Reports
in Advances of Physical Sciences, Vol.
1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025

**Two
components of the macroscopic general field**

**Sergey G. Fedosin**

PO box 614088, Sviazeva
str. 22-79, Perm, Perm Krai, Russia

E-mail: intelli@list.ru

The general field, containing
all the macroscopic fields in it, is divided into the mass component, the
source of which is the mass four-current, and the charge component, the source
of which is the charge four-current. The mass component includes the gravitational
field, acceleration field, pressure field, dissipation field, strong
interaction and weak interaction fields, other vector fields. The charge
component of the general field represents the electromagnetic field. With the
help of the principle of least action we derived the field equations, the
equation of the matter’s motion in the general field, the equation for the
metric, the energy and momentum of the system of matter and its fields, and
calibrated the cosmological constant. The general field components are related
to the corresponding vacuum field components so that the vacuum field generates
the general field at the macroscopic level.

*Keywords:** general field; vacuum field; acceleration
field; pressure field; dissipation field.*

**1. ****Introduction**

Most of the unified field
theories, such as the theory of everything, grand unified theory, loop quantum
gravity, string theory, and some other theories, are based on the quantum
approach and are intended to unite the fundamental interactions at the level of
elementary particles. There were known attempts to unite the physical fields at
the macroscopic level for sufficiently massive bodies, where gravitation
becomes the defining interaction. A variety of approaches are used for this
purpose. Thus, in the Kaluza-Klein theory [1-2] the
so far undiscovered fifth dimension and some scalar field were added to the
ordinary four-dimensional spacetime in order to derive both the gravitational
field equations and the equations that are equivalent to the Maxwell equations. The unified field theories, in four-dimensions and extra spacial dimensions, were considered in several important
papers [3-6].

In [7] it is assumed that the field
strength vector and the solenoidal vector of the unified field consist of the
sum of the field strengths and solenoidal vectors of all the known fields with
the corresponding coefficients. In this case the unified field obeys the
Maxwell equations, in which the source of the unified field strength is the sum
of the products of the fields’ charge densities by certain coefficients, and
the source of the unified field’s solenoidal vector is the sum of the products
of the currents with certain coefficients. As a result, each four-current,
associated with a particular field, contributes to the unified field vector
components. The drawback of this approach is lack of covariance of the
presented unified field equations.

The concept of the general
force vector field, which includes the electromagnetic and gravitational fields,
acceleration field, pressure field, dissipation field, strong interaction
field, weak interaction field, and other vector fields, was presented in [8]. The characteristic feature of this
approach is that the four-potential of the general field is given by the sum of
the four-potentials of all particular fields. An
exception is the four-potential of the electromagnetic field, which is included
in the sum of the four-potentials with a coefficient equal to the charge
density to mass density ratio . The equations derived from the
principle of least action fully describe the general field and its interaction
with the matter.

A certain limitation of this
approach is that the density ratio in the physical system under consideration is
assumed to be unchanged. Below, we present a more complete and universal
theory, in which the general field is resolved into two main components. The
source of the first component is the mass four-current , which generates such vector
fields as the gravitational field, acceleration field, pressure field, dissipation
field, macroscopic fields of strong and weak interactions. The second component
of the general field is the electromagnetic field, the source of which is the
charge four-current .

A characteristic feature of
macroscopic fields is that their description may not coincide with the field’s
description at the microscopic level. Thus, the general theory of relativity
(GTR) is considered to be quite satisfactory, although it is not related to the
quantum theory of gravity, and it does not follow from it. From the set of axioms of GTR it follows that the
gravitational field is described by the metric tensor and the stress-energy
pseudotensor, which are determined based on the space-time geometry [9]. This
approach, which was many times confirmed by experiments, leads to the idea that
the gravitational field is a tensor (metric) field. Thus, the gravitation in
GTR does not reduce to the vector field, and the tensor character of the field
directly preconditions the absence of the dipole component in the gravitational
radiation.

In order to describe the
gravitation we use the axiomatically constructed covariant
theory of gravitation (CTG) with the vector four-potential,
gravitational tensor and stress-energy tensor [10]. In this case the gravitation is an independent
physical field that does not require full reduction to geometry. In CTG for
each individual body the dipole gravitational radiation is admitted. Since for
detection of gravitational effects at least two bodies are required, their
dipole radiation mutually cancels out, and the system of bodies’ radiation
always occurs in a quadrupole way or by higher multipoles. Thus, with respect
to radiation the tensor field in GTR does not contradict the vector field in
CTG. The difference between the basic postulates of the both theories does not
prevent from describing in CTG all the standard effects of GTR [10], presenting
the expression for the metric tensor [11], explaining the Pioneer anomaly [12],
and considering for macroscopic systems the contribution of the gravitational
field in the Navier-Stokes equation [13], in
estimation of the parameters of cosmic bodies [14] and in the virial theorem
[15].

Although strong and weak
interactions at the level of elementary particles are usually described by
vector non-Abelian fields (nonlinear Yang-Mills fields with self-action), at
the macroscopic level we believe it is possible to describe these interactions
with the help of ordinary four-dimensional vector fields. Indeed, quantum
effects at the macrolevel as a rule disappear due to the large number of
interacting particles, as well as the contributions of individual particles
into the fields, only the average values of fields become important. As a
result, in order to describe the macroscopic fields it suffices to use simple
schemes in the form of equations for the four-potentials of Maxwellian type, at
least as a first approximation.

**2. The action function and its variation**

We assume that the two
components of the general field and the corresponding four-currents are
sufficiently independent of each other. This allows us to apply the
superposition principle in the principle of least action, so that each
component independently contributes to the Lagrangian. The action function for
the continuously distributed matter located in the general field (both in the
proper field and the external field) in the curved spacetime similarly as in [8], [16] takes
the following form:

(1)

where
is the Lagrangian , is the scalar curvature, is the cosmological constant, is the four-vector of the mass current, is the mass density of a point particle of matter
in the reference frame associated with the particle, is the four-velocity of the point particle, is the speed of light, is the four-potential of the
mass component of the general field,
described with the help of the scalar potential and the vector potential of this field, is the tensor of the mass component of the general
field, is the four-potential of the
electromagnetic field defined using the scalar potential and the vector potential , is the four-vector of the charge current, is the charge density of a point particle in the
reference frame associated with the particle, is the vacuum permittivity, is the
electromagnetic tensor, and are considered the constant coefficients.

In
(1) the first term under the integral sign is proportional to the energy density
associated with the curvature and the cosmological constant. The second term
defines the energy density of the mass four-current at the four-potential .
Similarly, the fourth term for the electromagnetic field defines the energy
density of the charge four-current at the four-potential . The third
and fifth terms are associated with the energy density of the general field’s
mass components and the energy density of the electromagnetic field,
respectively, and they do not vanish even in the empty space outside the
matter.

The
four-potential of the mass component of the general field is defined as a
generalized four-velocity in the form of the sum of the four-potentials of the
gravitational field [17],
acceleration field and pressure field [18], dissipation field [13] and fields
of strong and weak interactions [8],
respectively:

. (2)

From
(2) and the components of the four-potential it follows that the scalar and vector potentials of the mass component of the
general field are the sums of the respective scalar and vector potentials of
the fields under consideration.

The
tensor of the mass component of the general field is defined as the four-curl of the
four-potential . In view of
(2), the tensor is expressed in terms of the sum of the
tensors of the gravitational field, acceleration field, pressure field,
dissipation field, and fields of strong and weak interactions, respectively:

(3)

The
action function with the terms similar to the terms in (1) was varied in a
number of works, for example, in [16], [19]. For the action function
variation we can write the following:

, (4)

,

,

,

where
is the Ricci tensor, is the metric tensor variation, is the invariant four-volume, expressed in terms of the time coordinate
differential , the product of the space coordinate differentials and the square
root of the determinant of the metric tensor, taken with a
negative sign,

represents a variation of
four-coordinates [19, 20], due to which we obtain a variations of the mass
four-current and the charge four-current , is a variation of the four-potential of
the mass component of the general field, is a
variation of the four-potential of the electromagnetic field.

The
stress-energy tensor of the mass component of the general field and
the stress-energy tensor of the electromagnetic field are given by:

.

.

The
properties of the stress-energy tensor are described in [8].

**3. The equations for the fields and the motion of matter **

Summing up in (4) the terms
with the same variations and equating these sums to zero, we obtain the
corresponding equations. In particular, we find the equations for the general
field’s mass component with the field source in the form of the mass
four-current, as well as the equations of this field without sources, resulting
from antisymmetry of the tensor :

, . (5)

The
electromagnetic field equations have the standard form:

, . (6)

Applying
the covariant derivative to the equations of the field with the sources
in (5) and (6) gives continuity equations for the four-currents in the curved
spacetime:

, .

The
gauge condition of the four-potentials:

, .

The equations of motion are
obtained from (4) by equating the sum of the terms containing the variation to zero. In view of (3) we have:

. (7)

The tensor product can be expressed in terms of the
four-acceleration using the operator of the proper time
derivative [18]:

.

With this in mind, (7) is
transformed into the four-dimensional equation of motion of the viscous compressible
and charged fluid [13], with the addition from the density of the
four-forces, arising from the strong and weak interactions:

.

Because, due to the tensors’
properties, for the corresponding four-forces, the following relations hold
true:

, ,

then the equation of motion
(7) can be written using the divergence of the sum of the stress-energy
tensors:

. (8)

**4. ****The equation for the metric
and the relativistic energy**

The equation for the metric is
obtained by equating to zero the sum of the terms containing variation in (4):

(9)

Let us contract this equation
by multiplying by the metric tensor, given that , , , :

. (10)

Substituting (10) in (9)
simplifies the equation for the metric:

. (11)

Solution of equation (11) was
presented in [11] for the case
of gravitational and electromagnetic fields, and the coefficient can be found
by comparison with the experimental data.

Acting in the same way as in [18],
we will write the gauge condition of the cosmological constant . In view of (2) we obtain the following:

. (12)

In the complete physical
system all the field sources in the form of four-currents are taken into
account, and all the fields in such a system, acting on the matter, have the
internal origin. According to (12), each matter unit has its own value , depending on the four-potentials
of the two general field components and the respective four-currents. We can
assume that the initial state of the system under consideration was the
motionless, scattered in space, low-density matter in the form of individual
particles. Then under action of gravitation this matter was broken into
fragments and was brought together in much denser system. In this case, the
constant reflects the total energy density of the
matter in its proper fields in the initial state.

From calibration of (12) from
(10) the relation follows between the scalar curvature and the cosmological
constant:

.
(13)

After applying the covariant
derivative to all the terms in (11), taking into account
(8) and the equation as the property of the Einstein tensor, we
obtain the equation or the equivalent equation . From (13) the equation follows, and in view of (12) this leads to the
equation that must hold inside the matter:

.

The energy of the system,
consisting of the matter and the fields, can be calculated in the same way as
in [18].
In view of the calibration of (12), for the energy we obtain the following:

.

The energy depends only on the
time components of the four-potentials and four-currents and does not depend on
their space components. For the 4-momentum of the system we obtain: , where and denote the momentum of the system and the
velocity of the system’s center of mass.

**5. ****Conclusions**

We will divide all the known
fields into two groups. The first group includes the gravitational field,
acceleration field, pressure field, dissipation field, strong interaction
field, weak interaction field, and other vector fields, the source of which is
the mass four-current . This group of fields represents
the mass component of the general field. The second (charge) component of the
general field is the electromagnetic field, the source of which is the charge
four-current . All the field equations, the
stress-energy tensors, the equation of the matter motion, the equation for the
metric and the relativistic energy are found from the principle of least action
in the covariant form. As it was shown in [21, 22], the
potentials and strengths of all the fields obey the wave equation, and
therefore in some cases they have the same dependence of the coordinates and
time. In this case, the field potentials, field tensors and stress-energy
tensors can be found using the standard procedure [23].

Both components of the general
field are related to each other not only in the equations, but also in actual
processes. Thus, the charge four-current is always accompanied by the
corresponding mass four-current of the charge carriers, each of which has its
own mass and velocity. In turn, the gravitational field, pressure field and
other fields can generate some charge four-current. An example is the emergence
of magnetic fields in cosmic objects in the electrokinetic
model [24], due to the mechanism of charge separation and their simultaneous
rotation with the matter of the objects.

Dividing the general field to
two components is most naturally explained in the modernized Fatio-Le Sage’s model [25, 26], where the
vacuum field also has two components – the graviton field and the field of
charged particles (praons). At the same time, one physical mechanism can be
responsible both for the emergence of gravitational force [27] by means of the graviton field’s
action and for the emergence of electromagnetic interaction [28, 29] under the action of the field of
charged particles.

In this model, the fluxes of
the smallest particles of the vacuum field penetrate all bodies and perform the
electromagnetic and gravitational interaction between the remote matter
particles. The resulting different types of interaction between the matter
particles can be represented as the action of the pressure field, acceleration
field, dissipation field and other fields included in the general field. Thus
the vacuum field components lead to the general field components.

The common mechanism of
interaction of the vacuum and general fields helps to explain the reason for
the fact that all the partial fields as the components of the general field can
be described by the same Maxwell-type
equations. For example,
by solving the wave equations for the four-potentials of the acceleration field
and the pressure field it is possible to find the temperature and the pressure
inside the stars and planets [14] in good agreement with the calculations
performed by other methods. This approach to the four partial fields has been
successfully applied to estimation of the kinetic energy of the macroscopic
system’s particles performed by three different ways [15].

**References**

1. Kaluza T. Zum Unitätsproblem in der Physik.
Sitzungsber. Preuss. Akad. Wiss. Berlin. (Math. Phys.)
pp. 966-972 (1921).

2.
Klein O. Quantentheorie
und fünfdimensionale Relativitätstheorie.
Zeitschrift für Physik A,
Vol. 37** **(12),
pp. 895-906 (1926).

3. Hilbert D. Die Grundlangen der physic (Erste Mitteilung), Nachr. Koenigl. Gesellsch Wiss. Goettingen, Math-Phys. Kl., pp. 395-407 (1915).

4. Eddington A.S. A generalisaton of Weyl’s
theory of the electromagnetic and gravitational field. Proc. R. Soc. London, A
99, pp. 104-122 (1921).

5. Weyl H. Raum, Zeit und Matter, 4 edition,
Springer, Berlin (1923).

6. Einstein A. Riemann-Geometrie mit Auufrechterhaltung des Begriffs des Fernparallelismus. Sitzungsber. Preuss. Akad. Wiss. Vol. 17, 217-221
(1928).

7. Naumenko Yu. V. Yedinaya teoriya vektornykh poley (ot elektrodinamiki Maksvella k yedinoy teorii polya). Armavir, Armavirskoye poligraf-predpriyatiye, 2006. (In Russian).

8.
Fedosin S.G. The
concept of the general force vector field. OALib Journal,
Vol.
3, pp.
1-15 (2016). doi: 10.4236/oalib.1102459.

9.
Fedosin S.G. The General Theory of Relativity, Metric Theory of Relativity and
Covariant Theory of Gravitation. Axiomatization and Critical Analysis. International
Journal of Theoretical and Applied Physics (IJTAP), Vol. 4, No. I , pp. 9-26 (2014).

10.
Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. (Perm, 2009). ISBN 978-5-9901951-1-0.

11.
Fedosin S.G. The Metric Outside a Fixed Charged Body in the Covariant Theory of
Gravitation.
International Frontier Science Letters (IFSL), ISSN: 2349 – 4484, Vol. 1, No.
1, pp. 48- 55 (2014). doi: 10.18052/www.scipress.com/ifsl.1.41.

12.
Fedosin S.G. The Pioneer Anomaly in Covariant
Theory of Gravitation. Canadian Journal of Physics, Vol. 93, No. 11, pp. 1335-1342 (2015). doi: 10.1139/cjp-2015-0134.

13.
Fedosin S.G. Four-Dimensional Equation of Motion for Viscous
Compressible and Charged Fluid with Regard to the Acceleration Field, Pressure
Field and Dissipation Field. International Journal of
Thermodynamics, Vol. 18, No. 1, pp. 13-24, (2015). doi:
10.5541/ijot.5000034003.

14.
Fedosin S.G. Estimation of the physical
parameters of planets and stars in the gravitational equilibrium model. Canadian
Journal of Physics, Vol. 94, No. 4, pp. 370-379 (2016). doi: 10.1139/cjp-2015-0593.

15.
Fedosin S.G. The virial theorem and the kinetic energy of particles of a
macroscopic system in the general field concept. Continuum Mechanics
and Thermodynamics, Vol. 29, Issue 2, pp. 361-371 (2016). doi: 10.1007/s00161-016-0536-8.

16.
Fedosin S.G. The Principle of Least Action in Covariant Theory of Gravitation. Hadronic
Journal, Vol. 35, No. 1, pp. 35-70 (2012).

17.
Fedosin S.G. Fizika i filosofiia
podobiia ot preonov do metagalaktik. (Perm, 1999).
ISBN 5-8131-0012-1.

18.
Fedosin S.G. About the cosmological constant, acceleration field,
pressure field and energy. Jordan Journal of Physics, Vol. 9, No.
1, pp.
1-30 (2016).

19.
Fock V. A. (1964). "The Theory of Space, Time and Gravitation". Macmillan.

20.
Dirac P.A.M. General theory of relativity. Florida State University.
John Wiley & Sons, Inc., New York - London • Sydney • Toronto, 1975.

21.
Fedosin S.G. The Integral Energy-Momentum 4-Vector and Analysis of 4/3 Problem Based
on the Pressure Field and Acceleration Field. American Journal of
Modern Physics, Vol. 3, No. 4, pp. 152-167 (2014). doi:
10.11648/j.ajmp.20140304.12.

22.
Fedosin S.G. Relativistic
Energy and Mass in the Weak Field Limit. Jordan Journal of Physics, Vol. 8, No.
1, pp. 1-16 (2015).

23.
Fedosin S.G. The procedure of finding the stress-energy
tensor and vector field equations of any form. Advanced Studies in Theoretical Physics, Vol.
8, pp. 771-779 (2014). doi:10.12988/astp.2014.47101.

24.
Fedosin S.G. Generation of magnetic fields in cosmic objects: electrokinetic
model.
Advances in Physics Theories and Applications, Vol. 44, pp. 123-138 (2015).

25.
Sergey Fedosin, The physical theories and infinite hierarchical nesting
of matter, Volume 2, LAP LAMBERT Academic Publishing, pages: 420, ISBN-13:
978-3-659-71511-2. (2015).

26.
Fedosin S.G. The
force vacuum field as an alternative to the ether and quantum vacuum. WSEAS Transactions on Applied and
Theoretical Mechanics, Vol. 10, Art. #3, pp. 31-38 (2015).

27.
Fedosin S.G. The graviton field as the source of mass and gravitational force in the
modernized Le Sage’s model. Physical Science International Journal, Vol. 8, Issue 4, pp. 1-18
(2015). doi: 10.9734/PSIJ/2015/22197.

28.
Fedosin S.G. The
charged component of the vacuum field as the source of electric force in the
modernized Le Sage’s model. Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp.
971-1020 (2016). doi: 10.4314/jfas.v8i3.18.

29.
Fedosin S.G. The
substantial model of the photon. Journal
of Fundamental and Applied Sciences,
Vol. 9, No. 1, pp. 411-467 (2017). doi: 10.4314/jfas.v9i1.25.

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