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 vacuum permittivity.
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 can be considered as a source of electric 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]}
Consequently,
for the total mass-energy of general field a condition is obtained: .
Source:
http://sergf.ru/pmen.htm