The field mass-energy
limit of an arbitrary physical system is a certain boundary value of
the field’s mass-energy, which can be achieved in this system. Each field can
have several limiting values of its mass-energy, depending on what mass (or
mass-energy) it is compared with.
For a fixed uniformly charged
spherical body with chaotic motion of charges the general electromagnetic field
on the average is purely electric, and the total mass-energy of the field
inside and outside the body in the limit of special relativity is defined by
the formula: ^{[1]}
where is the electric field energy, is the speed of light, and are the electric charge and the radius of
the body, is the electric
constant.
The charge of an initially
neutral body emerges when either some of its charges are removed from the body,
or on the contrary external charges are transferred to the body. If the
transferred charges are electrons, then the absolute value of the body’s total
charge depends on the number of the electrons transferred: , where is the elementary charge. Accordingly, the
mass of all transferred electrons, contributing to the body’s charge, equals: where is the electron mass.
One of the limits of the
mass-energy of the electromagnetic field emerges on condition that . Hence we obtain the equality:
This equality is possible only at
a certain relation between the charge and the radius of the body. The electric field potential on the body’s surface should be great enough
by the absolute value so that the field’s mass-energy could exceed the mass of
the electrical charges that create the field. This potential can be expressed
from (1) in terms of the mass and charge of the electron and is equal to 852
kV. In this case, the contribution of the mass-energy of the electromagnetic
field into the total mass of the system and the contribution of the mass of the
charged particles, creating the total charge of the system, can have opposite
signs. ^{[2]} This means that when a sufficiently
large charge of the body is achieved, the electromagnetic field mass-energy can
start reducing the total mass of the system, consisting of the body and its
fields.
In the modernized Le Sage’s model, the charged component
of the vacuum field contained in electrogravitational
vacuum can be considered as a source of electrical force. ^{[3]} In this model, the
vacuum field consists of two components – the graviton field, causing the
gravitational forces, ^{[4]} and the field of
charged particles. Praons are
considered as the charged particles of the vacuum field, which are similar in
their properties to nucleons and neutron stars. Furthermore, according to the
theory of Infinite Hierarchical Nesting of
Matter and the similarity of matter
levels, a neutron star contains as many nucleons, as many praons are
contained in a neutron. The energy density of the field of charged particles in
the model of cubic distribution of fluxes of particles in space, assuming that
these particles fly into the cubic volume perpendicularly to cube faces, is
defined by the formula:
J/m^{3},
where m^{2} is the cross-section of interaction of the
charged particles of the vacuum field with the nucleons, which almost exactly
coincides with the geometrical cross-section of the nucleon.
On the other hand, the energy
density of the electric field reaches the maximum on the surface of the charged
sphere and is equal to:
Here denotes the electric field strength on the
sphere surface. The natural limit of the electric field energy density is the
energy density of the field of charged particles of the vacuum field. This
implies the condition , which gives the maximum possible value
of the electric field strength:
V/m.
One of the most highly charged
objects is the proton. Assuming that the proton radius is m in the
self-consistent model, ^{[5]} we can estimate the field
strength on the surface of the proton:
V/m.
The electric field strength of
the proton turns out to be almost five times less than the limiting value.
The mass-energy of the gravitational field for a uniform
spherical body can be calculated within the framework of the Lorentz-invariant theory of gravitation
(LITG):
where is
the gravitational field energy, is the body mass, is the gravitational
constant.
According to the general
relativity, the largest field must be located near a black hole, while the
gravitational radius of the black hole is related to its mass:
Consequently, for the black hole
in (2) we must have , and as an
estimate of the limiting mass-energy of the gravitational field we obtain the
relation: . Although this
calculation does not fully take into account the spacetime curvature, it allows
us to see that the mass-energy of the field can reach a considerable proportion
of the body mass.
Since in the general relativity
there is no evidence that the body mass is able to curve the spacetime to the
state of a black hole, the existence of such extreme objects is doubtful.
In the modernized Le Sage's
theory of gravitation, the graviton field has its energy density, which, in the
model of cubic distribution of flux of particles in space, equals: ^{[4]} ^{[3]}
J/m^{3},
where is the proton mass, m^{2} is the cross-section of interaction of the
gravitons with the matter, m^{3}•s^{–2}•kg^{–1} is the strong
gravitational constant.
The energy density of the
gravitational field of a stationary massive body reaches the minimum near the
surface and in LITG it equals:
where is the gravitational
field strength on the surface of the body.
The absolute value of the energy
density of the gravitational field cannot exceed the energy density of the
graviton field, , which allows us to estimate the maximum
possible absolute value of the gravitational field strength:
m/s^{2}.
For comparison, for a neutron
star with the mass of Solar masses and the radius km the absolute value of the gravitational
field strength on the surface is equal to:
m/s^{2}.
In cosmic bodies there are a number of fields at the same
time, including the gravitational and electromagnetic fields, pressure field, acceleration field, dissipation field, fields of strong and
weak interactions. All these fields are the components of the general field. ^{[6]}
^{[7]} Each field not only has its own energy, but also contributes to
the total relativistic energy of the system due to the interaction of one or
another field with the matter. In the Hamiltonian the field energy is defined
by the product of the field tensor by itself, and the energy of the field’s
interaction with the matter depends on the term with the product of the field’s
four-potential by the mass (charge) four-current. In the covariant theory of gravitation, in the
weak field limit the relativistic mass-energy of the system is calculated,
taking into account the contribution of the general field components – the
gravitational and electromagnetic fields, and the contribution of the pressure
field and acceleration field: ^{[2]}
Here the mass is an invariant inertial mass of the system
of a number of identical particles, which are under the action of their own
four fields, and are the total mass and charge of all
particles.
As the radius of the sphere,
inside which the particles are located, decreases, the mass remains unchanged, but and increase. This is
associated with the fact that the velocities of the particles’ motion inside
the sphere are increasing, and the mass is increasing due to changing of the Lorentz
factor. Furthermore it turns out that the mass is equal to the gravitational mass of the
system and . If we assume
that , then for the sum of the mass-energy of the
general field and the mass-energy of the particles in the general field we
obtain the following relation:
The contribution of the
electromagnetic field can usually be neglected in comparison with the
contribution of the gravitational field. Then, taking into account the formula
for the gravitational radius, we find:
Since for the known bodies , then the
mass-energy cannot exceed 15% of the gravitational
mass .
For relativistic uniform system
inertial mass is found to be: ^{[8]} ^{[9]}
Consequently, for the total mass-energy of general field a condition is
obtained: .
Source: http://sergf.ru/pmen.htm