Physics/Essays/Fedosin/General field

Components of general field

General field is a physical field, the components of which are the electromagnetic and gravitational fields, acceleration field, pressure field, dissipation field, strong interaction field, weak interaction field, and other vector fields acting on the matter and its particles. Thus, the general field is manifested through its components and it is not equal to zero, as long as at least one of these components exists. Fundamental interactions, which include electromagnetic, gravitational, strong and weak interactions, that occur in the matter, are part of the interactions described by the general field.

The concept of the general field appeared within the framework of the metric theory of relativity and covariant theory of gravitation as a generalization of the procedure for finding the stress-energy tensor and equations of a vector field of any kind. With the help of this procedure, based on the principle of least action the gravitational field equations were first derived, then the equations of acceleration and pressure fields, and then the equations of the field of energy dissipation due to viscosity. All these equations are similar in form to the Maxwell equations. This means that the nature of every vector field has something in common, which unites it with other fields. This implies the idea of a general field, which is described in articles by Sergey Fedosin.

The general field theory represents one of the variants of the non-quantum unified field theory and is one of the Grand Unified theories as well.

Notation of particular fields and field functions

Table 1 shows notation for all the fields, which are the components of the general field.

Table 1. Notation of field functions

Field function	Electromagnetic field	Gravitational field	Acceleration field	Pressure field	Dissipation field	Strong interaction field	Weak interaction field	General field
4-potential	$~A_{\mu }$	$~D_{\mu }$	$~U_{\mu }$	$~\pi _{\mu }$	$~\lambda _{\mu }$	$~g_{\mu }$	$~w_{\mu }$	$~s_{\mu }$
Scalar potential	$~\varphi$	$~\psi$	$~\vartheta$	$~\wp$	$~\varepsilon$	$~\phi$	$~\zeta$	$~\theta$
Vector potential	$~\mathbf {A}$	$~\mathbf {D}$	$~\mathbf {U}$	$~{\boldsymbol {\Pi }}$	$~{\boldsymbol {\Theta }}$	$~\mathbf {G}$	$~\mathbf {W}$	$~{\boldsymbol {\Phi }}$
Field strength	$~\mathbf {E}$	$~{\boldsymbol {\Gamma }}$	$~\mathbf {S}$	$~\mathbf {C}$	$~\mathbf {X}$	$~\mathbf {L}$	$~\mathbf {Q}$	$~\mathbf {T}$
Solenoidal vector	$~\mathbf {B}$	$~{\boldsymbol {\Omega }}$	$~\mathbf {N}$	$~\mathbf {I}$	$~\mathbf {Y}$	$~{\boldsymbol {\mu }}$	$~{\boldsymbol {\pi }}$	$~{\boldsymbol {\chi }}$
Field tensor	$~F_{\mu \nu }$	$~\Phi _{\mu \nu }$	$~u_{\mu \nu }$	$~f_{\mu \nu }$	$~h_{\mu \nu }$	$~\gamma _{\mu \nu }$	$~w_{\mu \nu }$	$~s_{\mu \nu }$
Stress-energy tensor	$~W^{\mu \nu }$	$~U^{\mu \nu }$	$~B^{\mu \nu }$	$~P^{\mu \nu }$	$~Q^{\mu \nu }$	$~L^{\mu \nu }$	$~A^{\mu \nu }$	$~T^{\mu \nu }$
Energy-momentum flux vector	$~\mathbf {P}$	$~\mathbf {H}$	$~\mathbf {K}$	$~\mathbf {F}$	$~\mathbf {Z}$	$~{\boldsymbol {\Sigma }}$	$~\mathbf {V}$	$~{\boldsymbol {\Xi }}$
Field constant	$~{\frac {1}{4\pi \varepsilon _{0}}}$	$~G$	$~\eta$	$~\sigma$	$~\tau$	$~\aleph$	$~\ell$	$~\varpi$

In Table 1, the vector $~\mathbf {P}$ is the Poynting vector, the vector $~\mathbf {H}$ is the Heaviside vector.

Mathematical description

In the covariant theory of gravitation the main representative of any vector field is its 4-potential, with the help of which all other field functions are expressed. Since the general field exists due to its components in the form of particular fields, the 4-potential of the general field is the sum of the 4-potentials of particular fields, in accordance with the superposition principle for the fields:

$~s_{\mu }={\frac {\rho _{0q}}{\rho _{0}}}A_{\mu }+D_{\mu }+U_{\mu }+\pi _{\mu }+\lambda _{\mu }+g_{\mu }+w_{\mu }.$

By its meaning the 4-potential $~s_{\mu }$ is a generalized 4-velocity.

Since the 4-potential of any field consists of the scalar and vector potentials, the scalar potential of the general field is the sum of the scalar potentials of particular fields, and the same applies to the vector potentials:

$~\theta ={\frac {\rho _{0q}}{\rho _{0}}}\varphi +\psi +\vartheta +\wp +\varepsilon +\phi +\zeta .$

$~{\boldsymbol {\Phi }}={\frac {\rho _{0q}}{\rho _{0}}}\mathbf {A} +\mathbf {D} +\mathbf {U} +{\boldsymbol {\Pi }}+{\boldsymbol {\Theta }}+\mathbf {G} +\mathbf {W} .$

The tensor of the general field is calculated as the 4-curl of the 4-potential. If we assume that the ratio of the charge density to the mass density $~{\frac {\rho _{0q}}{\rho _{0}}}$ in each considered matter unit is constant as the ratio of the charge to the mass of the unit, the tensor of the general field turns out to be the sum of tensors of particular fields:

$~s_{\mu \nu }=\nabla _{\mu }s_{\nu }-\nabla _{\nu }s_{\mu }={\frac {\rho _{0q}}{\rho _{0}}}F_{\mu \nu }+\Phi _{\mu \nu }+u_{\mu \nu }+f_{\mu \nu }+h_{\mu \nu }+\gamma _{\mu \nu }+w_{\mu \nu }.\qquad \qquad (1)$

The components of field tensors are their strengths and solenoidal vectors. Consequently, the general field strength in each matter unit (volume unit) is the sum of strengths of particular fields and the same applies to solenoidal vector of the general field:

$~\mathbf {T} ={\frac {\rho _{0q}}{\rho _{0}}}\mathbf {E} +{\boldsymbol {\Gamma }}+\mathbf {S} +\mathbf {C} +\mathbf {X} +\mathbf {L} +\mathbf {Q} .\qquad \qquad (2)$

$~{\boldsymbol {\chi }}={\frac {\rho _{0q}}{\rho _{0}}}\mathbf {B} +{\boldsymbol {\Omega }}+\mathbf {N} +\mathbf {I} +\mathbf {Y} +{\boldsymbol {\mu }}+{\boldsymbol {\pi }}.\qquad \qquad (3)$

Action, Lagrangian and energy

Within the covariant theory of gravitation the matter is characterized by the mass 4-current $~J^{\mu }=\rho _{0}u^{\mu }$, where $~u^{\mu }$ is the 4-velocity. While the charge 4-current is obtained with the help of the mass 4-current $~J^{\mu }$ from the following relation:

$~j^{\mu }={\frac {\rho _{0q}}{\rho _{0}}}J^{\mu }=\rho _{0q}u^{\mu }.$

Consequently, the energy density of interaction of the general field and the matter is given by the product of the 4-potential of the general field and the mass 4-current: $~s_{\mu }J^{\mu }$. Another tensor invariant, in the form $~s_{\mu \nu }s^{\mu \nu }$, is up to a constant factor proportional to the energy density of the general field. The action function containing the scalar curvature $~R$ and the cosmological constant $~\Lambda$, is given by the expression:

$~S=\int {Ldt}=\int (kR-2k\Lambda -{\frac {1}{c}}s_{\mu }J^{\mu }-{\frac {c}{16\pi \varpi }}s_{\mu \nu }s^{\mu \nu }){\sqrt {-g}}d\Sigma ,$

where $~L$ is the Lagrange function or Lagrangian; $~dt$ is the time differential of the coordinate reference frame; $~k$ and $~\varpi$ are the constants to be determined; $~c$ is the speed of light as a measure of the propagation speed of electromagnetic and gravitational interactions; $~{\sqrt {-g}}d\Sigma ={\sqrt {-g}}cdtdx^{1}dx^{2}dx^{3}$ is the invariant 4-volume expressed in terms of the time coordinate differential $~dx^{0}=cdt$, the product $~dx^{1}dx^{2}dx^{3}$ of space coordinates’ differentials and the square root $~{\sqrt {-g}}$ of the determinant $~g$ of the metric tensor, taken with a negative sign.

The variation $~\delta S$ of the action function consists of the sum of terms, including:

1) the variation $~\delta s_{\mu }$ of the 4-potential of the general field;

2) the variation of coordinates $~\xi ^{\mu }$, which creates the variation $~\delta J^{\mu }$ of the mass 4-current;

3) the variation $~\delta g_{\mu \nu }$ of the metric tensor.

Due to the principle of least action, the variation $~\delta S$ must vanish. This leads to the vanishing of the sums of all the terms, standing before the variations $~\delta s_{\mu }$, $~\xi ^{\mu }$ and $~\delta g_{\mu \nu }$, respectively. As a consequence, the equations of the general field, the four-dimensional equation of motion and the equation for the metric follow from this.

By definition, the integral of action should be the sum of the integrals over the 4-volume on all the elements of matter, and the entire volume occupied by fields. In many cases, the physical system contains elements of matter, in which the ratio of $~{\frac {\rho _{0q}}{\rho _{0}}}$ is different from the average. In this case, the equation for the field, the motion of matter and the metric will depend not only on local ratio $~{\frac {\rho _{0q}}{\rho _{0}}}$, but also on the ratio of charge to mass in other matter elements that is implemented through a total field of these elements.

The Lagrangian is a volume integral of the sum of terms with the dimension of the energy density and it is similar by its components to the Hamiltonian, which determines the system’s energy. Actually, the Hamiltonian is obtained from the Lagrangian by means of the Legendre transformation for a system of particles. As we know, the energy is determined up to a constant that means the energy is subject to gauge. For example, the energy of the electromagnetic field is gauged so that at infinity with respect to the charge the electromagnetic field energy density is equal to zero. Similarly, the system’s energy in the form of the Hamiltonian can be gauged. In the covariant theory of gravitation it is assumed that the cosmological constant $~\Lambda$ is a gauge term. By its meaning it up to a constant factor represents the energy density, that the system has after all the system’s matter is divided into separate particles and scattered to infinity. In this case, the energy of the particles’ interaction with each other by means of the fields disappears, and only the proper energy of the particles remains as the energy of their proper fields at zero temperature. The gauge condition of the cosmological constant has the following form:

$~ck\Lambda =-s_{\mu }J^{\mu }.$

When the gauge conditions of the cosmological constant is met, the system’s energy ceases to depend on the term with the scalar curvature and becomes uniquely defined:

$~E=\int {(s_{0}J^{0}+{\frac {c^{2}}{16\pi \varpi }}s_{\mu \nu }s^{\mu \nu }){\sqrt {-g}}dx^{1}dx^{2}dx^{3}},$

where $~s_{0}$ and $~J^{0}$ denote the time components of the 4-vectors $~s_{\mu }$ and $~J^{\mu }$.

The 4-momentum of the system is determined by the formula:

$~p^{\mu }=\left({\frac {E}{c}}{,}\mathbf {p} \right)=\left({\frac {E}{c}}{,}{\frac {E}{c^{2}}}\mathbf {v} \right),$

where $~\mathbf {p}$ and $~\mathbf {v}$ denote the system’s momentum and the velocity of motion of the center of mass.

Equations

The general field equations have the following form:

$~\nabla _{\nu }s^{\mu \nu }=-{\frac {4\pi \varpi }{c^{2}}}J^{\mu }.$

$\nabla _{\sigma }s_{\mu \nu }+\nabla _{\mu }s_{\nu \sigma }+\nabla _{\nu }s_{\sigma \mu }={\frac {\partial s_{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial s_{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial s_{\sigma \mu }}{\partial x^{\nu }}}=0.$

Thus, the only source of the general field is assumed to be the mass 4-current $~J^{\mu }$. The latter equation can be written more concisely using the Levi-Civita symbol or a totally antisymmetric unit tensor:

$~\varepsilon ^{\mu \nu \sigma \rho }{\frac {\partial s_{\mu \nu }}{\partial x^{\sigma }}}=0.$

Substituting (1), we can express the general field equations in terms of the tensors of particular fields:

$~\nabla _{\nu }\left({\frac {\rho _{0q}}{\rho _{0}}}F^{\mu \nu }+\Phi ^{\mu \nu }+u^{\mu \nu }+f^{\mu \nu }+h^{\mu \nu }+\gamma ^{\mu \nu }+w^{\mu \nu }\right)=-{\frac {4\pi \varpi }{c^{2}}}J^{\mu }.\qquad \qquad (4)$

$~\varepsilon ^{\mu \nu \sigma \rho }{\frac {\partial }{\partial x^{\sigma }}}\left({\frac {\rho _{0q}}{\rho _{0}}}F_{\mu \nu }+\Phi _{\mu \nu }+u_{\mu \nu }+f_{\mu \nu }+h_{\mu \nu }+\gamma _{\mu \nu }+w_{\mu \nu }\right)=0.\qquad \qquad (5)$

At equilibrium state, we can assume that equation (5) is satisfied separately for the tensor of each field, and not only for the entire sum of tensors of particular fields. Similarly, under the condition $~{\frac {\rho _{0q}}{\rho _{0}}}=const,$ equation (4) can be divided into seven separate equations, in which the mass 4-current $~J^{\mu }$ is the source of one or another particular field.

The gauge condition of the 4-potential of the general field:

$~\nabla ^{\mu }s_{\mu }=0.$

The continuity equation in the curved spacetime is written with the Ricci tensor $~R_{\mu \nu }$:

$~R_{\mu \nu }s^{\mu \nu }={\frac {4\pi \varpi }{c^{2}}}\nabla _{\mu }J^{\mu }.$

In Minkowski space of the special theory of relativity the left side of this equation becomes equal to zero, since the Ricci tensor becomes zero. Besides, the covariant derivative $~\nabla _{\mu }$ is transformed into the 4-gradient $~\partial _{\mu }$ so that the continuity equation is simplified:

$~\partial _{\mu }J^{\mu }={\frac {\partial J^{\mu }}{\partial x^{\mu }}}=0.$

The equation of motion of the matter unit in the general field is given by the formula:

$~s_{\mu \nu }J^{\nu }=0$.

Since $~J^{\nu }=\rho _{0}u^{\nu }$, and the general field tensor is expressed in terms of the tensors of particular fields, then the equation of motion can be presented using these tensors:

$~-u_{\mu \nu }J^{\nu }=F_{\mu \nu }j^{\nu }+\Phi _{\mu \nu }J^{\nu }+f_{\mu \nu }J^{\nu }+h_{\mu \nu }J^{\nu }+\gamma _{\mu \nu }J^{\nu }+w_{\mu \nu }J^{\nu }.$

Here $~u_{\mu \nu }$ is the acceleration tensor, $~F_{\mu \nu }$ is the electromagnetic tensor, $~\Phi _{\mu \nu }$ is the gravitational tensor, $~f_{\mu \nu }$ is the pressure field tensor, $~h_{\mu \nu }$ is the dissipation field tensor, $~\gamma _{\mu \nu }$ is the tensor of the strong interaction field, $~w_{\mu \nu }$ is the tensor of the weak interaction field.

The stress-energy tensor

The stress-energy tensor of the general field is determined from the principle of least action with the expression

$~T^{ik}={\frac {c^{2}}{4\pi \varpi }}\left(-g^{im}s_{nm}s^{nk}+{\frac {1}{4}}g^{ik}s_{mr}s^{mr}\right).$

With this tensor the equation of motion is written in a very simple form, as the equality to zero of the divergence of the tensor:

$~s_{\mu \nu }J^{\nu }=-\nabla ^{\nu }T_{\mu \nu }=0$.

The stress-energy tensor of the general field is included into the equation for the metric:

$~R^{ik}-{\frac {1}{4}}g^{ik}R={\frac {8\pi G\beta }{c^{4}}}T^{ik},\qquad \qquad (6)$

where $~G$ is the gravitational constant, $~\beta$ is a certain constant and the gauge condition of the cosmological constant is used.

The general field tensor $~s_{\mu \nu }$ has such components as the strength $~\mathbf {T}$ and solenoidal vector $~{\boldsymbol {\chi }}$ of the general field. The vector $~\mathbf {T}$, according to (2), is the sum of the strengths of particular fields, and the vector $~{\boldsymbol {\chi }}$, according to (3), consists of the solenoidal vectors of particular fields. The stress-energy tensor of the general field $~T^{ik}$ includes the tensor product $~s_{mr}s^{mr}$, so that the tensor $~T^{ik}$ contains the squared vectors $~\mathbf {T}$ and $~{\boldsymbol {\chi }}$. Substituting these vectors with the sums of the respective vectors of particular fields, we obtain the following:

$~T^{ik}=k_{1}W^{ik}+k_{2}U^{ik}+k_{3}B^{ik}+k_{4}P^{ik}+k_{5}Q^{ik}+k_{6}L^{ik}+k_{7}A^{ik}+cross\quad terms,\qquad \qquad (7)$

where $~k_{1}{,}k_{2}{,}k_{3}{,}k_{4}{,}k_{5}{,}k_{6}{,}k_{7}$ are some coefficients, $~W^{ik}$ is the electromagnetic stress-energy tensor, $~U^{ik}$ is the gravitational stress-energy tensor, $~B^{ik}$ is the acceleration stress-energy tensor, $~P^{ik}$ is the pressure stress-energy tensor, $~Q^{ik}$ is the dissipation stress-energy tensor, $~L^{ik}$ is the stress-energy tensor of the strong interaction field, $~A^{ik}$ is the stress-energy tensor of the weak interaction field.

As we can see, the stress-energy tensor of the general field $~T^{ik}$ contains not only stress-energy tensors of particular fields, but also the cross-terms with the products of strengths and solenoidal vectors of particular fields.

For example, if we consider only the gravitational field and the acceleration field, for the stress-energy tensor of the general field we obtain the following:

$~(T^{ik})_{g+u}={\frac {c^{2}}{4\pi \varpi }}\left(-g^{im}(\Phi _{nm}+u_{nm})(\Phi ^{nk}+u^{nk})+{\frac {1}{4}}g^{ik}(\Phi _{mr}+u_{mr})(\Phi ^{mr}+u^{mr})\right)=$

$~=-{\frac {G}{\varpi }}U^{ik}+{\frac {\eta }{\varpi }}B^{ik}+{\frac {c^{2}}{4\pi \varpi }}\left(-g^{im}(\Phi _{nm}u^{nk}+u_{nm}\Phi ^{nk})+{\frac {1}{4}}g^{ik}(\Phi _{mr}u^{mr}+u_{mr}\Phi ^{mr})\right).$

where $~\eta$ is a constant, which is part of the definition of the acceleration stress-energy tensor.

Dividing into two components

In the article the general field was divided into two main components. One of them is the mass component of the general field, the source of which is the mass four-current $~J^{\mu }$. The source of the second one – the charge component of the general field – is the charge four-current $~j^{\mu }$. The mass component of the general field contains the gravitational field, acceleration field, pressure field, dissipation field, fields of strong and weak interaction, and other vector fields. The charge component of the general field represents the electromagnetic field. As a result of dividing the general field into two components the field equations have become more independent of each other, since the invariability condition of the ratio of the invariant charge density to the invariant mass density $~\rho _{0q}/\rho _{0}$ is no longer required. To denote the field functions of the mass component of the general field, the same notation is used further, which are specified in Table 1 for the general field itself.

The four-potential of the charge component of the general field is the electromagnetic four-potential $~A_{\mu }=(\varphi /c,-\mathbf {A} )$. The 4-potential of the mass component of the general field is equal to the sum of the four-potentials of the corresponding fields:

$~s_{\mu }=D_{\mu }+U_{\mu }+\pi _{\mu }+\lambda _{\mu }+g_{\mu }+w_{\mu }.$

Similarly, the scalar and vector potentials of the mass component of the general field will equal:

$~\theta =\psi +\vartheta +\wp +\varepsilon +\phi +\zeta .$

$~{\boldsymbol {\Phi }}=\mathbf {D} +\mathbf {U} +{\boldsymbol {\Pi }}+{\boldsymbol {\Theta }}+\mathbf {G} +\mathbf {W} .$

Instead of (1), (2) and (3) for the tensor, field strength and solenoidal vector of the mass component of the general field, we will obtain the following:

$~s_{\mu \nu }=\nabla _{\mu }s_{\nu }-\nabla _{\nu }s_{\mu }=\Phi _{\mu \nu }+u_{\mu \nu }+f_{\mu \nu }+h_{\mu \nu }+\gamma _{\mu \nu }+w_{\mu \nu }.$

$~\mathbf {T} ={\boldsymbol {\Gamma }}+\mathbf {S} +\mathbf {C} +\mathbf {X} +\mathbf {L} +\mathbf {Q} .$

$~{\boldsymbol {\chi }}={\boldsymbol {\Omega }}+\mathbf {N} +\mathbf {I} +\mathbf {Y} +{\boldsymbol {\mu }}+{\boldsymbol {\pi }}.$

The tensor of the charge component of the general field is the electromagnetic tensor :$~F_{\mu \nu }=\nabla _{\mu }A_{\nu }-\nabla _{\nu }A_{\mu },$

consisting of the components of the electromagnetic field strength $~\mathbf {E}$ and the components of the magnetic field $~\mathbf {B}$.

The potentials, field strengths and solenoidal vectors of the particular fields for a spherical body were calculated in the article and in other articles.

The action function and the system’s energy are determined as follows:

$~S=\int {Ldt}=\int (kR-2k\Lambda -{\frac {1}{c}}s_{\mu }J^{\mu }-{\frac {c}{16\pi \varpi }}s_{\mu \nu }s^{\mu \nu }-{\frac {1}{c}}A_{\mu }j^{\mu }-{\frac {c\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }){\sqrt {-g}}d\Sigma .$

$~E=\int {(s_{0}J^{0}+A_{0}j^{0}+{\frac {c^{2}}{16\pi \varpi }}s_{\mu \nu }s^{\mu \nu }+{\frac {c^{2}\varepsilon _{0}}{4}}F_{\mu \nu }F^{\mu \nu }){\sqrt {-g}}dx^{1}dx^{2}dx^{3}}.$

By construction, the general field is a vector field, so each equation of vector field will be valid for it.

The equations for the tensors of the mass and charge components of the general field will be as follows:

$~\nabla _{\nu }s^{\mu \nu }=-{\frac {4\pi \varpi }{c^{2}}}J^{\mu }.$

$\nabla _{\sigma }s_{\mu \nu }+\nabla _{\mu }s_{\nu \sigma }+\nabla _{\nu }s_{\sigma \mu }={\frac {\partial s_{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial s_{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial s_{\sigma \mu }}{\partial x^{\nu }}}=0.$

$~\nabla _{\nu }F^{\mu \nu }=-{\frac {1}{c^{2}\varepsilon _{0}}}j^{\mu }.$

$\nabla _{\sigma }F_{\mu \nu }+\nabla _{\mu }F_{\nu \sigma }+\nabla _{\nu }F_{\sigma \mu }={\frac {\partial F_{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial F_{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial F_{\sigma \mu }}{\partial x^{\nu }}}=0.$

The gauge conditions of the four-potentials of the general field components are as follows:

$~\nabla ^{\mu }s_{\mu }=0.$

$~\nabla ^{\mu }A_{\mu }=0.$

The continuity equations for the corresponding four-currents in the curved spacetime are as follows:

$~R_{\mu \nu }s^{\mu \nu }={\frac {4\pi \varpi }{c^{2}}}\nabla _{\mu }J^{\mu }.$

$~R_{\mu \nu }F^{\mu \nu }={\frac {1}{c^{2}\varepsilon _{0}}}\nabla _{\mu }j^{\mu }.$

The equation of motion of the matter under the action of the fields is as follows:

$~F_{\mu \nu }j^{\nu }+s_{\mu \nu }J^{\nu }=F_{\mu \nu }j^{\nu }+\Phi _{\mu \nu }J^{\nu }+u_{\mu \nu }J^{\nu }+f_{\mu \nu }J^{\nu }+h_{\mu \nu }J^{\nu }+\gamma _{\mu \nu }J^{\nu }+w_{\mu \nu }J^{\nu }=0.$

The equation of motion can also be written using the stress-energy tensor of the electromagnetic field $~W_{\mu \nu }$ and the stress-energy tensor of the mass component of the general field $~T_{\mu \nu }$:

$~F_{\mu \nu }j^{\nu }+s_{\mu \nu }J^{\nu }=-\nabla ^{\nu }W_{\mu \nu }-\nabla ^{\nu }T_{\mu \nu }=0.$

These tensors with contravariant indices are defined as follows:

$~W^{\mu \nu }=c^{2}\varepsilon _{0}\left(-g^{\mu \alpha }F_{\beta \alpha }F^{\beta \nu }+{\frac {1}{4}}g^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right).$

$~T^{\mu \nu }={\frac {c^{2}}{4\pi \varpi }}\left(-g^{\mu \alpha }s_{\beta \alpha }s^{\beta \nu }+{\frac {1}{4}}g^{\mu \nu }s_{\alpha \beta }s^{\alpha \beta }\right).$

The equation for the metric:

$~R^{\mu \nu }-{\frac {1}{4}}g^{\mu \nu }R={\frac {8\pi G\beta }{c^{4}}}(W^{\mu \nu }+T^{\mu \nu }).\qquad \qquad (8)$

In the article it was shown that for the coefficients of the fields, which are part of the mass component of the general field, the following relation should hold:

$~\varpi =\eta +\sigma -G+\tau +\aleph +\ell ,$

where $~\eta$ is the acceleration field constant, $~\sigma$ is the pressure field constant, $~G$ is the gravitational constant, $~\tau$ is the dissipation field constant, $~\aleph$ is the constant of the macroscopic strong interaction field, $~\ell$ is the constant of the macroscopic weak interaction field.

For the case of the relativistic uniform system, the tensors of the fields, which are part of the mass component of the general field, are proportional to each other. With this in mind, the stress-energy tensor of the mass component of the general field is expressed in terms of the stress-energy tensors of the particular fields, while the cross terms disappear:

$~T^{\mu \nu }=U^{\mu \nu }+B^{\mu \nu }+P^{\mu \nu }+Q^{\mu \nu }+L^{\mu \nu }+A^{\mu \nu }.$

Particular solutions for the general field functions

In a stationary case we can assume that the energy in the system is distributed in accordance with the equipartition theorem. According to this theorem, for systems in thermal equilibrium under conditions, when quantum effects do not play a big role yet, any degree of freedom $~f$ of a particle, which is part of the energy as the power function $~f^{s}$, at the average has the same energy $~{\frac {1}{s}}kT$, where $~k$ is the Boltzmann constant, $~T$ is the temperature. The particles of the ideal gas have only three degrees of freedom of this kind – they are three components of velocity, which are a squared term of the kinetic energy ($~s=2$), therefore the average energy of the particle is $~{\frac {3}{2}}kT$.

In general case, the particles have their proper fields, and the strengths and solenoidal vectors of these fields are squared terms of the corresponding stress-energy tensors. With this in mind, it is assumed, that the equipartition theorem also holds for the field energy in the sense that the energy of the system in equilibrium tends to be distributed proportionally also between all the existing fields in the system. In equilibrium, we can expect that the particular fields as the components of the general field become relatively independent of each other. In this case, for each field their own field equations must hold, and the equations of the general field (4) and (5) are divided into sets of equations for each particular field. All these equations have a form similar to the Maxwell's equations.

The following solutions were calculated assuming that the cross-terms in (7) are equal to zero. This implies complete independence of particular fields so that not only the equations of particular fields are independent of each other, but also the way how the general field energy simply equals the sum of energies of particular fields. Since the particular fields do influence each other, these solutions can be considered as a first approximation to the real picture.

The metric

Outside the bodies there are only the electromagnetic and gravitational fields. The tensors of these fields only will contribute to the equation for the metric (8), while the scalar curvature $~R$ is zeroed. The metric around an isolated spherical body was calculated in the article. For the time component of the metric tensor there is obtained the following:

$~g_{00}=1-{\frac {\alpha R_{b}}{Mc^{2}r}}(c_{1}E_{g}+c_{2}E_{e})+{\frac {\beta R_{b}^{2}}{M^{2}c^{4}r^{2}}}(c_{1}E_{g}+c_{2}E_{e})^{2},$

where $~r$ is the distance from the center of the body to the point where the metric is defined; $~E_{g}=-{\frac {3GM^{2}}{5R_{b}}}$ is the energy of the gravitational field inside and outside of the body; $~E_{e}={\frac {3Q^{2}}{20\pi \varepsilon _{0}R_{b}}}$ is the energy of the electric field; $~M$, $~Q$ and $~R_{b}$ are the mass, charge and radius of the body; $~\varepsilon _{0}$ is the electric constant; $~\alpha {,}\beta$ are the coefficients to be determined; $~c_{1}{,}c_{2}$ are the numerical coefficients of the order of unity, in case of the uniform density of mass and charge of the body they are the same and equal approximately the value 5/3.

In this case it appears that the metric depends both on the ratio of the body radius to the radius vector to the observation point, and on the ratio of the total field energy to the rest energy of the body.

The relativistic energy

The energy of the system of particles with regard to the electromagnetic and gravitational fields, acceleration field and pressure field is calculated in the article. It is shown that in the center of mass frame the total energy and momentum of all the fields are equal to zero, and the system’s energy is formed only of the energy of particles under influence of these particular fields. Five mass values can be introduced for the system: the inertial mass $~M$; the gravitational mass $~m_{g}$; the total mass $~m'$ of all the particles of the body scattered at infinity; the mass $~m_{b}$ obtained by integrating over the volume the density $~\rho$ of the matter moving within the system; the auxiliary mass $~m$ obtained by integrating over the volume the density $~\rho _{0}$ of the matter, calculated in the reference frame associated with each particle. For these masses we obtain the relation:

$~m<m'=M<m_{b}=m_{g}.$

From the equality $~m'=M$ it follows that ideal spherical collapse is possible when the system’s energy does not change when the matter is compressed. In addition, the gravitational mass $~m_{g}$ appears to be larger than the system’s mass $~M$. This is due to the fact that the particles are moving inside the system and their energy is greater than if the particles were motionless at infinity and would not interact with each other.

Calculation shows that the energy of the electromagnetic field reduces the gravitational mass. Therefore, adding a number of charges to a certain body could lead to a situation when the gravitational mass of the body would begin to decrease, despite the additional mass of the introduced charges. This follows from the fact that the mass of the charges increases proportionally to their number, and the mass-energy of the electromagnetic field increases quadratically to the number of charges. We can calculate that if a body with the mass of 1 kg and the radius of 1 meter is charged up to the potential of 5 megavolt, it would decrease the gravitational mass of the body (excluding the mass of the added charges) at weighing in the gravity field by $~10^{-13}$ mass fraction, which is close to the modern accuracy of mass measurement.

If we take into account a more accurate relation for the field coefficients, we obtain for the masses another expression:

$~m'<M<m<m_{b}=m_{g}.$

Here, the gauge mass $~m'$ is related to the cosmological constant and represents the mass-energy of the matter’s particles in the four-potentials of the system’s fields; the inertial mass $~M$; the auxiliary mass $~m$ is equal to the product of the particles’ mass density by the volume of the system; the mass $~m_{b}$ is the sum of the invariant masses (rest masses) of the system’s particles, which is equal in value to the gravitational mass $~m_{g}$.

The conclusion that as the electric charge increases the system’s mass may decrease remains valid, however this applies not to the gravitational mass $~m_{g}$, but to the inertial mass $~M$ of the system.

The integral 4-vector, 4/3 problem and Poynting theorem

The 4/3 problem, according to which the field mass found through the field energy is not equal to the field mass determined through the field momentum, and the problem of neutrino energy in an ideal spherical collapse of a supernova, were considered in the article. It was shown that in a moving body the excess mass-energy of the gravitational and electromagnetic fields is compensated by a lack of the mass-energy of the acceleration field and pressure field. The result is achieved by integrating the equation of motion and by calculating the conserved integral 4-vector of the energy-momentum of the system. Since this integral 4-vector must be equal to zero, in contrast to the ordinary 4-vector of the energy-momentum of the system, it imposes restrictions on the constant $~\eta$, located in the acceleration stress-energy tensor, and the constant $~\sigma$ in the pressure stress-energy tensor. For these constants in case of the massive gravitationally bound system of particles and fields the relation is found which connects them with the gravitational constant and electric constant:

$~\eta =\sigma =3G-{\frac {3q^{2}}{4\pi \varepsilon _{0}m^{2}}}$,

where $~q$ and $~m$ are the charge and mass of the system, and their ratio within the assumptions made can be interpreted as the ratio of the charge density to mass density.

The solution of the wave equation for the acceleration field inside the system results in temperature distribution according to the formula:

$~T=T_{c}-{\frac {\eta M_{p}M(r)}{3kr}},$

where $~T_{c}$ is the temperature in the center; $~M_{p}$ is the particle mass, for which the mass of the proton is assumed (for the systems the basis of which is hydrogen or nucleons in atomic nuclei); $~M(r)$ is the mass of the system within the current radius $~r$; $~k$ is the Boltzmann constant.

Similarly, for the pressure distribution inside the system we obtain:

$~p=p_{c}-{\frac {2\pi \sigma \rho _{0}^{2}r^{2}\gamma _{c}}{3}},$

where $~p_{c}$ is the pressure in the center; $~\rho _{0}$ is the mass density in the co-moving frame of a particle; $~\gamma _{c}$ is the Lorentz factor in the center.

These formulas are well satisfied for various space objects, including gas clouds and Bok globules, the Earth, the Sun and neutron stars. The only significant discrepancy (58 times) has been found for the pressure in the center of the Sun. However, if we take into account the presence of thermonuclear reactions in the Solar core, which can be described by introducing the strong and weak interaction fields, then the increased pressure in the center of the Sun can be explained by the influence of these fields. In this case, for the constant $~\aleph$ in the stress-energy tensor of the strong interaction field we obtain the estimate: $~\aleph \approx 3G$, which coincides with the coefficients $~\eta$ and $~\sigma$ for the acceleration field and pressure field, respectively. In all cases the scalar potentials of particular fields inside the bodies change proportionally to the square of the radius, as it happens in case of the gravitational field.

In the article the 4/3 problem was explained using the generalized Poynting theorem. This theorem is applied to the four-tensors of the fields, which are part of the general field components. These tensors consist of the components of the strengths and solenoidal vectors of the corresponding fields, with the help of which the energy and momentum of these fields are found. As a result, we obtain a more exact relation between the coefficients of the fields inside the matter of massive bodies:

$~\eta +\sigma =G-{\frac {\rho _{0q}^{2}}{4\pi \varepsilon _{0}\rho _{0}^{2}}}=G-{\frac {q^{2}}{4\pi \varepsilon _{0}m^{2}}},$

which allows estimating the internal temperature, pressure and other parameters of cosmic bodies for the case of non-uniform density.

Due to the Poynting theorem, both the sum of the energy densities of all the fields inside the body and the sum of the vectors of the energy fluxes of all the fields inside this body become equal to zero. Outside the body, the energy flux of the gravitational or electromagnetic field accurately compensates for the change in the energy of the corresponding field in each selected volume. As a result, the 4/3 problem disappears inside the body, but it remains for fields outside the body. The solution of the 4/3 problem with the example of the electromagnetic field is reduced to the following: the requirement of the equality of the mass-energy associated with the time component $~W^{00}$ of the stress-energy tensor of the field and the mass-energy of the energy flux of this field in the tensor components $~W^{0i}$, is wrongful. The point is that these tensor components do not constitute a four-vector and therefore cannot contain the same mass-energy, as it occurs in the four-momentum.

Energy dissipation

One of the general field components is the dissipation field, it describes the energy, momentum and energy flux, which are associated with the processes of energy conversion of particular fields into thermal energy. In the real substance the interaction of the substance fluxes moving at different speeds can take place under the influence of the internal friction and viscosity. In such processes the velocities of the substance fluxes are equalized, their kinetic energy decreases, but the thermal energy increases and the total energy of the system does not change. It turns out that if we introduce the dissipation field as a vector field, similarly to all the other particular fields, then in case of appropriate choice of the scalar potential of the dissipation field, it allows us to obtain the Navier-Stokes equation in hydrodynamics and to describe the motion of viscous compressible and charged fluid.

If we assume the local equilibrium condition and the validity of the theorem of energy equipartition, then for each particular field it is possible to use with sufficient accuracy their own field equations. As a result, for the pressure, for which its field equations were previously not known, we obtain specific wave equations for the scalar and vector potentials of the pressure field and the equations for the strength and solenoidal vectors of the pressure field. Similar equations are valid for the dissipation field, electromagnetic and gravitational fields, acceleration field, etc. This allows us to close the system of equations for the moving fluid with the fields existing in this fluid and to make this system of equations basically solvable.

Virial theorem

According to this theorem, in each stationary physical system there is a relationship between the kinetic energy of the particles and the energy, associated with the acting forces from all the existing fields that together make up the general field. In case when in the physical system the pressure field, the acceleration field of particles, the electromagnetic and gravitational fields are taken into account, the virial theorem is expressed in the relativistic form as follows:

$~\langle W_{k}\rangle \approx -0.6\sum _{k=1}^{N}\langle \mathbf {F} _{k}\cdot \mathbf {r} _{k}\rangle .$

where $\mathbf {r} _{k}$ denotes the radius-vector of the $k$-th particle, $\mathbf {F} _{k}$ is the force acting on this particle, and the value $~W_{k}\approx \gamma _{c}T$ exceeds the kinetic energy of the particles $~T$ by a factor equal to the Lorentz factor $~\gamma _{c}$ of the particles at the center of the system.

In the weak fields, we can assume that $~\gamma _{c}\approx 1$, and then we can see that in the virial theorem the kinetic energy is related to the energy of the forces on the right-hand side of the equation not by the coefficient 0.5 as in the classical case, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering the pressure field and the acceleration field of particles inside the system. The expression for the scalar virial function is found:

$G_{V}=\sum _{k=1}^{N}\mathbf {p} _{k}\cdot \mathbf {r} _{k},$

where $\mathbf {p} _{k}$ is the momentum of the $k$-th particle, and it is shown that the derivative of this function is not equal to zero and should be considered as the material derivative. In addition, it is found out that, in contrast to the conclusions of classical mechanics, the energy associated with the acting forces from all the existing fields and included in the right-hand side of the virial theorem does not equal the potential energy of the system.

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:

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

where $~c$ is the speed of light, $~\eta$ is the acceleration field constant, $~\rho _{0}$ is the mass density of particles, $~r$ is the current radius.

The relation between the theorem and the cosmological constant, characterizing the physical system under consideration, is shown. The difference is explained between the kinetic energy and the energy of motion, the value of which is equal to half the sum of the Lagrangian and the Hamiltonian.

Binding energy of macroscopic bodies

The relativistic energy, total energy, binding energy, fields’ energy, pressure energy and the potential energy of the system of particles and four fields (the general field components) are calculated in the relativistic uniform model, and then are compared with the kinetic energy of particles and with the total energy of the gravitational and electromagnetic fields outside the system. Another result is the fact that the inertial mass of the system is less than the gravitational mass, which is equal to the total invariant mass of the particles that make up the system. It is also proved that as increasingly massive relativistic uniform systems are formed, the average density of these systems decreases in comparison with the average density of the particles or bodies making up these systems.

The model allows us to estimate the particles’ velocity $~v_{c}$ at the center of the sphere, the corresponding Lorentz factor $~\gamma _{c}$, the scalar potential $~\wp _{c}$ of the pressure field; to find the relationship between the field coefficients; to express the dependences of the scalar curvature and the cosmological constant in the matter as functions of the parameters of typical particles and field potentials. Besides, comparison of the cosmological constants inside a proton, a neutron star and in the observable Universe allows us to explain the problem of the cosmological constant arising in the Lambda-CDM model.

The essence of the general field

The general field is assumed to be the main source of the acting forces, energy and momentum, as well as the basis for calculating the metric of the system from the standpoint of non-quantum classical field theory.

Among all the fields unified by the general field, two fields – the electromagnetic and gravitational fields – act at a distance, while the rest fields act locally, at the pave of location of one or another matter unit. The proper vector potential of any field for one particle is proportional to the scalar potential of this field and the particle velocity, if the vector potential of this particle is zero in the reference system that accompanies the particle. For the electromagnetic and gravitational fields in the system with a number of particles the superposition principle holds, according to which the scalar potential at an arbitrary point equals the sum of scalar potentials of all the particles and the same is assumed to apply to the vector potential. Due to the different rules of the vector and scalar summation, the vector potential of the system ceases to depend on the scalar potential of the system of particles. The same situation should take place for other fields. For example, the pressure near the particle depends not only on the scalar potential of the pressure field in the co-moving frame and the particle velocity, but also on the total pressure from other particles in the system.

The scalar potentials of particular fields are proportional to the energy, appearing in the system during one or another interaction per unit mass (charge) of the matter, and have a dimension of the squared velocity. The vector potentials of particular fields have a dimension of velocity and allow us to take into account the additional energy, which appears due to motion. Since the 4-potential of a particular field consists of the scalar and vector potentials, then the sum of the 4-potentials of particular fields gives the 4-potential of the general field, which describes the total energy of all interactions in the system of particles and fields. This is why the general field exists as long as there is at least one of its components in the form of the particular field. From a philosophical point of view, the existence of only one particular field is impossible particular – there should always be other fields. For example, if there is a particle, whose motion is described by the acceleration field, then this particle must also have at least the gravitational field and a full set of proper internal fields inside the particle.

The most natural method of describing the emergence of the general field is provided by the Fatio-Le Sage's theory of gravitation. This theory provides a clear physical mechanism of emergence of the gravitational force, as a consequence of the influence of ubiquitous fluxes of gravitons, in the form of tiny particles like neutrinos or photons, on the bodies. The same mechanism can explain the electromagnetic interaction, if we assume the presence of praons – tiny charged particles in the fluxes of gravitons. Praons and neutral particles such as field quanta form a vacuum field contained in an electrogravitational vacuum. The fluxes of the particles of the vacuum field permeate all bodies and carry out electromagnetic and gravitational interaction by means of the field even between the bodies, which are distant from each other. The bodies can also exert direct mechanical action on each other, which can be represented by the pressure field. An inevitable consequence of the action of these fields is the deceleration of fast matter particles and bodies in the surrounding medium, which is described by the dissipation field. At last, the acceleration field is introduced for the kinematic description of the motion of particles and bodies, the forces acting on them, the energy and momentum of the motion.

For bodies of a spherical shape, the chaotically moving particles of their matter can be characterized by a certain average radial velocity and an average tangential velocity perpendicular to it, the values of which depend on the current radius. It can be assumed that the radial velocity gradient leads to the radial acceleration described with the help of the pressure field. The tangential velocity of the particles also causes the radial acceleration due to the centripetal force, which can be taken into account by the acceleration field. These radial accelerations with addition of the acceleration from the electric forces in the charged matter resist the acceleration from the gravitational forces that compress the matter of massive cosmic bodies.

As a result, the general field can be represented as a field, in which the neutral and charged bodies, under the action of the fluxes of neutral and charged particles of vacuum field, exchange energy and momentum with each other and with vacuum field. The energy and momentum of the general field can be associated with the energy and momentum acquired by the vacuum field during interaction with the matter, and in order to take into account the energy and momentum of the system we need to add the energy and momentum of the matter, arising from its interaction with vacuum field.

In the model of quark quasiparticles it is emphasized that quarks are not real particles but quasiparticles. In this regard, it is assumed that the strong interaction can be reduced to strong gravitation, acting at the level of atoms and elementary particles, with replacement of the gravitational constant by the strong gravitational constant. Based on the strong gravitation and the gravitational torsion field the gravitational model of strong interaction is substantiated. One of the consequences of this is that the gravitational and electromagnetic fields are represented as fundamental fields, acting at different levels of matter by means of the field quanta with different values of their spin and energy and with different penetrating ability in the matter.

The above-mentioned approach allowed calculating the proton radius in the self-consistent model and explaining the de Broglie wavelength. As for the weak interaction, from the standpoint of the theory of Infinite Hierarchical Nesting of Matter, it is reduced to the processes of matter transformation, which is under action of the fundamental fields, with regard to the action of strong gravitation. Similarly, the pressure and dissipation fields in principle could be reduced to the fundamental fields, if all the details of interatomic and intermolecular interactions were known. Due to the difficulties with such details, we have to attribute the existence of proper 4-potentials to the pressure field, energy dissipation field, strong interaction field and weak interaction field, and to approximate the influence of these fields in the matter with the help of these 4-potentials.

By analogy with the electromagnetic field, all fields included as components in the general field are considered as vector fields. For such fields, an integral field energy theorem is proved. This theorem is an analogue of the virial theorem and describes the connections between the various components of the field energy.

In the article, the concept of the general field was analyzed again and the main active component of the electrogravitational vacuum was determined in the form of fluxes of charged particles of the praons type. It is assumed that this component is responsible for electromagnetic and gravitational interactions, as well as for the action of other fields inside bodies.

See also

• Classical unified field theories
• Theory of everything
• Grand Unified Theory
• Fundamental interaction
• Covariant theory of gravitation
• Metric theory of relativity
• Acceleration field
• Pressure field
• Dissipation field
• Le Sage's theory of gravitation
• Strong gravitation
• Strong gravitational constant
• Gravitational model of strong interaction
• Model of quark quasiparticles
• Field mass-energy limit
• Equation of vector field

External links

• General field in Russian