2026-08-28
Temporal Field Projections
Origin of Electric, Magnetic, and Wave Fields
In the first part, the primary temporal field was introduced—a directed change of the global state \(J\) along time. It was also proposed to consider space not as a pre-existing container, but as a system of independent directions arising from the orthogonal splitting of this field. In this approach, the spatial coordinate is the accumulated value of the corresponding projection.
Now we must take the next step: to understand how different ways of organizing the temporal field create known physical fields. For this, it is not enough to immediately introduce the three-dimensional vectors \(\mathbf E\) and \(\mathbf B\). First, the corresponding components must be obtained in the geometry of the global operator \(J\), then mapped into space, and only then compared with observable quantities.
\[\tag{1}\boxed{\text{temporal field}\;\longrightarrow\;\text{component in geometry }J\;\longrightarrow\;\text{spatial projection}\;\longrightarrow\;\text{observable field}}\] The main hypothesis of this work is that the electric, magnetic, and wave fields are not independent primary entities. They arise as different spatial projections of a single temporal field, and the type of projection is determined by the internal dynamics, number of sheets, external motion, orientation, and closure of the wave structure.
1. Initial Temporal Field
The global state of Wave Electricity is written as
\[\tag{2}J(a,b)=\j^a(-\j)^b,\] where \(a\) describes the internal state, and \(b\) describes the external motion. For internal evolution
\[\tag{3}a=\varpi\tau,\qquad \omega=\pi\varpi\] the temporal field is determined by the derivative
\[\tag{4}\boxed{\mathcal T_J=\frac{dJ}{d\tau}}.\] For \(b=0\) and
\[\tag{5}J(\tau)=\j^{\varpi\tau}=\ep+\em e^{i\omega\tau}\] it has an explicit form
\[\tag{6}\boxed{\mathcal T_J=i\omega\em e^{i\omega\tau}}.\] This field is primary to spatial projections. Its modulus is determined by the internal frequency, and its directivity continuously changes in the phase plane:
\[\tag{7}|\mathcal T_J|=\omega,\qquad J\cdot\mathcal T_J=0.\] Next, the same field will be considered under different geometric parameters:
\[\tag{8}\boxed{\mathcal F_k=\Pi_k\!\left[\mathcal T_J\right],\qquad \Pi_k=\Pi_k(\dot a,N,b,\chi,\Omega,\dot b,\ldots)}\] Here \(N\) is the number of sheets, \(\chi\) is is the degree of opening, \(\Omega\) is the orientation of the internal plane, and \(\dot b\) characterizes the change in external motion.
2. Mean and difference fields of a multi-sheet structure
Let the global structure contain \(N\) sheets. Each leaf state corresponds to a temporal field. \[\tag{9}\mathcal T_{J,k}=\frac{dJ_k}{d\tau},\qquad k=1,2,\ldots,N.\]
\[\tag{55}\boxed{F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu}.\] The average component is:
\[\tag{10}\boxed{\overline{\mathcal T}_J=\frac1N\sum_{k=1}^{N}\mathcal T_{J,k}}.\] The deviation of each leaf from the average state defines the difference component:
\[\tag{11}\boxed{\delta\mathcal T_{J,k}=\mathcal T_{J,k}-\overline{\mathcal T}_J}.\] By definition, their sum cancels out:
\[\tag{12}\sum_{k=1}^{N}\delta\mathcal T_{J,k}=0.\] The middle part characterizes the overall internal motion of the structure and the associated energy scale. The difference parts characterize the differences between the sheets. With full internal compensation, they do not change the total energy, but can produce non-zero external projections.
\[\tag{13}\boxed{\overline{\mathcal T}_J\;\longrightarrow\;\text{total energy and motion},\qquad \delta\mathcal T_{J,k}\;\longrightarrow\;\text{fields of difference between sheets}}\] 3. Univalent closed structure
First, let's consider the simplest case:
\[\tag{14}N=1,\qquad b=0,\qquad \chi=0.\] Here, there is only one field \(\mathcal T_J\), so the average field coincides with the total field, and there is no difference component:
\[\tag{15}\overline{\mathcal T}_J=\mathcal T_J,\qquad \delta\mathcal T_J=0.\] If the internal rotation is symmetrically closed, its spatial projections over the full period are compensated:
\[\tag{16}\int_{\tau}^{\tau+T}\mathcal T_J(\tau')\,d\tau'=0.\] The local temporal field does not disappear. It determines the frequency and energy scale.
\[\tag{17}E=\hbar\omega.\] Thus, a univalent closed structure with a fixed center has a temporal field, but does not have an external field arising from the difference in sheets.
SpaceA sheet projection is not necessarily a force field. Without a sheet difference, external motion can manifest itself as a momentum of the structure itself, but not as an electrical or magnetic interaction.
4. External motion of a univalent structure
Let the center of a univalent structure begin to move:
\[\tag{18}b>0,\qquad b=\frac{\arcsin\beta}{\pi},\qquad \beta=\frac vc.\] Then
\[\tag{19}\Sin(\pi b)=\beta,\qquad \Cos(\pi b)=\sqrt{1-\beta^2}=\frac1\gamma.\] The accumulation of the external spatial coordinate is determined by the projection of the temporal motion:
\[\tag{20}\frac{dx}{dt}=c\Sin(\pi b)=v,\qquad x(t)-x(t_0)=\int_{t_0}^{t}v(t')\,dt'.\] The average component of the temporal field acquires a spatial manifestation associated with the motion of the center. After physical normalization, it corresponds to energy and momentum:
\[\tag{21}E=\gamma mc^2,\qquad \mathbf p=\gamma m\mathbf v.\] But since \(N=1\), the difference component is still absent. Therefore, external motion by itself does not create a charge field:
\[\tag{22}\boxed{N=1,\, b > 0\quad\longrightarrow\quad\text{energy and momentum, but not a charge field}}\] 5. Two-sheeted temporal field of an electron
For an electron, a single wave successively passes through two close internal branches. In deep idempotent splitting, such a state can be represented by the operator
\[\tag{23}J_e(\tau)=\j^a\ast\j_{\prp}^{a/2}.\] We denote the two sheet states by \(J_+\) and \(J_-\). They correspond to the fields
\[\tag{24}\mathcal T_+=\frac{dJ_+}{d\tau},\qquad \mathcal T_-=\frac{dJ_-}{d\tau}.\] We introduce the mean and difference components:
\[\tag{25}\overline{\mathcal T}_J=\frac{\mathcal T_++\mathcal T_-}{2},\qquad \boxed{\delta\mathcal T_J=\frac{\mathcal T_+-\mathcal T_-}{2}}.\] The mean field characterizes the overall internal dynamics and does not depend on which sheet the wave passes through. The difference field exists only due to its two-sheet nature. It is not yet a fully formed electric field in three-dimensional space, but serves as its geometric precursor:
\[\tag{26}\boxed{\mathcal E_J\equiv\delta\mathcal T_J}.\] It is \(\mathcal E_J\) that contains information about the difference between the two branches and the direction of their deep bypass. To obtain the observable field, it is necessary to determine how this difference is projected onto an external object.
6. Spatial Projection of Two Sheets
The geometric derivation of the difference spatial projection is discussed in detail in the paper "Geometric Origin of Electric Force". Here we will reproduce its main steps, relating them to the temporal field.
Let the effective radii of the two branches be equal
\[\tag{27}r_+=r_0+\frac{\Delta r}{2},\qquad r_-=r_0-\frac{\Delta r}{2}.\] For an external point located at a distance \(\ell\), two projection coefficients arise:
\[\tag{28}P_\pm(\ell)=\frac{r_\pm}{\sqrt{\ell^2+r_\pm^2}}.\] The difference between the two projections is
\[\tag{29}\boxed{\Delta P(\ell)=P_+(\ell)-P_-(\ell)}.\] In the far field
\[\tag{30}\ell\gg r_+,r_-\] we obtain
\[\tag{31}\boxed{\Delta P(\ell)\approx\frac{\Delta r}{\ell}}.\] The temporal field defines the directional energy scale \(E_*\), and \(\Delta P\) identifies its uncompensated spatial part:
\[\tag{32}\Delta E_{\mathrm{proj}}(\ell)=E_*\Delta P(\ell).\] For a consistent state, the potential energy is taken to be
\[\tag{33}U(\ell)=-E_*\Delta P(\ell)\approx-E_*\frac{\Delta r}{\ell}.\] Its external spatial gradient creates a force:
\[\tag{34}\boxed{F_\ell=-\frac{dU}{d\ell}\approx-E_*\frac{\Delta r}{\ell^2}}.\] Thus, two different mechanisms act sequentially here: the internal sheet difference creates uncompensated projection, and the derivative of this projection with respect to the external distance creates a force field.
\[\tag{35}\boxed{\delta\mathcal T_J\;\longrightarrow\;\Delta P(\ell)\;\longrightarrow\;U(\ell)\;\longrightarrow\;-\frac{dU}{d\ell}}\] 7. From the difference projection to the electric field
For an electron, the observed energy scale is
\[\tag{36}E_*=m_ec^2.\] If the effective branch separation is equal to the classical electron radius,
\[\tag{37}\Delta r=r_e,\] then the spatial force coefficient becomes
\[\tag{38}E_*\Delta r=m_ec^2r_e=\alpha_{\mathrm{fs}}\hbar c.\] From the definition of the fine structure constant:
\[\tag{39}\alpha_{\mathrm{fs}}\hbar c=\frac{e^2}{4\pi\varepsilon_0}.\] Consequently, the magnitude of the force in the far field coincides with the Coulomb magnitude:
\[\tag{40}\boxed{|F_\ell|=\frac{e^2}{4\pi\varepsilon_0\ell^2}}.\] Only after obtaining the dependence \(1/\ell^2\) and the correct coefficient, can the geometric component \(\mathcal E_J\) be Compare with the electric field:
\[\tag{41}\boxed{\mathcal E_J\overset{\Pi_{\mathrm E}}{\longrightarrow}\mathbf E}.\] The electric field is a spatial difference projection of a two-sheeted temporal field. The two-sheeted nature creates a source of difference, and the dependence on external distance transforms this difference into an interaction field.
The condition \(\Delta r=r_e\), as well as the sign of the consistent energy, remain physical principles of the model for now. They have yet to be derived directly from the algebra of the global operator.
8. Sign of a leaf traversal
Opposite directions of a deep traversal can be represented as
\[\tag{42}J_{e^-}=\j^a\ast\j_{\prp}^{a/2},\qquad J_{e^+}=\j^a\ast\j_{\prp}^{-a/2}.\] The common component is preserved, but the difference component changes orientation:
\[\tag{43}\overline{\mathcal T}_{e^-}=\overline{\mathcal T}_{e^+},\qquad \delta\mathcal T_{e^+}=-\delta\mathcal T_{e^-}.\] After spatial mapping, this should lead to opposite directions of the electric field:
\[\tag{44}\mathbf E_{e^+}=-\mathbf E_{e^-}.\] Thus, the identical mass of the electron and positron is associated with the middle part of the temporal field, and the opposite charges are associated with the orientation of its deep difference component. However, a rigorous derivation of the charge sign from the conjugation rule \(J\) remains a separate problem.
9. Internal Circulation and Magnetic Moment
The electron's center at rest has \(b=0\), but the charge wave continues to circulate along an internal closed trajectory. Therefore, it is necessary to distinguish between the immobility of the center and the absence of internal motion:
\[\tag{45}b=0,\qquad \dot a\ne0.\] The oriented circulation of the differential field creates an internal moment component:
\[\tag{46}\boxed{\mathcal E_J+\text{internal circulation}\;\longrightarrow\;\mathcal M_J^{\mathrm{int}}}.\] After mapping into three-dimensional space, it manifests itself as a magnetic moment:
\[\tag{47}\mathcal M_J^{\mathrm{int}}\overset{\Pi_\mu}{\longrightarrow}\boldsymbol\mu.\] The geometry of the normal magnetic moment is discussed in the paper "Electron Spin", and the influence of additional transitions between the two branches is discussed in the paper "Anomalous Magnetic Moment of the Electron".
It is important to separate the intrinsic magnetic moment from the magnetic field of a translationally moving charge:
\[\tag{48}\boxed{\boldsymbol\mu\;\longleftarrow\;\text{internal circulation},\qquad \mathbf B_{\mathrm{move}}\;\longleftarrow\;\text{external motion of the center}}\] 10. External Motion of a Two-Sheet Field
Now we begin to move the center of the two-sheet structure. The external operator acts not on the existing three-dimensional vector \(\mathbf E\), but on its internal source \(\mathcal E_J\):
\[\tag{49}\mathcal E_J(0)\;\longrightarrow\;\mathcal E_J(0)(-\j)^b.\] In the rest frame, the charge projection has only a time component. We denote the corresponding scalar potential by \(\Phi_J\) and construct an internal four-dimensional object.
\[\tag{50}\mathbb A_J^{(0)}=\left(\frac{\Phi_J}{c},\mathbf 0\right).\] When \(b>0\), the temporal component receives a spatial projection along the direction of motion. Taking into account
\[\tag{51}\beta=\Sin(\pi b),\qquad \gamma=\frac1{\Cos(\pi b)}\] we obtain
\[\tag{52}\boxed{\mathbb A_J(b)=\left(\frac{\gamma\Phi_J}{c},\frac{\gamma\Phi_J}{c}\boldsymbol\beta\right)}.\] Its spatial part is
\[\tag{53}\boxed{\mathbf A_J=\frac{\gamma\Phi_J}{c^2}\mathbf v}.\] This transformation preserves the space-time norm:
\[\tag{54}\left(\frac{\gamma\Phi_J}{c}\right)^2-|\mathbf A_J|^2=\left(\frac{\Phi_J}{c}\right)^2.\] Formula (53) is the geometric precursor of the vector potential. It shows that external motion creates a spatial component of the existing charge projection.
11. Antisymmetric gradient and magnetic component
The spatial component \(\mathbf A_J\) alone does not yet constitute a magnetic field. To obtain the field, we must study its variation in space. Let's combine the scalar and spatial components into a four-potential \(A_\mu\). The complete field object is defined by the antisymmetric derivative:Its mixed time-spatial components correspond to the electric field, and its mixed space-spatial components correspond to the magnetic field:
\[\tag{56}F_{0i}\;\longleftrightarrow\;E_i,\qquad F_{ij}\;\longleftrightarrow\;B_k.\] In three-dimensional notation:
\[\tag{57}\mathbf E=-\nabla\Phi-\frac{\partial\mathbf A}{\partial t},\qquad \boxed{\mathbf B=\nabla\times\mathbf A}.\] For uniform motion in a quasi-stationary region, we can write
\[\tag{58}\mathbf A=\frac{\Phi}{c^2}\mathbf v,\qquad \mathbf v=\operatorname{const},\] where the Lorentz factor is already included in the laboratory potential \(\Phi\). Then
\[\tag{59}\mathbf B=\nabla\times\left(\frac{\Phi}{c^2}\mathbf v\right)=\frac1{c^2}\nabla\Phi\times\mathbf v.\] Since in this region \(\mathbf E=-\nabla\Phi\), we obtain
\[\tag{60}\boxed{\mathbf B=\frac1{c^2}\mathbf v\times\mathbf E}.\] Thus, the magnetic field appears not as a result of a formal vector product, but earlier—as the spatial component of a moving two-sheet temporal field. The vector product only maps the direction of the already obtained orthogonal component into 3D.
\[\tag{61}\boxed{\delta\mathcal T_J\;\longrightarrow\;\Phi_J\;\overset{b>0}{\longrightarrow}\;\mathbf A_J\;\longrightarrow\;\nabla\times\mathbf A\;\longrightarrow\;\mathbf B}\] For a uniformly moving point charge in the far nonrelativistic region, this yields
\[\tag{62}\mathbf B=\frac{\mu_0q}{4\pi}\frac{\mathbf v\times\mathbf n_\ell}{\ell^2}.\] The magnetic field is a vortex spatial projection of a moving two-sheeted temporal field. The two-sheeted nature creates an electrical difference, and external motion converts part of this difference into a spatial potential.
12. Geometry and Three-Dimensional Field Direction
Prior to formula (55), components within space-time geometry were considered. Now let's introduce three-dimensional directions:
\[\tag{63}\mathbf n_\ell=\frac{\mathbf r-\mathbf r_0}{|\mathbf r-\mathbf r_0|},\qquad \mathbf n_v=\frac{\mathbf v}{v}.\] The electric projection is directed along \(\mathbf n_\ell\):
\[\tag{64}\mathbf E=E\mathbf n_\ell.\] The magnetic projection is orthogonal to both the external motion and the electric projections:
\[\tag{65}\boxed{\mathbf n_B=\mathbf n_v\times\mathbf n_\ell}.\] Therefore
\[\tag{66}\mathbf B\perp\mathbf v,\qquad \mathbf B\perp\mathbf E.\] The magnetic component is absent along the direction of motion, and is maximum in the transverse direction:
\[\tag{67}B=\frac{\beta E}{c}\sin\vartheta,\] where \(\vartheta\) is the angle between \(\mathbf v\) and \(\mathbf E\).
13. Closed and Open Structures
So far, we have considered a localized structure in which the internal wave returns to the center. It is convenient to characterize the closure by a functional.
\[\tag{68}\mathcal C(T)=\int_0^T e^{i\alpha(t)}\j^{\varpi t}\,dt.\] For a closed state.
\[\tag{69}\boxed{\mathcal C(T)=0}.\] When the closure is broken, an uncompensated transfer occurs:
\[\tag{70}\boxed{\mathcal C(T)\ne0}.\] Therefore, closure determines not the presence of a temporal field, but the way it is spatially Manifestations:
\[\tag{71}\boxed{\text{closure}\;\longrightarrow\;\text{localized projection},\qquad \text{opening}\;\longrightarrow\;\text{spatial transfer}}\] The geometry of the opening of a closed motion into a propagating wave is discussed in more detail in the article "Geometric Origin of a Propagating Wave".
14. Two leaves as two parts of a period
In a closed electron, a single wave sequentially passes through two leaves:
\[\tag{72}J_+\;\longrightarrow\;J_-\;\longrightarrow\;J_+.\] After passing one leaf, the internal revolution is complete, but the full leaf state is not yet restored. Complete closure occurs after both leaves:
\[\tag{73}T_{\mathrm w}=2T_{\mathrm{leaf}}.\] When openingThe leaves do not vanish or average. Their sequence unfolds along the direction of propagation:
\[\tag{74}\boxed{J_+\;\longrightarrow\;\text{first half of the period},\qquad J_-\;\longrightarrow\;\text{second half of the period}}\] For propagation with the velocity \(c\), the full period is converted into a wavelength:
\[\tag{75}\boxed{\lambda=cT_{\mathrm w}=2cT_{\mathrm{leaf}}}.\] Thus, the internal discrete index of the leaf becomes the phase of the propagating waves:
\[\tag{76}\{+,-\}\;\longrightarrow\;\left\{0\le\varphi<\pi,\quad \pi\le\varphi<2\pi\right\}.\] 15. Electromagnetic wave
Introduce the coordinate along the direction of propagation
\[\tag{77}\xi=\mathbf n\cdot\mathbf r-ct.\] The difference geometric components become functions \(\xi\):
\[\tag{78}\mathcal E_J=\mathcal E_J(\xi),\qquad \mathcal B_J=\mathcal B_J(\xi).\] The transition to the former second sheet corresponds to a change in orientation after half a period:
\[\tag{79}\mathcal E_J\!\left(\xi+\frac\lambda2\right)=-\mathcal E_J(\xi),\qquad \mathcal B_J\!\left(\xi+\frac\lambda2\right)=-\mathcal B_J(\xi).\] After a full period, the state is repeated:
\[\tag{80}\mathcal E_J(\xi+\lambda)=\mathcal E_J(\xi),\qquad \mathcal B_J(\xi+\lambda)=\mathcal B_J(\xi).\] After mapping into three-dimensional space, the propagating components must satisfy
\[\tag{81}\mathbf E\perp\mathbf B,\qquad \mathbf E\perp\mathbf n,\qquad \mathbf B\perp\mathbf n\] and the free electromagnetic wave relation
\[\tag{82}\boxed{E=cB}.\] The direction of energy transfer is determined by the Poynting vector:
\[\tag{83}\mathbf S=\frac1{\mu_0}\mathbf E\times\mathbf B\parallel\mathbf n.\] Both fields change sign simultaneously after half a period, so the direction of \(\mathbf S\) is preserved. The average electric field over a period is zero, but the average energy does not vanish:
\[\tag{84}\langle\mathbf E\rangle_T=0,\qquad \langle E^2\rangle_T\ne0.\] An electromagnetic wave is a spatially unfolded two-sheet projection of the temporal field. Two sheets of a closed structure become two consecutive halves of a full period.
16. Spin and Polarization — An Independent Feature
The transformation of two sheets into two parts of a period should not be confused with the orientation of the internal plane. States
\[\tag{85}J(a,0),\qquad J(0,a)\] can define two spin orientations of a closed particle. Upon opening, the plane orientation must transform into the orientation of the transverse projection of the wave:
\[\tag{86}\boxed{\text{spin orientation}\;\longrightarrow\;\text{wave polarization}}\] Therefore, the features perform different functions:
\[\tag{87}\boxed{\text{two-sheetedness}\;\longrightarrow\;\text{period structure},\qquad \text{plane orientation}\;\longrightarrow\;\text{polarization}}\] This transition is discussed in detail in the paper "Transition of Electron Spin to Photon Polarization".
17. Partial Uncoupling and Acceleration
Between a completely closed particle and a free wave, we can formally introduce a degree of uncoupling
\[\tag{88}0\le\chi\le1.\] Then
\[\tag{89}\chi=0\;\longrightarrow\;\text{closed structure},\qquad 0<\chi<1\;\longrightarrow\;\text{transition state},\qquad \chi=1\;\longrightarrow\;\text{free wave}.\] If the state parameters change, the total derivative acquires additional terms:
\[\tag{90}\boxed{\frac{dJ}{dt}=\dot a\frac{\partial J}{\partial a}+\dot b\frac{\partial J}{\partial b}+\dot\chi\frac{\partial J}{\partial\chi}+\ldots}.\] The first term describes the internal evolution, the second describes the change in external orientation, and the third describes the transition between closed and open modes. During acceleration
\[\tag{91}\dot b\ne0\] a variable uncompensated component associated with radiation may arise. However, for physical identification, it is also necessary to derive the radiation power and its dependence on acceleration. Therefore, the connection between \(\dot b\ne0\) and radiation remains a hypothesis here.
18. Deeper Splittings
The two-sheeted nature is only the first nontrivial level. For \(N>2\), several independent difference components arise:
\[\tag{92}\delta\mathcal T_{J,1},\quad\delta\mathcal T_{J,2},\quad\ldots,\quad\delta\mathcal T_{J,N-1}.\]Each of them can have its own spatial mapping rule:
\[\tag{93}\mathcal F_k=\Pi_k\!\left[\delta\mathcal T_{J,k}\right].\] When closed, such components can manifest themselves as additional internal properties, and when opened, as a more complex multiphase or modulated wave structure. Until the geometry reproduces a specific observed law and its coefficient, such projections cannot be identified with known or new physical interactions.
19. General Classification
The resulting system can be reduced to the following main modes.
Single-sheet structure, \(b=0\), closure. There is a temporal field associated with the internal frequency and energy; There is no external difference projection.
Single-sheet structure, \(b>0\), closure. The spatial projection of the mean field manifests itself as the energy and momentum of the center's motion.
Two-sheet structure, \(b=0\), closure. The difference between the sheets creates an electric projection, and the internal circulation creates its own magnetic moment.
Two-sheet structure, \(b>0\), closure. The external movement of the electric difference creates a spatial potential and a magnetic field of the moving charge.
Open two-sheet structure. The two sheets become two parts of a period, and the electric and magnetic components are transported as an electromagnetic wave.
Varying External motion and partial opening. A possible radiation regime arises, requiring a separate quantitative derivation.
Multisheet structure. Additional difference fields appear, the physical meaning of which should be determined only after obtaining the observed laws.
\[\tag{94} \boxed{\begin{aligned}\overline{\mathcal T}_J&\;\longrightarrow\;\text{energy and momentum},\\ \delta\mathcal T_J&\;\longrightarrow\;\text{electric projection},\\ \delta\mathcal T_J,\, b>0&\;\longrightarrow\;\text{magnetic projection},\\ \delta\mathcal T_J,\, \chi>0&\;\longrightarrow\;\text{electromagnetic wave}.\end{aligned}}\] 20. What has been obtained, and what remains a hypothesis
From the geometry of two nearby branches, the difference in the coefficients \(\Delta P(\ell)\) is directly obtained. In the far field, it has the dependence \(1/\ell\), and the external gradient of the corresponding potential energy has the dependence \(1/\ell^2\). Under the condition \(\Delta r=r_e\), the coefficient coincides with the Coulomb coefficient.
From the space-time transformation of the charge projection at \(b>0\), the spatial component of the potential arises. Its antisymmetric gradient leads to a standard relationship between the magnetic and electric fields of a uniformly moving charge.
At the same time, the following remain hypotheses:
— obtaining \(\Delta r=r_e\) directly from the algebra \(J\);
— rigorous derivation of the charge sign from the direction of deep bypass;
— complete derivation of the electromagnetic field transformations only from the operator \(( -\j)^b\);
— quantitative relationship of partial disconnection with radiation power;
— derivation of Maxwell's equations from the geometry of \(J\);
— the physical meaning of deeper levels of splitting.
— rigorous derivation of the charge sign from the direction of deep bypass;
— complete derivation of the electromagnetic field transformations only from the operator \(( -\j)^b\);
— quantitative relationship of partial disconnection with radiation power;
— derivation of Maxwell's equations from the geometry of \(J\);
— the physical meaning of deeper levels of splitting.
This distinction is necessary: the geometric classification specifies a possible mechanism for the origin of the fields, but each physical identification requires the reproduction of the form of the law, dimension, coefficient, and observable orientation.
Conclusion
The temporal field \(\mathcal T_J=dJ/d\tau\) is the common source of all the spatial manifestations considered. For a univalve structure, its middle part determines the internal energy, and with external motion, the energy and momentum of the center.
The two-sheeted structure creates a differential temporal field. The difference in the spatial projections of the two branches leads to a potential \(1/\ell\) and a field \(1/\ell^2\), which, with the necessary geometric normalization, coincides with the electric field. When the center moves, the same charge projection acquires a spatial component; its vortex manifests itself as a magnetic field.
When the two-sheeted structure is opened, the two-sheeted structure does not disappear, but unfolds during the free wave period. The two sheets become two consecutive halves of a period, the orientation of the internal plane transforms into polarization, and the associated electric and magnetic projections transfer energy in space.
\[\boxed{\text{known fields differ not in their primary source, but in the geometry of the projection of a single temporal field}}\] Thus, mass, electric field, magnetic moment, magnetic field of motion, and electromagnetic wave do not form a set of independent constructs.y, but successive manifestations of one geometry: temporal dynamics, sheet splitting, spatial projection, external movement and opening.