**Field**** ****energy**** ****theorem**

**Field energy theorem**, also called the integral
theorem of field energy or Fedosin’s theorem, defines
in a covariant four-dimensional form in a curved spacetime the relation between
components of the energy of any vector field, which has the corresponding
four-potential and the tensor of this field. According to this theorem, the
integral tensor invariant of the field energy in an arbitrary volume of a
system can be determined in terms of the particles’ energy in the field’s
four-potential in the given volume, in terms of the time derivative of the
volume integral of the product of the four-potential by the field tensor, and
in terms of the integral over the surface of the given volume taken of the
product of the four-potential by the field tensor.

For the electromagnetic (gravitational) field, with the help of the theorem
we can introduce the concepts of the kinetic and potential energies of the
field and relate them to each other with a simple numerical coefficient in the
case of a stationary system. In contrast to the virial
theorem, where the kinetic energy of the
system’s particles is half as large in the absolute value as the system’s
potential energy, according to the field energy theorem the kinetic energy of
the electromagnetic (gravitational) field is twice as large in the absolute
value as the entire potential energy of the field. This allows us to explain
why the electrostatic energy of a system of charged particles can be found by
two seemingly unrelated ways – using either the scalar field potential or the
integral over the entire volume of the tensor invariant, which is part of the
stress-energy tensor of the electromagnetic field.

The field energy theorem was proved by Fedosin in 2018 and was published in
2019. ^{[1]} The theorem can be applied to such vector fields as the __acceleration field__, __pressure field__, gravitational
field, electromagnetic field, __dissipation field__, and __general field__, etc. As a
rule, in real systems there are several fields at the same time. Under the
assumption that the conditions of the fields’ superposition and independence of
the fields from each other hold true, the theorem can be applied to each vector
field separately. The verification of the theorem for an ideal __relativistic uniform system__ containing
non-rotating and randomly moving particles shows complete coincidence in all
significant terms for each field.

**Contents**

- 1 Electromagnetic
field
- 2 Gravitational
field
- 3 Acceleration field and pressure field
- 4 Significance
of the theorem
- 5 References
- 6 See
also
- 7 External
links

**Electromagnetic**** field**

The field energy theorem has the following form:

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 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 .

Equation (1) is significantly simplified in case when the electromagnetic
field is considered not just in a limited volume, but in the entire infinite
volume that goes beyond the limits of a closed physical system. Then the last
integral on the right-hand side of (1) vanishes despite the infinite size of
the surrounding surface ,
since far from the system’s charges, at infinity, both the four-potential and the field tensor are equal to zero.

As a rule, in electrostatics, as well as in the relativistic uniform
system, the product is equal to zero, so the first integral on
the right-hand side of (1) also vanishes. In this case, the following remains
in (1):

If we denote

then (2) is rewritten as follows:

Here, the energy can be considered as the field’s kinetic
energy associated with the four-current . In this case, the energy should be considered as the field’s potential
energy, found in terms of the components of the electromagnetic tensor, that
is, in terms of the electric field strength and the magnetic field induction.

It can be noted that (3), with an accuracy up to a numerical coefficient,
resembles the virial theorem in the following form:

where is the kinetic energy of the system’s
particles, denotes the potential energy of the system.

In the theory of vector fields, the contribution of the charges and the
electromagnetic field to the relativistic energy of a physical system is given
by the expression: ^{[2]}

where is the charge density of the matter element
in the comoving reference frame, is the scalar potential at the location of the
element of charged matter, is the time component of the four-velocity of
the matter element.

Suppose relation (2) holds for a physical system, while the global vector
potential of the electromagnetic field is zero everywhere in the matter or such
that it does not make a contribution to the energy . Then taking into account (3) we will obtain:

In this case, we can see that the total electromagnetic energy of the
particles and the field can be found by two different ways – either in terms of
the energy or in terms of the energy . In electrostatics, the
energy is expressed in terms of the electric field
strength and is defined by the time component of the stress-energy tensor of
the electromagnetic field, and the energy depends only on the distribution of the
scalar field potential and the four-current.

**Gravitational field**

The field energy theorem for the gravitational field is written as follows:

where is the __gravitational constant__; is the __gravitational
four-potential__;
is the mass four-current; is the __gravitational tensor__.

All the conclusions made with respect to the electromagnetic field hold
true for the gravitational field. For example, in the system with fixed masses
and in the __relativistic uniform system__, the
following relation will be valid:

The expressions for the kinetic and potential energy of the gravitational
field are as follows:

The contribution of the particles and the gravitational field to the energy
of the physical system has the form:

where is the mass density of the matter element in
the comoving reference frame, is the gravitational scalar potential at the
location of the matter element.

If the vector potential of the gravitational field does not make a
contribution to the system’s energy, then the equation of type (4) is again
satisfied, but now with respect to the energy of the gravitational field.

**Acceleration field and pressure field**

For the __acceleration
field__ and the __pressure
field__, the integral theorem of field energy is written as follows:

where is the acceleration field constant; is the four-potential of the acceleration
field; is the __acceleration tensor__; is the pressure field constant; is the four-potential of the pressure field; is the __pressure field tensor__.

The characteristic feature of the acceleration field and the pressure field
is that these fields act only within the limits of the system’s matter.
Therefore, the surface integrals on the right-hand side of the expressions for
the field energy are taken over the outer surface of the volume, in which the
matter is present.

**Significance of the theorem**

The significance of the theorem lies in the fact that in many cases it
allows us to simplify much the calculation of the system’s relativistic energy.
According to its meaning, the field energy theorem describes the relations
between the components of the fields’ energy and thus it differs from the virial
theorem related to the components of the
particles’ energy. The theorem additionally allows us to interrelate and
differentiate the role of the four-potentials, field tensors and stress-energy
tensors within the framework of the theory of vector fields. For example, it is
known that the equation of motion of the system’s particles can be written in
terms of any of these quantities, ^{[3]} while the time component of
the equation of motion represents the generalized Pointing theorem. ^{[4]}

The equation for the metric with appropriate gauging can only be expressed
in terms of the stress-energy tensors of the fields, ^{[5]} just as the
four-dimensional integral vector describing the energy of the system’s fields and
defining the vector of the flux of this energy. However, the integral vector is
not a real four-vector, and although it is conserved in closed systems, it
cannot replace the four-momentum of a physical system. ^{[6]}

On the other hand, the stress-energy tensors of the fields are completely
absent in the Lagrangian, in the Hamiltonian, in the energy, in the momentum,
in the four-momentum and in the generalized four-momentum of the system. ^{[7]}
This means that the fields’ energy densities and the three-dimensional vectors
of the fields’ energy fluxes, such as the Poynting vector, that are contained
in the stress-energy tensors do not allow us to calculate either the
generalized four-momentum or the system’s four-momentum. For this purpose, it
is necessary to use the four-potentials and the field tensors. ^{[8]}

**References**

1. 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.

2. 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.

3. 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.

4. 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.

5. 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.

6. Fedosin S.G. The covariant additive integrals
of motion in the theory of relativistic vector fields. Bulletin of Pure and
Applied Sciences, Vol. 37 D (Physics), No. 2, pp. 64-87 (2018). http://dx.doi.org/10.5958/2320-3218.2018.00013.1.

7. Fedosin S.G. Generalized four-momentum for
continuously distributed matter. Preprint, 2018.

8. Fedosin S.G. Что мы должны понимать под
4-импульсом физической системы? Preprint, 2018.

**See**** also**

**External links**

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