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Gravitation theory of elementary particles

It is a translation of article “Теория гравитации элементарных частиц” this website with the help the www.translate.ru resource Internet.
I tried to do my best with this text. May be my English is not very good. When I had the possibility to improve my English text, I would replace it.

Contents:

1. Introduction to the gravitation theory of elementary particles
1.1. The base of the gravitation theory of elementary particles
1.2. The gravitational field created by elementary particles
1.3. Choice of units system
1.4. Introduction to the gravitation theory of elementary particles - a result
2. The gravitational field of the elementary particle with a quantum number L>0 (except a photon)
3. The gravitational field of an annular domain (elementary particle)
4. The gravitational field created by an annular domain of the free not moving elementary particle beyond its limits
5. Components of the gravitational field created by an annular domain of the free not moving elementary particle beyond its limits
6. The gravity potential of the field created by an annular domain of the free not moving elementary particle beyond its limits
7. Nature of inertial properties of the elementary particles (electromagnetic nature)
8. The field of a gravitational dipole
8.1. The field of the rotating gravitational dipole, its gravity waves
8.2. Field of an oscillating gravitational dipole, its gravity waves
9. The gravity waves created by thermal agitation a crystal lattice of atoms
10. The gravity waves of the elementary particles
11. Gravitational waves, black holes, and LIGO (Laser Interferometer Gravitational-Wave Observatory).

1. Introduction to the gravitation theory of elementary particles

(to contents)

Physics of the 20th century, studying elementary particles, established existence of the electromagnetic fields in them:

• the constant electric field,
• the constant magnetic field,
• the electromagnetic field (is variable).

1.1. The base of the gravitation theory of elementary particles

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In the base of the gravitation theory of elementary particles are:

• Law of universal gravitation,
• Classical electrodynamics,
• Einstein's formula,
• Field theory of elementary particles,
• Geometry laws,
• The law of conservation of energy, etc.

1.2. The gravitational field created by elementary particles

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So, let we have an elementary particle with the electromagnetic fields which are (listed above) and a gravitational field. The fantastic fields which are thought up by fantastic theories we will leave them to authors.
According to classical electrodynamics, for this electromagnetic field it is possible to write down the energy density equations. Further calculations will be kept in the Gaussian unit system accepted in classical electrodynamics and convenient for understanding of physics. The interested persons can transfer the equations in to SI independently.
Density of the electric field energy (both constant and variable) will be:

(I-1)

1.3. The choice of the units system

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When I on the website “Wikiznanie” in the article “Intensity of a Gravitational Field”, the section “Intensity of a Gravitational Field and Field Theory of Elementary Particles” have given only four formulas from given above (No.: I-7, I-10, I-11, I-12), I have received the following:
“To attention of experts: formulas (...) aren't true, correction is required”.
“The SI system electric field (E) has dimension of volts on meter, and intensity of a magnetic field (H) - amperes on meter. And you mix them in one line so as if it is sizes of one physical dimension. Correct please. Fedosin”
It is about the equations following from the equation (I-3) - consequences of the law of conservation of energy.
So what should be trusted: to the fundamental law of the nature - to the Law of conservation of energy or the International units system (SI) accepted now? Which of them misleads?

According to the Law of conservation of energy, in the equation (I-3) we have put density of two types of electromagnetic energy - and the energies are summarize. If we multiply each of equation elements by identical coefficient of dv, dimensional volume - we will receive addition of two types of electromagnetic energy: electric and magnetic:

(I-13)

1.4. Introduction to the gravitation theory of elementary particles - a result

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I, extremely in detail, have proved the equations of the gravitation theory of elementary particles. All this is obvious and elementary things. Any, knowing physics, can be convinced in truth of the presented equations.

Gravitation theory of elementary particles

As all bodies in the Universe consist of elementary particles, except of course fantastic (dark matter and other representatives of mathematical fairy tales), conclusions from this theory will have global value. The developed theory isn’t relativistic as in elementary particles the special theory of relativity (STR) doesn’t work.

2. The gravitational field of an elementary particle with quantum number L>0 (except a photon)

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The field theory of elementary particles, having established an electromagnetic structure of elementary particles, I have determined by that the gravitation nature as ability of the electromagnetic fields energy to an attraction.

According to the law of universal gravitation, classical electrodynamics, the field theory of elementary particles (in abbreviated form the field theory) and Einstein’s formula, a gravitational field intensity of the based elementary particle is equal:

(1)

where G - is a gravitational constant, E - is the electric field intensity of an elementary particle (both constant, and variable), H - is the magnetic field intensity of an elementary particle (both constant, and variable), with - the velocity of light, R - is the distance from a supervision point to a point of integration, n - is a single vector from a supervision point in an integration point.

Lack of anti-gravitation in the nature follows from the equation (1). By what don’t do E or H, and the sum of their squares will ALWAYS be positive, except for of course fantastic “theories”, but to the physicist those don’t interest.

But in practice, it is difficult to use the equation (1) as it is required to know the sizes E and H in each point of space. The field theory helps to overcome this difficulty. We will start.

Let us should solve a problem of finding of the gravitational field intensity created by the based elementary particle. About an elementary particle are known: the size of the rest mass with a set of quantum numbers and other characteristics, location of the center, orientation a back.

We will place the beginning of coordinates in the center of an elementary particle, and we will direct axis Z along a vector a back. Then, according to the field theory of elementary particles, the ring area of rotation of the variable electromagnetic field of an elementary particle will lie in the plane (X, Y), and the center will coincide with the beginning of coordinates. Coordinate (Z) will mean height of an arrangement of a required point over the plane (or under the plane when Z is less than zero). The weight which is contained in the variable electromagnetic field will rotate on the average radius of r_0~, determined by the field theory. As the radius of rotation is replaced with an average, we can instead of volume distribution of substance in ring area use linear, having avoided, on the one hand, difficult integrals, and on the other hand, considerable losses of accuracy, at determination intensity of a gravitational field outside ring area.
So, let we have a thin material thread, the mass of m_0~ and radius r_0~, placed in the plane (X, Y) with the center at the beginning of coordinates (see rice 1). We will consider other part of the rest mass (m₀) of an elementary particle (we will call it m₀₌) later. In a distant zone of weight will develop and will yield right result, and in a near zone it will allow to minimize a deviation (from the real field).


Fig. 1 Approximation of ring area of an elementary particle

According to the field theory of elementary particles:

(2)

where L - is the main quantum number of an elementary particle in the field theory, ħ - is the Plank constant divided on 2π, k_~ - is the coefficient specifying what part of total internal energy of the based free elementary particle is concentrated in her variable electromagnetic field.

(3)

(4)

Where m₀₌ - is the weight size concluded in constant fields (electric and magnetic) the based free elementary particle. For the elementary particle which is a part of an atomic nucleus, this size will be size defects less than the mass of a particle, the corresponding binding energy of a kernel.

The linear substance density (σ) in ring area of an elementary particle:

(5)

We will determine coordinates of the integration point of ring area of an elementary particle

(6)

(7)

where ϕ - is a corner between axis X and a vector from a point of the beginning of coordinates in integration point (x₀, y₀, 0).

Distance from a point (x₀, y₀, 0) to a point (x, y, z) will be:

(8)

The segment mass of ring area is creating a gravitational field will be:

(9)

And the size of the gravitational field intensity created by him in a point (x, y, z):

(10)

For further integration we will spread out dГ_~ vector to components. dГ_~ projection on the plane (X, Y) it is equal:

(11)

DГ_~ projection on axis Z:

(12)

DГ_~ projection on axis X:

(13)

DГ_~ projection on axis Y:

(14)

Now it is possible to carry out integration:

(15)

(16)

(17)

We introduce one more designation. The ring area radius of an elementary particle:

(18)

We will note that r is a distance from the center of ring area to a point in which intensity of the field (x, y, z), is defined

The equations (15)-(17) are fair outside the ring area (when r > r_~) creating the basic (more than 90%) a gravitational field of an elementary particle, limited to r_~ radius (fig. 2 see). But there is still an area of space in ring area (when r < r_~) creating a gravitational field. We will consider as well this area of space.
So, we look for intensity of the gravitational field created by substance with an average ρ_0~ density in a ring with r_~ radius. In fig. 2 the cross section of an elementary particle is displayed (in the plane (X, Z)) when the plane is carried out through a supervision point. As the ring area of an elementary particle has the central symmetry, and also symmetry in relation to axis Z, we can use it and simplify a task. We will solve a problem on the example of protons and neutrons as in them more than 99% of mass of substance are concentrated.


Fig. 2 the Cross section of an elementary particle, with quantum number L=3/2 (a proton, a neutron)


Fig. 3 Ring area of an elementary particle, with quantum number L=3/2 (a proton, a neutron). The top view which is a little reduced.

According to the second of the Pappus - Guldin theorems, the volume of the geometrical figure formed by rotation of a circle with r_~ radius around the center of an elementary particle will be:

(19)

From here it is possible to determine average density (ρ_0~) as the relation of the weight concentrated in the variable electromagnetic field of an elementary particle to the volume occupied by her:

(20)

According to the first theorem of the Pappus-Guldin theorems, the surface area of rotation is equal to the length work of the rotating line at length of a circle as which radius serves the distance from an axis to a line barycenter. For the gravitational field created by the variable electromagnetic field of an elementary particle, length of the line will be equal in our case 2πr_~, and circle length 2πr_0~.

(21)

Now it is possible to define the gravitational field intensity arising on a surface of the area occupied by the rotating variable electromagnetic field she will be:

(22)

The vector will be perpendicular surfaces and is directed to a source. As we see, the size of the gravitational field intensity created by the variable electromagnetic field of an elementary particle on a surface of ring area in which the field rotates is proportional to a cube of size of the rest mass.

Having substituted in (22) known the rest mass of a proton, the calculated parameter k_~ (0.9221), quantum number L (3/2), and others, we will receive the size approximately equal to the maximum intensity of a gravitational field of the free based proton:

At a neutron a little big size will turn out: 9.5 • 10^-7 m/sec², and at the based electron: 4.8 • 10^-16 m/sec². Exact values can be defined by a formula (45), they will differ from presented less than for 1 percent. I hope, the level of influence of gravitation on a microcosm is well visible.
For the determination of the gravitational field intensity in ring area of an elementary particle (see fig. 2), we will use the Pappus-Guldin theorems again. The volume of the geometrical figure formed by rotation of a circle with r radius (smaller, than r_~) around the center of an elementary particle, will be:

(23)

and her surface area:

(24)

The average weight concluded in volume of V_~r will be:

(25)

From where, the gravitational field intensity arising on a surface of a geometrical figure will be:

(26)

where n - is a single vector, perpendicular surfaces of a geometrical figure.

The vector is perpendicular surfaces and is directed to a source. As we see, at aspiration of r to zero, the gravitational field intensity created by the variable electromagnetic field of an elementary particle will aspire to zero.

The elementary particle has one more source of a gravitational field: m₀₌ - is the weight concluded in constant electric and magnetic fields of an elementary particle. We will divide her into three parts:

where:

• m₌₀ - is the mass of constant fields of an elementary particle concluded in ring area (rotations of the variable electromagnetic field;
• m_=e - is the weight concluded in external constant electric field of an elementary particle;
• m_=m - is the weight concluded in an external constant magnetic field of an elementary particle (as show calculations of a proton and neutron, creating the main gravitational fields of substance, for these particles of m_=m >> m_=e).

According to classical electrodynamics and Einstein’s formula, m₌₀ it is equal:

(27)

Where E_m - that is the maximum intensity of constant electric field of an elementary particle on a surface of ring area. For a constant magnetic field of the loaded elementary particles it is possible to enter similar to H_m and to simplify expression, but for a magnetic field of neutral elementary particles it won’t turn out - there more difficult structure of the field.

For simplicity of record of the equations, we introduce one more coefficient:

(28)

where k₌₀ - the coefficient (size about 0.001) indicating the part of the rest mass of an elementary particle concentrated in ring area in a constant electric and constant magnetic field.

For the gravitational field created by this part of mass of a constant electric and magnetic field we will receive the result similar (15)-(17),

(29)

(30)

(31)

Similarly in a near zone it is necessary in (26) to replace one m_0~ with m_=o:

(32)

We can put the similar equations and we will receive expressions for part of the rest mass

The sizes m_=e and m_=m can be determined, knowing in each point of space of the size E and H (for this purpose it is necessary to use the field theory of elementary particles).

(33)

(34)

But for definition of the rest of a gravitational field of an elementary particle, it isn’t obligatory to do it. It is possible to enter density of electromagnetic substance (ρ₌) the sums constant of electric and magnetic fields

(35)

Then, according to the law of universal gravitation, we will receive a vector of the gravitational field intensity created ρ₌:

(36)

where (x₀, y₀, z₀) - coordinates of a point of a source, the considered gravitational field, (x, y, z) - supervision point coordinates.

But it are required for us making Г₌. The size of a projection of Г₌ on the plane (X, Y) it is equal:

(37)

Projection of Г₌ on axis Z:

(38)

Projection of Г₌ on axis X:

(39)

Projection of Г₌ on axis Y:

(40)

It is time to put components of vectors of the gravitational field intensity. For a gravitational field of an elementary particle, outside ring area we will have:

(41)

(42)

(43)

And the size of a vector of the gravitational field intensity will be:

(44)

For a gravitational field of an elementary particle, in ring area we will have:

(45)

The mathematical expressions having a little general with a formula of the gravitational field intensity of dot gravitational object have turned out:

(46)

on the basis of which, many mathematical fairy tales have been created.

If in the equations (41)-(43) to put r₀=0 (field of the dot object which is not existing in the nature), then the semblance will turn out (46):

(47)

But it is only part of a gravitational field. There still will be a gravitational field created by a constant electric and constant magnetic field of an elementary particle out of ring area, and this field extends in infinity, under laws of classical electrodynamics.

Now we will deal with symmetry which is possessed by a gravitational field of an elementary particle.

The formulas 41 - 43 follows from the equations that the gravitational field of an elementary particle won’t change if to change particle backs for opposite. For elementary particles with zero to backs it belongs to change of the direction of a vector of the internal rotary moment, on opposite. At any other change of orientation of an elementary particle also her gravitational field will change. Therefore: the gravitational field of an elementary particle doesn’t possess spherical symmetry.
But the gravitational field of an elementary particle remains symmetric concerning the straight line passing through the center of an elementary particle and perpendicular plane of rotation of the variable electromagnetic field (parallel or anti-parallel to a vector of the internal rotary moment). I.e. the size of the gravitational field intensity of an elementary particle is identical to all points equidistant from the center of an elementary particle and, along with it, equidistant from a symmetry axis. - The structure of elementary particles is reflected in structure of their fields.

It is possible to invent many mathematically beautiful theoretical constructions, but the gravitational field of substance is created by elementary particles of which it finally consists.

3. The gravitational field of ring area

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We will consider a case of some distribution of weight in ring area of the variable electromagnetic field (other than constant density). Let density will be equal ρ_~(x).
Then the weight size (dm_~r) which is contained in a ring segment of radius x and dx thickness

(48)

In this case, the weight concluded in volume of V_~r will be:

(49)

Then the equation (26) will correspond, and the gravitational field intensity arising on a surface of a geometrical figure will be:

(50)

We have received replacement of the equation (26) for a case of some distribution of the substance density in the variable electromagnetic field of an elementary particle, in ring area. In this case the equation (45) for components of the gravitational field intensity of an elementary particle created by electromagnetic fields of internal area of radius of r on a surface of a geometrical figure in ring area will also correspond:

(51)

and

(52)

where E_~ - is the intensity of variable electric field of an elementary particle, H_~ - is the intensity of a variable magnetic field of an elementary particle.
Isn’t necessary on the equation (51), seeing 1/r, to cost any fantastic theories about "strong" gravitational fields in elementary particles. Here just the physics and when you substitutes distribution (52) in integral works - euphoria will thaw, and the gravitational field will become just gravitational field again.

Now there is a little mathematics.
We will allocate from the equation (26) m_~r/S_~r and we will paint it.

(53)

where S_r - is the area of a circle of radius of r, l_r - is the circle length of r radius. We won’t reduce it further, and we will stop on it. Then the equation (26) can be rewritten:

(54)

Persons interested can be convinced (having painted further) that

(55)

Where the r-vector is directed to a source of a gravitational field, as well as is the n-vector.

I.e. the Pappus - Guldin theorem having reduced circle length, I have transferred a problem of finding of a gravitational field in ring area of an elementary particle from three-dimensional option to two-dimensional option that simplifies integrals and facilitates her decision.

Now we will solve one mathematical problem. Let us have a thin ring, thickness of dr, average radius r₀ and mass of m_k. We will look what gravitational field it creates outside and inside. We will place a ring in the plane (X, Y) with the center at the beginning of coordinates (see fig. 4).

Fig. 4 Thin ring
where:

(56)

(57)

(58)

(59)

(60)

(61)

(62)

r₁ and r₂ - are the distances from dm_k to x₁ and x₂ respectively, dϕ - is the angular size of the considered segment.
The size of the gravitational field intensity created by the dm_k segment in two-dimensional space in a point (x, 0) will be:

(63)

dГ projection to axis X for points of x₁ and x₂, will be:

(64)

And the projection to axis X of the gravitational field intensity created by a thin ring will be:

(65)

In a point of x_i=r₀ the difference of two infinity - not a physical case turns out, in the nature there are no infinitely big fields.
We introduce designations:

(66)

(67)

The first integral can be transformed and reduced to the second:

(68)

Then contents of a square bracket of the equation (65) will correspond:

(69)

At long distances when x_i > > r₀ turns out:

(70)

as had to be for a gravitational field of the infinitely long cylinder. The second member of a square bracket gives-π/x_i on infinity and that will be in a near zone, at small distances - we will look.

(71)

We will paint it

(72)

(73)

Then:

(74)

(75)

We will substitute it in (71), we will receive:

(76)

Then the equation (69) will correspond:

(77)

If to substitute in him ϕ=2π, we will receive that intensity of a gravitational field in a ring (in any point) is equal to zero, and outside of a ring is equal - Gm_k/R. The thin material ring, dr thickness, has created a gravitational field, intensity Г_xi in each point (i) of space, outside a ring. The relation of the size change of the gravitational field intensity to the size of distance on which this change has happened is called a gradient. In our case (for a source of a gravitational field), we will have:

(78)

We will paint m_k. We introduce substance density (σ). Then:

(79)

Then (78) will correspond:

(80)

I.e. the gravitational field intensity in a source of gravitation is created by the substance density (his electromagnetic field). In three-dimensional space it will be similar, only 2π (a full corner of a circle) it will be replaced on 4π (a full space angle of the sphere), and area density (σ) will be replaced with volume density (ρ).
But the gradient from mathematical function is equal to her first derivative, and then the gradient of the gravitational field intensity in ring area of an elementary particle (a scope of the equation 51) will be:

(81)

In the equation (81) the last two (negative) members point to sources of gravitation of an elementary particle (the variable electromagnetic field and a constant electric and constant magnetic field), and the first (positive) member reflects feature of distribution of the field in space, ρ_~ - is defined in (52).

In the nature there are no dot sources of gravitation - the gravitational field is created by the electromagnetic field distributed in space. And as electromagnetic fields submit to laws of electromagnetism, the same laws also influence structure of the created gravitational field. Having placed an elementary particle in a constant external electric or magnetic field, we will change structure of her constant electromagnetic fields and consequently and the gravitational field created by them.
In the nature there is also no uniform substance. Each atom of substance consists of a massive kernel and electrons of a cover, and each kernel has a certain structure and consists of the protons with neutrons which aren’t spherical educations. Having evenly smeared atom on the space occupied by it, we lose a gravitational field of a center (which energy is the main part of energy of a gravitational field of elementary particles of atom), and mathematical fairy tales begin then.

4. The gravitational field created by an annular domain of the free not moving elementary particle beyond its limits

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Thanks to symmetry of a gravitational field, equation (41) - (42) it is possible to simplify. We introduce new designation:

(82)

5. Components of the gravitational field created by ring area of the free not moving elementary particle beyond her limits

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We will define intensity of a gravitational field in the plane of an elementary particle. The equation (86) will give zero, and here the equation (85) will yield more interesting result.

(93)

If in denominators of integrals in (93) it was possible to replace sin ϕ with its average value in the range of 0 - 2π (i.e. on zero), then integrals sharply would become simpler:

(94)

where R - is distance from a supervision point to the center of an elementary particle, and r_0~ - is the radius of an elementary particle in the field theory, is defined in (2). I remind that a certain integral from a sine in the range of 0 - 2π gives zero.

We have received the expression similar (89) saying that in a gravitational field of an elementary particle there is spherically a symmetric component. The most interesting in (94) is the fraction denominator at which there is a square of radius of an elementary particle from the field theory of elementary particles (lying in the base of the developed theory of gravitation together with Einstein’s formula and the law of universal gravitation) - the nature limits a gravitational field.

We introduce new designation:

(95)

For a gravitational field outside an elementary particle β will be less than 1, and at long distances to aspire in zero. Then the equation (93) will correspond:

(96)

The uncertain integral from the sub integral function similar (96), according to the Internet to a resource (http://matematikam.ru/), has the following decision:

(97)

The equation (96) can be painted, it will turn out:

(96-2)

Can seem that the analog of the equation (46) has turned out, but it is illusion: in everyone β, it agrees (95), there is R and rejection of this fact leads to mathematical manipulations over laws of the nature.

In figure 5 the family of curves of sub integral function (97) is presented, at the fixed values β (0.5, 0.25, 0.1, 0.05, 0.025, 0.01), received with the help the Internet of a resource (http://www.yotx.ru/). To bigger value β there is corresponds the big size of a maximum on graphics.

Fig. 5 is the family of curves of function (97) in the range of integration of 0 - 2π.

As we see, a certain integral from any of curves will give the value exceeding 2π that together with (89) tells about existence in a gravitational field of an elementary particle spherically symmetric component, defined:

(98)

Even for spherically components of a gravitational field of an elementary particle, the mathematical expression differing from (46) has turned out symmetric.
Besides, in a gravitational field of an elementary particle is available also asymmetric a component, determined by structure of her electromagnetic sexes which are defined as:

(99)

(100)

(101)

6. The gravity potential of the field created by ring area of the free not moving elementary particle beyond her limits

(to contents)

In physics there is such concept as gravitational potential (ϕ): “the scalar function of coordinates and time characterizing a gravitational field in classical mechanics equal to the relation of potential energy of a material point (in a gravitational field) to the mass of this point”. Knowing intensity of a gravitational field in any his point, it is possible to determine also the gravitational potential of some point of the field as:

(102)

7. Nature of inertial properties of elementary particles

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The field theory of elementary particles, having established an electromagnetic structure of elementary particles, I have determined by that the gravitation nature, having confirmed existence in the nature of an electromagnetic form of gravitation. But if electromagnetic matter creates around itself gravitation, i.e. electromagnetic weight is gravitational, and then there is one more question: whether electromagnetic weight is a source of inertia of elementary particles. We will deal with this question.

Let in some elementary volume of space of dv there is uniform electric field intensity of E.
According to Classical electrodynamics, his energy will be:

(107)