265a) [1]) and 4-vector K (formula (265) [1], written down there in a metric with the signature (+ + + –)) assume in our designations the following form:

where c – the dimension matching factor, dependent on the system of units used.

Equality of kinematic values g₃ 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 M-kinematics – the M-motions satisfying an additional condition G=0, we shall for distinctness call hyperbolic motions.

Minkowski has singled out in M-kinematics 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 M-kinematics, determined by a constancy (preservation) of a 3-vector of acceleration a along all world way in family of (instantaneously) accompanying inertial (Galilean) frames of reference, – is singled out in M-kinematics only by considerations of symmetry in the family of vectors of acceleration a_K' , which have purely kinematic nature.

Definition 1  M-motion in conformity with kinematic conditions:

will be called hyperbolic or uniformly accelerated motion, or – G-motion, and kinematics of G-motions will be called G-kinematics.

From the equation (6.GII) follows, that k is a constant preserving along all G-motion. At such k the equation (6.GI) coincides with the 4-vector covariant condition, singling out uniformly accelerated motion, and conformable to its 3-vector condition b=0 in family of the own (instantaneously-accompanying) frames of reference, correlated to the flock of all points of the world way. Thus, any G-motion 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

that is fair for any G-motion. We shall name such motions M-motions, preserving square of 4-acceleration. Substituting WW into the equation (6.GI) instead of k, we receive the equation of hyperbolic motion G=0. Thus, any G-motion is hyperbolic motion.

The system of the equations (6.G) is logically «closer» to the definition of uniformly accelerated motion in M-kinematics 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 M-motions, in definition (1) for singling out G-motions from M-motions 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 G-motions forms the single-parametric family of Gk-motions with various values of the constant k=WW. If we limit ourselves to the studying of Gk₀-motions with the fixed value k₀=–W₀², it is expedient to redefine the system of units of length and time measuring according to the conditions:

In such natural and complete, for Gk₀-motions with the constant k₀, a system of units UW₀ 4-vectors of Gk₀-motions demonstrate quite clearly the remarkable symmetries peculiar to them. In order to be convinced in it, we shall write down all values of Gk₀-kinematics in their complete system of units UW₀ and give that kinematics a new name.

Definition 2  M-motion, meeting the kinematic conditions:

we will name the hyperbolic motion or H-motion, and the kinematics of H-motions – H-kinematics.

Under the system of equations (7.H), which completely determines (singling out) H-motions among M-motions, the it system of equations (7.M), made of definition of 4-vector W in M-kinematics and the basic equation of M-kinematics, is written down clearly. This «liberty» is allowed for giving better presentation to the presence of symmetry in the pair of 4-vectors U and W, that describe H-motion.

If not to redefine the system of units according to (7.UW₀), but at once to define H-motion as G-motion with a constant k₀=–1, it is possible to receive the definition of H-motion formally coincident with the definition (2). But then the fact remains unseen that any Gk₀-motions, with the constant k₀ chosen at random, automatically turn into H-motions by simple conversion into their own kinematically complete (absolute) system of units determined by the correlations (7.UW₀).

Structure and properties of values of H-kinematics

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 4-vectors U and the acceleration W, corresponding to the moment of the proper time s:

where: U₀ and W₀ – vertex values of 4-vectors 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 4-vectors (U,W) of H-kinematics, corresponding to arbitrary value s, is the result of:

• linear homogeneous transformation of the vertex pair of 4-vectors (U₀,W₀), corresponding to the value s=0;

• hyperbolic turn of the vertex pair (U₀,W₀) at the «angle» s.

It is characteristic for H-kinematics that combinations of 4-vectors U±W are isotropic: – (U±W)²=0, while the 4-vectors U and W are equal to the half-sum and the half-difference of the isotropic 4-vectors U+W and U–W. The r-components u and w of 4-vectors U and W which are laid off from a common point O draw at their motion hyperbolas in a commom plane uOw, whereas the r-components u±w of isotropic 4-vectors U±W draw the asymptotes of these hyperbolas, intersecting in a point O. Such a hyperbolic behaviour of the r-components of 4-vectors U and W at their H-motion gave cause for naming such a motion as hyperbolic one.

The characteristic property of H-motions reflecting symmetry peculiar to them, is the following sets of equations for 4-vectors describing H-motion:

Solutions (8.U)&(8.W) may be obtained by way that clearly illustrates the structure of H-kinematics. Let's expand the velocity 4-vector U(s) in Taylor series in powers of proper time s:

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 re-arranging components we obtain – with regard to determination of hyperbolic functions – the required solution:

Acting similarly with W(s), – we obtain the resolution (8.W).

The geometry of H-motions is also singled out by the fact that the values of 4-vector total differentials dR and dW coincide and 4-vector R turns out to be a shift (translation) of 4-vector W on a (common for all the way) constant 4-vector, equal to the difference R₀–W₀ of values of these 4-vectors in the vertex (a turning or return point) of H-motion:

where: R₀ and W₀ – vertex values of 4-vectors R and W, corresponding to the zero value of world parameter of evolution (invariant proper time) s=0. The vertex 4-vectors and their vertex components resemble «initial» values, still which they are not, as any H-motion «begins» with value of world parameter s=–¥, passes through the vertex value at s=0 and «comes to an end» at s=+¥.

In M-kinematics Minkowski's world metrics is set with the help of the square of differential of 4-radius-vector, which is equated to the square of differential of its proper invariant time:

In H-kinematics there exist its proper, not depending on M-kinematics, 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 4-vectors U and W:

In H-kinematics there exist following remarkable equations for the differentiation operator D including t-components t, l and p 4-vectors R, U, W and the differentials of these components:

Comparison of H-motions with harmonic oscillations

It is useful to trace similarity and difference in the description of H-motions 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:

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:

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

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 H-dual 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 E-dual symmetry.

H-duality is called hyperbolic in view of possibility of representation of H-motions, 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.

E-duality 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.

E-duality 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 H-motions with Lorentz transformations

In H-dual transformations (8.U)&(8.W) it is possible to see the formal features of «Lorentz's transformations» of U- and a W-components of a vertex (1+3)×2-state vector (U₀,W₀), which describes H-motion, towards a new «frame reference» with a parameter of evolution (of duality) s. Indeed, – transformations of Lorentz (L-transformations) of components of the 1×2-vector (t₀,x₀) into t- and x-components in a new frame of reference look the following way:

where: Ĵ – a parameter of velocity (speed) of the L-transformation, connected (chĴ=l) with the Lorentz-factor l  (of L-transformation); relative speed V of the new frame of reference is connected with l ratio l²(1–V²)=1. As an occasion for analogy the fact serves, that both H-dual transformations (8.U)&(8.W), and L-transformations (10.t)&(10.x) are set by equally arranged hyperbolic 2×2 matrixes H(0,s) and L(0,Ĵ).

When comparing H-motions with harmonic oscillations it was noticed that the structure of hyperbolic matrix H(0,s), describing the transformations of the pair of 4-vectors (U,W) at H-motion, is the reflection of H-dual symmetry of such transformations. Just as that, – the hyperbolic structure of matrix of L-transformation L(0,Ĵ) is determined by the H-dual equation of L-transformation:

where: Ĵ – the angular parameter of duality of L-transformation, by which differentiation is made.

All that proves that the L-transformations as well as the H-transformations are the elements of the same, common for them, group – the L-group (group of Lorentz).

Heuristic considerations at parade of revealed SYMMETRIES
in the structures of field and of motion of its sources

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 ML-equations (Maxwell-Lorentz 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:

it was possible to guess the indication of 4-vector character of Hamilton's 4-operator (¶/¶t,–Ñ) and the hyperbolic signature (+ – – –) of metrics and scalar product of 4-vectors.

•  The same D'Alambertian operator affected both the scalar potential φ, and the 3-vector potential A in the left parts of the ML-equations.

•  It indicated the naturalness of their association in 4-vector potential (φ,A) and formation of uniform 4-vector density of a current of charges (ρ,ρv) entering the ML-equations:

Electromagnetic potentials of Lorentz's theory: the scalar potential φ and the vector one A, they also have simple four-dimensional interpretation. As it was mentioned by Minkowski first (look [2], Minkowski I), they can be joined in one vector of four-dimensional world – four-dimensional 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 4-orthogonality condition of corresponding pairs of 4-vectors:

This natural scheme of construction of 4-vector 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 ML-equations.

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/¶t²–Ѳ)φ=0 preserve the form at the transition to new space-time 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 ML-equations 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 ML-equations together with the symmetries, which were set by Lorentz transformations.

Hyperbolic mixing of t- and r-components of 4-vectors during L-transformations, reminding Euclidean mixing of spatial coordinates of 3-vector (r-component) at spatial turn of a coordinate reference point, – was for Minkowski a final stroke that made him accept the physical reality of 4-vectors in electrodynamics and its physical geometry.

Hyperbolic mixing of time and spatial coordinates during L-transformations, reminding the mixing of coordinates of Euclidean 3-vector at spatial turn of a coordinate reference point, has served Minkowski as the convincing basis for their unification in uniform into space-time – 4-vector 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 L-transformations by transition from N-kinematics to M-kinematics in the four-dimensional 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 E-dual symmetry in the part of Maxwell equations that described of free radiation field. In the solutions themselves, reperesenting electric waves, this E-dual 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. M-motions of sources of a field are still too wide for the maintenance of H-dual 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 M-motion 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 4-vector World of Minkowski essentially narrowed both allowable M-motions of sources in comparison with N-motions 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 L-transformations from L-group.

In SubQFT there is a need of additional symmetrization of the equations of Maxwell concerning the H-transformations from L-group 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]:

A really new thing here is the replacement of the 4-operator div by 4-operator rot, that is the transition in the left part of the «dual» equations from [32] from the 4-vector to the tensor of the 3-rd rank and the assumption of the coincidence of its corresponding components with the components of the 4-vector ρ₀W. It is clear, that it cardinally changes all the physical interpretation of the *ML-equations. To the substantiation and analysis of thus symmetrized system of the equations of Maxwell:

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 H-motion is naturally singled out from set of H-motions only by the additional symmetries of its kinematics. H-motion 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 3-vector of acceleration a₀. General 4-motion in set of proper Lorentz frames, being realized by sequence of infinitesimal Lorentz transformations, comes to hyperbolic rotation of t-axis according r-axes, whereas – at such a transformation no Euclidean rotations of the very r-axes take place. Spatial r-axes undergo strictly parallel shift at their transformation. At this very motion of r-axes, which is realized by Lorentz transformations, parallel shift of both spatial r-axes and permanent unit 3-vector of Newton acceleration a₀, resolved along these axes, take place.

Let us absolutely arbitrarily fix one of these instantaneously concomitant Lorentz frames K₀ and describe in it H-motion along all H-way. In this frame K₀ at the moment of its fixation s₀=0 (freezing of its velocity of motion) 4-vectors U and W took on values U₀=(1,0) and W₀=(0,a₀). When reevaluating in K₀ the values of U and W from all other instantaneously concomitant Lorentz frames (performing corresponding Lorentz tranformations), we will get the values of these 4-vectors (chs,a₀shs) and (shs,a₀chs) in K₀, fitting with the moment of proper time s (ar the angle of hyperbolic turn). Spatial r-component of these 4-vectors is in strictly one-dimensional motion along permanent direction a₀. It is characteristic for H-motion that the just obtained picture of hyperbolic motion in K₀ doesn't depend in the least on its concrete choice from corresponding set. Description of H-motion is invariant relatively to the choice of concrete frame K₀ from set of all instantaneously concomitant (to arbitrary point of H-way) 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 H-mouvements

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 4-vectors U and W, corresponding to the moment of the proper time s, in the form (8.U)&(8.W) chosen before:

where: U₀ and W₀ – vertex values of 4-vectors U и W; l and t – t-components of 4-vectors U and W; v₀ and a₀ – vertex values of 3-vectors of Newton's velocity v and acceleration a; l₀ – vertex Lorentz factor or the vertex value of t-component of 4-vector U₀; the vertex values of t-components of 4-vectors R₀ and W₀ are chosen equal to zero (t₀=p₀=0).

The choice of zero values for vertex t-components (t₀=p₀=0) of vertex 4-vectors R₀ and W₀ right away leads both to the equation t=p (to complete identification of values of t-components of 4-vectors R and W) and orthogonality of vertex 3-vector Newton velocity v₀ and acceleration a₀. The latter is connected to the presence of the equation p₀=l₀⁴v₀a₀. The necessity of such a choice is caused by the attempt to obtain strictly spherically symetric H-motions for subcurrents of electron. In order to ensure such a symmetry it is strictly necessary to fulfil the condition p₀=0, bringing with it v₀a₀=0 (and vice versa). The choice t₀=0 is not of of fundamental importance but leads to the simplification of notation of formulas.

The r-components u and w of 4-vectors U and W which are laid off from a common point O draw at their motion hyperbolas in a commom plane uOw, whereas the r-components u±w of isotropic 4-vectors U±W draw the asymptotes of these hyperbolas, intersecting in a point O. The angle of inclination of asymptotes ±α to w-axis of symmetry of w-hyperbolas, described at the motion w, is connected to vertex Lorent factor l₀ by the ratio l₀cosα=1. On w-axis lies a₀. Accordingly, — on u-axis symmetry of u-hyperbolas, described at the motion u, – lies v₀. Both vertex 3-vectors v₀ and a₀, and u- and w-axes themselves are (Euclidean) orthogonal. Expansion of r-components U and W by Newton 3-vectors of vertex velocity v₀ and acceleration a₀ (by u- and w-axes) looks like:

where: i and j – unit vector (orts) on u- and w-axes. The expansions of Newton velocity 3-vectors v and of acceleration a look like:

In H-kinematics a point of view is preferable, in accordance with which – fundamental (basic, determinant, initial, primary, base) value appears to be the very pair of 4-vectors (U,W), whereas the 4-radius vector R is secondary and can be reconstructed on their basis, taking into consideration corresponding conditions. Formal status of R in H-kinematics is similar to role of 4-vector potential (φ,A) in electrodynamics.

In view of the identity of H-kinematics r–r₀=w–w₀ for reconstruction of r-component R it is enough to choose its 3-spatial vertex direction. With the condition of spherical symmetry only its vertext w-axis direction is compatible r₀=r₀j. The reconstruction in this way of expansion of r-component R at u- and w-axes finally looks like:

The very procedure of 3-vector r reconstruction, as well as its expansion (17.r) evidence that r at its motion draws a hyperbola. Both this r-hiperbola and its asymptotes result from the spatial shift of w-hyperbola and its asymptotes by constant 3-vector r₀–w₀.

For the determination of position of asymptotes of r-hyperbolas it is useful, addiotionally to their angle of inclination to w-axis ±α, to determine aiming parameter o equal to (the shortest) distance of asymptotes from point O. Aiming parameter o is connected with vertex parameters v₀ and r₀ by the correlation:

With the help of symbols «±» and «m» there are described values of at once two qualitatively different spherically symmetrical subfamilies of H-motions, 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 three-dimensional rotations around u-, w- and z-axes. The third spatial z-axis is laid from point O at right angle to the plane uOw in the corresponding direction.

Spherically symmetrical set of Z-mouvements*

For trajectories of motion (17.r) there takes place a relationship between Lorentz-factor l(t) and the distance to the center of symmetry r(t) of an arbitrarily selected point

that shows that the Lorentz-factor l and the value of Newton velocity v, connected with it, are non-izotropic and depend on an individual parameter of the trajectory. If to set up, in volitional way, a connection between the vertex parameters

this additional symmetrization of spherically symmetrical H-kinematics will result in substantial simplification of its structure and, in particular:

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 single-parametric (depending, for instance, only on the aiming parameter o).

Definition 3*  Spherically symmetrical H-motions isotropic everywhere by vv (Euclid norm of 3-vector v) we call Z-motions and kinematics of Z-motions – Z-kinematics.

In Z-motion the magnitude of W displacement for obtaining Z 4-radius-vector coincides with the vertex Lorentz-factor (20*) along the axis OY

which is a result of Z-symmetrization of vector R(n). The use of a new letter Z for description of 4-radius-vector is justified by change of its nature. «Space-Time» of Z-, H-, G-, M-kinematics 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.

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