Geometrical Aspects of the Electromagnetic Field

Abstract

The space–time is based on space as three-dimensional sphere, space rotations also as three-dimensional sphere and time which is homeomorphic to the Euclidean three-dimensional space. There are four basic exchanges among them and four induced exchanges, which lead to the basic interactions in the nature. This geometrical approach enables to obtain new viewpoint especially on the basic interactions and their geometrical interpretations. More attention is devoted to the electromagnetic interactions, where the magnetic field of the spinning bodies is studied separately from the electromagnetic field of the charged bodies. It is also considered the gravitational interaction in order to emphasize the similarities between them and the properties which separate them.

1. Introduction

The space and time were subject of interest from the old civilizations up to the present time. They were separated many centuries ago. Even in the Newton theory they are still separated and the time flow was considered as uniform phenomena in the universe. Remarkable approach in understanding the space and time was done by the well known scientist and philosopher Roger Boscovich (1711–1787), who was not well understood at that time. He made distinction between the real space–time and the space–time according to our observations (Ref. [1]). More than one century before the Special Relativity, he wrote that there does not exist an absolute space in rest, i.e. about the relativity among the moving systems.

Basic argument in the Spacial Relativity is the assumption that there does not exist strong distinction between the space and time, which is obvious from the Lorentz transformations. In the recent references this idea was generalized between the space (S), space rotations (SR) and the time (T) [2, 3, 4, 5, 6]. For each small body besides its three spatial coordinates, can be jointed also three degrees of freedom about its rotation in the space and also three degrees of freedom for the velocity of the considered body. The basic assumption is that these six degrees of freedom are of the same importance as the basic three spatial coordinates and so they can be considered as dimensions. The spaces $S$ and $\mathit{SR}$ are both homeomorphic to $S^3$ and can be considered as the group of quaternions with module 1, which is locally isomorphic to $\mathit{SO}\left(3,\mathrm{ℝ}\right)$ . Note that if the space $\mathit{SR}$ is not admitted, we would not be able to turn. The time $T$ is homeomorphic to ${\mathrm{ℝ}}^3$ , and if it is not admitted, everything would be static. Each two of these three sets may interfere analogously to the space and time in the Special Relativity.

It is deduced in Refs. [2, 3] that the general form of three-dimensional time of one point is given by

$$\mathbf{n}t+\frac{\mathbf{v}\times \mathbf{r}}{c^2},$$ E1

where $t$ is the 1-parameter time, $\mathbf{n}$ is the unit vector of its velocity, $\mathbf{v}$ is its velocity and $\mathbf{r}$ is the radius vector of the considered point. The last term gives possibility for a new view of the quantum mechanics, where the quantization of the angular momentum is indeed quantization of the three-dimensional time.

The group of Lorentz transformations $O_{+}^{\uparrow }\left(1,3\right)$ is known to be isomorphic to the complex Lie group $\mathit{SO}\left(3,\mathrm{ℂ}\right)$ . If we consider this complex group as a Lie group of real $6\times 6$ matrices, then that group is the required Lie group which connects the spaces $S$ and $T$ . This Lie group of $6\times 6$ transformations has Lie algebra consisting of matrices of type

$$\left[\begin{array}{cc}X& Y\\ -Y& X\end{array}\right],$$ E2

where $X$ and $Y$ are antisymmetric $3\times 3$ matrices. This temporal Lie group is denoted by $G_{\mathrm{t}}$ . Moreover, in Ref. [4] the Lorentz transformations are converted as transformations $S\times T\to S\times T$ via the Lie group $G_{\mathrm{t}}$ .

The Lie group which connects the spaces $S$ and $\mathit{SR}$ has Lie algebra which consists of matrices of type

$$\left[\begin{array}{cc}X& Y\\ Y& X\end{array}\right],$$ E3

where $X$ and $Y$ are antisymmetric $3\times 3$ matrices. This Lie group is generated by the following six matrices of type

$$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& cos\psi & 0& 0& 0& sin\psi \\ 0& 0& cos\psi & 0& -sin\psi & 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& sin\psi & 0& cos\psi & 0\\ 0& -sin\psi & 0& 0& 0& cos\psi \end{array}\right],$$

(translation along the $x$ -axis)

$$\left[\begin{array}{cccccc}cos\psi & 0& 0& 0& 0& -sin\psi \\ 0& 1& 0& 0& 0& 0\\ 0& 0& cos\psi & sin\psi & 0& 0\\ 0& 0& -sin\psi & cos\psi & 0& 0\\ 0& 0& 0& 0& 1& 0\\ sin\psi & 0& 0& 0& 0& cos\psi \end{array}\right],$$

(translation along the $y$ -axis)

$$\left[\begin{array}{cccccc}cos\psi & 0& 0& 0& sin\psi & 0\\ 0& cos\psi & 0& -sin\psi & 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& sin\psi & 0& cos\psi & 0& 0\\ -sin\psi & 0& 0& 0& cos\psi & 0\\ 0& 0& 0& 0& 0& 1\end{array}\right],$$ E4

(translation along the $z$ -axis)

$$\left[\begin{array}{cccccc}1& 0& 0& 0& 0& 0\\ 0& cos\psi & sin\psi & 0& 0& 0\\ 0& -sin\psi & cos\psi & 0& 0& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& cos\psi & sin\psi \\ 0& 0& 0& 0& -sin\psi & cos\psi \end{array}\right],$$

(rotation around the $x$ -axis)

$$\left[\begin{array}{cccccc}cos\psi & 0& -sin\psi & 0& 0& 0\\ 0& 1& 0& 0& 0& 0\\ sin\psi & 0& cos\psi & 0& 0& 0\\ 0& 0& 0& cos\psi & 0& -sin\psi \\ 0& 0& 0& 0& 1& 0\\ 0& 0& 0& sin\psi & 0& cos\psi \end{array}\right],$$

(rotation around the $y$ -axis)

$$\left[\begin{array}{cccccc}cos\psi & sin\psi & 0& 0& 0& 0\\ -sin\psi & cos\psi & 0& 0& 0& 0\\ 0& 0& 1& 0& 0& 0\\ 0& 0& 0& cos\psi & sin\psi & 0\\ 0& 0& 0& -sin\psi & cos\psi & 0\\ 0& 0& 0& 0& 0& 1\end{array}\right],$$ E5

(rotation around the $z$ -axis),

where $\psi$ is an arbitrary parameter. This is space group which connects the spaces $S$ and $\mathit{SR}$ and is denoted by $G_{\mathrm{s}}$ . The group is isomorphic to the spin group Spin(4) [7].

Analogously as the speed of light is a coefficient of proportionality between the length and the time, in case of the spaces $S$ and $\mathit{SR}$ we have the following situation. The elements of the space $S$ are measured in length while elements of spatial rotations $\mathit{SR}$ are measured in angles. Consequently there exists a local constant as a coefficient of proportionality between $S$ and $\mathit{SR}$ . This local constant is called radius of range $R$ . It depends mainly of the mass of the particles, galaxies and the universe. Now the connection between $\mathit{SR}$ and $T$ is determined, and the group which connects the spaces $\mathit{SR}$ and $T$ is the temporal group $G_t$ . The radius of range is of the same importance as the velocity of light, which is a coefficient of proportionality between the spaces $S$ and $T$ . While the Lorentz transformations tend to Galilean transformations for small velocities, if the radius of range is close to infinity, then the group $G_s$ tends to the affine group of translations and rotations in the Euclidean space.

The three-dimensional time was investigated also by another authors, for example in Refs. [8, 9, 10, 11, 12, 13, 14, 15].

2. Basic four exchanges among the spaces $S$ , ${SR}$ and $T$

The spaces $S$ , $\mathit{SR}$ and $T$ are not separated. Analogously as there does not exist a strong distinction between the space and time in the Special Relativity, there exist similar exchanges in this case. In [16] are given four basic exchanges and four generalizations of them. Let the elements of the spaces $S$ , $\mathit{SR}$ and $T$ are denoted respectively by $s$ , $r$ and $t$ . The basic four cases are presented on Figure 1 and they are

$$1.r\to s,2.s\to r,3.r\to t,4.s\to t.$$ E6

Figure 1.

Spaces $S$ , $\mathit{SR}$ and $T$ and the possible exchanges between them.

Here the notation $x\to y$ denotes that if there is a constraint of $x\in X$ , then it converts into $y\in Y$ . The composite exchanges $s\to r\to t$ and $r\to s\to t$ are also admitted, while the exchange of type $x\to y\to x$ is not admitted because we may not return back to $x$ , if $x$ was constrained previously. The exchanges $t\to s$ and $t\to r$ are not admitted, because it is impossible to constraint the time. For example, we can not conduct the speed of time without constraints of $S$ and $\mathit{SR}$ . Since the exchange $s\to r\to s$ is not possible, the case 3. must be of type $s\to r\to t$ , and since the exchange $r\to s\to r$ is not possible, the case 4. must be of type $r\to s\to t$ .

The second case $\left(s\to r\right)$ means that if the space displacement is constrained (completely or partially), then it induces a spatial rotation. It can be interpreted if we consider a rigid body, which moves with a velocity v. Assume that one point $O$ of the rigid body is constrained to move. Then the body will start to rotate around the point $O$ . Each point $P$ of the body intends to rotate with angular velocity ${\mathbf{w}}_{\mathrm{P}}$ . This means that the case $s\to r$ occurs. However, the rigid body will rotate globally with an averaged angular velocity $<\mathbf{w}>$ , and it depends on the distribution of the mass. So at the point $P$ the angular velocity ${\mathbf{w}}_{\mathrm{P}}-<\mathbf{w}>$ is constrained. It will cause a change of the speed of time if ${\mathbf{w}}_{\mathrm{P}}-<\mathbf{w}>\ne 0$ , and this is the third case $r\to t$ .

The first case $r\to s$ means that if the rotation is not permitted, then the particles will be displaced, which is the induced spin motion. This induced spin velocity, or shortly spin velocity, was considered in some previous papers, for example in [17]. Let a solid body moves including some rotations, then each particle intends to rotate according to its spatial trajectory. But that rotation is often constrained (completely or partially), because we consider a solid body, and the particles of the body do not have freedom to rotate as they intend to rotate. So, the constrained space rotation induces the spin velocity. This is the reason for circular motion of the spinning bodies when they rotate on a horizontal plane or in free fall motion. Moreover, the spin velocity of the Earth is considered in Ref. [17], and as a consequence the change of the Earth’s angular velocity with period of 6 months was deduced with accuracy 5%, compared with the measurements. In the recent Ref. [6] the spin velocity is determined via the curvature and torsion of the trajectory of the considered particle.

The spin velocity is non-inertial and it will be denoted by capital letter $\mathbf{V}$ in order to distinguish from the classical inertial velocity $\mathbf{v}$ . The main property of the inertial velocities is that temporal axes are inclined, i.e. no-parallel, analogously as the temporal axes are mutually non-parallel in case of two mutually moving inertial coordinate systems. It is also important that if a coil moves with spin velocity in magnetic field, there will not appear an electromotive force, because for electromotive force it is necessary to have inertial velocity between the magnetic field and the coil. Indeed, the spin velocity means simply displacement in the space and there does not appear change of speed of time by the coefficient $\sqrt{1-V^2/c^2}$ . But, if the spin velocity is constrained completely or partially, then the constrained part converts into inertial velocity $\mathbf{v}=-\mathbf{V}/\sqrt{1+V^2/c^2}$ , which is indeed the fourth case $s\to t$ . This velocity $\mathbf{v}$ can be non-observable, but it is important that the speed of time is changed at that position. Then the coil in magnetic field will produce electromotive force via the inertial velocity.

We give now another important examples of the case 4.

Let a non-rotating small body initially rests with respect to the Earth on infinite distance. When the body comes at the surface of the Earth, it is not permitted to go further. This constraint causes time displacement, such that the time will be slower. More precisely, if the velocity at the surface is equal to $v$ , then the constraint for the space displacement will induce slower time for coefficient $\lambda =\sqrt{1-\frac{v^2}{c^2}}$ and this is the case 4 ( $s\to t$ ). Using that $v\approx \sqrt{2\mathit{GM}/R}$ , the time on the surface of the Earth is slower for coefficient $\lambda \approx \sqrt{1-\frac{2\mathit{GM}}{{\mathit{Rc}}^2}}\approx 1-\frac{\mathit{GM}}{{\mathit{Rc}}^2}$ , which is also well known from the General Relativity up to approximation of $c^{-2}$ .

In this example it was not important that the particle moved under gravitation. So let us consider now two charged bodies with charges $e_1$ and $e_2$ , masses $m_1$ and $m_2$ , and $m_2<<m_1$ . Assume that the second body is not rotating and emits waves with frequency ${\nu }_0$ on infinity distance ( Figure 2 ). If the second body is placed close to the first body on distance $r$ between their centers and assume that the distance $r$ remains a constant. In case of free fall motion (if $e_1{e}_2<0$ ) the velocity of the second body on distance $r$ would be $\sqrt{\frac{-e_1{e}_2}{2{\mathit{\pi \epsilon }}_0{\mathit{rm}}_2}}$ . Then according to the case 4, analogously to the gravitational case now we have slower speed of time and the new frequency is given by

Figure 2.

The frequency $\nu$ changes when the distance $r$ changes.

$$\nu ={\nu }_0\left(1+\frac{e_1{e}_2}{4{\mathit{\pi \epsilon }}_0{\mathit{rm}}_2{c}^2}\right).$$ E7

This formula is true also in case $e_1{e}_2>0$ .

3. The induced (extended) four exchanges among the spaces $S$ , ${SR}$ and $T$

The basic cases 1-4 describe the changes of the positions, rotations and temporal coordinates of a chosen particle in case of some constraint. The induced four exchanges, will be denoted by 1*, 2*, 3*, and 4* and they are analogous to the cases 1-4 respectively. In cases 1*-4* we need to describe the fields which appear in case of constraints and to describe the interactions between the particles. Now the basic role play the group structures, especially their non-commutativity. As a consequence some interactions appear in these cases. The non-commutativity leads to some angles. They are functions of the spatial coordinates, and so instead of differentiation by $t$ we should apply the operator rot. Half of this quantity is admitted, and half is not admitted. That half which is not admitted will be converted into acceleration by multiplication by a constant $K$ . This principle of admitted and unadmitted angular velocities is general and further in each section it will be used. It is shown and supported by the following example.

Example 1. Let us consider the vector field $\mathbf{v}\left(x,y,z\right)=\left(-\mathit{yw},\mathit{xw},0\right)$ . Since $\mathrm{rot}\mathbf{v}=\left(\mathrm{0,0,2}w\right)$ , obviously half of it is admitted as angular velocity $\mathbf{w}=\left(0,0,w\right)$ , and the other half which in not admitted converts into space displacement, which is observed as centripetal acceleration $\left(a_x,a_y,a_z\right)=\mathbf{v}\times \left(-\mathbf{w}\right)$ . These two effects do not occur simultaneously, but they change permanently in a very short interval $\mathrm{\Delta }t$ as a process of discretization, because these two effects are mutually dependent: in the first time interval $\mathrm{\Delta }t$ the body rotates around its axis with angular velocity $2w$ , in the next interval $\mathrm{\Delta }t$ it falls toward the center with centripetal acceleration $\mathbf{v}\times \left(-2\mathbf{w}\right)$ , and it periodically repeats. So, we observe that the average angular velocity is $\left(0,0,w\right)$ , and the average centripetal acceleration is $\mathbf{v}\times \left(-\mathbf{w}\right)$ .

In the cases 1* and 2* we should not use the Lorentz transformation and should not use the operator $\partial /\partial t$ . The time exists in these cases, but caused by another external reasons and it should not be actively used. The motions in these two case are analogous to the induced spin motions.

The cases 3* and 4* lead to the electromagnetic and gravitational interactions. They are more sophisticated interactions and will be considered in more details in the next sections. It is convenient to consider the gravitation together with the electromagnetic interaction because we need to emphasize the similarity and also where these two interactions differ. To the end of this section we give a brief view of the case 1*.

Let us consider two nucleons with masses $m_1$ and $m_2$ , momentums of inertia $I_1$ and $I_2$ , radiuses of range $R_1$ and $R_2$ , and centers at $O_1$ and $O_2$ ( Figure 3 ). If $X$ is arbitrary point of the second nucleon, then the composition of the translations for vectors $\mathbf{d}=\overrightarrow{O_1{O}_2}$ and $\overrightarrow{O_{2}X}=\left(a,b,c\right)$ is non-commutative in the Lie group $G_s$ and contains also a rotation for an angle. The operator $\frac{1}{2}\mathrm{rot}$ should be applied and yields [16]

Figure 3.

The strong interaction is a consequence of non-commutativity of translations for the vectors d and $\left(a,b,c\right)$ in $S\times \mathit{SR}$ .

$$\frac{1}{2}\mathrm{rot}\phi =-\frac{1}{2}sin\frac{d}{R_1}\cdot sin\frac{d}{R_2}\cdot \frac{R_1{R}_2}{d^2}\frac{\mathbf{d}}{d^2}.$$ E8

Assume that the two nucleons have opposite spins, and assume that the rotation is constrained (or not admitted). Then the space displacement appears as a force, if we multiply the previous value by the constant $\mu v_0^2$ , where $\mu =m_1{m}_2/\left(m_1+m_2\right)$ is the reduced mass, which appears in binary systems. The velocity $v_0$ is a local constant, analogously as the radius of range is a local constant. In case of the nucleons, this constant is close to $c$ , but at this moment we are unable to prove it. In the next sections about the cases 3* and 4*, where the time has role, the corresponding constant takes value $c$ . The force of the second nucleon toward the first nucleon is equal to

$${\mathbf{f}}_{2,1}=-\frac{m_1{m}_2{v}_0^2}{2\left(m_1+m_2\right)}sin\frac{d}{R_1}\cdot sin\frac{d}{R_2}\cdot \frac{R_1{R}_2}{d^3}\frac{\mathbf{d}}{d}.$$ E9

Assume that the space displacement is constrained (or not admitted). Then it appears rotation of the two nucleons. The admitted value (8) induces that the two nucleons also have opposite angular momentums and we determine now the magnitude of this angular momentum. This can be obtained by multiplication with $I_1{I}_2{v}_0/\left(I_1+I_2\right)$ and hence the required angular momentum is equal to

$$\mathcal{L}=\pm \frac{I_1{I}_2{v}_0}{2\left(I_1+I_2\right)}sin\frac{d}{R_1}\cdot sin\frac{d}{R_2}\cdot \frac{R_1{R}_2}{d^3}\frac{\mathbf{d}}{d}.$$ E10

The force (9) can be applied for two galaxies and then it avoids collisions between them, because the forces appear to be also repulsive on some distances.

The following special case, where $m_2/m_1\approx 0$ , $I_2/I_1\approx 0$ and $R_2\approx \infty$ , is of special interest. It means that the second body is much smaller and can be considered as a test particle. The first body can be arbitrary, for example it can be the massive body at the center of a galaxy, while the test body can be arbitrary star. If the angular velocity is constrained, the required space displacement is observed by the acceleration

$$\mathbf{a}=-sin\frac{d}{R}\cdot \frac{v_0^{2}R}{2d^2}\frac{\mathbf{d}}{d},$$ E11

where the local constant $v_0$ is much smaller compared with that constant in case of the nucleons. If the space displacement is constrained, then we obtain the angular velocity

$$\mathbf{w}=-sin\frac{d}{R}\cdot \frac{v_{0}R}{2d^2}\frac{\mathbf{d}}{d}.$$ E12

The angular velocity (12) is the reason that the ecliptic plane of planetary orbits is rotated with respect to the galactic plane. The acceleration (11) has important role for the motion of the stars in the galaxies such that we do not need to introduce dark matter and to bind the stars on a distance at most $\mathit{\pi R}$ , where the acceleration is attractive [16].

It is interesting that R. Boscovich in his Ref. [1] considered also four basic cases between the space and time which are related to one point and analogous to them also four cases which are related for several points. He also comments which combinations, i.e. compositions among these cases are possible and which are not possible. There is an interesting analogy between his comments and the results in this Chapter based on the Lie groups and their properties.

4. Magnetic fields of the spinning bodies

First we give the following preliminaries about the magnetic fields. While in case of gravitation and inertial forces we have motion in the space $S\times T$ , the electromagnetic interaction occurs in the space $\mathit{SR}\times T$ where the antisymmetric matrix $F_{\mathit{ij}}$ consisting of electric and magnetic field behaves as a tensor in 3 + 1 = 4 dimensions or alternatively in 3 + 3 = 6 dimensions. But however, we observe motion and angular velocity of the charged particles in the observable $S\times T$ space. If $\mathbf{E}=0$ , according to Ref. [18] or (38), the observed angular velocity is equal to $\mathtt{\Omega }=\frac{e}{\mathit{mc}}\mathbf{H}$ , where $m$ and $e$ are the mass and the charge of the electron or the considered charged body. This formula assumes that the density of mass is proportional to the density of charge at each point. This is satisfied in case of the electron. But in case of proton and neutron these two densities are no globally proportional, and so there appear the factors $g_{\mathrm{p}}\approx 2.792$ and $g_{\mathrm{n}}\approx -1.913$ in front of $\frac{e}{\mathit{mc}}\mathbf{H}$ .

Further let us consider again the case $r\to t$ . If the angular velocity of the electron may take arbitrary value $\mathtt{\Omega }$ , i.e. it is not constrained, then an electron in magnetic field would move with a constant velocity $\left(\mathbf{v}\right)$ and should rotate with the angular velocity $\mathtt{\Omega }=\frac{e}{\mathit{mc}}\mathbf{H}$ . The force $\frac{e}{c}\mathbf{v}\times \mathbf{H}$ as a part of the Lorentz force would disappear. Assuming that the unadmitted angular velocity is $\mathtt{\Omega }=\frac{e}{\mathit{mc}}\mathbf{H}$ for $\mathbf{E}=0$ , then according to the acceleration $\mathbf{v}\times \mathbf{w}$ , i.e. “temporal part” of the Coriolis acceleration, we obtain that the induced displacement is caused by the force

$$\mathbf{f}=m\left(-\mathtt{\Omega }\right)\times \mathbf{v}=m\frac{-e}{\mathit{mc}}\mathbf{H}\times \mathbf{v}=\frac{e}{c}\mathbf{v}\times \mathbf{H}.$$ E13

So, as a consequence of the case $r\to t$ we obtain the Lorentz force in magnetic field. Now the general formula for the Lorentz force in case of arbitrary electromagnetic field is a consequence of the Lorentz covariance of the Lorentz force among the inertial coordinate systems.

Indeed the magnetic field $\mathbf{H}$ manages partially to rotate the charged particles. The charge $e$ may be defined as a coefficient of proportionality between the unadmitted angular velocity $\mathtt{\Omega }$ and $\mathbf{H}/\left(\mathit{mc}\right)$ , which are collinear.

Finally we can resume the following. A charged body in a magnetic field $\mathbf{H}$ should rotate with angular velocity $\mathtt{\Omega }=\frac{e}{\mathit{mc}}\mathbf{H}$ . On the other hand, the Lorentz force is a consequence of the argument that the unadmitted angular velocity of the charged body is the reason of the existence of the Lorentz force. If we assume that the unadmitted angular velocity is also $\mathtt{\Omega }=\frac{e}{\mathit{mc}}\mathbf{H}$ , we obtain in (13) a satisfactory result. Hence we come to the conclusion that both admitted and unadmitted angular velocity are equal to $\mathtt{\Omega }=\frac{e}{\mathit{mc}}\mathbf{H}$ . So the total angular velocity (admitted and unadmitted) is equal to

$${\mathtt{\Omega }}_{\mathrm{t}}=\frac{2e}{\mathit{mc}}\mathbf{H}.$$ E14

In order to study the magnetic field of the spinning bodies, it is necessary to refer to the precessions of gyroscopes. In Refs. [18, 19, 20] the precessions in case of Gravity Probe B (GPB) experiment were considered. The total precession (geodetic precession and frame dragging) of the gyroscope with respect to the chosen coordinate system is deduced to be

$${\mathtt{\Omega }}_{\mathrm{gyr}}=2\sum _{\mathrm{a}}\left(\mathbf{v}-{\mathbf{v}}_{\mathrm{a}}\right)\times \nabla \frac{{\mathit{Gm}}_{\mathrm{a}}}{r_{\mathrm{a}}{c}^2}-\sum _{\mathrm{a}}G\left[{\mathbf{J}}_{\mathrm{a}}-3{\widehat{\mathbf{n}}}_{\mathrm{a}}\left({\widehat{\mathbf{n}}}_{\mathrm{a}}\cdot {\mathbf{J}}_{\mathrm{a}}\right)\right]/r_{\mathrm{a}}^3{c}^2,$$ E15

where $a=\mathrm{1,2,3},\cdots$ denotes the $a$ -th celestial spherical and homogeneous body which causes this precession, $\mathbf{v}$ denotes the velocity of the observer, ${\mathbf{v}}_{\mathrm{a}}$ denotes velocity of the barycentre of the $a$ -th celestial body, ${\mathbf{J}}_{\mathrm{a}}$ is its angular momentum, $m_{\mathrm{a}}$ is its mass, $r_{\mathrm{a}}$ is the distance from it to the observer, and ${\widehat{\mathbf{n}}}_{\mathrm{a}}$ is the unit radial vector to the gyroscope. In non-integrated form this angular velocity is given by

$${\mathtt{\Omega }}_{\mathrm{gyr}}=2\sum _{\mathrm{i}}\frac{\left(\mathbf{v}-{\mathbf{u}}_{\mathrm{i}}\right)\times {\mathbf{a}}_{\mathrm{i}}}{c^2},$$ E16

where ${\mathbf{a}}_{\mathrm{i}}$ is the acceleration toward the $i$ -the particle and ${\mathbf{u}}_{\mathrm{i}}$ is the velocity of the $i$ -th particle. The coefficient “2” is analogous to the coefficient “2” in case of Coriolis force, i.e. half of this value corresponds to the changes in the space and other half corresponds to the changes in time. According to Ref. [20] it also appears a precession of the local coordinate system close to the massive bodies, which in non-integrated form is given by

$${\mathtt{\Omega }}_{\mathrm{lcs}}=\frac{1}{2}\sum _{\mathrm{i}}\frac{\left(\mathbf{v}-{\mathbf{u}}_{\mathrm{i}}\right)\times {\mathbf{a}}_{\mathrm{i}}}{c^2}.$$ E17

It is analogous to the Thomas precession. The precession (17) leads to apparent precession of the celestial bodies on the sky. It is necessary to consider this precession, because otherwise the total precession (geodetic plus frame dragging) would not be Lorentz covariant. This precession is unknown for the General Relativity. C.M. Will in his book ([21], p. 95) noticed that the term for the frame dragging deduced in General Relativity is not Lorentz covariant, but he did not offer any solution of the problem.

Analogously to (17), in integrated form this precession is given by

$${\mathtt{\Omega }}_{\mathrm{lcs}}=\frac{1}{2}\sum _{\mathrm{a}}\left(\mathbf{v}-{\mathbf{v}}_{\mathrm{a}}\right)\times \nabla \frac{{\mathit{Gm}}_{\mathrm{a}}}{r_{\mathrm{a}}{c}^2}-\frac{1}{4}\sum _{\mathrm{a}}G\left[{\mathbf{J}}_{\mathrm{a}}-3{\widehat{\mathbf{n}}}_{\mathrm{a}}\left({\widehat{\mathbf{n}}}_{\mathrm{a}}\cdot {\mathbf{J}}_{\mathrm{a}}\right)\right]/r_{\mathrm{a}}^3{c}^2.$$ E18

The precessions (15) and (18) have basic theoretical role, but in case of experimental measurements, as GPB experiment, it is measured the relative precession of the gyroscope (15) with respect to the local coordinate system (18), i.e. it is measured their subtraction

$${\mathtt{\Omega }}_{\mathrm{rel}}=\frac{3}{2}\sum _{\mathrm{a}}\left(\mathbf{v}-{\mathbf{v}}_{\mathrm{a}}\right)\times \nabla \frac{{\mathit{Gm}}_{\mathrm{a}}}{r_{\mathrm{a}}{c}^2}-\frac{3}{4}\sum _{\mathrm{a}}G\left[{\mathbf{J}}_{\mathrm{a}}-3{\widehat{\mathbf{n}}}_{\mathrm{a}}\left({\widehat{\mathbf{n}}}_{\mathrm{a}}\cdot {\mathbf{J}}_{\mathrm{a}}\right)\right]/r_{\mathrm{a}}^3{c}^2.$$ E19

So the geodetic precession now remains the same as in case of the General Relativity, while the precession of the frame dragging is 25% less than the value from the General Relativity.

The relative precession (19) is confirmed by the GPB experiment according to the following comments.

	geodet. Prec. (mas/yr)	frame drag. (mas/yr)
gyroscope 1	$-6588.6\pm 31.7$	$-41.3\pm 24.6$
gyroscope 2	$-6707.0\pm 64.1$	$-16.1\pm 29.7$
gyroscope 3	$-6610.5\pm 43.2$	$-25.0\pm 12.1$
gyroscope 4	$-6558.7\pm 33.2$	$-49.3\pm 11.4$
predict. From GR	−6606.1	−39.2
predict. From (19)	−6606.1	−29.4

The final results from the measurements of the geodetic precession and the frame dragging are given by the previous table published in [22]. In the last row now for comparison it is added the prediction according to Eq. (19). There is no doubt about the formula for the geodetic precession, which is −6606.1 mas/yr. The third gyroscope gives the most close result to this value. It means that it has the best of the required performances, for example it should be homogeneous and spherical body. So it is naturally to expect that this gyroscope gives the most precise results about the frame dragging. The measured value for the frame dragging is $\left(-25.0\pm 12.1\right)\mathrm{mas}/\mathrm{yr}$ and it is closer to the prediction of −29.3 mas/yr according to Eq. (19), than the prediction from the General Relativity, which is −39.2 mas/yr. So, the GPB experiment confirms the precession (18), which will be essential in deducing the magnetic field of the spinning bodies.

Now we are ready to deduce the magnetic field of the spinning bodies. It is analogous to the deduction of the spin velocities. Let us consider separately local coordinate systems for all particles of the Earth, or arbitrary spinning celestial body. We use the previous formulas where the celestial bodies are the atoms of the spinning body (Earth) as micro celestial bodies. Using that the particles of the $a$ -th celestial body move with average velocity ${\mathbf{v}}_{\mathrm{a}}$ of the barycentre of that celestial body, and using that ${\mathbf{J}}_{\mathrm{a}}\approx 0$ for each such small particle, i.e. its self angular momentum is almost 0, according to (18) we obtain that each local coordinate system intends to rotate with average angular velocity $\frac{1}{2}{\sum }_{\mathrm{a}}\left(\mathbf{v}-{\mathbf{v}}_{\mathrm{a}}\right)\times \nabla \frac{{\mathit{Gm}}_{\mathrm{a}}}{r_{\mathrm{a}}{c}^2}$ . But globally, all local coordinate systems rotate with the same angular velocity (18), because the Earth is a solid celestial body. So for each local coordinate system the non-inertial constrained angular velocity is $-\frac{1}{4}{\sum }_{\mathrm{a}}G\left[{\mathbf{J}}_{\mathrm{a}}-3{\widehat{\mathbf{n}}}_{\mathrm{a}}\left({\widehat{\mathbf{n}}}_{\mathrm{a}}\cdot {\mathbf{J}}_{\mathrm{a}}\right)\right]/r_{\mathrm{a}}^3{c}^2$ . Half of the corresponding inertial angular velocity, which is with opposite sign, will be admitted although non-observable, and the other half of the inertial angular velocity from the spatial part is non-admitted and it induces magnetic field. Hence the magnetic field of the Earth is a sum of large number of such small magnetic fields caused by each particle of the Earth. In order to determine the magnetic field, the angular velocity $\frac{1}{4}{\sum }_{\mathrm{a}}G\left[{\mathbf{J}}_{\mathrm{a}}-3{\widehat{\mathbf{n}}}_{\mathrm{a}}\left({\widehat{\mathbf{n}}}_{\mathrm{a}}\cdot {\mathbf{J}}_{\mathrm{a}}\right)\right]/r_{\mathrm{a}}^3{c}^2$ should be multiplied by $m/\left(2e\right)$ according to (14). Here $m$ is the mass of each particle of the Earth and $e$ is the charge which “corresponds” to $m$ .

The quotient $m/e$ we chose to be the universal constant $1/\sqrt{4{\mathit{\pi \epsilon }}_{0}G}$ , such that ${\mathit{Gm}}^2/R^2=e^2/\left(4{\mathit{\pi \epsilon }}_0{R}^2\right)$ , i.e. the Newton force is equal to the Coulomb force up to the sign. Hence the magnetic field measured from arbitrary inertial coordinate system is given by $\mathbf{H}=c\mathbf{B}$ , where

$$\mathbf{B}=\frac{-G}{8c^2{r}^3\sqrt{4{\mathit{\pi \epsilon }}_{0}G}}\left[\mathbf{J}-3\widehat{\mathbf{n}}\left(\widehat{\mathbf{n}}\cdot \mathbf{J}\right)\right].$$ E20

Let the spinning axis is chosen as $z$ -axis and if we put $\widehat{\mathbf{n}}=\left(cos\phi ,0,sin\phi \right)$ , then

$$\mathbf{B}=\frac{-J\sqrt{G}}{8c^2{r}^3\sqrt{4{\mathit{\pi \epsilon }}_0}}\left(-3cos\phi sin\phi \mathrm{,0,1}-3{sin}^2\phi \right).$$ E21

If $r$ is a constant, for example $r$ is radius of the Earth, then

$$B=\mid \mathbf{B}\mid =\frac{J\sqrt{G}}{8c^2{r}^3\sqrt{4{\mathit{\pi \epsilon }}_0}}\sqrt{1+3{sin}^2\phi }.$$ E22

The calculations lead to the following values: 0.295G on the equator ( $\phi =0$ ) and 0.59G on the poles ( $\phi =\pi /2$ ), which fits with the measured values.

This magnetic field is the same in arbitrary inertial coordinate system. So, this magnetic field may not be a part of electromagnetic tensor field. It means that in a moving coordinate system with arbitrary velocity close to the Earth an electric field E will not appear, and electromotive force can not be obtained.

The previous conclusion is deduced if the spinning body (Earth) is electro-neutral body. If there are some free electrons on the Earth and inside, since they are rotating together with the Earth they will produce a small quantity of magnetic field, and this magnetic field may be part of tensor of electromagnetic field, because it is caused by charged particles. The telluric current is not caused by the geomagnetic field, but from the electrostatic properties of the Earth, atmosphere and solar radiation.

In general case, as we will see in the next section, the electromagnetic field is a consequence of 3*. Hence the magnetic field of spinning bodies belongs to the case 3*. It is a specific case because the produced magnetic field can not be a part of the tensor of the electromagnetic field and consequently there is no magnetic induction.

At the end of this section let us return to the admitted half of the inertial angular velocity $\frac{1}{4}{\sum }_{\mathrm{a}}G\left[{\mathbf{J}}_{\mathrm{a}}-3{\widehat{\mathbf{n}}}_{\mathrm{a}}\left({\widehat{\mathbf{n}}}_{\mathrm{a}}\cdot {\mathbf{J}}_{\mathrm{a}}\right)\right]/r_{\mathrm{a}}^3{c}^2$ . Although this admitted half is constrained in case of the Earth, it is not constrained for the gyroscope. So, the coefficient −1 from (15) becomes now $-1+\frac{1}{8}=-\frac{7}{8}$ , and hence the coefficient $-\frac{3}{4}$ from (19) becomes $-\frac{5}{8}$ . The new prediction for the frame dragging is now $-39.2\times \frac{5}{8}=-24.5\mathrm{mas}/\mathrm{yr}$ , which almost the same with the average measured value −25 mas/yr for the third gyroscope.

5. Electromagnetic fields of the charged bodies

Let us return to the Eq. (7) with the same notations. Then the first charged body rotates the second charged body. We assume as an Axiom 1 that the angle of rotation is equal to ( Figure 4 )

$$\overrightarrow{\theta }=\frac{e_1{e}_2}{4{\mathit{\pi \epsilon }}_0{r}^2{m}_2{c}^2}\mathbf{r},$$ E23

where $\mathbf{r}=\left(x,y,z\right)$ is the radius vector from the center of the first charged body toward the second body. This axiom will be used for deriving the Coulomb’s force. So, this geometrical assumption gives us a more fundamental understanding of the charge as a physical phenomena.

Figure 4.

The second charged particle is rotated by the first charged particle (and vice versa).

Let us consider two charged particles with centers at $O_1$ and $O_2$ and let $X$ be a close point of $O_2$ of the second particle with radius vector $\left(x+a,y+b,z+c\right)$ , where $a$ , $b$ , and $c$ are small quantities of length. We can use again the Figure 3 , although it was related for strong interaction. Then it causes displacement of the point $X$ by the vector $\theta \frac{\left(x,y,z\right)}{r}\times \left(x+a,y+b,z+c\right)=\theta \frac{\left(x,y,z\right)}{r}\times \left(a,b,c\right)$ , and it corresponds to angle $\theta \frac{\left(x,y,z\right)}{r^2}\times \left(a,b,c\right)$ .

This is a consequence since the matrices of Lorentz boosts in general do not commute with the matrices of rotation from Section 1. Moreover, we can use Galilean transformations instead of Lorentz transformations and choose the coordinate system such that the angle of rotation is $\left(0,0,\theta \right)$ and obtain

$$\left[\begin{array}{cccc}cos\theta & -sin\theta & 0& 0\\ sin\theta & cos\theta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& b\\ 0& 0& 1& c\\ 0& 0& 0& 1\end{array}\right]$$

$${\left[\begin{array}{cccc}cos\theta & -sin\theta & 0& 0\\ sin\theta & cos\theta & 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]}^{-1}{\left[\begin{array}{cccc}1& 0& 0& a\\ 0& 1& 0& b\\ 0& 0& 1& c\\ 0& 0& 0& 1\end{array}\right]}^{-1}\approx \left[\begin{array}{cccc}1& 0& 0& a^{\prime }\\ 0& 1& 0& b^{\prime }\\ 0& 0& 1& c^{\prime }\\ 0& 0& 0& 1\end{array}\right],$$

where $\left(a^{\prime },b^{\prime },c^{\prime }\right)=\left(0,0,\theta \right)\times \left(a,b,c\right)$ .

The unadmitted translation leads to the Coulomb’s acceleration/force

$$\mathbf{a}=\frac{c^2}{2}\mathrm{rot}\left(\theta \frac{\left(x,y,z\right)}{r^2}\times \left(a,b,c\right)\right)=\theta c^2\frac{\left(x,y,z\right)}{r^2}=\frac{e_1{e}_2}{4{\mathit{\pi \epsilon }}_0{r}^3{m}_2}\left(x,y,z\right),$$ E24

$$\mathbf{f}=\frac{e_1{e}_2}{4{\mathit{\pi \epsilon }}_0{r}^3}\left(x,y,z\right).$$ E25

Hence the electric field caused by the first charged body is

$$\mathbf{E}=\frac{e_1}{4{\mathit{\pi \epsilon }}_0{r}^3}\left(x,y,z\right).$$ E26

Since the angle of rotation observed orthogonally to the plane of the angle, remains unchanged, the charge remains unchanged in all inertial coordinate systems.

The special case when $e_2/m_2=\sqrt{4{\mathit{\pi \epsilon }}_{0}G}$ is of special interest. Then (23) leads to

$${\theta }_{\ast }=\frac{e\sqrt{G}}{\sqrt{4{\mathit{\pi \epsilon }}_0}{\mathit{rc}}^2},$$ E27

and the corresponding acceleration is given by

$${\mathbf{a}}_{\ast }=\frac{e\sqrt{G}}{\sqrt{4{\mathit{\pi \epsilon }}_0}{r}^3}\left(x,y,z\right).$$ E28

This is extremely small acceleration. If we compare the acceleration of the electron toward arbitrary charged particle, the acceleration (28) is smaller for coefficient

$$\frac{m_e}{e}\frac{\sqrt{G}}{{\left(4{\mathit{\pi \epsilon }}_0\right)}^{3/2}}=4.8994\cdot {10}^{-22}.$$ E29

We introduce now Axiom 2, which will be used for deriving the Biot-Savart law. This axiom states that any elementary particle (not composite) with mass $m$ has charge $e$ such that

$$\mid e\mid \ge m\sqrt{4{\mathit{\pi \epsilon }}_{0}G},$$ E30

i.e. $\sqrt{4{\mathit{\pi \epsilon }}_{0}G}$ is the minimal possible value for the quotient $e/m$ . This axiom does not refer to composite particles, and also arbitrary electro-neutral particle should contain both electro-positive part and electro-negative part. This constraint is analogous to the constraint $\mid v\mid \le c$ . The local coordinate system, considered as local frame, can be treated as a border case: $m=0$ , $e=0$ and $e/m=\sqrt{4{\mathit{\pi \epsilon }}_{0}G}$ . So, (27) and (28) can be written as ${\theta }_{\mathrm{lcs}}=\frac{e\sqrt{G}}{\sqrt{4{\mathit{\pi \epsilon }}_0}{\mathit{rc}}^2}$ and ${\mathbf{a}}_{\mathrm{lcs}}=\frac{e\sqrt{G}}{\sqrt{4{\mathit{\pi \epsilon }}_0}{r}^3}\left(x,y,z\right)$ . The Axiom 2 is in accordance with the previous section, where we used the universal constant $1/\sqrt{4{\mathit{\pi \epsilon }}_{0}G}$ for the quotient $m/e$ .

If the charged particle moves with velocity $\mathbf{u}$ , then we have angular velocity

$${\mathbf{w}}_{\mathrm{lcs}}=\mathbf{u}\times \frac{e\sqrt{G}}{\sqrt{4{\mathit{\pi \epsilon }}_0}{r}^3{c}^2}\left(x,y,z\right),$$ E31

which is rather different than (17). Here it is used that the acceleration together with the components of the angular velocity multiplied by $c$ form a tensor ${\varphi }_{\mathit{ij}}$ , which is analogous to the tensor of the electromagnetic field $F_{\mathit{ij}}$ . Indeed it is an antisymmetric tensor since it is the Lie algebra of $G_t$ as $4\times 4$ matrix group with imaginary temporal coordinate. Analogously to the deduction of the magnetic field of the spinning bodies, now we should multiply this angular velocity by the universal constant $m/e=1/\sqrt{4{\mathit{\pi \epsilon }}_{0}G}$ , and obtain the Biot-Savart law (33). Moreover, since ${\varphi }_{\mathit{ij}}$ is a tensor, $-\mathbf{v}\times \mathbf{w}$ is acceleration, which in electromagnetism means that $\left(-\mathbf{v}\right)\times \frac{\mathbf{H}}{c}$ is electric field. So, for a charged body in electric field E and magnetic field H, in local system where the charged body rests, appears also an electric field $\frac{\mathbf{v}}{c}\times \mathbf{H}$ and we obtain the Lorentz force

$$\mathbf{f}=e\left(\mathbf{E}+\frac{\mathbf{v}}{c}\times \mathbf{H}\right).$$ E32

This is the case 3*, because the non-compatibilities arising from the space rotations in the temporal group $\mathit{SR}\times T$ ( $G_t$ ) lead to some changes in the same space $\mathit{SR}\times T$ , which is observed as Coulomb’s force and Lorentz force. The electromagnetic tensor field $F_{\mathit{ij}}$ in the Lie group $\mathit{SR}\times T$ , plays the main role among the inertial coordinate systems.

At the end of this section we give also a deeper view of the electromagnetic field by returning to the gyroscopes. Let us consider a charged particle with mass $m$ and charge $e$ and let us find the magnetic field caused by the angular velocity (16). Half of it is admitted and the other half is non-admitted and it causes the required magnetic field. It means constraint of both geodetic precession and frame dragging. While the coefficient $m/e$ was universal constant when it was applied to (18), in this case $m/e$ is determined by the mass and charge of the considered charged particle. The magnetic field appears, because for the charged particles, it is not admitted to rotate completely. In order to simplify the calculation, we consider only one body ( $a=1$ ) with point mass $m$ , charge $e$ and hence $J=0$ . First let us consider the border case $e/m=\frac{1}{\sqrt{4{\mathit{\pi \epsilon }}_{0}G}}$ . Analogously to the previous section, first we multiply the Eq. (16) by $\frac{1}{2}\frac{1}{\sqrt{4{\mathit{\pi \epsilon }}_{0}G}}$ and replace $\mathbf{v}=0$ because we measure the magnetic field with respect to us. Further in general case, for arbitrary mass and charge, using the proportionality $B\sim e$ , the Eq. (16) should additionally be multiplied by $\frac{e}{m}\frac{1}{\sqrt{4{\mathit{\pi \epsilon }}_{0}G}}$ . It means that in the last two steps the Eq. (16) is multiplied by $\frac{1}{2}\frac{e}{m}\frac{1}{4{\mathit{\pi \epsilon }}_{0}G}$ , and hence it becomes

$$\mathbf{B}=\frac{\mathbf{H}}{c}=\mathbf{u}\times \nabla \frac{e}{4{\mathit{\pi \epsilon }}_0{\mathit{rc}}^2},$$ E33

which is the Biot-Savart law. So the Biot-Savart law is a consequence from constraint in angular velocity of the body with electric charge $e$ . If we have electric and magnetic fields E and H, having in mind the formula (33) and $\mathtt{\Omega }=\frac{e}{\mathit{mc}}\mathbf{H}$ (when E = 0), its generalization leads to

$$\mathtt{\Omega }=\frac{e}{\mathit{mc}}\left(\mathbf{H}-\frac{1}{c}\left(\mathbf{v}\times \mathbf{E}\right)\right).$$ E34

Note that while the magnetic field of the spinning bodies was obtained from Eq. (17), the Biot-Savart law was obtained from Eq. (16). Analogously as in case of GPB experiment where the subtraction ${\mathtt{\Omega }}_{\mathrm{gyr}}-{\mathtt{\Omega }}_{\mathrm{lcs}}$ was measured, here the measurable magnetic induction can be obtained from the subtraction of the magnetic inductions (20) and (33).

6. Gravitation and general equations of motions

Although the gravitation is well studied phenomena, now we explain the mechanism of the attraction. The gravitation field caused by a mass $M$ induces that on distance $r$ from the center the lengths seem to be larger for the coefficient $1+\gamma \frac{\mathit{GM}}{{\mathit{rc}}^2}$ , if the terms of order $c^{-4}$ and smaller are neglected. Here $\gamma$ is the known PPN parameter and takes value 1.

Let us consider a body with mass $M$ and center at point $O$ , and let us consider also a test body with negligible mass and center at $A$ and radius vector $\overrightarrow{\mathit{OA}}=\left(x,y,z\right)$ . Let $B$ be a close point to $A$ of the test body with radius vector $\overrightarrow{\mathit{OB}}=\left(x+a,y+b,z+c\right)$ , where $a$ , $b$ , and $c$ are small quantities of length ( Figure 5 ). The vector product of almost unit vectors

Figure 5.

Observation of lengths in gravitational field.

$$\frac{\left(x,y,z\right)}{r}\text{and}\frac{\left(x+a,y+b,z+c\right)}{r},$$

where $r=\sqrt{x^2+y^2+z^2}$ , leads approximately to the angle

$$\angle \mathit{AOB}=\frac{\left(x,y,z\right)}{r}\times \frac{\left(x+a,y+b,z+c\right)}{r}.$$ E35

The vector $\left(a,b,c\right)$ , far from the massive bodies is observed as

$$\left(a\left(1+\gamma \frac{\mathit{GM}}{{\mathit{rc}}^2}\right),b\left(1+\gamma \frac{\mathit{GM}}{{\mathit{rc}}^2}\right),c\left(1+\gamma \frac{\mathit{GM}}{{\mathit{rc}}^2}\right)\right)$$

and instead of (35) we have the angle

$$\angle \mathit{AO}{B}^{\prime }\approx \frac{\left(x,y,z\right)}{r}\times \frac{\left(x+a\left(1+\frac{\mathit{\gamma GM}}{{\mathit{rc}}^2}\right),y+b\left(1+\frac{\mathit{\gamma GM}}{{\mathit{rc}}^2}\right),z+c\left(1+\frac{\mathit{\gamma GM}}{{\mathit{rc}}^2}\right)\right)}{r}.$$ E36

The subtraction between the vectors (35) and (36) is given by

$$\angle \mathit{BO}{B}^{\prime }=\gamma \frac{\mathit{GM}}{{\mathit{rc}}^2}\frac{\left(\mathit{yc}-\mathit{zb},\mathit{za}-\mathit{xc},\mathit{xb}-\mathit{ya}\right)}{r^2}.$$

Half of this angle is admitted and the other half is not admitted. The unadmitted half induces acceleration given by

$$\mathbf{g}=\gamma \frac{c^2}{2}\mathrm{rot}\angle \mathit{BO}{B}^{\prime }=-\gamma c^2\frac{\mathit{GM}}{{\mathit{rc}}^2}\frac{\left(x,y,z\right)}{r^2},$$

i.e.

$$\mathbf{g}=-\gamma \mathbf{r}\frac{\mathit{GM}}{r^3}.$$ E37

Now we are able to explain why the two different deductions of the Biot-Savart law from the previous section led to identical result. It is sufficient to consider only the special case when $e/m=\sqrt{4{\mathit{\pi \epsilon }}_{0}G}$ and to prove that the gravitational acceleration a from (16) is identical with the Coulomb acceleration. So, we need to prove that the gravitational acceleration ${\mathbf{a}}_{\text{grav}}=\frac{c^2}{2}\mathrm{rot}\angle \mathit{BO}{B}^{\prime }$ from (37) is identical with the Coulomb’s acceleration ${\mathbf{a}}_{\mathrm{el}}=\frac{c^2}{2}\mathrm{rot}\left(\theta \frac{\left(x,y,z\right)}{r^2}\times \left(a,b,c\right)\right)$ from (24), neglecting the terms of order $c^{-4}$ . Indeed it is easy to verify that $\angle \mathit{BO}{B}^{\prime }=\theta \frac{\left(x,y,z\right)}{r^2}\times \left(a,b,c\right)$ neglecting the terms of order $c^{-4}$ . This argument makes the Axiom 1 more evident.

Shortly we discuss what happens with the admitted half of the angle $\angle \mathit{BO}{B}^{\prime }$ . It leads to the total precession of the gyroscopes known as geodetic precession and frame dragging. It is analogous to the magnetic field in case of the electromagnetism. For example the geodetic precession is given by $\mathtt{\Omega }=-\frac{3}{2}\left(\mathbf{v}\times \mathbf{a}\right)/c^2$ . The coefficient 3/2 now is not the same with $1$ in case of the magnetitic field, because the coefficient 3/2 includes the PPN parameter $\gamma =1$ , while in case of the electromagnetism $\gamma =0$ , because the rotation caused by the charged bodies does not changes the Euclidean metric $g_{11}=g_{22}=g_{33}=1$ .

There are two different approaches for the equations of motion: (i) The motion of the particles is according to the metric of the massive body and (ii) the metric caused by the massive body should not be used for motion of the particles. The first approach is widely used in the General Relativity. The second approach is proposed by Logunov [23]. It is also used in [24] and there for equations of motion is used a spin connection in the space–time. It is based on the antisymmetric tensor ${\varphi }_{\mathit{ij}}$ with respect to the Lorentz transformations, which can be used now. We mentioned that it is analogous to the tensor of electromagnetic field, where instead of electric field we have acceleration and instead of magnetic field we have angular velocity multiplied by $c$ . So, this approach is analogous to the electromagnetism, and this connection can be used also for rotating systems. Moreover, in [18] the equations of motion are converted into high-dimensional space–time. Using the gravitational acceleration and the Lorentz covariance, the precession (16) occurs.

The change of the metric for the time in case of electromagnetism is obvious from (7), but in case of electromagnetism we do not use the metric for motion of the charged bodies. On the other hand the experience shows that the metric in gravitation can be used for equations of motion. It occurs because the force in gravitation is proportional to the mass, but not with the charge as in electromagnetism.

This is the case 4*, because the non-compatibilities arising from the lengths in the temporal group $S\times T$ , i.e. $G_t$ , lead to some changes in the same space $S\times T$ , which is observed as gravitational force and precessions (geodetic precession and frame dragging). The gravitational force is very close to the inertial forces, and so in the General Relativity the Equivalence Principle was introduced.

Having in mind the previous comments we present now the equations of motions and precessions in case of gravitation, and on the other hand they can be used for motions and precessions of the charged particles in electromagnetic field. It refers to test bodies with negligible mass. The following equations of motion are done with respect to the three-dimensional time and they are in agreement of the gravitational experiments known from the General Relativity. The general $3\times 3$ complex matrix equation for motion and precession of test particles is given by

$$\frac{\mathrm{d}\left(exp\left(\mathit{iA}+C\right)-exp\left(-\mathit{iA}-C\right)\right)}{\mathrm{d}s}=\frac{1}{\sqrt{1-v^2/c^2}}S+\frac{\mathbf{v}}{c\sqrt{1-v^2/c^2}}\times S,$$ E38

where $A$ is antisymmetric $3\times 3$ matrix which corresponds to motion with velocity, i.e.

$$\begin{array}{l}{e}^{\mathit{iA}}=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]-\frac{1}{\left(1+\gamma \right)\left(c^2-v^2\right)}\left[\begin{array}{ccc}{v}_x^2& v_x{v}_y& v_x{v}_z\\ v_x{v}_y& v_y^2& v_y{v}_z\\ v_x{v}_z& v_y{v}_z& v_z^2\end{array}\right]-\\ -\frac{i}{c\sqrt{1-\frac{v^2}{c^2}}}\left[\begin{array}{ccc}0& v_z& -v_y\\ -v_z& 0& v_x\\ v_y& -v_x& 0\end{array}\right],\end{array}$$

$C$ is antisymmetric $3\times 3$ matrix which corresponds to space rotation, i.e. $\mathtt{\Omega }=d\mathbf{C}/\mathit{dt}$ , $\mathrm{d}s=c\mathrm{d}t\sqrt{1-v^2/c^2}$ , and $\mathbf{v}$ is 3-vector of velocity. The matrix $S$ is defined by $S=Q^T\mathit{\varphi Q}$ , where the tensor $\varphi$ as a complex $3\times 3$ matrix is given by

$$\varphi =\left[\begin{array}{ccc}0& \frac{w_z}{c}-i\frac{a_z}{c^2}& -\frac{w_y}{c}+i\frac{a_y}{c^2}\\ & & \\ -\frac{w_z}{c}+i\frac{a_z}{c^2}& 0& \frac{w_x}{c}-i\frac{a_x}{c^2}\\ & & \\ \frac{w_y}{c}-i\frac{a_y}{c^2}& -\frac{w_x}{c}+i\frac{a_x}{c^2}& 0\end{array}\right].$$ E39

The matrix $Q$ depends on the relative velocity of the particle with respect to the source of the field, and becomes $I$ if the relative velocity is $0$ . It is determined in [18]. The matrix $\mathbf{v}\times S$ , for each antisymmetric matrix $S$ is defined by $f^{-1}\left(\mathbf{v}\times \left(s_{23},s_{31},s_{12}\right)\right)$ , where $f\left(L\right)=\left(L_{23},L_{31},L_{12}\right)$ for each antisymmetric matrix $L$ .

The PPN parameter $\gamma$ has the same role in case of equations of motion (35) as the matrices $Q$ and $Q^T$ . Indeed if $\gamma =1$ , then the matrices $Q$ and $Q^T$ should be included (in case of gravitation) and if $\gamma =0$ (in case of electromagnetism), then $Q$ and $Q^T$ should be omitted. This is noticed in [24], where instead of $3\times 3$ matrices $Q$ is considered its equivalent $4\times 4$ matrix $P$ . In case of electromagnetism the matrix $S$ should be the following

$$S=-\frac{e}{{\mathit{mc}}^2}\left[\begin{array}{ccc}0& E_z+{\mathit{iH}}_z& -E_y-{\mathit{iH}}_y\\ -E_z-{\mathit{iH}}_z& 0& E_x+{\mathit{iH}}_x\\ E_y+{\mathit{iH}}_y& -E_x-{\mathit{iH}}_x& 0\end{array}\right].$$ E40

Then the Lorentz force (32) and the angular velocity of the charged body (34) can be obtained.

7. Conclusions

This research gives a global view of the interactions in the nature, especially the electromagnetic. It is based of the non-commutativity of the structural Lie groups $G_t$ and $G_s$ . The duality is one of the main principle of this model. If the non-inertial velocity (or angular velocity) is constrained, then it transforms into inertial velocity (angular velocity), which belongs to the temporal part of the space–time. Also the admitted and non-admitted quantities are equal. Indeed, both these quantities are mutually dependent. It makes the model more abstract, but it leads to satisfactory results and gives more fundamental view of the interactions. It also gives new view of the charges, such that a charged particle rotates the other charged particles for a small angle $\theta$ . The magnetic field of the spinning bodies is separated from the electromagnetic field of the charged particles. It also explains why the charge remains the same among the inertial systems. This view helps us more clearly to see the similarity between the electromagnetism and gravitation and also to see where they differ. In some future research should be considered the electromagnetic waves from this geometrical viewpoint.