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
stressenergy tensor and equations of a vector field of any kind. ^{[1]} With the help of this procedure, based on the
principle of least action the gravitational field equations were first derived,
^{[2]} ^{[3]} ^{[4]} then the equations of acceleration and pressure
fields, ^{[5]} and then the equations of the field of
energy dissipation due to viscosity. ^{[6]} 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. ^{[7] [8]}
The general field theory represents one of the variants
of the nonquantum unified field theory and is one of the Grand Unified
theories as well.
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 
4potential 








Scalar potential 








Vector potential 








Field strength 








Solenoidal vector 








Field tensor 








Stressenergy tensor 








Energymomentum flux
vector 








Field constant 








In Table 1, the vector is
the Poynting vector, the vector is
the Heaviside vector.
In the covariant theory of gravitation the main representative of any
vector field is its 4potential, 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 4potential of the general field is the
sum of the 4potentials of particular fields, in accordance with the superposition principle for the fields:
By its meaning the 4potential is
a generalized 4velocity. ^{[9]}
Since the 4potential 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:
The tensor of the general field is calculated as the
4curl of the 4potential. If we assume that the ratio of the charge density to
the mass density 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:
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:
Within the covariant
theory of gravitation the matter is characterized by the mass
4current , where is the 4velocity. While the charge 4current is obtained with the help of
the mass 4current from the following relation:
Consequently, the energy density of interaction of the
general field and the matter is given by the product of the 4potential of the
general field and the mass 4current: . Another tensor invariant, in the form , is up to a constant factor
proportional to the energy density of the general field. The action function
containing the scalar curvature and the cosmological constant , is given by the expression:
where is
the Lagrange function or Lagrangian; is
the time differential of the coordinate reference frame; and are the constants to be determined; is
the speed of light as a measure of the propagation speed of electromagnetic and
gravitational interactions; is the
invariant 4volume expressed in terms of the time coordinate differential , the product of space coordinates’ differentials and the square root of the determinant of the metric tensor, taken with a negative sign.
The variation of
the action function consists of the sum of terms, including:
1) the variation of
the 4potential of the general field;
2) the variation of coordinates , which creates the variation of
the mass 4current;
3) the variation of
the metric tensor.
Due to the principle of least action, the variation must vanish. This leads to the vanishing of
the sums of all the terms, standing before the variations , and ,
respectively. As a consequence, the equations of the general field, the
fourdimensional 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 4volume 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 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 , 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 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:
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:
where and denote the time components of the 4vectors
and .
The 4momentum of the system is determined by the
formula:
where and denote the system’s momentum and the velocity of motion of the center of
mass.
The general field equations have the following form:
Thus, the only source of the general field is assumed
to be the mass 4current . The latter equation can be written more
concisely using the LeviCivita symbol or a totally antisymmetric unit tensor:
Substituting (1), we can express the general field
equations in terms of the tensors of particular fields:
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 equation
(4) can be divided into seven separate equations, in which the mass 4current is the source of one or another
particular field.
The gauge condition of the 4potential of the general
field:
The continuity equation in the curved spacetime is
written with the Ricci tensor :
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 is
transformed into the 4gradient so
that the continuity equation is simplified:
The equation of motion of the matter unit in the general
field is given by the formula:
.
Since , 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 and the 4acceleration :
Here is
the acceleration tensor, is the electromagnetic tensor, is
the gravitational tensor, is the pressure field tensor, is
the dissipation field tensor, is
the tensor of the strong interaction field, is
the tensor of the weak interaction field.
The stressenergy tensor of the general field is determined
from the principle of least action with the expression
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:
.
The stressenergy tensor of the general field is included
into the equation for the metric:
where is
the gravitational constant, is
a certain constant and the gauge condition of the cosmological constant is
used.
The general field tensor has such components as the strength and solenoidal vector of the general field. The vector , according to (2), is the sum of the
strengths of particular fields, and the vector ,
according to (3), consists of the solenoidal vectors of particular fields. The
stressenergy tensor of the general field includes the tensor product , so that the tensor contains the squared vectors and .
Substituting these vectors with the sums of the respective vectors of
particular fields, we obtain the following:
where are some coefficients, is
the electromagnetic stressenergy tensor, is
the gravitational stressenergy tensor,
is the acceleration stressenergy tensor,
is the pressure stressenergy tensor,
is the dissipation
stressenergy tensor, is the stressenergy tensor of the strong interaction field, is
the stressenergy tensor of the weak interaction field.
As we can see, the stressenergy tensor of the general
field contains not only stressenergy tensors of
particular fields, but also the crossterms 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 stressenergy tensor of the general field
we obtain the following:
where is
a constant, which is part of the definition of the acceleration stressenergy
tensor.
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 of
a particle, which is part of the energy as the power function , at the average has the same energy , where is
the Boltzmann constant, 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 (), therefore the average energy of the
particle is .
In general case, the particles have their proper fields,
and the strengths and solenoidal vectors of these fields are squared terms of
the corresponding stressenergy tensors. With this in mind, it is assumed, ^{[7]} 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
crossterms 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.
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 (6), being a part of stressenergy tensor of the
general field поля . The metric around an isolated spherical
body was calculated in the article, ^{[10]} where the
crossterms in (7) were not taken into account. For the time component of the
metric tensor we obtained the following:
where is
the distance from the center of the body to the point where the metric is
defined; is the energy of the gravitational
field inside and outside of the body; is the
energy of the electric field; , and are the mass, charge and radius of the body, is the vacuum permittivity; are the coefficients to be determined; 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 energy of the system of particles with regard to the
electromagnetic and gravitational fields, acceleration field and pressure field
is calculated in the article. ^{[11]} 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 ; the gravitational mass ; the total mass of
all the particles of the body scattered at infinity; the mass obtained by integrating over the volume the
density of the matter moving within the system; the
auxiliary mass obtained by integrating over the volume the
density of the matter, calculated in the reference frame associated with each
particle. For these masses we obtain the relation:
From the equality 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 appears to be larger than the system’s mass . 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 massenergy 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 mass fraction, which is close to the modern
accuracy of mass measurement.
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. ^{[1}^{2}^{]} It was shown that in a moving body the excess massenergy of the gravitational and
electromagnetic fields is compensated by a lack of the massenergy of the acceleration field and pressure field. The
result is achieved by integrating the equation of motion and by calculating the
conserved integral 4vector of the energymomentum of the system. Since this
integral 4vector must be equal to zero, in contrast to the ordinary 4vector
of the energymomentum of the system, it imposes restrictions on the constant , located in the acceleration stressenergy tensor, and the
constant in the pressure
stressenergy 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 vacuum permittivity:
,
where and 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:
where is
the temperature in the center; 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); is
the mass of the system within the current radius ; is
the Boltzmann constant.
Similarly, for the pressure distribution inside the
system we obtain:
where is
the pressure in the center; is
the matter density in the comoving frame; 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. ^{[7]} In this case, for the constant in
the stressenergy tensor of the strong interaction field we obtain the
estimate: , which coincides with the coefficients and 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.
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 NavierStokes
equation in hydrodynamics and to describe the motion of viscous compressible and charged fluid. ^{[6]}
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.
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 nonquantum 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.
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 comoving 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 4potential of a particular field consists of
the scalar and vector potentials, then the sum of the 4potentials of
particular fields gives the 4potential 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 FatioLe Sage's
theory of gravitation. This theory provides a clear physical mechanism of
emergence of the gravitational force, ^{[13]} ^{[14]} ^{[15] [16]} 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. ^{[3]} ^{[17]} Praons and neutral particles in the form of
field quanta form vacuum field. 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.
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. ^{[3]} ^{[4]} 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 abovementioned approach allowed calculating the
proton radius in the selfconsistent model and explaining the de Broglie wavelength. ^{[18]}
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 4potentials 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
4potentials.

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