# Physics/Essays/Fedosin/General field

General field is a physical field, the components of which are electromagnetic and gravitational fields, acceleration field, pressure field, dissipation field, strong interaction field, weak interaction field, and other vector fields acting on 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 matter, are part of interactions described by the general field.
The concept of general field appeared within the framework of metric theory of relativity and covariant theory of gravitation as a generalization of procedure for finding stress-energy tensor and equations of a vector field of any kind. With the help of this procedure, based on principle of least action the gravitational field equations were first derived,
then equations of acceleration and pressure fields, and then equations of field of energy dissipation due to viscosity. All these equations are similar in form to Maxwell equations. This means that 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 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 general field.
In Table 1, the vector $~\mathbf {P}$ is Poynting vector, the vector $~\mathbf {H}$ is Heaviside vector.

## Mathematical description

In 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 general field is sum of the 4-potentials of particular fields, in accordance with 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 4-potential of any field consists of scalar and vector potentials, the scalar potential of general field is sum of scalar potentials of particular fields, and the same applies to vector potentials:
    $~\theta ={\frac {\rho _{0q}}{\rho _{0}}}\varphi +\psi +\vartheta +\wp +\varepsilon +\phi +\zeta .$
    $~{\boldsymbol {\Phi }}={\frac {\rho _{0q}}{\rho _{0}}}\mathbf {A} +\mathbf {D} +\mathbf {U} +{\boldsymbol {\Pi }}+{\boldsymbol {\Theta }}+\mathbf {G} +\mathbf {W} .$
Tensor of general field is calculated as 4-curl of 4-potential. If we assume that ratio of charge density to mass density $~{\frac {\rho _{0q}}{\rho _{0}}}$ in each considered matter unit is constant as the ratio of charge to mass of the unit, the tensor of general field turns out to be the sum of tensors of particular fields:
    $~s_{\mu \nu }=\nabla _{\mu }s_{\nu }-\nabla _{\nu }s_{\mu }={\frac {\rho _{0q}}{\rho _{0}}}F_{\mu \nu }+\Phi _{\mu \nu }+u_{\mu \nu }+f_{\mu \nu }+h_{\mu \nu }+\gamma _{\mu \nu }+w_{\mu \nu }.\quad (1)$
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 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 covariant theory of gravitation matter is characterized by the mass 4-current $~J^{\mu }=\rho _{0}u^{\mu }$, where $~u^{\mu }$ is 4-velocity. While the charge 4-current is obtained with the help of 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 general field and matter is given by product of 4-potential of general field and 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 energy density of general field. Action function containing scalar curvature $~R$ and 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 Lagrange function or Lagrangian; $~dt$ is time differential of reference frame; $~k$ and $~\varpi$ are constants to be determined; $~c$ is the speed of light as a measure of propagation speed of electromagnetic and gravitational interactions; $~{\sqrt {-g}}d\Sigma ={\sqrt {-g}}cdtdx^{1}dx^{2}dx^{3}$ is invariant 4-volume expressed in terms of time coordinate differential $~dx^{0}=cdt$, the product $~dx^{1}dx^{2}dx^{3}$ of space coordinates’ differentials and square root $~{\sqrt {-g}}$ of determinant $~g$ of metric tensor, taken with negative sign.
The variation $~\delta S$ of action function consists of the sum of terms, including:
1) variation $~\delta s_{\mu }$ of 4-potential of general field;
2) variation of coordinates $~\xi ^{\mu }$, which creates variation $~\delta J^{\mu }$ of mass 4-current;
3) variation $~\delta g_{\mu \nu }$ of metric tensor.
Due to principle of least action, the variation $~\delta S$ must vanish. This leads to vanishing of the sums of all the terms, standing before variations $~\delta s_{\mu }$, $~\xi ^{\mu }$ and $~\delta g_{\mu \nu }$, respectively. As a consequence, equations of general field, four-dimensional equation of motion and equation for metric follow from this.
By definition, integral of action should be a sum of integrals over 4-volume on all elements of matter, and entire volume occupied by fields. In many cases, physical system contains elements of matter, in which the ratio of $~{\frac {\rho _{0q}}{\rho _{0}}}$ is different from the average. In this case, equations for field, motion of matter and metric will depend not only on local ratio $~{\frac {\rho _{0q}}{\rho _{0}}}$, but also on ratio of charge to mass in other matter elements that is implemented through a total field of these elements.
Lagrangian is a volume integral of sum of terms with dimension of energy density and it is similar by its components to Hamiltonian, which determines system’s energy. Actually, the Hamiltonian is obtained from the Lagrangian by means of 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 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 Hamiltonian can be gauged. In covariant theory of gravitation it is assumed that cosmological constant $~\Lambda$ is a gauge term. By its meaning it up to a constant factor represents the energy density, that a 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 fields disappears, and only proper energy of the particles remains as 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 condition of cosmological constant is met, the system’s energy ceases to depend on the term with 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 time components of 4-vectors $~s_{\mu }$ and $~J^{\mu }$.

### 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 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 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 tensor of each field, and not only for entire sum of tensors of particular fields. Similarly, under condition $~{\frac {\rho _{0q}}{\rho _{0}}}=const,$ equation (4) can be divided into seven separate equations, in which mass 4-current $~J^{\mu }$ is a source of one or another particular field.
The gauge condition of 4-potential of general field is:
    $~\nabla ^{\mu }s_{\mu }=0.$
In curved spacetime, field equations give the equality:
    $~{\frac {4\pi \varpi }{c^{2}}}\nabla _{\mu }J^{\mu }=R_{\mu \nu }s^{\mu \nu }=0.$
The second part of this equality vanishes due to symmetry of Ricci tensor $~R_{\mu \nu }$ and antisymmetry of tensor $~s^{\mu \nu }$. This yields continuity equation of the form $~\nabla _{\mu }J^{\mu }=0$. In Minkowski spacetime of special relativity, covariant derivative $~\nabla _{\mu }$ becomes a 4-gradient $~\partial _{\mu }$, so that continuity equation is simplified:
    $~\partial _{\mu }J^{\mu }={\frac {\partial J^{\mu }}{\partial x^{\mu }}}=0.$
Equation of motion of matter unit in general field is given by the formula:
    $~s_{\mu \nu }J^{\nu }=0$.
Since $~J^{\nu }=\rho _{0}u^{\nu }$, and general field tensor is expressed in terms of tensors of particular fields, then 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 acceleration tensor, $~F_{\mu \nu }$ is electromagnetic tensor, $~\Phi _{\mu \nu }$ is gravitational tensor, $~f_{\mu \nu }$ is pressure field tensor, $~h_{\mu \nu }$ is dissipation field tensor, $~\gamma _{\mu \nu }$ is tensor of strong interaction field, $~w_{\mu \nu }$ is tensor of weak interaction field.

### Stress-energy tensor

Stress-energy tensor of 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 equation of motion is written in a very simple form, as equality to zero of divergence of the tensor:
    $~s_{\mu \nu }J^{\nu }=-\nabla ^{\nu }T_{\mu \nu }=0$.
Stress-energy tensor of general field is included into equation for metric:
    $~R^{ik}-{\frac {1}{4}}g^{ik}R={\frac {8\pi G\beta }{c^{4}}}T^{ik},\qquad \qquad (6)$
where $~G$ is gravitational constant, $~\beta$ is a certain constant and gauge condition of cosmological constant is used.
The general field tensor $~s_{\mu \nu }$ has such components as strength $~\mathbf {T}$ and solenoidal vector $~{\boldsymbol {\chi }}$ of general field. The vector $~\mathbf {T}$, according to (2), is sum of strengths of particular fields, and the vector $~{\boldsymbol {\chi }}$, according to (3), consists of solenoidal vectors of particular fields. The stress-energy tensor of general field $~T^{ik}$ includes tensor product $~s_{mr}s^{mr}$, so that the tensor $~T^{ik}$ contains squared vectors $~\mathbf {T}$ and $~{\boldsymbol {\chi }}$. Substituting these vectors with the sums of 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 electromagnetic stress-energy tensor, $~U^{ik}$ is gravitational stress-energy tensor, $~B^{ik}$ is acceleration stress-energy tensor, $~P^{ik}$ is pressure stress-energy tensor, $~Q^{ik}$ is dissipation stress-energy tensor, $~L^{ik}$ is stress-energy tensor of strong interaction field, $~A^{ik}$ is stress-energy tensor of weak interaction field.
As we can see, the stress-energy tensor of general field $~T^{ik}$ contains not only stress-energy tensors of particular fields, but also cross-terms with products of strengths and solenoidal vectors of particular fields.
For example, if we consider only gravitational field and acceleration field, for stress-energy tensor of 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 definition of acceleration stress-energy tensor.

## Dividing into two components

In the article general field was divided into two main components. One of them is mass component of general field, the source of which is mass four-current $~J^{\mu }$. The source of second one – charge component of general field – is charge four-current $~j^{\mu }$. The mass component of general field contains gravitational field, acceleration field, pressure field, dissipation field, fields of strong and weak interaction, and other vector fields. The charge component of general field represents electromagnetic field. As a result of dividing the general field into two components field equations have become more independent of each other, since invariability condition of ratio of invariant charge density to invariant mass density $~\rho _{0q}/\rho _{0}$ is no longer required. To denote field functions of mass component of general field, the same notation is used further, which are specified in Table 1 for general field itself.
Four-potential of charge component of general field is electromagnetic four-potential $~A_{\mu }=(\varphi /c,-\mathbf {A} )$. Four-potential of mass component of general field is equal to the sum of four-potentials of corresponding fields:
    $~s_{\mu }=D_{\mu }+U_{\mu }+\pi _{\mu }+\lambda _{\mu }+g_{\mu }+w_{\mu }.$
Similarly, scalar and vector potentials of mass component of general field 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 tensor, field strength and solenoidal vector of mass component of general field, we 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 charge component of general field is electromagnetic tensor :$~F_{\mu \nu }=\nabla _{\mu }A_{\nu }-\nabla _{\nu }A_{\mu },$
consisting of components of electromagnetic field strength $~\mathbf {E}$ and components of magnetic field $~\mathbf {B}$.
Potentials, field strengths and solenoidal vectors of particular fields for a spherical body were calculated in the article and in other articles.
Action function and 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 is valid for it.
Equations for tensors of mass and charge components of general field are as follows:
    $~\nabla _{\nu }s^{\mu \nu }=-{\frac {4\pi \varpi }{c^{2}}}J^{\mu }.$
    $\nabla _{\sigma }s_{\mu \nu }+\nabla _{\mu }s_{\nu \sigma }+\nabla _{\nu }s_{\sigma \mu }={\frac {\partial s_{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial s_{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial s_{\sigma \mu }}{\partial x^{\nu }}}=0.$
    $~\nabla _{\nu }F^{\mu \nu }=-{\frac {1}{c^{2}\varepsilon _{0}}}j^{\mu }.$
    $\nabla _{\sigma }F_{\mu \nu }+\nabla _{\mu }F_{\nu \sigma }+\nabla _{\nu }F_{\sigma \mu }={\frac {\partial F_{\mu \nu }}{\partial x^{\sigma }}}+{\frac {\partial F_{\nu \sigma }}{\partial x^{\mu }}}+{\frac {\partial F_{\sigma \mu }}{\partial x^{\nu }}}=0.$
Gauge conditions of four-potentials of general field components have the form:
    $~\nabla ^{\mu }s_{\mu }=0.$
    $~\nabla ^{\mu }A_{\mu }=0.$
Continuity equations for corresponding four-currents in curved spacetime are as follows:
    $~\nabla _{\mu }J^{\mu }=0.$
    $~\nabla _{\mu }j^{\mu }=0.$
Equation of motion of matter under action of fields is:
    $~F_{\mu \nu }j^{\nu }+s_{\mu \nu }J^{\nu }=F_{\mu \nu }j^{\nu }+\Phi _{\mu \nu }J^{\nu }+u_{\mu \nu }J^{\nu }+f_{\mu \nu }J^{\nu }+h_{\mu \nu }J^{\nu }+\gamma _{\mu \nu }J^{\nu }+w_{\mu \nu }J^{\nu }=0.$
The equation of motion can also be written using stress-energy tensor of electromagnetic field $~W_{\mu \nu }$ and stress-energy tensor of mass component of 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:
    $~W^{\mu \nu }=c^{2}\varepsilon _{0}\left(-g^{\mu \alpha }F_{\beta \alpha }F^{\beta \nu }+{\frac {1}{4}}g^{\mu \nu }F_{\alpha \beta }F^{\alpha \beta }\right).$
    $~T^{\mu \nu }={\frac {c^{2}}{4\pi \varpi }}\left(-g^{\mu \alpha }s_{\beta \alpha }s^{\beta \nu }+{\frac {1}{4}}g^{\mu \nu }s_{\alpha \beta }s^{\alpha \beta }\right).$
Equation for metric is:
    $~R^{\mu \nu }-{\frac {1}{4}}g^{\mu \nu }R={\frac {8\pi G\beta }{c^{4}}}(W^{\mu \nu }+T^{\mu \nu }).\qquad \qquad (8)$
In the article it was shown that for coefficients of fields, which are part of mass component of general field, the following relation should hold:
    $~\varpi =\eta +\sigma -G+\tau +\aleph +\ell ,$
where $~\eta$ is acceleration field constant, $~\sigma$ is pressure field constant, $~G$ is gravitational constant, $~\tau$ is dissipation field constant, $~\aleph$ is constant of macroscopic strong interaction field, $~\ell$ is constant of macroscopic weak interaction field.
For the case of relativistic uniform system, tensors of fields, which are part of mass component of general field, are proportional to each other. With this in mind, stress-energy tensor of mass component of general field is expressed in terms of stress-energy tensors of 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 general field functions

In a stationary case we can assume that energy in a system is distributed in accordance with the equipartition theorem. According to this theorem, for systems in thermal equilibrium under conditions where quantum effects do not yet play a major role, any degree of freedom $~f$ of a particle that contributes to energy in the form of a power function $~f^{s}$, has, on average, the same energy $~{\frac {1}{s}}kT$, where $~k$ is Boltzmann constant, $~T$ is temperature. Particles of ideal gas have only three such degrees of freedom - these are three components of velocity that enter quadratically into kinetic energy ($~f=3$, $~s=2$), therefore the average energy of a particle is equal to $~{\frac {3}{2}}kT$.
In general case, particles have their proper fields, and strengths and solenoidal vectors of these fields are squared terms of corresponding stress-energy tensors. With this in mind, it is assumed, that equipartition theorem also holds for field energy in the sense that energy of a system in equilibrium tends to be distributed proportionally also between all the existing fields in the system. In equilibrium, we can expect that particular fields as components of general field become relatively independent of each other. In this case, for each field their own field equations must hold, and equations of general field (4) and (5) are divided into sets of equations for each particular field. All these equations have a form similar to Maxwell's equations.
The following solutions were calculated assuming that cross-terms in (7) are equal to zero. This implies complete independence of particular fields so that not only 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.

### Metric

Outside bodies there are only electromagnetic and gravitational fields. Tensors of these fields only contribute to equation for metric (8), while the scalar curvature $~R$ is zeroed. Metric around an isolated spherical body was calculated in the article. For time component of 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 distance from center of the body to the point where metric is defined; $~E_{g}=-{\frac {3GM^{2}}{5R_{b}}}$ is energy of gravitational field inside and outside of the body; $~E_{e}={\frac {3Q^{2}}{20\pi \varepsilon _{0}R_{b}}}$ is energy of electric field; $~M$, $~Q$ and $~R_{b}$ are the mass, charge and radius of the body; $~\varepsilon _{0}$ is electric constant; $~\alpha {,}\beta$ are coefficients to be determined; $~c_{1}{,}c_{2}$ are numerical coefficients of the order of unity, in case of 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.
Сomponents of metric tensor in matter of a massive spherical body, taking into account gravitational and electromagnetic fields, acceleration field and pressure field, were found in the article. Although dependences of metric tensor components on the current radius inside and outside the body are different, on surface of the body corresponding components must coincide with each other. This allows us to find some of unknown coefficients and clarify the components of metric outside body in the following form.
    $~(g_{00})_{o}=-{\frac {1}{(g_{11})_{o}}}=1+{\frac {2Gm\gamma _{c}\beta }{c^{2}r}}+$
    $~+{\frac {2G\beta }{c^{4}r}}\left(m\psi _{a}+{\frac {1}{2}}m_{g}(\psi -\psi _{a})-{\frac {Gm^{2}\gamma _{c}}{2a}}+q\varphi _{a}+{\frac {1}{2}}q_{b}(\varphi -\varphi _{a})+{\frac {q^{2}\gamma _{c}}{8\pi \varepsilon _{0}a}}+m\wp _{c}\right),$
where $~\gamma _{c}$ is Lorentz factor of particle motion at the center of sphere; the quantities $~m={\frac {4\pi a^{3}\rho _{0}}{3}}$ and $~q={\frac {4\pi a^{3}\rho _{0q}}{3}}$ are auxiliary quantities; $~\rho _{0}$ is invariant mass density of particles of matter inside the sphere; $~\rho _{0q}$ is invariant charge density of particles of matter moving inside the sphere; $~\psi _{a}=-{\frac {Gm_{g}}{a}}$ is gravitational potential on surface of the sphere with radius $~a$ and gravitational mass $~m_{g}$; $~\psi =-{\frac {Gm_{g}}{r}}$ is gravitational potential outside the sphere; $~\varphi _{a}={\frac {q_{b}}{4\pi \varepsilon _{0}a}}$ is electric potential on surface of sphere with electric charge of the sphere $~q_{b}$; $~\varphi ={\frac {q_{b}}{4\pi \varepsilon _{0}r}}$ is electric potential outside the sphere; $~\wp _{c}$ is pressure field potential at the center of the sphere.

### Relativistic energy and mass

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 energy of a physical system was found in the article, through scalar potentials $~\varphi ,\psi ,\vartheta ,\wp$, through vector potentials $~\mathbf {A} ,\mathbf {D} ,\mathbf {U} ,\mathbf {\Pi }$, through tensors of fields $~F_{\mu \nu },\Phi _{\mu \nu },u_{\mu \nu },f_{\mu \nu }$ and particle velocities $~\mathbf {v}$ :
$~\varphi ,\psi ,\vartheta ,\wp$, через векторные потенциалы полей $~\mathbf {A} ,\mathbf {D} ,\mathbf {U} ,\mathbf {\Pi }$, и через тензоры полей $~F_{\mu \nu },\Phi _{\mu \nu },u_{\mu \nu },f_{\mu \nu }$:
    $~E={\frac {1}{c}}\int \operatorname {\big \rho _{0q}\varphi +\rho _{0}\psi +\rho _{0}\vartheta +\rho _{0}\wp -\mathbf {v} \cdot {\frac {\partial }{\partial \mathbf {v} }}\left(\rho _{0q}\varphi +\rho _{0}\psi +\rho _{0}\vartheta +\rho _{0}\wp \right)+$
    $~+v^{2}{\frac {\partial }{\partial \mathbf {v} }}\left(\rho _{0q}\mathbf {A} +\rho _{0}\mathbf {D} +\rho _{0}\mathbf {U} +\rho _{0}\mathbf {\Pi } \right)\operatorname {\big ]} u^{0}{\sqrt {-g}}dx^{1}dx^{2}dx^{3}+$
    $~+\int \left({\frac {1}{4\mu _{0}}}F_{\mu \nu }F^{\mu \nu }-{\frac {c^{2}}{16\pi G}}\Phi _{\mu \nu }\Phi ^{\mu \nu }+{\frac {c^{2}}{16\pi \eta }}u_{\mu \nu }u^{\mu \nu }+{\frac {c^{2}}{16\pi \sigma }}f_{\mu \nu }f^{\mu \nu }\right){\sqrt {-g}}dx^{1}dx^{2}dx^{3}+$
    $~+\sum _{n=1}^{N}\left(\mathbf {v} _{n}\cdot {\frac {\partial L_{f}}{\partial \mathbf {v} _{n}}}\right).$
The last term in this expression describes contribution to energy from fields of the system. This term depends on velocities of particles $~\mathbf {v} _{n}$, and the entire substance of the system is divided into $~N$ parts so that each part is a point particle or a small element of matter. The quantity $~L_{f}$ is Lagrange function, containing tensor invariants of fields
The 4-momentum of a system is defined by the formula:
    $~P_{\mu }=\left({\frac {E}{c}}{,}-\mathbf {P} \right),$
where $~\mathbf {P}$ denotes momentum of the system.

### Integral vector, 4/3 problem and Poynting theorem

The 4/3 problem, according to which field mass found through field energy is not equal to field mass determined through field flux, 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 excess mass-energy of gravitational and electromagnetic fields is compensated by a lack of mass-energy of acceleration field and pressure field. The result is achieved by integrating equation of motion and by calculating conserved integral vector of the system. Since this integral vector must be equal to zero, in contrast to ordinary 4-vector of energy-momentum of the system, it imposes restrictions on the constant $~\eta$, located in acceleration stress-energy tensor, and the constant $~\sigma$ in pressure stress-energy tensor. For these constants in case of massive gravitationally bound system of particles and fields a relation is found which connects them with gravitational constant and electric constant:
    $~\eta =\sigma =3G-{\frac {3q^{2}}{4\pi \varepsilon _{0}m^{2}}}$,
where $~q$ and $~m$ are charge and mass of the system, and their ratio within the assumptions made can be interpreted as ratio of charge density to mass density.
Solution of wave equation for 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 temperature at the centre; $~M_{p}$ is particle mass, for which the mass of proton is assumed (for systems basis of which is hydrogen or nucleons in atomic nuclei); $~M(r)$ is mass of the system within current radius $~r$; $~k$ is Boltzmann constant.
Similarly, for 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 pressure at the centre; $~\rho _{0}$ is mass density in co-moving frame of a particle; $~\gamma _{c}$ is Lorentz factor at the centre.
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 pressure at the centre of the Sun. However, if we take into account the presence of thermonuclear reactions in the Solar core, which can be described by introducing strong and weak interaction fields, then increased pressure at the centre of the Sun can be explained by influence of these fields. In this case, for constant $~\aleph$ in stress-energy tensor of strong interaction field we obtain the estimate: $~\aleph \approx 3G$, which coincides with coefficients $~\eta$ and $~\sigma$ for acceleration field and pressure field, respectively. In all cases scalar potentials of particular fields inside the bodies change proportionally to square of radius, as it happens in case of gravitational field.
In the article the 4/3 problem was explained using generalized Poynting theorem. This theorem is applied to four-tensors of fields, which are part of general field components. These tensors consist of components of strengths and solenoidal vectors of corresponding fields, with the help of which energy and momentum of these fields are found. As a result, we obtain a more exact relation between the coefficients of fields inside 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 internal temperature, pressure and other parameters of cosmic bodies for the case of non-uniform density.
Due to Poynting theorem, both the sum of energy densities of all fields inside a body and the sum of vectors of energy fluxes of all fields inside this body become equal to zero. Outside the body, energy flux of gravitational or electromagnetic field accurately compensates for change in energy of 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. Solution of 4/3 problem with example of electromagnetic field is reduced to the following: requirement of equality of mass-energy associated with time component $~W^{00}$ of stress-energy tensor of the field and mass-energy of energy flux of this field in 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 four-momentum. The same results were obtained in an article in which components of integral vector inside and outside a moving spherical body were calculated.

### Energy dissipation

One of general field components is dissipation field, it describes energy, momentum and energy flux, which are associated with processes of energy conversion of particular fields into thermal energy. In real substance interaction of the substance fluxes moving at different speeds can take place under influence of internal friction and viscosity. In such processes velocities of the substance fluxes are equalized, their kinetic energy decreases, but thermal energy increases and total energy of system does not change. It turns out that if we introduce dissipation field as a vector field, similarly to all other particular fields, then in case of appropriate choice of scalar potential of dissipation field, it allows us to obtain Navier-Stokes equation in hydrodynamics and to describe motion of viscous compressible and charged fluid.
If we assume local equilibrium condition and validity of 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 pressure, for which its field equations were previously not known, we obtain specific wave equations for scalar and vector potentials of pressure field and equations for strength and solenoidal vectors of the pressure field. Similar equations are valid for dissipation field, electromagnetic and gravitational fields, acceleration field, etc. This allows us to close system of equations for moving fluid with 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 kinetic energy of particles and energy, associated with acting forces from all existing fields that together make up the general field. In case when in physical system pressure field, acceleration field of particles, electromagnetic and gravitational fields are taken into account, virial theorem is expressed in 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 radius-vector of $k$-th particle, $\mathbf {F} _{k}$ is the force acting on this particle, and the value $~W_{k}\approx \gamma _{c}T$ exceeds kinetic energy of particles $~T$ by a factor equal to Lorentz factor $~\gamma _{c}$ of particles at centre of the system.
In weak fields, we can assume that $~\gamma _{c}\approx 1$, and then we can see that in virial theorem kinetic energy is related to energy of forces on the right-hand side of equation not by the coefficient 0.5 as in classical case, but rather by the coefficient close to 0.6. The difference from the classical case arises due to considering pressure field and acceleration field of particles inside the system. Expression for scalar virial function is found:
    $G_{V}=\sum _{k=1}^{N}\mathbf {p} _{k}\cdot \mathbf {r} _{k},$
where $\mathbf {p} _{k}$ is momentum of the $k$-th particle, and it is shown that derivative of this function is not equal to zero and should be considered as material derivative. In addition, it is found out that, in contrast to conclusions of classical mechanics, the energy associated with acting forces from all the existing fields and included in the right-hand side of virial theorem does not equal potential energy of the system.
Analysis of integral theorem of generalized virial makes it possible to find, on basis of field theory, a formula for root-mean-square speed of typical particles of a system without using 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 acceleration field constant, $~\rho _{0}$ is mass density of particles, $~r$ is current radius.
Relation between the theorem and cosmological constant, characterizing the physical system under consideration, is shown. Difference is explained between kinetic energy and energy of motion, the value of which is equal to half the sum of Lagrangian and Hamiltonian.

### Binding energy of macroscopic bodies

Relativistic energy, total energy, binding energy, fields’ energy, pressure energy and potential energy of a system of particles and four fields (the general field components) are calculated in relativistic uniform model, and then are compared with kinetic energy of particles and with total energy of gravitational and electromagnetic fields outside the system. Another result is the fact that inertial mass of the system is less than gravitational mass, which is equal to total invariant mass of particles that make up the system. It is also proved that as increasingly massive relativistic uniform systems are formed, average density of these systems decreases in comparison with average density of particles or bodies making up these systems.
The model allows us to estimate particles’ velocity $~v_{c}$ at the centre of a sphere, corresponding Lorentz factor $~\gamma _{c}$, scalar potential $~\wp _{c}$ of pressure field; to find relationship between the field coefficients; to express dependences of scalar curvature and cosmological constant in matter as functions of parameters of typical particles and field potentials. Besides, comparison of cosmological constants inside a proton, a neutron star and in observable Universe allows us to explain the problem of cosmological constant arising in Lambda-CDM model.

## Essence of general field

General field is assumed to be the main source of acting forces, energy and momentum, as well as basis for calculating metric of a system from the standpoint of non-quantum classical field theory.
Of all the fields that are united by general field, two fields, electromagnetic and gravitational fields, act at a distance, while the remaining fields act locally at location of a particular element of matter. Proper vector potential of any field for one particle is proportional to scalar potential of this field and the particle velocity, if vector potential of this particle is zero in reference system that comoving with the particle. For electromagnetic and gravitational fields in a system with a number of particles superposition principle holds, according to which the scalar potential at an arbitrary point equals the sum of scalar potentials of all particles and the same is assumed to apply to vector potential. Due to different rules of vector and scalar summation, vector potential of the system ceases to depend on scalar potential of the system of particles. The same situation should take place for other fields. For example, pressure near a particle depends not only on scalar potential of pressure field in co-moving frame and the particle velocity, but also on total pressure from other particles in the system.
Scalar potentials of particular fields are proportional to the energy, appearing in the system during one or another interaction per unit mass (charge) of matter, and have dimension of squared velocity. Vector potentials of particular fields have dimension of velocity and allow us to take into account additional energy, which appears due to motion. Since 4-potential of a particular field consists of the scalar and vector potentials, then sum of 4-potentials of particular fields gives 4-potential of general field, which describes energy of all interactions in a 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 particular field. From philosophical point of view, existence of only one particular field is impossible – there should always be other fields. For example, if there is a particle, whose motion is described by acceleration field, then this particle must also have at least gravitational field and a full set of proper internal fields inside the particle.
The most natural method of describing the emergence of general field is provided by Fatio-Le Sage's theory of gravitation. This theory provides a clear physical mechanism of emergence of gravitational force, as a consequence of impact on bodies of ubiquitous fluxes s of gravitons in the form of tiny particles like neutrinos or photons. The same mechanism can explain electromagnetic interaction, if we assume presence of praons – tiny charged particles in fluxes of gravitons. Praons and neutral particles such as field quanta form a vacuum field contained in electrogravitational vacuum. Fluxes of 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 action of these fields is deceleration of fast matter particles and bodies in surrounding medium, which is described by dissipation field. At last, acceleration field is introduced for kinematic description of motion of particles and bodies, the forces acting on them, the energy and momentum of the motion.
For bodies of a spherical shape, 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 current radius. It can be assumed that radial velocity gradient leads to radial acceleration described with the help of pressure field. Tangential velocity of the particles also causes the radial acceleration due to centripetal force, which can be taken into account by acceleration field. These radial accelerations with addition of acceleration from electric forces in charged matter resist the acceleration from gravitational forces that compress matter of massive cosmic bodies.
As a result, the general field can be represented as a field, in which neutral and charged bodies, under action of 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 general field can be associated with the energy and momentum acquired by the vacuum field during interaction with matter, and in order to take into account energy and momentum of a system we need to add the energy and momentum of matter, arising from its interaction with vacuum field.
In model of quark quasiparticles it is emphasized that quarks are not real particles but quasiparticles. In this regard, it is assumed that strong interaction can be reduced to strong gravitation, acting at the level of atoms and elementary particles, with replacement of gravitational constant by strong gravitational constant. Based on strong gravitation and 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 field quanta with different values of their spin and energy and with different penetrating ability in matter.
The above-mentioned approach allowed calculating the proton radius in self-consistent model and explaining de Broglie wavelength. As for the weak interaction, from the standpoint of theory of Infinite Hierarchical Nesting of Matter, it is reduced to processes of matter transformation, which is under action of fundamental fields, with regard to action of strong gravitation. Similarly, the pressure and dissipation fields in principle could be reduced to fundamental fields, if all details of interatomic and intermolecular interactions were known. Due to difficulties with such details, we have to attribute existence of proper 4-potentials to pressure field, energy dissipation field, strong interaction field and weak interaction field, and to approximate influence of these fields in matter with the help of these 4-potentials.
By analogy with electromagnetic field, all fields included as components in general field are considered as vector fields. For such fields, an integral field energy theorem is proved. This theorem is an analogue of virial theorem and describes connections between various components of field energy.
In the article, the concept of general field was analysed again and the main active component of electrogravitational vacuum was determined in the form of fluxes of charged particles of praons type. It is assumed that this component is responsible for electromagnetic and gravitational interactions, as well as for action of other fields inside bodies. Based on this approach, it was possible to explain the operating principle of a spaceship engine, which uses the energy of space vacuum for its movement.