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Relativistic uniform system

 

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Relativistic uniform system is an ideal physical system, in which the mass density (or any other physical quantity) depends on the Lorentz factor of the system’s particles, but is constant in the reference frames associated with the moving particles.

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Difference from the classical uniform system

In classical physics, the ideal uniform body model is widely used, in which the mass density is constant throughout the volume of the body or is given as the volume-averaged quantity. This model simplifies solution of physical problems and allows us to quickly estimate different physical quantities. For example, the body mass is calculated by simply multiplying the mass density by the body volume, which is easier than integrating the density over the volume in case of dependence of the density on the coordinates. The disadvantage of the classical model is that the majority of real physical systems are far from this ideal uniformity.

The use of the concept of the relativistic uniform system is based on the special theory of relativity (STR) and is the next step towards a more precise description of physical systems. In STR particular importance is given to invariant physical quantities, which can be calculated in each inertial reference frame and are equal to the values that these quantities have in the proper reference frame of the body. For example, multiplication of the invariant mass by the four-velocity gives the four-momentum of the body containing the invariant energy, and the multiplication of the corresponding invariant quantities by the four-velocity allows us in the case of motion of solid point particles to find the four-potentials of any vector fields and to develop their complete theory. [1] Another example is that for determination of the four-velocity or four-acceleration as a rule the operator of proper-time-derivative is used instead of the time derivative. Therefore, the use of the invariant mass density and charge density of the moving particles that make up the system does not only conform to the principles of STR but also significantly simplifies solution of the relativistic equations of motion.

Field functions for the bodies of spherical shape

Field equations are most easily solved in case of spherical symmetry in the absence of general rotation of particles. In this case all the physical quantities depend only on the current radius, which starts at the center of the sphere. Below the solutions are provided of the equations for different fields in the framework of STR, including the scalar potentials, field strengths and solenoidal vectors. Due to the random motion of the particles in the system, the vector field potentials become equal to zero. This leads to zeroing of the solenoidal vectors of fields, including the magnetic field and the gravitational torsion field.

Acceleration field

The four-potential   Описание: ~U_{\mu }=\left({\frac  {\vartheta }{c}},-{\mathbf  U}\right)   of the acceleration field includes the scalar potential  Описание: ~\vartheta  and the vector potential Описание: ~{\mathbf  U}. Applying the four-curl to the four-potential gives the acceleration tensor  Описание: ~u_{{\mu \nu }}=\nabla _{\mu }U_{\nu }-\nabla _{\nu }U_{\mu } . In the curved spacetime the acceleration field equation with the field sources is derived from the principle of least action: [1] 

Описание: ~\nabla ^{\nu }u_{{\mu \nu }}=-{\frac  {4\pi \eta }{c^{2}}}J_{\mu }.

This equation after expressing the acceleration tensor Описание: ~u_{{\mu \nu }}  in terms of the four-potential turns into the wave equation for finding the four-potential of the acceleration field:

Описание: ~\nabla ^{\nu }\nabla _{\mu }U_{\nu }-\nabla ^{\nu }\nabla _{\nu }U_{\mu }=-{\frac  {4\pi \eta }{c^{2}}}J_{\mu },

 

which, taking into account the calibration condition of the four-potential Описание: ~\nabla ^{\mu }U_{\mu }=0  ,  can be transformed as follows:

Описание: ~\nabla ^{\nu }\nabla _{\nu }U_{\mu }+R_{{\mu \nu }}U^{\nu }={\frac  {4\pi \eta }{c^{2}}}J_{\mu },

 

where Описание: ~c  is the speed of light, Описание: ~\eta   is the acceleration field coefficient, Описание: ~J_{\mu }=g_{{\mu \nu }}J^{\nu }=g_{{\mu \nu }}\rho _{0}u^{\nu }  is the mass four-current with the covariant index, Описание: ~g_{{\mu \nu }}  is the metric tensor, Описание: ~R_{{\mu \nu }}  is the Ricci tensor, Описание: ~u^{\nu }  is the four-velocity, Описание: ~\rho _{0}  is the invariant mass density of the particles in the comoving reference frames, which is the same for all the particles.

In Minkowski spacetime within the framework of STR, the covariant derivatives of the form Описание: ~\nabla _{\mu }  turn into the partial derivatives of the form Описание: ~\partial _{\mu }, while the result of the action of the partial derivatives does not depend on the order of their action. As a consequence of the calibration of the 4-potential, the equality holds: Описание: ~\partial ^{\nu }\partial _{\mu }U_{\nu }=\partial _{\mu }\partial ^{\nu }U_{\nu }=0.  As a result, the four-potential of the acceleration field can be found from the wave equation:

Описание: ~\partial ^{\nu }\partial _{\nu }U_{\mu }={\frac  {4\pi \eta }{c^{2}}}J_{\mu }. 

This equation can be divided into two equations – one for the scalar potential and the other for the vector potential of the acceleration field. In this system under consideration the vector potential is equal to zero, and the scalar potential of the acceleration field is given by:

Описание: ~\vartheta =cg_{{0\mu }}u^{\mu }=\gamma 'c^{2},

where Описание: ~g_{{0\mu }}  are the time components of the metric tensor, Описание: ~\gamma '  is the Lorentz factor of the particles in the reference frame K'  associated with the center of the sphere.

Since the scalar potential of the stationary system does not depend on time, the wave equation for the scalar potential turns into the Poisson equation: [2]

Описание: ~\triangle \vartheta =-4\pi \eta \rho _{0}\gamma '

and the following formula is obtained for the Lorentz factor of the particles: [3]

Описание: ~\gamma '={\frac  {c\gamma _{c}}{r{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\approx \gamma _{c}-{\frac  {2\pi \eta \rho _{0}r^{2}\gamma _{c}}{3c^{2}}},\qquad \qquad (1)

where Описание: ~\gamma _{c}  is the Lorentz factor of the particles at the center of the sphere, Описание: ~r  is the current radius.

The acceleration field strength and the corresponding solenoidal vector are expressed by the formulas:

Описание: ~{\mathbf  S}=-\nabla \vartheta -{\frac  {\partial {\mathbf  U}}{\partial t}}={\frac  {c^{2}\gamma _{c}{\mathbf  r}}{r^{3}}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-r\cos \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx {\frac  {4\pi \eta \rho _{0}\gamma _{c}{\mathbf  r}}{3}}\left(1-{\frac  {4\pi \eta \rho _{0}r^{2}}{10c^{2}}}\right).

 

Описание: ~{\mathbf  N}=\nabla \times {\mathbf  U}=0.

Pressure field

The four-potential  Описание: ~\pi _{\mu }=\left({\frac  {\wp }{c}},-{\mathbf  \Pi }\right) of the pressure field includes the scalar potential  Описание: ~\wp   and the vector potential  Описание: ~{\mathbf  \Pi }, and obeys the calibration condition: Описание: ~\nabla ^{\mu }\pi _{\mu }=0 .

The pressure field equation with the field sources, the pressure field tensor Описание: ~f_{{\mu \nu }}  and the equation for finding the four-potential of the pressure field have the form: [1]

Описание: ~\nabla ^{\nu }f_{{\mu \nu }}=-{\frac  {4\pi \sigma }{c^{2}}}J_{\mu },\quad f_{{\mu \nu }}=\nabla _{\mu }\pi _{\nu }-\nabla _{\nu }\pi _{\mu },\quad \nabla ^{\nu }\nabla _{\nu }\pi _{\mu }+R_{{\mu \nu }}\pi ^{\nu }={\frac  {4\pi \sigma }{c^{2}}}J_{\mu },

 

where  Описание: ~\sigma   is the pressure field coefficient.

In STR the latter equation turns into the wave equation:

Описание: ~\partial ^{\nu }\partial _{\nu }\pi _{\mu }={\frac  {4\pi \sigma }{c^{2}}}J_{\mu }.

In the stationary case the potentials do not depend on time and the time component of the wave equation turns into the Poisson equation for the scalar potential of the pressure field:

Описание: ~\triangle \wp =-4\pi \sigma \rho _{0}\gamma '.

The solution of this equation inside the sphere with particles is as follows: [3]

Описание: ~\wp =\wp _{c}-{\frac  {\sigma c^{2}\gamma _{c}}{\eta }}+{\frac  {\sigma c^{3}\gamma _{c}}{\eta r{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\approx \wp _{c}-{\frac  {2\pi \sigma \rho _{0}r^{2}\gamma _{c}}{3}}.

where  Описание: ~\wp _{c}  is the scalar potential at the center of the sphere. This potential is approximately equal to: [4]

~\wp _{c}\approx {\frac  {3\sigma m}{10a}}\left(1+{\frac  {9}{2{\sqrt  {14}}}}\right),

where the acceleration field constant  ~\eta  and the pressure field constant  ~\sigma  are expressed by the formulas:

~\eta ={\frac  {3}{5}}\left(G-{\frac  {\rho _{{0q}}^{2}}{4\pi \varepsilon _{0}\rho _{0}^{2}}}\right),\qquad \qquad \sigma ={\frac  {2}{5}}\left(G-{\frac  {\rho _{{0q}}^{2}}{4\pi \varepsilon _{0}\rho _{0}^{2}}}\right).

 

The strength of the pressure field and the corresponding solenoidal vector are found as follows:

Описание: ~{\mathbf  C}=-\nabla \wp -{\frac  {\partial {\mathbf  \Pi }}{\partial t}}={\frac  {\sigma c^{2}\gamma _{c}{\mathbf  r}}{\eta r^{3}}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-r\cos \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx {\frac  {4\pi \sigma \rho _{0}\gamma _{c}{\mathbf  r}}{3}}\left(1-{\frac  {4\pi \eta \rho _{0}r^{2}}{10c^{2}}}\right).

 

Описание: ~{\mathbf  I}=\nabla \times {\mathbf  \Pi }=0.

Gravitational field

The gravitational four-potential  Описание: ~D_{\mu }=\left({\frac  {\psi }{c}},-{\mathbf  D}\right)  of the gravitational field is made up with the use of the scalar Описание: ~\psi   and vector Описание: ~{\mathbf  D}  potentials. The calibration condition of the four-potential is: Описание: ~\nabla ^{\mu }D_{\mu }=0 .

The gravitational field equation with the field sources, the gravitational tensor  Описание: ~\Phi _{{\mu \nu }}  and the equation for finding the four-potential of the gravitational field in the covariant theory of gravitation have the form: [5] [6]

Описание: ~\nabla ^{\nu }\Phi _{{\mu \nu }}={\frac  {4\pi G}{c^{2}}}J_{\mu },\quad \Phi _{{\mu \nu }}=\nabla _{\mu }D_{\nu }-\nabla _{\nu }D_{\mu },\quad \nabla ^{\nu }\nabla _{\nu }D_{\mu }+R_{{\mu \nu }}D^{\nu }=-{\frac  {4\pi G}{c^{2}}}J_{\mu },

 

where Описание: ~G is the gravitational constant.

In STR the latter equation is simplified and becomes the wave equation:

Описание: ~\partial ^{\nu }\partial _{\nu }D_{\mu }=-{\frac  {4\pi G}{c^{2}}}J_{\mu }.

From the wave equation in the stationary case, the Poisson equation follows for the scalar potential inside the sphere with randomly moving particles in the framework of the Lorentz-invariant theory of gravitation (LITG):

Описание: ~\triangle \psi _{i}=4\pi G\rho _{0}\gamma '.

The right-hand side of this equation contains the Lorentz factor Описание: ~\gamma ', which depends on the radius according to (1). In addition, the internal scalar potential near the surface of the sphere must coincide with the scalar potential of the external field of the system, in view of the standard potential gauge, that is with equality of the potential to zero at infinity.

As a result, the dependence of the scalar potential on the current radius differs from the dependence in the classical case of the uniform sphere with the radius Описание: ~a and is equal to it only approximately: [3]

Описание: ~\psi _{i}=-{\frac  {Gc^{2}\gamma _{c}}{\eta r}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-r\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx -{\frac  {2\pi G\rho _{0}\gamma _{c}(3a^{2}-r^{2})}{3}}.

 

For the gravitational field strength and the gravitational torsion field inside the sphere we obtain the following: [7]

Описание: ~{\mathbf  \Gamma }_{i}=-\nabla \psi _{i}-{\frac  {\partial {\mathbf  D}_{i}}{\partial t}}=-{\frac  {Gc^{2}\gamma _{c}{\mathbf  r}}{\eta r^{3}}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-r\cos \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx -{\frac  {4\pi G\rho _{0}\gamma _{c}{\mathbf  r}}{3}}\left(1-{\frac  {4\pi \eta \rho _{0}r^{2}}{10c^{2}}}\right).

 

Описание: ~{\mathbf  \Omega }_{i}=\nabla \times {\mathbf  D}_{i}=0.

 

The solutions for the external gravitational field potential and for the field strength Описание: ~\Gamma _{o} according to LITG are as follows:

Описание: ~\psi _{o}=-{\frac  {Gc^{2}\gamma _{c}}{\eta r}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx -{\frac  {Gm\gamma _{c}}{r}}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right).

 

Описание: ~{\mathbf  \Gamma }_{o}=-\nabla \psi _{o}-{\frac  {\partial {\mathbf  D}_{o}}{\partial t}}=-{\frac  {Gc^{2}\gamma _{c}{\mathbf  r}}{\eta r^{3}}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx -{\frac  {Gm\gamma _{c}{\mathbf  r}}{r^{3}}}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right).

Here, the auxiliary mass Описание: ~m  is equal to the product of the mass density Описание: ~\rho _{0}  by the volume of the sphere:  Описание: ~m={\frac  {4\pi \rho _{0}a^{3}}{3}}. From the expressions for the potential and strength of the external gravitational field we can see that the role of the gravitational mass is played by the mass  Описание: ~m_{g}\approx m\gamma _{c}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right).  Since Описание: ~\gamma _{c}>1  then the relation  Описание: ~m_{g}>m  is satisfied.

To understand the difference between these masses we should calculate the total relativistic mass  Описание: ~m_{b}  of the particles moving inside the sphere. For the motion of particles there should be some voids between them. Both the average accelerations and the average velocities of the particles inside the sphere are functions of the current radius. Dividing the particles’ velocities by their acceleration, we can find the dependence of the average period of the oscillatory motion of particles on the radius. Finally, multiplying the velocity by the average period of motion, we can obtain an estimate of the size of the voids between the particles.

In order to calculate the volume of the sphere, it is necessary to sum up the volumes of all the typical particles moving inside the sphere, as well as the volumes of the voids between them. Suppose now that the sizes of the typical particles are much larger than the voids between the particles, and the volume of the voids is substantially less than the total volume of the particles. In this case, we can use the approximation of continuous medium, so that the unit of the mass of matter inside the sphere will be given by the approximate expression  Описание: ~dm\approx \rho _{0}\gamma 'dV, where  Описание: ~\rho _{0}  is the mass density in the reference frames associated with the particles, Описание: ~\gamma '  is the Lorentz factor of the moving particles, the product  Описание: ~\rho _{0}\gamma '  gives the mass density of the particles from the viewpoint of an observer, who is stationary with respect to the sphere, and the volume element  Описание: ~dV  inside the sphere corresponds to the volume of the particle from the viewpoint of this observer. This leads to the fact that the total volume of the particles moving inside the sphere becomes approximately equal to the volume of the sphere. For the mass, in view of the Lorentz factor (1), the following relation is obtained:

 

Описание: ~m_{b}=\int dm=\int \rho _{0}\gamma 'dV={\frac  {c^{2}\gamma _{c}}{\eta }}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx m\gamma _{c}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right).

 

This implies the equality of the gravitational mass Описание: ~m_{g}  and the total relativistic mass Описание: ~m_{b}  of the particles moving inside the sphere. The both masses are greater than the mass  Описание: ~m. By the method of its calculation, the mass Описание: ~m_{b}  is equal to the sum of the invariant masses of the particles that make up the system.

The external torsion field is equal to zero:

Описание: ~{\mathbf  \Omega }_{o}=\nabla \times {\mathbf  D}_{o}=0.

Electromagnetic field

The electromagnetic four-potential  Описание: ~A_{\mu }=\left({\frac  {\varphi }{c}},-{\mathbf  A}\right)  of the electromagnetic field includes the scalar potential  Описание: ~\varphi   and the vector potential Описание: ~{\mathbf  A}. The covariant Lorentz calibration for four-potential is:  Описание: ~\nabla ^{\mu }A_{\mu }=0  .  For a fixed uniformly charged spherical body with random motion of charges the total electromagnetic field on the average is purely electric and the vector potential is equal to zero.

The electromagnetic field equation with the field sources, the electromagnetic tensor Описание: ~F_{{\mu \nu }}  and the equation for finding the four-potential are expressed as follows:

Описание: ~\nabla ^{\nu }F_{{\mu \nu }}=-{\frac  {1}{\varepsilon _{0}c^{2}}}j_{\mu },\quad F_{{\mu \nu }}=\nabla _{\mu }A_{\nu }-\nabla _{\nu }A_{\mu },\quad \nabla ^{\nu }\nabla _{\nu }A_{\mu }+R_{{\mu \nu }}A^{\nu }={\frac  {1}{\varepsilon _{0}c^{2}}}j_{\mu },

 

where Описание: ~\varepsilon _{0}  is the electric constant, Описание: ~j_{\mu }  is the electromagnetic four-current.

The latter equation in STR turns into the wave equation:

Описание: ~\partial ^{\nu }\partial _{\nu }A_{\mu }={\frac  {1}{\varepsilon _{0}c^{2}}}j_{\mu }.

Due to the absence of time-dependence in the case under consideration, the wave equation becomes the Poisson equation for the scalar potential Описание: ~\varphi _{i}  inside the sphere:

Описание: ~\triangle \varphi _{i}=-{\frac  {\rho _{{0q}}\gamma '}{\varepsilon _{0}}},

where Описание: ~\rho _{{0q}}  is the charge density in the reference frames associated with the charges.

The dependence of the scalar potential on the current radius in the general case differs from the dependence in the classical case of the potential of a uniformly charged sphere with the radius Описание: ~a, coinciding with it only in the first approximation: [8]

Описание: ~\varphi _{i}={\frac  {\rho _{{0q}}c^{2}\gamma _{c}}{4\pi \varepsilon _{0}\eta \rho _{0}r}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-r\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx {\frac  {\rho _{{0q}}\gamma _{c}(3a^{2}-r^{2})}{6\varepsilon _{0}}}.

 

The field strength of the electric field and the magnetic field inside the sphere have the form:

Описание: ~{\mathbf  E}_{i}=-\nabla \varphi _{i}-{\frac  {\partial {\mathbf  A}_{i}}{\partial t}}={\frac  {\rho _{{0q}}c^{2}\gamma _{c}{\mathbf  r}}{4\pi \varepsilon _{0}\eta \rho _{0}r^{3}}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-r\cos \left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx {\frac  {\rho _{{0q}}\gamma _{c}{\mathbf  r}}{3\varepsilon _{0}}}\left(1-{\frac  {4\pi \eta \rho _{0}r^{2}}{10c^{2}}}\right).

 

Описание: ~{\mathbf  B}_{i}=\nabla \times {\mathbf  A}_{i}=0.

 

Outside the system under consideration the charge density is equal to zero and the Poisson equation for the scalar potential turns into the Laplace equation:

Описание: ~\triangle \varphi _{o}=0.

The solution for the external electric field potential, corresponding to the potential gauge and the Maxwell's equations for the electric field strength Описание: ~E_{o}  is given by:

Описание: ~\varphi _{o}={\frac  {\rho _{{0q}}c^{2}\gamma _{c}}{4\pi \varepsilon _{0}\eta \rho _{0}r}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx {\frac  {q\gamma _{c}}{4\pi \varepsilon _{0}r}}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right).

 

Описание: ~{\mathbf  E}_{o}=-\nabla \varphi _{o}-{\frac  {\partial {\mathbf  A}_{o}}{\partial t}}={\frac  {\rho _{{0q}}c^{2}\gamma _{c}{\mathbf  r}}{4\pi \varepsilon _{0}\eta \rho _{0}r^{3}}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx {\frac  {q\gamma _{c}{\mathbf  r}}{4\pi \varepsilon _{0}r^{3}}}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right).

 

The external magnetic field is equal to zero:

Описание: ~{\mathbf  B}_{o}=\nabla \times {\mathbf  A}_{o}=0.

In these expressions, the charge Описание: ~q  is an auxiliary quantity equal to the product of the charge density Описание: ~\rho _{{0q}}  by the volume of the sphere:  Описание: ~q={\frac  {4\pi \rho _{{0q}}a^{3}}{3}}.

In this case, the following quantity serves as the total charge of the system:

Описание: ~q_{b}=\int \rho _{{0q}}\gamma 'dV={\frac  {\rho _{{0q}}c^{2}\gamma _{c}}{\eta \rho _{0}}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx q\gamma _{c}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right),

 

while Описание: ~q_{b}>q.  The charge   Описание: ~q_{b}  is calculated in the same way as the mass Описание: ~m_{b}  and has the meaning of the sum of the charges of all the system’s particles.

Tensor field invariants

The knowledge of the field strengths and the solenoidal components of fields allows us to find the tensor components of the corresponding fields with the covariant indices. To pass on to the field tensors with the contravariant indices we need to know the metric tensor. In STR the metric tensor does not depend on the coordinates and time, is uniquely defined, and in Cartesian coordinates consists of zeros and unities. As a result, it is easy to find the tensor field invariants  Описание: ~u_{{\mu \nu }}u^{{\mu \nu }},   Описание: ~f_{{\mu \nu }}f^{{\mu \nu }},     Описание: ~\Phi _{{\mu \nu }}\Phi ^{{\mu \nu }} and  Описание: ~F_{{\mu \nu }}F^{{\mu \nu }}, where  Описание: ~u_{{\mu \nu }},  Описание: ~f_{{\mu \nu }},   Описание: ~\Phi _{{\mu \nu }}  and   Описание: ~F_{{\mu \nu }}   are the acceleration tensor, the pressure field tensor, the gravitational tensor and the electromagnetic tensor, respectively.

The tensor field invariants are included in the Lagrangian, the Hamiltonian. the action function and the relativistic energy of the system, and they are located there inside the integrals over the space volume. In addition, they are included in the corresponding stress-energy tensors of the fields. [2]  Since in the system under consideration the solenoidal vectors are zero, the tensor invariants depend only on the field strengths:

Описание: ~u_{{\mu \nu }}u^{{\mu \nu }}=-{\frac  {2}{c^{2}}}(S^{2}-c^{2}N^{2})=-{\frac  {2}{c^{2}}}S^{2}.

 

Описание: ~f_{{\mu \nu }}f^{{\mu \nu }}=-{\frac  {2}{c^{2}}}(C^{2}-c^{2}I^{2})=-{\frac  {2}{c^{2}}}C^{2}.

 

Описание: ~\Phi _{{\mu \nu }}\Phi ^{{\mu \nu }}=-{\frac  {2}{c^{2}}}(\Gamma ^{2}-c^{2}\Omega ^{2})=-{\frac  {2}{c^{2}}}\Gamma ^{2}.

 

Описание: ~F_{{\mu \nu }}F^{{\mu \nu }}=-{\frac  {2}{c^{2}}}(E^{2}-c^{2}B^{2})=-{\frac  {2}{c^{2}}}E^{2}.

 

The volume integrals of the tensor invariants multiplied by the corresponding factors were calculated in the article. [7]  For the acceleration field and the pressure field the integrals are taken only over the volume of the sphere:

Описание: ~\int \limits _{{r=0}}^{{a}}{\frac  {c^{2}}{16\pi \eta }}u_{{\mu \nu }}u^{{\mu \nu }}dV=-{\frac  {c^{4}\gamma _{c}^{2}}{2\eta }}\left[{\frac  {a}{2}}+{\frac  {c}{4{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {2a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-{\frac  {c^{2}}{4\pi \eta \rho _{0}a}}\sin ^{2}\left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx -{\frac  {\eta m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right).

 

Описание: ~\int \limits _{{r=0}}^{{a}}{\frac  {c^{2}}{16\pi \sigma }}f_{{\mu \nu }}f^{{\mu \nu }}dV=-{\frac  {\sigma c^{4}\gamma _{c}^{2}}{2\eta ^{2}}}\left[{\frac  {a}{2}}+{\frac  {c}{4{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {2a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-{\frac  {c^{2}}{4\pi \eta \rho _{0}a}}\sin ^{2}\left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx -{\frac  {\sigma m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right).

 

The gravitational and electromagnetic fields of the system are present not only inside but also outside the sphere, where they extend to infinity, while the field strengths of the internal and external fields behave differently. The field strengths  Описание: ~{\mathbf  \Gamma }_{i}  and  Описание: ~{\mathbf  E}_{i}  are substituted respectively into the integrals of the tensor invariants of these fields taken over the volume of the sphere, which gives the following:

Описание: ~-\int \limits _{{r=0}}^{{a}}{\frac  {c^{2}}{16\pi G}}\Phi _{{\mu \nu }}\Phi ^{{\mu \nu }}dV=

Описание: ~={\frac  {Gc^{4}\gamma _{c}^{2}}{2\eta ^{2}}}\left[{\frac  {a}{2}}+{\frac  {c}{4{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {2a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-{\frac  {c^{2}}{4\pi \eta \rho _{0}a}}\sin ^{2}\left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx {\frac  {Gm^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right).

 

Описание: ~\int \limits _{{r=0}}^{{a}}{\frac  {c^{2}\varepsilon _{0}}{4}}F_{{\mu \nu }}F^{{\mu \nu }}dV=

Описание: ~=-{\frac  {\rho _{{0q}}^{2}c^{4}\gamma _{c}^{2}}{8\pi \varepsilon _{0}\eta ^{2}\rho _{0}^{2}}}\left[{\frac  {a}{2}}+{\frac  {c}{4{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {2a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-{\frac  {c^{2}}{4\pi \eta \rho _{0}a}}\sin ^{2}\left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx -{\frac  {q^{2}\gamma _{c}^{2}}{40\pi \varepsilon _{0}a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right).

 

 

Into the volume integrals of the tensor invariants of the gravitational and electromagnetic fields of the system outside the sphere the field strengths  Описание: ~{\mathbf  \Gamma }_{o}  and  Описание: ~{\mathbf  E}_{o}  are substituted, respectively:

Описание: ~-\int \limits _{{r=a}}^{{\infty }}{\frac  {c^{2}}{16\pi G}}\Phi _{{\mu \nu }}\Phi ^{{\mu \nu }}dV=

Описание: ~={\frac  {Gc^{4}\gamma _{c}^{2}}{2\eta ^{2}a}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]^{2}\approx {\frac  {Gm^{2}\gamma _{c}^{2}}{2a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right).

 

Описание: ~\int \limits _{{r=a}}^{{\infty }}{\frac  {c^{2}\varepsilon _{0}}{4}}F_{{\mu \nu }}F^{{\mu \nu }}dV=

Описание: ~=-{\frac  {\rho _{{0q}}^{2}c^{4}\gamma _{c}^{2}}{8\pi \varepsilon _{0}\eta ^{2}\rho _{0}^{2}a}}\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]^{2}\approx -{\frac  {q^{2}\gamma _{c}^{2}}{8\pi \varepsilon _{0}a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right).

 

Energies of particles in the field potentials

All the four fields act on the particles inside the sphere, and therefore each particle of the system acquires the corresponding energy in a particular field. The energy of the particle in the field is calculated as the volume integral of the product of the effective mass density  Описание: ~\rho =\rho _{0}\gamma '  by the corresponding scalar potential, and for the electric field the energy is determined as the volume integral of the product of the effective charge density  Описание: ~\rho _{q}=\rho _{{0q}}\gamma '   by the scalar potential  Описание: ~\varphi , where the Lorentz factor Описание: ~\gamma '  from (1) is used. In STR the energies of particles in the acceleration field, pressure field, gravitational and electric fields in the uniform relativistic spherical system, in view of the expressions for the field potentials [7] and the corrections to calculations, [8] [9] [10]  [11]  are, respectively:

Описание: ~\int \rho \vartheta dV=\rho _{0}c^{2}\int \gamma '^{2}dV={\frac  {c^{4}\gamma _{c}^{2}}{\eta }}\left[{\frac  {a}{2}}-{\frac  {c}{4{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {2a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx

Описание: ~\approx mc^{2}\gamma _{c}^{2}-{\frac  {3\eta m^{2}\gamma _{c}^{2}}{5a}}\left(1-{\frac  {2\eta m}{7ac^{2}}}\right).

 

Описание: ~\int \rho \wp dV=\rho _{0}\int \gamma '\wp dV={\frac  {c^{2}\gamma _{c}}{\eta }}\left(\wp _{c}-{\frac  {\sigma c^{2}\gamma _{c}}{\eta }}\right)\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]+

Описание: ~+{\frac  {\sigma c^{4}\gamma _{c}^{2}}{\eta ^{2}}}\left[{\frac  {a}{2}}-{\frac  {c}{4{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {2a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx m\wp _{c}\gamma _{c}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right)-{\frac  {3\sigma m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {13\eta m}{28ac^{2}}}\right).

 

Описание: ~\int \rho \psi _{i}dV=\rho _{0}\int \gamma '\psi _{i}dV=

Описание: ~={\frac  {Gc^{4}\gamma _{c}^{2}}{\eta ^{2}}}\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]-

 

Описание: ~-{\frac  {Gc^{4}\gamma _{c}^{2}}{\eta ^{2}}}\left[{\frac  {a}{2}}-{\frac  {c}{4{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {2a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx -{\frac  {6Gm^{2}\gamma _{c}^{2}}{5a}}\left(1-{\frac  {4\eta m}{7ac^{2}}}\right).

 

Описание: ~\int \rho _{q}\varphi _{i}dV=\rho _{{0q}}\int \gamma '\varphi _{i}dV=

Описание: ~=-{\frac  {\rho _{{0q}}^{2}c^{4}\gamma _{c}^{2}}{4\pi \varepsilon _{0}\eta ^{2}\rho _{0}^{2}}}\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\left[{\frac  {c}{{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)-a\cos \left({\frac  {a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]+

Описание: ~+{\frac  {\rho _{{0q}}^{2}c^{4}\gamma _{c}^{2}}{4\pi \varepsilon _{0}\eta ^{2}\rho _{0}^{2}}}\left[{\frac  {a}{2}}-{\frac  {c}{4{\sqrt  {4\pi \eta \rho _{0}}}}}\sin \left({\frac  {2a}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)\right]\approx {\frac  {3q^{2}\gamma _{c}^{2}}{10\pi \varepsilon _{0}a}}\left(1-{\frac  {4\eta m}{7ac^{2}}}\right).

 

It should be noted that all the fields, which contain particles, are not the fields arising from the external sources, but are generated by the particles themselves. As a result, the particles’ energies calculated above in the scalar potentials of the fields are twice as large as the potential energy of one or another interaction. For example, in order to calculate the electrostatic energy of the system of two charges, it is sufficient to take the potential of the first charge at the location of the second charge and to multiply it by the value of the second charge. But if we use the formula for the energy in the form of an integral, then the electrostatic energy will be taken into account twice, because the term is added, which contains the potential of the second charge at the location of the first charge multiplied by the value of the first charge. On the other hand, the electrostatic energy must consist of two components that take into account both the energy of particles in each other’s fields and the energy of the electric field itself.

Instead, in electrostatics, the electrostatic energy is calculated either through the scalar potential or through the field strength by integrating the time component of the stress-energy tensor over the volume. Both methods provide the same result, but the connection between the field energy and the energy of particles in the potential is lost in this case, and it is not clear why these energies should coincide.

Relation between the field coefficients

For the four fields under consideration the equation of motion of matter in the concept of the general field is as follows: [12]

Описание: ~u_{{\mu \nu }}J^{\nu }+f_{{\mu \nu }}J^{\nu }+\Phi _{{\mu \nu }}J^{\nu }+F_{{\mu \nu }}j^{\nu }=0,

 

where Описание: ~J_{\mu }  is the mass four-current, Описание: ~j^{\nu }  is the electromagnetic four-current.

The components of the field tensors are the field strengths and the corresponding solenoidal vectors, but in the physical system under consideration the latter are equal to zero. As a result, the space component of the equation of motion is reduced to the relation:

Описание: ~{\mathbf  S}+{\mathbf  C}+{\mathbf  \Gamma }_{i}+{\frac  {\rho _{{0q}}}{\rho _{0}}}{\mathbf  E}_{i}=0.

If we substitute here the expression for the field strengths inside the sphere, we obtain the relation between the field coefficients: [13]

Описание: ~\eta +\sigma =G-{\frac  {\rho _{{0q}}^{2}}{4\pi \varepsilon _{0}\rho _{0}^{2}}}=G-{\frac  {q^{2}}{4\pi \varepsilon _{0}m^{2}}}.\qquad \qquad (2)

The same is obtained for the time component of the equation of motion, which leads to the generalized Poynting theorem. [9]

 

Relation between the energies of the internal and external fields

In article [14] it was found that the energy of particles in the gravitational field inside the stationary sphere is up to a sign two times greater than the total energy associated with the tensor invariants of the gravitational field inside and outside the body. A similar situation takes place in the system under consideration with the random motion of particles and zero solenoidal vectors both for the gravitational [10]  and electromagnetic fields. [8]  In particular, we can write the following:

Описание: ~\int \limits _{{r=0}}^{{a}}\rho \psi _{i}dV=2\int \limits _{{r=0}}^{{a}}{\frac  {c^{2}}{16\pi G}}\Phi _{{\mu \nu }}\Phi ^{{\mu \nu }}dV+2\int \limits _{{r=a}}^{{\infty }}{\frac  {c^{2}}{16\pi G}}\Phi _{{\mu \nu }}\Phi ^{{\mu \nu }}dV=2\int \limits _{{r=0}}^{{\infty }}{\frac  {c^{2}}{16\pi G}}\Phi _{{\mu \nu }}\Phi ^{{\mu \nu }}dV.

 

Описание: ~\int \limits _{{r=0}}^{{a}}\rho _{q}\varphi _{i}dV=-2\int \limits _{{r=0}}^{{a}}{\frac  {c^{2}\varepsilon _{0}}{4}}F_{{\mu \nu }}F^{{\mu \nu }}dV-2\int \limits _{{r=a}}^{{\infty }}{\frac  {c^{2}\varepsilon _{0}}{4}}F_{{\mu \nu }}F^{{\mu \nu }}dV=-2\int \limits _{{r=0}}^{{\infty }}{\frac  {c^{2}\varepsilon _{0}}{4}}F_{{\mu \nu }}F^{{\mu \nu }}dV.

 

These expressions relate the energy of particles in the scalar field potentials with the energy found with the help of the field strengths.

 

Relativistic energy

In the curved spacetime the system’s energy for the continuously distributed matter is given by the formula: [2] [4]

Описание: ~E_{r}={\frac  {1}{c}}\int {(\rho _{0}\vartheta +\rho _{0}\wp +\rho _{0}\psi +\rho _{{0q}}\varphi )u^{0}{\sqrt  {-g}}dx^{1}dx^{2}dx^{3}+}

Описание: ~+\int {\left({\frac  {c^{2}}{16\pi \eta }}u_{{\mu \nu }}u^{{\mu \nu }}+{\frac  {c^{2}}{16\pi \sigma }}f_{{\mu \nu }}f^{{\mu \nu }}-{\frac  {c^{2}}{16\pi G}}\Phi _{{\mu \nu }}\Phi ^{{\mu \nu }}+{\frac  {c^{2}\varepsilon _{0}}{4}}F_{{\mu \nu }}F^{{\mu \nu }}\right){\sqrt  {-g}}dx^{1}dx^{2}dx^{3}}.

 

This formula is valid in the case when we can assume that the potentials and strengths of the fields at each point of space do not depend on the velocities of the motion of the individual particles of the system.

In the STR the metric tensor determinant is Описание: ~g=-1, the time component of the four-velocity is  Описание: ~u^{0}=c\gamma ', and in order to calculate the energy of the spherical system with particles, taking into account the fields’ energies, we can use the above-mentioned energies of particles in the field potentials and the energies in the form of the tensor invariants of the fields:

Описание: ~E_{r}\approx mc^{2}\gamma _{c}^{2}-{\frac  {3\eta m^{2}\gamma _{c}^{2}}{5a}}\left(1-{\frac  {2\eta m}{7ac^{2}}}\right)+m\wp _{c}\gamma _{c}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right)-{\frac  {3\sigma m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {13\eta m}{28ac^{2}}}\right)-

Описание: ~-{\frac  {6Gm^{2}\gamma _{c}^{2}}{5a}}\left(1-{\frac  {4\eta m}{7ac^{2}}}\right)+{\frac  {3q^{2}\gamma _{c}^{2}}{10\pi \varepsilon _{0}a}}\left(1-{\frac  {4\eta m}{7ac^{2}}}\right)-{\frac  {\eta m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right)-{\frac  {\sigma m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right)+

Описание: ~+{\frac  {Gm^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right)-{\frac  {q^{2}\gamma _{c}^{2}}{40\pi \varepsilon _{0}a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right)+{\frac  {Gm^{2}\gamma _{c}^{2}}{2a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right)-{\frac  {q^{2}\gamma _{c}^{2}}{8\pi \varepsilon _{0}a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right).

 

The expression for the energy is simplified if we use the relation between the field coefficients (2):

Описание: ~E_{r}\approx mc^{2}\gamma _{c}^{2}-{\frac  {3\eta m^{2}\gamma _{c}^{2}}{5a}}\left(1-{\frac  {2\eta m}{7ac^{2}}}\right)+m\wp _{c}\gamma _{c}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right)-{\frac  {3\sigma m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {13\eta m}{28ac^{2}}}\right)-

Описание: ~-{\frac  {6Gm^{2}\gamma _{c}^{2}}{5a}}\left(1-{\frac  {4\eta m}{7ac^{2}}}\right)+{\frac  {3q^{2}\gamma _{c}^{2}}{10\pi \varepsilon _{0}a}}\left(1-{\frac  {4\eta m}{7ac^{2}}}\right)+{\frac  {Gm^{2}\gamma _{c}^{2}}{2a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right)-{\frac  {q^{2}\gamma _{c}^{2}}{8\pi \varepsilon _{0}a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right).

 

Taking into account the relations between the energies of the internal and external fields also simplifies the expression for the system’s energy:

Описание: ~E_{r}\approx mc^{2}\gamma _{c}^{2}-{\frac  {3\eta m^{2}\gamma _{c}^{2}}{5a}}\left(1-{\frac  {2\eta m}{7ac^{2}}}\right)+m\wp _{c}\gamma _{c}\left(1-{\frac  {3\eta m}{10ac^{2}}}\right)-{\frac  {3\sigma m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {13\eta m}{28ac^{2}}}\right)-

 

Relation between the energy and the cosmological constant

In the approach under consideration, the relativistic energy of the system is not an absolute value and requires gauging. For this purpose the cosmological constant  Описание: ~\Lambda   is used. The gauge condition for the four main fields is related to the sum of the products of the fields’ four-potentials by the corresponding four-currents and has the following form: [2] [4]

~-ck\Lambda =A_{\mu }j^{\mu }+(D_{\mu }+U_{\mu }+\pi _{\mu })J^{\mu },\qquad \qquad (5)

 

where for large cosmic systems  Описание: ~-ck={\frac  {c^{4}}{16\pi G\beta }},  and  Описание: ~\beta   is the constant of the order of unity.

Within the framework of the STR the gauge condition has the following form:

Описание: ~-ck\Lambda =\gamma \rho _{{0q}}(\varphi -{\mathbf  A}\cdot {\mathbf  v})+\gamma \rho _{{0}}(\psi -{\mathbf  D}\cdot {\mathbf  v}+\vartheta -{\mathbf  U}\cdot {\mathbf  v}+\wp -{\mathbf  \Pi }\cdot {\mathbf  v}).

 

If we divide the system’s particles and remove them to infinity and leave there at rest, the terms with the products of the vector field potentials by the velocity of particles Описание: ~{\mathbf  v}  would vanish, and the Lorentz factor of an arbitrary particle would be  Описание: ~\gamma =1. On the right-hand side we will have only the sum of the terms specifying the energy density of the particles located in the potentials of their proper fields. Since  Описание: ~\vartheta \approx \gamma _{c}c^{2},  we see that the cosmological constant for each system’s particle is up to the multiplier Описание: ~-ck  equal to the rest energy density of this particle with a certain addition from its proper fields. Then the integral over the volume of all the particles gives a certain energy:

Описание: ~-ck\int \Lambda dV=m'c^{2},

where the gauge mass  Описание: ~m'  is related to the gauge condition of the energy.

In the process of gravitational clustering the particles that were initially far from each other are united into closely bound systems, in which the field potentials increase manyfold. In the system under consideration Описание: ~\gamma =\gamma ',  the solenoidal vectors of the fields are considered equal to zero due to the random motion of the particles, which gives the following:

Описание: ~m'c^{2}=\int [\gamma '\rho _{{0q}}\varphi _{i}+\gamma '\rho _{{0}}(\psi _{i}+\vartheta +\wp )]dV.

 

The expression on the right-hand side is part of the relativistic energy of the system, so that the energy can be written as follows:

Описание: ~E_{r}=Mc^{2}\approx m'c^{2}-{\frac  {\eta m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right)-{\frac  {\sigma m^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right)+

Описание: ~+{\frac  {Gm^{2}\gamma _{c}^{2}}{10a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right)-{\frac  {q^{2}\gamma _{c}^{2}}{40\pi \varepsilon _{0}a}}\left(1-{\frac  {3\eta m}{7ac^{2}}}\right)+{\frac  {Gm^{2}\gamma _{c}^{2}}{2a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right)-{\frac  {q^{2}\gamma _{c}^{2}}{8\pi \varepsilon _{0}a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right).

 

The mass Описание: ~M  is related to the relativistic energy of the generally stationary system and is the inertial mass of the system. In view of (2), the energy will be equal to:

Описание: ~E_{r}=Mc^{2}\approx m'c^{2}+{\frac  {Gm^{2}\gamma _{c}^{2}}{2a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right)-{\frac  {q^{2}\gamma _{c}^{2}}{8\pi \varepsilon _{0}a}}\left(1-{\frac  {3\eta m}{5ac^{2}}}\right).

 

This shows that the relativistic energy of this system is equal to the gauge mass-energy  Описание: ~m'c^{2}, from which the gravitational and electromagnetic energy of the fields outside the system should be subtracted.

 

Virial theorem and the kinetic energy of particles

In article [15] the kinetic energy of the particles of the system under consideration is estimated by three methods: from the virial theorem, from the relativistic definition of energy and using the generalized momenta and the proper fields of the particles. In the limit of low velocities, all these methods give for the kinetic energy the following:

Описание: ~E_{k}\approx {\frac  {0.3608\eta m^{2}\gamma _{c}}{a}}.

The possibility to use the generalized momenta to calculate the energy of the particles’ motion is associated with the fact that despite zeroing of the vector potentials and the solenoidal vectors on the large scale, in the volume of each randomly moving particle these potentials and vectors are not equal to zero. As a result, the energy of motion of the system’s particles can be found as the half-sum of the scalar products of the vector field potentials by the particles’ momentum, while for the electromagnetic field we should take not the momentum, but the product of the charge by the velocity and the Lorentz factor.

If we square the equation for Описание: ~\gamma '  in (1), we can obtain the dependence of the squared velocity of the particles’ random motion on the current radius:

Описание: ~{v'}^{2}\approx v_{c}^{2}-{\frac  {4\pi \eta \rho _{0}r^{2}}{3}}.

On the other hand, we can assume that  Описание: ~{\mathbf  v}'={\mathbf  v}_{r}+{\mathbf  v}_{\perp },   where Описание: ~{\mathbf  v}_{r}  denotes the averaged velocity component directed along the radius, and Описание: ~{\mathbf  v}_{\perp }  is the averaged velocity component perpendicular to the current radius. In addition, from statistical considerations, it follows that

Описание: ~{v'}^{2}=v_{r}^{2}+v_{\perp }^{2}=3v_{r}^{2}.

 

This implies the dependence of the radial velocity on the radius:

Описание: ~v_{r}\approx {\frac  {v_{c}}{{\sqrt  3}}}\left(1-{\frac  {2\pi \eta \rho _{0}r^{2}}{3v_{c}^{2}}}\right).

 

Next, from the virial theorem we find the squared velocity of the particles at the center of the sphere:

Описание: ~v_{c}^{2}\approx {\frac  {3\eta m}{5a}}\left(1+{\frac  {9}{{\sqrt  {56}}}}\right)\approx {\frac  {1.3216\eta m}{a}}.

 

This makes it possible to estimate the Lorentz factor in the center:

~\gamma _{c}={\frac  {1}{{\sqrt  {1-{\frac  {v_{c}^{2}}{c^{2}}}}}}}\approx 1+{\frac  {v_{c}^{2}}{2c^{2}}}+{\frac  {3v_{c}^{4}}{8c^{4}}}\approx 1+{\frac  {3\eta m}{10ac^{2}}}\left(1+{\frac  {9}{2{\sqrt  {14}}}}\right)+{\frac  {27\eta ^{2}m^{2}}{200a^{2}c^{4}}}\left(1+{\frac  {9}{2{\sqrt  {14}}}}\right)^{2}.

 

In the ordinary interpretation of the virial theorem the time-averaged kinetic energy of the system of particles must be two times less than the averaged energy associated with the forces Описание: ~{\mathbf  F}_{i}   holding the particles at the radius-vectors  Описание: ~{\mathbf  r}_{i} :

Описание: ~\langle W_{k}\rangle _{m}=-0.5\langle \sum _{{i=1}}^{{N}}{\mathbf  F}_{i}\cdot {\mathbf  r}_{i}\rangle .

 

However, in the relativistic uniform system this equation is changed:

Описание: ~\langle W_{k}\rangle \approx -0.6\langle \sum _{{i=1}}^{{N}}{\mathbf  F}_{i}\cdot {\mathbf  r}_{i}\rangle ,

 

while the quantity Описание: ~W_{k}  exceeds the kinetic energy of particles, Описание: ~W_{k}\approx \gamma _{c}E_{k},  and it becomes equal to it only in the limit of low velocities.

In contrast to the classical case, the total time derivative of the virial in the stationary system is other than zero due to the virial’s dependence on the radius:

Описание: ~{\frac  {dG_{V}}{dt}}\approx {\mathbf  v}\cdot \nabla {G_{V}}\approx {\frac  {0.1216\eta m^{2}\gamma _{c}^{2}}{a}}.

 

An analysis of the integral theorem of generalized virial makes it possible to find, on the basis of field theory, a formula for the root-mean-square speed of typical particles of a system without using the notion of temperature: [16]

v_{{\mathrm  {rms}}}=c{\sqrt  {1-{\frac  {4\pi \eta \rho _{0}r^{2}}{c^{2}\gamma _{c}^{2}\sin ^{2}{\left({\frac  {r}{c}}{\sqrt  {4\pi \eta \rho _{0}}}\right)}}}}}.

 

Extreme objects

In formula (2) for the gravitational field strength  Описание: {\displaystyle ~\mathbf {\Gamma } _{o}}  outside the body there is a quantity  Описание: {\displaystyle ~A=\sin \delta -\delta \cos \delta } , where  Описание: {\displaystyle ~\delta ={\frac {a}{c}}{\sqrt {4\pi \eta \rho _{0}}}} . As was shown in article, [11] at the value  Описание: {\displaystyle ~\delta =\delta _{0}=4.494}  radians the gravitational field strength  Описание: {\displaystyle ~\mathbf {\Gamma } _{o}}  vanishes and the gravitational acceleration disappears. Therefore, in real physical objects the following condition must hold:  Описание: {\displaystyle ~\delta ={\frac {a}{c}}{\sqrt {4\pi \eta \rho _{0}}}<\delta _{0}} . If the angle  Описание: {\displaystyle ~\delta }  is increased, then the quantity  Описание: {\displaystyle ~A}  would first increase, and then would begin to decrease and even change its sign. So, at  Описание: {\displaystyle ~\delta ={\frac {\pi }{2}}}  we will have  Описание: {\displaystyle ~A=1} , at  Описание: {\displaystyle ~\delta =\pi }  we will have  Описание: {\displaystyle ~A=\pi } , at  Описание: {\displaystyle ~\delta ={\frac {3\pi }{2}}}  we will have  Описание: {\displaystyle ~A=-1} .

Let us now consider the observable Universe, which on a scale 100 Mpc or more can be considered as a relativistic uniform system. The total mass-energy density of the Universe is close to the critical value  Описание: {\displaystyle ~\rho _{c}\approx 10^{-26}}  kg/m3 and the size of the Universe can be estimated as the Hubble length  Описание: {\displaystyle ~R_{H}=c/H_{0}\approx 10^{26}}  m, where  Описание: {\displaystyle ~H_{0}}  is the Hubble parameter.

Using the approximate equality  Описание: {\displaystyle ~\eta \approx {\frac {3}{5}}G}  according to, [13] we find the value  Описание: {\displaystyle ~\delta _{U}={\frac {R_{H}}{c}}{\sqrt {4\pi \eta \rho _{c}}}\approx 1.7>{\frac {\pi }{2}}}  radians. Since the angle  Описание: {\displaystyle ~\delta _{U}}  is sufficiently large, then for modeling of the gravitational field of the Universe it is necessary to use refined formulas with sines and cosines. For example, if we take the size of the observable Universe equal to  Описание: {\displaystyle ~2.64R_{H}} , then we will have  Описание: {\displaystyle ~\delta _{U}=\delta _{0}} , and the gravitational field at the boundaries of the Universe will tend to zero. This is what we observe in the form of a large-scale cellular structure consisting of clusters of galaxies. The reason for the gravitation action weakening is assumed to be graviton scattering by the particles of the space medium. [17]

Another extreme object is a proton, in which the mass density in the entire volume changes approximately by 1.5 times. As a result, in the first approximation a proton is a relativistic uniform system. The proton radius  Описание: {\displaystyle ~r_{p}}  is of the order of 0.873 fm[18] and the average density is of the order of  Описание: {\displaystyle ~\rho _{p}=6\cdot 10^{17}}  kg/m3. As a gravitational constant at the atomic level the strong gravitational constant  Описание: {\displaystyle ~G_{s}}  should be used. An estimate of the quantity  Описание: {\displaystyle ~\delta }  for a proton at  Описание: {\displaystyle ~\eta \approx {\frac {3}{5}}G_{s}}  gives:  Описание: {\displaystyle ~\delta _{p}={\frac {r_{p}}{c}}{\sqrt {4\pi \eta \rho _{p}}}\approx 2.4<\delta _{0}}  radians. This shows that a proton is an extreme object from the point of view of weakening of its gravitational field.

In article, [11] a method is provided for estimating the Lorentz factor of the matter’s motion at the center of a proton, which gives  Описание: {\displaystyle ~\gamma _{c}=1.9} . In addition, the radius of action of the strong gravitation in the matter with the critical mass density  Описание: {\displaystyle ~\rho _{c}\approx 10^{-26}}  kg/m3 in the observable Universe is estimated:  Описание: {\displaystyle ~r_{G}<1.3\cdot 10^{7}}  m. On a large scale in the Universe not the strong gravitation, but the ordinary gravitation is acting with the radius of action of the order of the Hubble length.

Let us suppose that  Описание: {\displaystyle ~r_{G}}  corresponds to the radius of a certain black hole for the strong gravitation, calculated by the Schwarzschild formula:  Описание: {\displaystyle ~r_{G}={\frac {2G_{s}m}{c^{2}}}} . If the mass is  Описание: {\displaystyle ~m={\frac {4\pi \rho _{c}r_{G}^{3}}{3}}} , then for the radius of a black hole with such mass we obtain Описание: {\displaystyle ~r_{G}=c{\sqrt {\frac {3}{8\pi G_{s}\rho _{c}}}}=2.7\cdot 10^{6}}  m, and the mass is  Описание: {\displaystyle ~m=8\cdot 10^{-7}}  kg. The Schwarzschild formula admits a black hole for the strong gravitation at small mass of the order of the proton mass, large mass density and a radius smaller than the proton radius. In addition, substitution of the mass  Описание: {\displaystyle ~m}  and the radius  Описание: {\displaystyle ~r_{G}}  into the Schwarzschild formula formally corresponds to a black hole with a large radius and low density  Описание: {\displaystyle ~\rho _{c}}. However, for an external observer, such a black hole would rather correspond not to a black hole, but to an object, containing strongly rarefied hydrogen gas of the cosmic space. Similarly, the Metagalaxy with the radius of order of  Описание: {\displaystyle ~r_{H}}  and the mass density  Описание: {\displaystyle ~\rho _{c}}  is not a black hole, although it corresponds to the Schwarzschild formula for the ordinary gravitation. Hence, in accordance with the theory of infinite nesting of matter, the conclusion follows – at each level of matter the corresponding gravitation forms only one type of the most compact and stable object. So, at the level of nucleons a proton appears under the action of the strong gravitation, and at the level of stars the ordinary gravitation generates a neutron star. If we multiply the radius of a neutron star by the coefficient of similarity in size  Описание: {\displaystyle ~P=1.4\cdot 10^{19}} , which is equal to the ratio of the stellar radius to the proton radius, we will obtain the radius of the order of  Описание: {\displaystyle 1.7\cdot 10^{23}}  m. This radius must correspond to a compact object of a neutron star-type at the level of metagalaxies, which can emerge under the action of gravitation at this matter level. In the first approximation, the gravitational constant for metagalaxies is determined with the help of the similarity theory:  Описание: {\displaystyle ~G_{M}={\frac {GPS^{2}}{\Phi }}=3\cdot 10^{-50}}  m3•s–2•kg–1, where  Описание: {\displaystyle ~S=0.23}  is the coefficient of similarity in velocities,  Описание: {\displaystyle ~\Phi =1.62\cdot 10^{57}}  is the coefficient of similarity in mass.

By analogy with the case of a proton, a neutron star is also considered as a relativistic uniform system. For a star with the mass of 1.35 Solar masses, the radius  Описание: {\displaystyle ~R_{s}=12}  km and the average density  Описание: {\displaystyle ~\rho _{s}\approx 3.7\cdot 10^{17}}  kg/m3, at  Описание: {\displaystyle ~\eta \approx {\frac {3}{5}}G}  we obtain the angle  Описание: {\displaystyle ~\delta _{s}={\frac {R_{s}}{c}}{\sqrt {4\pi \eta \rho _{s}}}\approx 0.546}  radians. With this in mind, if we substitute into (3) the stellar mass instead of Описание: {\displaystyle ~m_{b}}  and the stellar radius instead of  Описание: {\displaystyle ~a} , we can estimate the Lorentz factor at the center of the star:  Описание: {\displaystyle ~\gamma _{cs}=1.04} . This allows us to estimate the temperature at the center of the star:  Описание: {\displaystyle ~T_{s}\approx 2.8\cdot 10^{11}}  K, which is close enough to calculation of the temperature at the center of a newly formed star. [13]

Thus, the dependences of the gravitational field inside and outside the bodies in article [11] are in good agreement with the conclusions of the Le Sage’s theory of gravitation and the theory of Infinite Hierarchical Nesting of Matter, with the strong gravitation at the level of nucleons and with the concept of a dynamic force vacuum field.

 

Cosmological constant and scalar curvature

According to (5), outside the body, where the four-currents are equal to zero, the cosmological constant  {\displaystyle ~\Lambda }  becomes equal to zero. In addition, the scalar curvature {\displaystyle ~R}  also becomes equal to zero. [4] Inside the body the relation  {\displaystyle ~R=2\Lambda }  holds true, so that in the matter with higher density both the scalar curvature and the cosmological constant increase. These quantities can be calculated using (5) as the averaged values for typical particles of the physical system. For the cosmic space we obtain approximately the following: {\displaystyle ~\Lambda _{0}\approx {\frac {16\pi G\rho _{0}}{c^{2}}}\approx 10^{-52}} m-2, where the average mass density is {\displaystyle ~\rho _{0}\approx 2.7\cdot 10^{-27}} kg/m3.

A similar formula for a proton gives the following: {\displaystyle ~\Lambda \approx {\frac {16\pi G\rho _{p}}{c^{2}}}\approx 2.2\cdot 10^{-8}} m-2. However, for a proton in the calculations we should use the strong gravitational constant  {\displaystyle ~G_{s}} . In this case, we find: {\displaystyle ~\Lambda _{p}\approx {\frac {16\pi G_{s}\rho _{p}}{c^{2}}}\approx 5.1\cdot 10^{31}} m-2. The obtained value is almost 84 orders of magnitude greater than the value of the cosmological constant for the cosmic space. The difference between the cosmological constants for the cosmic space and for a proton is associated with the averaging procedure: the cosmological constant inside a proton is large, but in the cosmic space the matter containing protons, neutrons and electrons is very rarefied, the main place is occupied by the void, so that the cosmological constant averaged over the entire space becomes a small value. Thus one of the paradoxes of the general theory of relativity is solved, in which the cosmological constant is associated with zero vacuum energy and therefore it must be very large, but in fact the cosmological constant turns out to be a small value.

For the relativistic uniform system with four fields acting in it, the average value  {\displaystyle ~{\stackrel {-}{\Lambda }}}  of the cosmological constant in the matter is constant and can be written as follows:

{\displaystyle ~-ck{\stackrel {-}{\Lambda }}={\frac {G\rho _{0}c^{2}\gamma _{c}}{\eta }}\cos \left({\frac {a}{c}}{\sqrt {4\pi \eta \rho _{0}}}\right)-{\frac {\rho _{0q}^{2}c^{2}\gamma _{c}}{4\pi \varepsilon _{0}\eta \rho _{0}}}\cos \left({\frac {a}{c}}{\sqrt {4\pi \eta \rho _{0}}}\right)+\rho _{0}\wp _{c}-{\frac {\sigma \rho _{0}c^{2}\gamma _{c}}{\eta }}.}

This expression can be simplified by using the scalar potential of the gravitational field  {\displaystyle ~\psi _{a}=-{\frac {Gm_{g}}{a}}} and the scalar potential of the electric field  {\displaystyle ~\varphi _{a}={\frac {q_{b}}{4\pi \varepsilon _{0}a}}} on the surface of the body at  {\displaystyle ~r=a}  :

{\displaystyle ~-ck{\stackrel {-}{\Lambda }}\approx \rho _{0}\psi _{a}-{\frac {Gm\rho _{0}\gamma _{c}}{2a}}+\rho _{0}c^{2}\gamma _{c}+\rho _{0q}\varphi _{a}+{\frac {q\rho _{0q}\gamma _{c}}{8\pi \varepsilon _{0}a}}+\rho _{0}\wp _{c}.}

 

 

References

  1. 1.0 1.1 1.2 Fedosin S.G. The procedure of finding the stress-energy tensor and vector field equations of any form. Advanced Studies in Theoretical Physics, Vol. 8, no. 18, 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101.
  2. 2.0 2.1 2.2 2.3 Fedosin S.G. About the cosmological constant, acceleration field, pressure field and energy. Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304.
  3. 3.0 3.1 3.2 Fedosin S.G. The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based on the Pressure Field and Acceleration Field. American Journal of Modern Physics. Vol. 3, No. 4, pp. 152-167 (2014). http://dx.doi.org/10.11648/j.ajmp.20140304.12 .
  4. 4.0 4.1 4.2  4.3  Fedosin S.G. Energy and metric gauging in the covariant theory of gravitation. Accepted by Aksaray University Journal of Science and Engineering, September 2018.
  5. Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennostmaterii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. ISBN 978-5-9901951-1-0. (in Russian).
  6. Fedosin S.G. The Principle of Least Action in Covariant Theory of Gravitation. Hadronic Journal, Vol. 35, No. 1, pp. 35-70 (2012). http://dx.doi.org/10.5281/zenodo.889804.
  7. 7.0 7.1 7.2 Fedosin S.G. Relativistic Energy and Mass in the Weak Field Limit. Jordan Journal of Physics. Vol. 8, No. 1, pp. 1-16 (2015). http://dx.doi.org/10.5281/zenodo.889210.
  8. 8.0 8.1 8.2 Fedosin S.G. The electromagnetic field in the relativistic uniform model. Preprint, May 2016.
  9.  9.0 9.1 Fedosin S.G. The generalized Poynting theorem for the general field and solution of the 4/3 problem. Preprint, February 2016.
  10. 10.0 10.1 Fedosin S.G. The gravitational field in the relativistic uniform model within the framework of the covariant theory of gravitation. 5th Ulyanovsk International School-Seminar “Problems of Theoretical and Observational Cosmology” (UISS 2016), Ulyanovsk, Russia, September 19-30, 2016, Abstracts, p. 23, ISBN 978-5-86045-872-7.
  11. 11.0 11.1 11.2 11.3 Fedosin S.G. The Gravitational Field in the Relativistic Uniform Model within the Framework of the Covariant Theory of Gravitation. International Letters of Chemistry, Physics and Astronomy, Vol. 78, pp. 39-50 (2018). http://dx.doi.org/10.18052/www.scipress.com/ILCPA.78.39.
  12. Fedosin S.G. The Concept of the General Force Vector Field. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.
  13. 13.0 13.1 13.2 Fedosin S.G. Estimation of the physical parameters of planets and stars in the gravitational equilibrium model. Canadian Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). http://dx.doi.org/10.1139/cjp-2015-0593.
  14. Fedosin S.G. The Hamiltonian in Covariant Theory of Gravitation. Advances in Natural Science, Vol. 5, No. 4, pp. 55-75 (2012). http://dx.doi.org/10.3968%2Fj.ans.1715787020120504.2023.
  15. Fedosin S.G. The virial theorem and the kinetic energy of particles of a macroscopic system in the general field concept. Continuum Mechanics and Thermodynamics, Vol. 29, Issue 2, pp. 361-371 (2016). https://dx.doi.org/10.1007/s00161-016-0536-8.
  16. Fedosin S.G. The integral theorem of generalized virial in the relativistic uniform model. Continuum Mechanics and Thermodynamics (2018). https://dx.doi.org/10.1007/s00161-018-0715-x.
  17. Fedosin S.G. Cosmic Red Shift, Microwave Background, and New Particles. Galilean Electrodynamics, Vol. 23, Special Issues No. 1, pp. 3-13 (2012). http://dx.doi.org/10.5281/zenodo.890806.
  18. Fedosin S.G. The radius of the proton in the self-consistent model. Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). http://dx.doi.org/10.5281/zenodo.889451.

See also

External links

 

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