**Equation of vector field**

**Equation of vector field** is a field
equation that relates field’s four-potential or tensor with
the field source in the form of the corresponding four-current or tensor. Due
to the tensor representation the field equations are expressed in a covariant
form and are valid in curved spacetime.

Vector fields include such fields as the electromagnetic field,
__acceleration field__, __pressure field__, __dissipation field__, macroscopic field of strong
interaction, macroscopic field of weak interaction. Within the framework of the
__covariant theory of gravitation__, the gravitational field
is also considered a vector field.

All these fields can be considered as the __general field__ components
represented in the Lagrangian and in the Hamiltonian of an arbitrary physical
system by the corresponding term with the energy of the particles’ motion and
by the term with the field energy. ^{[1]} ^{[2]} The __relativistic uniform system__ is an
example of a physical system, in which equations of vector field have a
complete and exact solution for all vector fields.

**Contents**

- 1 Types
of equations
- 2 Standard equations
- 3 Wave equations
- 4 Equation
of motion
- 5 Generalized
Poynting theorem
- 6 Equation
for the metric
- 7 Field energy
theorem
- 8 Integral
equations
- 9 See
also
- 10 References
- 11 External
links

**Types**** of
equations**

The standard field equation is a differential equation derived using the principle of least action.
As a rule, such an equation contains the covariant derivative
acting on the field tensor. The second covariant derivative turns the field
equation into the wave equation for the field tensor. If the field tensor is
expressed in terms of the four-potential, then the field equation can be
transformed into the wave equation for the four-potential.

The equation of motion of matter particles can be considered such a type of
field equations, in which the fields act on the field sources and cause them to
move.

The generalized Poynting theorem defines the balance of energy and momentum
at any point of the system, and is formulated as a tensor gauge condition in
the form of equality to zero of the divergence of the total stress-energy
tensor of all the fields acting in the system. Just in the same way, there is a
continuity equation
as a gauge condition for the four-currents, when equality to zero of the
divergence of the mass and charge four-currents in the form and defines the local balance of the mass and
charge at each point. This means that the mass (charge) density in a certain
unit volume changes either when the mass (charge) flux from this volume arises,
or when the metric in the given volume changes.

The equation for the metric is obtained by varying the action function with
respect to the metric tensor and contains the Ricci
tensor, scalar curvature, cosmological
constant, and the fields’ stress-energy tensors. The result of the equation’s
solution represents the metric tensor components as functions of time and
coordinates.

The __field energy theorem__ is expressed
using the integral equation of field energy, generalizes the virial
theorem with respect to the fields and presents it in
the curved spacetime.

The integral field equations are obtained by integrating the standard field
equations over the four-dimensional spacetime. This allows us to formulate
several theorems with respect to the tensor components and the fields’
four-potentials, and to determine a number of new quantities that characterize
the system as a whole.

**Standard equations**

Each vector field is described by two equations, one of which contains the
field sources, and the other imposes restrictions on the type of the field,
regardless of the field sources. ^{[3]} ^{[4]}

The electromagnetic field equations:

where is the electromagnetic tensor, is the charge four-current, is the charge density in the comoving
reference frame, is the four-velocity
of the matter element, is the magnetic constant
(vacuum permeability),
is the speed of light.

The latter equation can be written using the dual electromagnetic field
tensor:

where and the Levi-Civita
symbol
is used.

The gravitational field equations: ^{[5]}

where is the __gravitational tensor__,
is the mass four-current, is the mass density of the matter in the
comoving reference frame, is the __gravitational constant__,
is the speed
of light as the speed of
gravitation and the limiting propagation speed of the
gravitational perturbation.

The acceleration field equations:

where is the __acceleration tensor__, is the acceleration field constant determined
in each problem.

The pressure field equations:

where is the __pressure field tensor__, is the pressure field constant.

The dissipation field equations: ^{[6]}

where is the __dissipation field tensor__, is the dissipation field constant.

**Wave equations**

In the wave equations the sources of fields are the mass and charge
four-currents, so that in case of motion of the masses and charges, wave
phenomena are observed in the propagation of the fields’ potentials and
strengths in the spacetime. As a result, each wave equation contains the
four-dimensional scalar d’Alembert operator , which can act both on the four-potential
and on the field tensor. In some cases, the solution of wave equations is
simpler than the solution of field equations, allowing us to directly find the
field potentials that are part of the four-potentials.

The wave equations for the four-potentials of the above-mentioned fields
are as follows:

where is the four-potential of the electromagnetic
field, is the Ricci tensor,
is the __gravitational
four-potential__,
is the four-potential of the acceleration
field, is the four-potential of the pressure field, is the four-potential of the dissipation
field.

The components of the fields’ four-potentials are not arbitrary functions
and must be gauged. In the Lorentz gauge the divergences of the four-potentials
are equal to zero:

If in the standard field equations we take the
divergence of both sides of the equations and apply the Lorentz gauge, the
gauge conditions would also arise for the field tenors:

The wave equations for the field tensors: ^{[7]}

where is the curvature
tensor.

**Equation of motion**

The equation of motion of matter particles can be expressed in terms of the
field tensors: ^{[7]}

Here is the strong interaction field tensor, is the weak interaction field tensor. The
acceleration field tensor can be expressed in terms of the four-potential in
the form The subsequent application of the __operator
of proper-time-derivative__ allows us to put the left-hand side of the equation
in a form similar to the equation of motion in classical mechanics.

The first term on the right-hand side defines the density of
electromagnetic Lorentz force in the four-dimensional form, the second term is
the gravitational force density expressed using the gravitational field tensor,
the third term describes the pressure force density, the force densities from
the remaining fields are also represented using the corresponding tensors and
the mass four-current. The total sum on the right-hand side of the equation of
motion is the density of the total __four-force__ acting in
the system.

In the equation of motion, the tensors of all the fields can be expressed
in terms of the corresponding four-potentials. For the four fields acting in
the system this gives the following:

**Generalized Poynting theorem**

The stress-energy tensor of
the system is the sum of the tensors of individual components of the __general
field__:

where is the electromagnetic
stress–energy tensor, is the __gravitational
stress-energy tensor__, is the __acceleration
stress-energy tensor__, is the __pressure stress-energy
tensor__, is the __dissipation stress-energy
tensor__, is the stress-energy tensor of the strong
interaction field, is the stress-energy tensor of the weak
interaction field.

The generalized Poynting theorem is written as equality to zero of the
divergence of the stress-energy tensor of the general field:

where is the Christoffel symbol.

The obtained expression for the space components of this equation in the
system’s matter defines the balance of the fields’ energy and momentum, and is
nothing but the differential equation of the matter’s motion under the action
of the forces, generated by the fields, written in a covariant form. ^{[7]}
Outside the matter’s limits the equality to zero of the space components of
this equation means that changes in the fluxes of the electromagnetic and
gravitational fields occur only in the presence of spatial gradients of the
respective components of the stress-energy tensors of these fields. As for the
time components of the equation, for them the generalized Poynting theorem
defines the balance of the energy and energy fluxes of all the fields in any
selected volume of the system. ^{[8]}

At the same time there is a difference between the total energy density of
all the fields at a certain point and the relativistic energy density at this
point, since the relativistic energy also takes into account the particles’
energy and is calculated in a different way, without the use of the
stress-energy tensor of
the system. Consequently, the Poynting theorem applies to the local rates of
change of the field energy over time and the rates of change of the energy
fluxes in space, while the law of conservation of energy-momentum deals with
the energy and momentum of the system’s fields and particles. The relativistic
energy of the system in the center-of-momentum frame defines the __invariant
energy__
equal to the product of the inertial mass by the speed of light squared. Thus,
conservation of the energy in a closed system leads to conservation of the
inertial mass of the system. There is also the gravitational mass of the
system, which is determined by the gravitational interaction with other bodies
(for example, when weighting), and in the __covariant theory of
gravitation__
the gravitational mass turns out to be greater than the inertial mass. ^{[9]}
The difference between the masses follows from the fact that the gravitational
mass is determined not from the relativistic energy or momentum, but from the
sum of the invariant masses of all the system’s particles. In a closed system
there is no exchange of the matter’s particles, energy and information with the
environment, and therefore the gravitational mass is also conserved.

The generalized Poynting theorem can be represented in an integral form.
Integrating over the covariant four-volume and taking
into account the divergence theorem, we find:

With this in mind, we obtain:

Here and the vector is a three-dimensional unit normal vector
directed outward from the two-dimensional surface surrounding an arbitrarily selected
volume in the system under consideration.

**Equation for the metric**

The equation for the metric is written in terms of the tensor :
^{[10]}

where is
the Ricci tensor, is the scalar
curvature, is the metric tensor, is a certain constant.

In the vector field theory, all four-vectors and tensors are gauged,
including four-currents, four-potentials and field tensors. For example, the
four-velocity gauging has the form: Applying here the operator of
proper-time-derivative, we arrive at the four-acceleration gauging: where the __four-acceleration__ is
For the four-momentum
gauging, we have: , where is the inertial mass of the system.

The four-displacement
gauging: where is the four-dimensional spacetime interval.
The metric tensor gauging: , where is the Kronecker delta.

The curvature tensor is not an exception, from this tensor, using the
contraction with the metric tensor, we first obtain the Ricci tensor, and then
the scalar curvature. The gauge condition for the scalar curvature is as
follows: The cosmological constant is also gauged as follows: This allows us to eliminate in the expression
for the relativistic energy both the scalar curvature and the cosmological
constant in the only possible way. ^{[4]}

**Field energy theorem**

The __field energy theorem__ for the
electromagnetic field has the following form: ^{[11]}

where is the magnetic constant;
is the electromagnetic four-potential; is the electromagnetic four-current; is the electromagnetic field tensor; is the element of the invariant volume,
expressed in terms of the product of the differentials of the space coordinates
and in terms of the square root of the determinant of the metric tensor, taken with a negative
sign; is the speed of light; the last integral on
the right-hand side is the surface integral of the second kind taken over the
two-dimensional surface , surrounding the volume under consideration; is the outward-directed three-dimensional normal
vector to the surface .

Similarly, we obtain the following for the gravitational field,
acceleration field and pressure field:

The above expressions are the integral equations, relating the
four-potentials, four-currents and tensors of the respective fields.

**Integral equations**

For the electromagnetic field, the integral equations in the curved
spacetime have been considered in the article. ^{[12]}

Integrating the standard equation of the electromagnetic field over the
four-dimensional volume and applying the divergence theorem
gives the following equation:

For the index , taking into account the equality , we obtain from (1) the Gauss's law
in the covariant notation:

where is an orthonormal element of the
two-dimensional surface surrounding the charge
; represents the electric flux
through the closed surface; cyclically repeated three-dimensional indices , which do not coincide with each other.

Now let us assume that in (1) the indices
:

This equation is valid for any three-dimensional volume and the two-dimensional surface surrounding it. If we split the volume by a certain plane perpendicularly to the
axis ,
then at the index we can pass on from integration over and to integration over the cross-section area
and the contour of this cross-section:

The obtained covariant expression represents the theorem on the magnetic
field circulation (Ampère's circuital law),
and can be considered here as the flux of the
electric field through the cross-section surface in the direction of the
axis .
Hence it follows that the magnetic field circulation in the contour occurs not
only due to changing over time of the flux of the electric field through the
contour, but also occurs when the contour area changes with the constant
electric field.

Integration of the electromagnetic field equation with the dual
electromagnetic field tensor over the four-dimensional volume, taking into
account the divergence theorem, leads to the following equation:

Hence, with the index , taking into account the dual tensor
components, we obtain the Gauss's law for magnetism:

where the covariant quantity is the flux of the magnetic field through the
closed surface.

If in equation (2) we assume that the index , then when the volume under consideration is
split by a plane perpendicular to the axis
, for the obtained cross-section and the
contour surrounding it, we have the following equation:

This integral equation represents the theorem on the electric field
circulation and the Faraday's law of induction in a covariant form, where the quantity is the magnetic flux
through the surface bounded by the conductive contour , and the quantity is the electromotive force.

If we write the four-potential
of the electromagnetic field with a contravariant index in terms of the
components in the form and denote the flux of the vector potential
over a closed two-dimensional surface as follows

then the equation will hold true

which means that when the volume integral of scalar potential changes over
time, the flux of the vector potential appears.

The integral equations, similar to those provided above for the
electromagnetic field, must also be valid for other vector fields.

**See**** also**

- General field
- Acceleration field
- Pressure field
- Dissipation field
- Covariant theory of gravitation
- Four-force
- Field energy theorem
- Fedosin's theorem

**References**

1. Fedosin S.G. The Concept of the
General Force Vector Field. OALib Journal, Vol. 3, pp. 1-15 (2016), e2459. http://dx.doi.org/10.4236/oalib.1102459.

2. Fedosin S.G. Two components of the macroscopic general field. Reports in
Advances of Physical Sciences, Vol. 1, No. 2, 1750002, 9 pages (2017). http://dx.doi.org/10.1142/S2424942417500025.

3. Fedosin S.G. The procedure of finding the
stress-energy tensor and vector field equations of any form.
Advanced Studies in Theoretical Physics, Vol. 8, No. 18, pp. 771-779 (2014). http://dx.doi.org/10.12988/astp.2014.47101.

4. ^{4.0 4.1 }Fedosin S.G. About the cosmological
constant, acceleration field, pressure field and energy.
Jordan Journal of Physics. Vol. 9, No. 1, pp. 1-30 (2016). http://dx.doi.org/10.5281/zenodo.889304.

5. Fedosin S.G. The Principle of Least Action in
Covariant Theory of Gravitation.
Hadronic Journal, Vol. 35, No. 1, pp. 35-70 (2012). http://dx.doi.org/10.5281/zenodo.889804.

6. 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). http://dx.doi.org/10.5541/ijot.5000034003.

7. ^{7.0 7.1 7.2 }Fedosin S.G. Equations of Motion in the Theory of
Relativistic Vector Fields. International Letters of Chemistry, Physics and
Astronomy, Vol. 83, pp. 12-30 (2019). https://doi.org/10.18052/www.scipress.com/ILCPA.83.12.

8. Fedosin S.G. The generalized Poynting theorem for the general field and
solution of the 4/3 problem. International Frontier Science Letters, Vol. 14,
pp. 19-40 (2019). https://doi.org/10.18052/www.scipress.com/IFSL.14.19.

9. Fedosin S.G. The
Mass Hierarchy in the Relativistic Uniform System. Bulletin of Pure and Applied
Sciences, Vol.38 D (Physics), No. 2, pp. 73-80 (2019). http://dx.doi.org/10.5958/2320-3218.2019.00012.5.

10. Fedosin S.G. Energy and
metric gauging in the covariant theory of gravitation. Aksaray
University Journal of Science and Engineering, Vol. 2, Issue 2, pp. 127-143
(2018). http://dx.doi.org/10.29002/asujse.433947.

11. Fedosin S.G. The Integral Theorem of
the Field Energy. Gazi
University Journal of Science. Vol. 32, No. 2, pp. 686-703 (2019). http://dx.doi.org/10.5281/zenodo.3252783.

12. Fedosin S.G. On the
Covariant Representation of Integral Equations of the Electromagnetic Field.
Progress In Electromagnetics Research C, Vol. 96, pp. 109-122 (2019). https://doi.org/10.2528/PIERC19062902.

**External**** links**

Source: http://sergf.ru/uven.htm