# Field mass-energy limit

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.

## Electromagnetic field

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: 

$~m_{ef}={\frac {W_{e}}{c^{2}}}={\frac {3q^{2}}{20\pi \varepsilon _{0}ac^{2}}},$ where $~W_{e}$ is the electric field energy, $~c$ is the speed of light, $~q$ and $~a$ are the electric charge and the radius of the body, $~\varepsilon _{0}$ 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 $~N$ of the electrons transferred: $~|q|=Ne$ , where $~e$ is the elementary charge. Accordingly, the mass of all transferred electrons, contributing to the body’s charge, equals: $~m_{e}=Nm_{0e}={\frac {|q|m_{0e}}{e}},$ where $~m_{0e}$ is the electron mass.

One of the limits of the mass-energy of the electromagnetic field emerges on condition that $~m_{e}=m_{ef}$ . Hence we obtain the equality:

$~m_{0e}={\frac {3|q|e}{20\pi \varepsilon _{0}ac^{2}}}={\frac {3e|\varphi _{a}|}{5c^{2}}}.\qquad (1)$ This equality is possible only at a certain relation between the charge $~q$ and the radius $~a$ of the body. The electric field potential $~\varphi _{a}={\frac {q}{4\pi \varepsilon _{0}a}}$ 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.  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.  In this model, the vacuum field consists of two components – the graviton field, causing the gravitational forces,  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:

$~\varepsilon _{cq}={\frac {e^{2}}{\varepsilon _{0}\vartheta ^{2}}}=4\cdot 10^{32}$ J/m3,

where $~\vartheta =2.67\cdot 10^{-30}$ m2 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:

$~\varepsilon _{e}={\frac {\varepsilon _{0}E_{a}^{2}}{2}}={\frac {\varepsilon _{0}\varphi _{a}^{2}}{2a^{2}}}.$ Here $~E_{a}$ 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 $~\varepsilon _{e}<\varepsilon _{cq}$ , which gives the maximum possible value of the electric field strength:

$~E_{a}<{\frac {{\sqrt {2}}e}{\varepsilon _{0}\vartheta }}=9.58\cdot 10^{21}$ V/m.

One of the most highly charged objects is the proton. Assuming that the proton radius is $~a_{p}=8.73\cdot 10^{-16}$ m in the self-consistent model,  we can estimate the field strength on the surface of the proton:

$~E_{p}={\frac {e}{4\pi \varepsilon _{0}a_{p}^{2}}}=1.89\cdot 10^{21}$ V/m.

The electric field strength of the proton turns out to be almost five times less than the limiting value.

## Gravitational field

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):

$~m_{gf}={\frac {|W_{g}|}{c^{2}}}={\frac {3Gm^{2}}{5ac^{2}}},\qquad (2)$ where $~W_{g}=-{\frac {3Gm^{2}}{5a}}$ is the gravitational field energy, $~m$ is the body mass, $~G$ 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:

$~r_{g}={\frac {2Gm}{c^{2}}}.$ Consequently, for the black hole in (2) we must have $~a\approx r_{g}$ , and as an estimate of the limiting mass-energy of the gravitational field we obtain the relation: $~m_{gf}=0.3m$ . 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:  

$~\varepsilon _{c}={\frac {4\pi Gm_{p}^{2}}{\sigma ^{2}}}={\frac {4\pi G_{a}m_{p}^{2}}{\vartheta ^{2}}}=7.4\cdot 10^{35}$ J/m3,

where $~m_{p}$ is the proton mass, $~\sigma =5.6\cdot 10^{-50}$ m2 is the cross-section of interaction of the gravitons with the matter, $~G_{a}=1.514\cdot 10^{29}$ m3•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:

$~\varepsilon _{g}=-{\frac {\Gamma _{a}^{2}}{8\pi G}},$ where $~\Gamma _{a}$ 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, $~|\varepsilon _{g}|<\varepsilon _{c}$ , which allows us to estimate the maximum possible absolute value of the gravitational field strength:

$~|\Gamma _{a}|<{\frac {4{\sqrt {2}}\pi Gm_{p}}{\sigma }}=3.54\cdot 10^{13}$ m/s2.

For comparison, for a neutron star with the mass $~m_{s}=1.35$ of Solar masses and the radius $~a_{s}=12$ km the absolute value of the gravitational field strength on the surface is equal to:

$~|\Gamma _{s}|={\frac {Gm_{s}}{a_{s}^{2}}}=1.24\cdot 10^{12}$ m/s2.

## General field

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.   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: 

$~M=m_{b}-{\frac {3Gm_{b}^{2}}{10ac^{2}}}+{\frac {3q_{b}^{2}}{40\pi \varepsilon _{0}ac^{2}}}.$ Here the mass $~M$ is an invariant inertial mass of the system of a number of identical particles, which are under the action of their own four fields, $~m_{b}$ and $~q_{b}$ are the total mass and charge of all particles.

As the radius of the sphere, inside which the particles are located, decreases, the mass $~M$ remains unchanged, but $~m_{b}$ and $~q_{b}$ increase. This is associated with the fact that the velocities of the particles’ motion inside the sphere are increasing, and the mass $~m_{b}$ is increasing due to changing of the Lorentz factor. Furthermore it turns out that the mass $~m_{b}$ is equal to the gravitational mass of the system and $~m_{b}>M$ . If we assume that $~m_{b}=M+m_{f}$ , 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:

$~m_{f}={\frac {3Gm_{b}^{2}}{10ac^{2}}}-{\frac {3q_{b}^{2}}{40\pi \varepsilon _{0}ac^{2}}}.$ 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:

$~m_{f}\approx {\frac {3m_{b}r_{g}}{20a}}.$ Since for the known bodies $~a>r_{g}$ , then the mass-energy $~m_{f}$ cannot exceed 15% of the gravitational mass $~m_{b}$ .

For relativistic uniform system inertial mass is found to be:  

$~M=m_{b}-m_{f}\approx m_{b}-{\frac {0.3392Gm_{b}^{2}}{ac^{2}}}+{\frac {0.0848q_{b}^{2}}{\pi \varepsilon _{0}ac^{2}}}.$ Consequently, for the total mass-energy of general field a condition is obtained: $~m_{f}<0.17m_{b}$ .