computer

Klein–Gordon equation is very controversial. In one place of Wikipedia is writen: "related to the Schrödinger equation", in another place: "The wave function cannot therefore be interpreted as a probability amplitude. The conserved quantity is instead interpreted as electric charge, and the norm squared of the wave function is interpreted as a charge density. The equation describes all spinless particles with positive, negative, and zero charge." Completely arbitrary assumption. Other thing, there is no first time derivative, only second, so this expression is incomplete and requires intermediate values. Also can not be simulated on a computer. And its stationary state has the unique spherically symmetrical solution, like sin(k • r) / r, with some constant k and distance from center r. But all-space intregral of this function, of its square, of its first and second derivatives, and of derivatives squares is infinite, and can not be represented as finite charge or probability.

I had not "abandoned" Hypothesis about the formation of particles from fields, but at first it was about velocity fundamental field, then somehow switched to the equations of quantum physics, and I was afraid the administration of the forum would be dissatisfied. Four-vectors have too unclear notation. I try to write equations in the most direct and comprehensible way. But the biggest problem is other: long-action. Gravitoelectromagnetism (in the Wikipedia article at least) is described as follows: div E = - 4 • π • g0 • ρ div B = 0 E' = c2 • rot B - 4 • π • g0 • J B' = - rot E First two equations represent the pure long-action. Arbitrary assumption, that some "charge" or even density generates field over the whole Universe. In electromagnetism charge density is really proportional to div E and changes with E'. But in gravitation theories mass or enerdy density is "external" value to these equations.

Excuse me, studiot, I haven't understood about what thread You are writing. Probably, two weeks ago I wasn't present at this forum.

Forth dimension is the time. Such approach is convenient to write some laws briefly, including electromagnetic, but curls anyway are existing and used. Gradient and divergency defined in a world with any dimensions amount (except zero-dimensional). Both gravitoelectromagnetism and relativity (modified Poisson's equation) use mass or energy density in long-action manner. It is always laplacian or divergency of some operator, not cause of first derivative change.

All the most known theories of gravity are built on the principle of long-range action. When approximately "point" body or mass density (also electromagnetic energy) distributed in space creates gravitational potential. This article attempts to substantiate gravitational phenomena on the principle of proximity (locality). The cause of interactions is some spatial state of fundamental fields, the consequence is change of these fields over time (first derivative in time at local point of continuum). Presumably, following fundamental gravitational fields exist: (SI units in parentheses are m-metre, s-second, k-kilogram, A-Ampere) scalar potential g (m2/s2) vector potential G (m/s) scalar strain f (m2/s3) vector strain F (m/s2) The gravitational constant g0 = 6.6742^-11 (m3/s2/k) is also used, local energy density u (k/m/s2), for example electromagnetic = ε0/2 • E2 + μ0/2 • H2 and Poynting vector S (k/s3) = [E × H] Time derivatives are expressed as follows: g' = - f - c2 • div G G' = - F - grad g f' = - c2 • div grad g + fu • u F' = c2 • rot rot G - fs • S The constants fu (m3/s2/k) and fs (m/k) are positive, signs are selected so that scalar potential g becomes negative in presence of positive density u in vicinity of point. The equations are similar to electromagnetic equations expressed in potentials: a' = - c2 • div A A' = - E - grad a E' = c2 • rot rot A In stationary state, for example, during formation of gravitational fields by stable elementary particle or single celestial body: S = 0, G = 0, f = 0 F = - grad g div grad g = fu • u / c2 = 4 • π • g0 • ρ, according to Newton's potential Hence we get at ρ = u / c2: fu = 4 • π • g0 The effect of gravitational fields on other fundamental ones can manifest itself as a curvature of space, and direct effect on velocity vector V, mentioned in this topic: With zero u and S, following types of "pure" gravitational waves can exist: 1. Longitudinal potential-potential: g' = - c2 • div G, G' = - grad g 2. Longitudinal with phase shift of 90 degrees: g' = - f, f' = - c2 • div grad g 3. Transverse: g' = - c2 • div G, G' = - F - grad g, F' = c2 • rot rot G Transverse ones are probably easier to detect in experiments.

In this work are investigated details of an elementary electric and magnetic dipole radiation at distances much greater than size of the emitting element. Debatable conclusions are drawn. Formulas are given in a cylindrical coordinate system (ρ,φ,z) r2 = ρ2 + z2 Given this, it is possible to write expressions differently for ρ and z For example, 2 - 3 · ρ2 / r2 = 3 · z2 / r2 - 1 For all values ∂/∂φ = 0 (cylindrical symmetry) Time derivatives are denoted by a quote ' Electric elementary dipole Charge oscillates along z-axis near zero point with frequency ω, amplitude of dipole moment is P0. Dipole moment: Pz = P0 · cos(ω·t) Auxiliary functions: COS = cos(ω·(t - r/c)), SIN = sin(ω·(t - r/c)) Scalar potential: a = P0 / (4·π·ε0) · z / r2 · (1 / r · COS - ω/c · SIN) a' = - P0 / (4·π·ε0) · ω · z / r2 · (ω/c · COS + 1 / r · SIN) Vector potential: Az = - P0 · μ0/(4·π) · ω / r · SIN Az' = - P0 · μ0/(4·π) · ω2 / r · COS div A = ∂Az/∂z = P0 · μ0/(4·π) · ω · z / r2 · (ω/c · COS + 1 / r · SIN) a' = - c2 · div A Scalar potential gradient: ∂a/∂ρ = P0 / (4·π·ε0) · ρ · z / r3 · {(ω2/c2 - 3 / r2) · COS + ω/c · 3 / r · SIN} ∂a/∂z = P0 / (4·π·ε0) / r2 · {1 / r · (ω2/c2 · z2 + 1 - 3 · z2 / r2) · COS + ω/c · (3 · z2 / r2 - 1) · SIN} Magnetic induction: Bφ = - ∂Az/∂ρ = - P0 · μ0/(4·π) · ω · ρ / r2 · (ω/c · COS + 1 / r · SIN) Bφ' = - P0 · μ0/(4·π) · ω2 · ρ / r2 · (1 / r · COS - ω/c · SIN) Electric field: Eρ = - ∂a/∂ρ = - P0 / (4·π·ε0) · ρ · z / r3 · {(ω2/c2 - 3 / r2) · COS + ω/c · 3 / r · SIN} Ez = - Az' - ∂a/∂z = P0 / (4·π·ε0) / r · {(ω2/c2 · ρ2 / r2 - 1 / r2 + 3 · z2 / r4) · COS + ω/c / r · (1 - 3 · z2 / r2) · SIN} Electric field curl: ∂Eρ/∂z - ∂Ez/∂ρ = P0 / (4·π·ε0) · ω2/c2 · ρ / r2 · (1 / r · COS -ω/c · SIN) Bφ' = - (∂Eρ/∂z - ∂Ez/∂ρ) as it should be in equations of electromagnetic field. div E = ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z = 0 (checked) Magnetic field annular curl: Jρ = - 1/μ0 · ∂Bφ/∂z = - P0 / (4·π) · ω · ρ · z / r3 · {ω/c · 3 / r · COS - (ω2/c2 - 3 / r2) · SIN} Jz = 1 / μ0 · (∂Bφ/∂ρ + Bφ / ρ) = P0 / (4·π) · ω / r · {ω/c / r · (1 - 3 · z2 / r2) · COS - (ω2/c2 · ρ2 / r2 - 1 / r2 + 3 · z2 / r4) · SIN} Eρ' = - P0 / (4·π·ε0) · ω · ρ · z / r3 · {ω/c · 3 / r · COS - (ω2/c2 - 3 / r2) · SIN} = Jρ/ε0 Ez' = P0 / (4·π·ε0) · ω / r · {ω/c / r · (1 - 3 · z2 / r2) · COS - (ω2/c2 · ρ2 / r2 - 1 / r2 + 3 · z2 / r4) · SIN} = Jz/ε0 as it should be in equations of electromagnetic field. Magnetic dipole An annular current with small radius R changes direction according to periodic law. Magnetic moment is directed along z-axis: Mz = M0 · cos(ω·t), где M0 = π · R2 · I0, I0 is current amplitude. Auxiliary functions: COS = cos(ω·(t - r/c)), SIN = sin(ω·(t - r/c)) Vector potential: Aφ = M0 · μ0/(4·π) · ρ / r2 · (1 / r · COS - ω/c · SIN) Electric field: Eφ = - Aφ' = M0 · μ0/(4·π) · ω · ρ / r2 · (ω/c · COS + 1 / r · SIN) Eφ' = M0 · μ0/(4·π) · ω2 · ρ / r2 · (1 / r · COS - ω/c · SIN) Magnetic induction: Bρ = - ∂Aφ/∂z = - M0 · μ0/(4·π) · ρ · z / r3 · {(ω2/c2 - 3 / r2) · COS + ω/c · 3 / r · SIN} Bz = ∂Aφ/∂ρ + Aφ / ρ = M0 · μ0/(4·π) / r2 · {(ω2/c2 · ρ2 / r + 2 / r - 3 · ρ2 / r3) · COS - ω/c · (2 - 3 ·ρ2 / r2) · SIN} Bρ' = - M0 · μ0/(4·π) · ω · ρ · z / r3 · {ω/c · 3 / r · COS - (ω2/c2 - 3 / r2) · SIN} = - (- ∂Eφ/∂z) Bz' = - M0 · μ0/(4·π) · ω / r2 · {ω/c · (2 - 3 · ρ2 / r2) · COS + (ω2/c2 · ρ2 / r + 2 / r - 3 · ρ2 / r3) · SIN} = - (∂Eφ/∂ρ + Eφ / ρ) as it should be in equations of electromagnetic field. Magnetic field curl: Jφ = 1/μ0 · (∂Bρ/∂z - ∂Bz/∂p) = M0 · μ0/(4·π) · ω2/c2 · ρ / r2 · {1 / r · COS - ω/c · SIN} Eφ' = Jφ/ε0 (checked) as it should be in equations of electromagnetic field. Conclusions Although divergence of electric field div(E) is zero everywhere (charge density is zero), scalar potential is urgently needed to describe radiation of electric dipole. To express time derivative a' is required vector potential A. At long distances, there is no question of lagging potentials of forcibly oscillating system, waves must propagate "by themselves" in wave zone. It begs the conclusion that potentials are objective physical reality, fundamental fields in vacuum, and are not mathematical abstractions. To describe dipole radiation, three fundamental fields are sufficient: a' = - c2 · div A A' = - E - grad a E' = c2 · rot rot A At the same time, Laplacian div grad (a) is fundamentally different from local charge density ε0 · div E, these are different quantities. Laplacian of scalar potential can be locally not zero in electric dipole radiation, unlike divergence of electric field. Formally, both of these quantities are "conserved", since it is possible to express derivatives in time as minus divergence of some known "flow" or current. But with respect to electric dipole, laplasian of scalar potential is preserved only globally, when positive density is emitted in one direction along z-axis, in opposite direction the same modulo negative goes. It cannot be said that scalar potential has significant value only in near zone of forced generation and lagging potentials. In far wave zone, its intensity, like time derivative, decreases proportionally to 1 / r along z-axis, the same applies to its gradient in some directions (ρ · z / r3). Electric and magnetic field decrease on average with distance as 1 / r, respectively, energy density decreases as 1 / r2. That is, integral of energy density throughout space is infinite, and elementary dipoles cannot be used as basis for representing field objects with finite energy. The more time emitter works, more energy it loses with waves, without restrictions on final value.

In terms of field, rather m = E / c^2 (or u / c^2 in the notation I used). But the difference is only what value is primary. Of course, there is a deep connection between mass and energy.

It seems that the authors of such statements are referring to a "very simple" structure. For example, an electric field of the form Eρ ~ ρ / s3 , Eφ = 0 , Ez ~ z / s3 and magnetic Hρ ~ 3 · ρ · z / s5 , Hφ = 0 , Hz ~ 2 / s3 - 3 · ρ2 / s5 with some effective size parameter R, s2 = R2 + r2 because we can not divide by zero at the center if we write ρ / r3 or z / r3 But it is impossible to find detailed descriptions of experiments that would give rise to unambiguous conclusions about a very simple structure. The same goes for the "size" of electron. Electron microscopy resolution is about 10-9 m, even with advanced software processing. Estimation of 10-13 or 10-15 m arises from theoretical reasoning, as if all the mass of electron was in the squares of electric and magnetic field. A system of objects may have "excess" energy. If two electrons are forcibly approached, energy will be stored in the grown square of the electric field within and between them. Then released on repulsion and converted into excess of magnetic field square, if electrons are released.

Thanks for clarifying. It seems that due to this there is also an attraction of massive bodies, they slide into a common gravitational pit. Although inertia when a moving body is exposed to an electric or magnetic field may have a different origin. I had not seen detailed descriptions of the internal structure of particles or formulas suitable for computer simulation within the framework of quantum theories. Explanations, suitable for implementation, end with the Schrödinger equation or some of its modifications, which is really useful in the modeling of atoms and molecules. The curl of the electric field is zero in some stable formations, like electrons, where the field can be expressed as a gradient of the scalar potential. But in a radio wave the curl is not zero.

Magnetic poles exist in "large" objects, where many electrons are oriented in one direction. At the pure field level, there are only closed magnetic field lines, as in the example I gave: Hρ = 3 · ρ · z / s5 Hφ = 0 Hz = 2 / s3 - 3 · ρ2 / s5 It is in a some sense mathematically simplest dipole, with minimal integer powers of s (s2 = R2 + ρ2 + K · z2). Тhe magnetic field lines are always closed, since it is formed only by the rotor (curl) of E. Even if in the early stages of the formation of the Universe were not closed, they quickly became so, since natural fields are in motion and pass into each other. Regarding the pages of the excerpt from the book you cited, I want to note: It's not clear what mass means. If it is a measure of inertia, the ability to change direction of motion under the influence of forces, but light is deflected under the influence of huge attractions in cosmic galaxies, and photon mass is not zero. If mass has a close relationship with internal energy, it would be logical to call mass density simply a quantity u / c2. The difference between particles and waves is more technical than philosophical. It is only necessary to find mechanisms for "self-assembly" of fields into more stable formations, with sufficient for this internal energies. Although there are few truly stable particles: electrons (positrons), protons (+anti), photons and neutrinos. Muons and neutrons still have quite a long lifetime. The rest it seems to be very short-term field spikes.

studiot, if you mean "infinitely small" points-particles by sources of field, it's a simplified abstraction from electrodynamics. Of course, in the macroscopic, and even in the microscopic world sources are very important. But I talk about "pure" field theory, where is explained the very internal structure of particles. Here is more accurate level. These three vectors CAN exist everywhere, but at the time of observation all or part of them may be zero at a certain point. As an example: near the pole of a permanent magnet only the magnetic field is non-zero, E and V are equal to zero. Тhe experimental ratio E = hv is true for photons, and there is a need to find out how photons are arranged. What is "v" in this case. If we consider a monochromatic wave, it is infinite in space and carries infinite energy. Obviously, there are no such formations in the real world. If a photon has restricted dimensions and form-factors, like amplitude E0 and positive constant R in my article, it is necessary to explain, why photons are the way they are. In the real world most photons are emitted during electron transitions between energy levels, and they can "inherit" the amplitude E0 of electrons, and differ only by R, that experimenters conventionally call frequency or wavelength. Maybe I was wrong and the neutral particles I described are actually photons, not neutrinos.

Hypothesis about the formation of particles from fields The hypothesis is an extension of field theory and an attempt to explain the internal structure of elementary particles. Basic equations Presumably, in three-dimensional space there is a field formed by vectors of electric intensity E = (Ex, Ey, Ez), magnetic intensity H = (Hx, Hy, Hz), and velocity V = (Vx, Vy, Vz). Also later in this article, the vectors of electrical induction in vacuum D = ε0 · E and magnetic induction in vacuum B = μ0 · H can be used. E and H are "energy carriers" local density of energy is expressed as follows: u = ε0/2 · E2 + μ0/2 · H2 where E2 = Ex2 + Ey2 + Ez2 and H2 = Hx2 + Hy2 + Hz2 Law of energy conservation: time derivative u′ = - div W where W = (Wx, Wy, Wz) is the energy flux vector. In this case, W = [E × H] + ε0 · (E · V) · E The scalar product EV = E · V = Ex · Vx + Ey · Vy + Ez · Vz expresses the cosine of the angle between E and V. In more detail, Wx = Ey · Hz - Ez · Hy + ε0 · EV · Ex Wy = Ez · Hx - Ex · Hz + ε0 · EV · Ey Wz = Ex · Hy - Ey · Hx + ε0 · EV · Ez Respectively, div W = H · rot E - E · rot H + ε0 · E · grad EV + ε0 · EV · div E Derivatives of the magnetic and electric field by time: H′ = - 1/μ0 · rot E E′ = 1/ε0 · rot H - grad EV - V · div E In this case, div E is proportional to the local charge density q with a constant positive multiplier: q ~ div E, in the SI measurement system q = ε0 · div E. Having performed the necessary transformations, we get: u′ = ε0/2 · (2 · Ex · Ex′ + 2 · Ey · Ey′ + 2 · Ez · Ez′) + μ0/2 · (2 · Hx · Hx′ + 2 · Hy · Hy′ + 2 · Hz · Hz′) = Ex · (∂Hz/∂y - ∂Hy/∂z - ε0 · ∂EV/∂x - ε0 · Vx · div E) + Ey · (∂Hx/∂z - ∂Hz/∂x - ε0 · ∂EV/∂y - ε0 · Vy · div E) + Ez · (∂Hy/∂x - ∂Hx/∂y - ε0 · ∂EV/∂z - ε0 · Vz · div E) - Hx · (∂Ez/∂y - ∂Ey/∂z) - Hy · (∂Ex/∂z - ∂Ez/∂x) - Hz · (∂Ey/∂x - ∂Ex/∂y) = E · rot H - H · rot E - ε0 · E · grad EV - ε0 · EV · div E = - div W A term in the form of "grad EV" for E′ arises from the need to make an adequate expression of the energy conservation law, and although in the "natural" structures discussed below E is everywhere perpendicular to V, that is, EV = 0, it can play a role in maintaining the stability of field formations. Velocity derivative by time From the point of the energy-flux view, the time derivative V′ can be any expression, but should not contain a common multiplier V or 1 - V2/c2, since when approaching zero or the speed of light, the vector would practically cease to change locally, which contradicts many experimental facts and theoretical studies. The most likely are the two-membered constituents for V′, where one part contains V as a multiplier in the scalar or vector product, the second does not. For example, the pure field similarity of the Lorentz forces is of interest: V′ ~ (D · V2 - [H × V]) · div E where V2 = Vx2 + Vy2 + Vz2 The expressions D · V2 and H × V have the same dimension, A/s in SI, and after multiplying by the div E, it is still necessary to enter a coefficient to convert the resulting units into acceleration m/s2. The numerical value of the coefficient will probably have to be determined experimentally. Although there are no strict restrictions on the absolute value of V, as we shall see later, for field formations common in nature, it is uncharacteristically |V| > c, and the speed of light is achieved at a mutually perpendicular arrangement of E, H, and V, when the local "E-energy" is equal to "H-energy", that is, E2 ~ 1/ε0, H2 ~ 1/μ0. The exception is artificially created or simulated on the computer situations. Another hypothetical set of terms for the velocity derivative over time is V′ ~ W - u · V. In the models of particles discussed below, in this case, there is a "longitudinal" effect on the velocity vector, in contrast to the "transverse" one under the influence of an electric and magnetic field, with the mutual perpendicularity of all three vectors. If indeed V′ ~ W - u · V, then although there is still no hard limit |V| ≤ c, the unlimited increase of the velocity in the absolute value is more explicitly limited by the member u · V with a negative sign. If the magnetic or electric field somehow disappears, the velocity will rush to zero, although the energy density may remain non-zero. Modulus of V reaches its maximum value (= c) when E and H are perpendicular and ε0/2 · E2 = μ0/2 · H2. When the charged particle is in an external electric field, like created by another particle in the vicinity, due to the multiplier V2 in the expression V′ ~ (D · V2 - [H × V]) · div E is independent of the sign of V, and the presence of significant velocities close to the speed of light inside the particle, the total acceleration acts in one direction (on average, although internal deformations may occur). In an external magnetic field the velocity vector is involved in the first degree, in any projection about half of the currents are directed in one direction, and about half in the opposite direction, so only internal deformations occur. The shift of a particle as a whole is observed when it moves in an external magnetic field. Let us consider the alleged structure of some elementary particles. To do this, we will use a cylindrical coordinate system (ρ,φ,z), where ρ2 = x2 + y2, φ is the angle counted from the positive direction of the x-axis counterclockwise if it is directed to the right, the y-axis upwards, and the z-axis is directed towards us (the right coordinate system). Also, for the particles under consideration, we will set the condition of cylindrical symmetry, that is, ∂/∂φ = 0 for any variables. Neutral particles moving at the speed of light Consider the motion of an "almost point" charge, spherically symmetrical at rest, along the z-axis with a constant velocity V = Vz. If the field structure moves as a whole unit, then ∂/∂t = - Vz · ∂/∂z for all variables. As mentioned earlier, with cylindrical symmetry, ∂/∂φ = 0. Let the scalar potential at rest to be a = E0 / s, where s2 = R2 + ρ2 + K · z2 Here, the letter "a" is used to avoid confusion with the angle φ, the constant E0 expresses the amplitude of the field, the constant R characteristic dimensions (similarity of the wavelength or frequency of the wave), the constant K shows the likely deformation of the field along the axis of motion, as "compression" or "stretching" of the lines of force. Note that ∂s/∂ρ = ρ / s, ∂s/∂z = K · z / s The vector potential A = Az = E0 / s · Vz / c2 is directed along the z-axis, Aρ = 0, Aφ = 0. H = 1/μ0 · rot A forms closed rings around z-axis, Hρ = 1/μ0 · (- ∂Aφ/∂z) = 0 Hφ = 1/μ0 · (∂Aρ/∂z - ∂Az/∂ρ) = E0/μ0 · Vz / c2 · ρ / s3 = E0·ε0 · Vz · ρ / s3 Hz = 1/μ0 · (∂Aφ/∂ρ + Aφ / ρ) = 0, since ε0 · μ0 = 1 / c2. Let be d = 1/μ0 · div A = 1/μ0 · ∂Az/∂z = - E0/μ0 · Vz / c2 · K · z / s3 = - E0·ε0 · Vz · K · z / s3 Obviously a′ = - 1/ε0 · d = - c2 · div A, since a′ = - Vz · ∂a/∂z = E0 · Vz · K · z / s3 that corresponds to the classical field equations. E we will find from the field equation A′ = - E - G where G = grad a Gρ = ∂a/∂ρ = - E0 · ρ / s3 Gφ = 0 Gz = ∂a/∂z = - E0 · K · z / s3 Eρ = - Aρ′ - Gρ = Vz · ∂Aρ/∂z - Gρ = E0 · ρ / s3 Eφ = 0 Ez = - Az′ - Gz = Vz · ∂Az/∂z - Gz = E0 · (1 - Vz2 / c2) · K · z / s3 As can be seen, when Vz raises to the speed of light, Ez "disappears" and only the non-zero radial Eρ, perpendicular to the z-axis, remains. The time derivatives G′, d′, H′ are computed trivially, and the time derivative E′ requires special attention. Let be J = rot H, Jρ = - ∂Hφ/∂z = E0·ε0 · 3 · Vz · K · ρ · z / s5 Jφ = 0 Jz = ∂Hφ/∂ρ + Hφ / ρ = E0·ε0 · Vz · (2 / s3 - 3 · ρ2 / s5) div E = ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z = E0· {2 / s3 - 3 · ρ2 / s5 + (1 - Vz 2 / c2) · K · (1 / s3 - 3 · K · z2 / s5)} Must be observed equality Eρ′ = - Vz · ∂Eρ/∂z = E0 · 3 · Vz · K · ρ · z / s5 According to a hypothetical equation, and since E is perpendicular to V (E · V = 0), and Vρ = 0: Eρ′ = 1/ε0 · Jρ - ∂EV/∂ρ - Vρ · div E = 1/ε0 · E0·ε0 · 3 · Vz · K · ρ · z / s5 = E0 · 3 · Vz · K · ρ · z / s5 Also must be observed equality Ez′ = - Vz · ∂Ez/∂z = - E0 · Vz · (1 - Vz2 / c2) · K · (1 / s3 - 3 · K · z2 / s5) According to the hypothetical equation, and since E is perpendicular to V (EV = 0): Ez′ = 1/ε0 · Jz - ∂EV/∂z - Vz · div E = 1/ε0 · E0·ε0 · Vz · (2 / s3 - 3 · ρ2 / s5) - Vz · E0 · {2 / s3 - 3 · ρ2 / s5 + (1 - Vz 2 / c2) · K · (1 / s3 - 3 · K · z2 / s5)} = - E0 · Vz · (1 - Vz 2 / c2) · K · (1 / s3 - 3 · K · z2 / s5) The equations for E′ are true for any Vz, E0, R and K, but if Vz = c and Ez = 0, the all-space integral of div E multiplied by volume unit is zero: ∫-∞+∞∫02·π∫0∞ (2 / s3 - 3 · ρ2 / s5) · ρ ∂ρ ∂φ ∂z = 0 That is, when accelerating to the speed of light, the whole field formation will be charged neutrally, although locally the charge density changes. Probably a similar structure, relatively simple, have neutrinos, and we will also use the given example of the direct motion of a particle to check the adequacy of expressions for V′. If V′ ~ (D · V2 - [H × V]) · div E then Vρ′ ~ (ε0 · Eρ · Vz2 - Hφ · Vz) · div E = (ε0 · E0 · ρ / s3 · Vz2 - E0·ε0 · Vz · ρ / s3 · Vz) · div E = 0 and the radial velocity remains zero. If D · div E would be used in the "similarity of the Lorentz forces" without multiplication by V2, like the classical effect of an electric field on charge, equality would not be observed. Also Vz′ ~ (ε0 · Ez · Vz2 + Hφ · Vρ) · div E = {ε0 · E0 · (1 - Vz2 / c2) · K · z / s3 · Vz2} · div E Vz′ = 0 and the velocity Vz remains constant only at Vz = c. Apparently, this is related to the fact that experimentally detected neutrino-like particles move at the speed of light, while the movement of massive particles at low speeds is a much more complex process at the field level. When Vz = c and the electrical energy density uE = ε0/2 · E2 is equal to the magnetic energy density uH = μ0/2 · H2, the equality W = u · V is also fulfilled, which is an argument in favor of considering the hypothesis V′ ~ W - u · V. The same fact occurs in field structures with zero electric field divergence (electromagnetic waves and presumably photons). Stable charged particles with cylindrical symmetry Probably, the basis of electrons and other leptons is the E, H, V field in the form of closed rings of energy flow and velocity vector. The structure of the particle is not similar to the classical "infinitely thin" circuit with an electric current, where on the elements of the ring the electric and magnetic field differ from zero, due to the fields created by other parts of the circular current. On the line of "main circuit", the electric and magnetic field is zero, whereas the charge density (~ div E) is close to the local maximum. In computer modeling in a cylindrical coordinate system, the following field values can be a good initial approximation (with s2 = R2 + ρ2 + z2😞 vector potential A = Aφ ~ ρ3 / s5 Hρ ~ - ∂Aφ/∂z ~ 5 · ρ3 · z / s7 Hφ = 0 Hz ~ ∂Aφ/∂ρ + Aφ / ρ ~ 4 · ρ 2 / s5 - 5 · ρ 4 / s7 Jρ = 0 Jφ = ∂Hρ/∂z - ∂Hz/∂ρ ~ - 8 · ρ / s5 + 10 · ρ3 / s7 + 35 · ρ3 · R2 / s9 Jz = 0 scalar potential a ~ ρ 4 / s5 Eρ ~ - ∂a/∂ρ ~ - 4 · ρ 3 / s5 + 5 · ρ 5 / s7 Eφ = 0 Ez ~ - ∂a/∂z ~ 5 · ρ 4 · z / s7 div E = ∂Eρ/∂ρ + Eρ / ρ + ∂Ez/∂z ~ - 16 · ρ2 / s5 + 20 · ρ4 / s7 + 35 · ρ4 · R2 / s9 Note that Eρ · Hρ + Ez · Hz = 0, E is perpendicular to H everywhere. Near the center of the particle is a region where the values div E and rot H are opposite in sign to those found in the rest of space. Meanwhile, V has the same sign everywhere. If the conditional magnetic dipole is directed along the z-axis, with a positive multiplier for A and H, and the total charge of the particle is positive, then near the center there will be a region with a negative divergence E and a negative rotor H, but at the great distance these values are positive. The velocity is positive everywhere, that is, it is directed counterclockwise with the direction of the z-axis towards us and the x-axis (the start of the counting φ) to the right. div E and rot H must change the sign synchronously so that equality is observed: E′ = 1/ε0 · rot H - grad EV - V · div E = 0 since in a more or less stable particle all fields derivatives in time are zero. EV = 0, V is perpendicular to E and H, that is, we are talking about a mutually perpendicular triple of vectors in any combination. A model with a field arrangement closer to the z-axis, for example: A = Aφ ~ ρ / s3 Hρ ~ 3 · ρ · z / s5 Hφ = 0 Hz ~ 2 / s3 - 3 · ρ2 / s5 a ~ ρ2 / s3 Eρ ~ - 2 · ρ / s3 + 3 · ρ3 / s5 Eφ = 0 Ez ~ 3 · ρ 2 · z / s5 where on the z-axis there is a pronounced maximum of Hz and J = Jφ = ∂Hρ/∂z - ∂Hz/∂ρ ~ 15 · R2 · ρ / s7 does not fit, because rot H is positive everywhere, and div E changes sign in the central part. Models with a spherically symmetric scalar potential and electric field are even more inadequate: a ~ 1 / s Eρ ~ ρ / s3 Eφ = 0 Ez ~ z / s3 div E ~ 3 · R2 / s5 there E is not even perpendicular to H, if we take A and H from the previous model. If V′ ~ (D · V2 - [H × V]) · div E then the multiplier V2 prevents the destruction of the particle due to electrical repulsion near the z-axis, although exactly onto it can be also D = 0. With V = Vφ, Vρ = 0 and Vz = 0, there must inevitably be a zone with zero velocity near the z-axis, since the values of the existing quantities in the physical world are finite and only smooth functions with continuous derivatives are permissible. Precise solutions in degrees of s for real lepton-like particles seem impossible, numerical simulations are required. In the first time after setting the initial approximation, occurs a rapid adaptation of the fields to more accurate values, further stability depends on the adequacy of the model and the accumulation of numerical errors. Let's try to estimate the magnitudes of what order are the fields at a considerable distance from the z-axis in the perpendicular plane (z = 0, s ≈ r, r2 = ρ2 + z2). From experimental data and works on classical physics it is known that E = Eρ ~ 1 / r2, H = Hz ~ - 1 / r3, u ≈ E2 ~ 1 / r4 D · V2 - [H × V] will tend to zero if V = Vφ ~ 1 / r The same order of Vφ ~ 1 / r follows from the equation W - u · V = 0, given that W = Wφ = - Eρ · Hz ~ 1 / r5 Further directions of research All the above mentioned equations are linear in E and H. That is, when these vectors are both multiplied by the same number, they remain true. Since the particles observed in the physical world have strictly defined charges and masses (internal energies), it is logical to assume that the expression V′ may contain nonlinear terms with relatively small factors. Although due to statistical factors, greater resistance to random disturbances, some field formations may be more stable than others. Excuse me, I thought the nice HTML-file here would be turned into plain text, so attached it. The main differences from the more classical approach are: 1) Velocity vector V had introduced as an "independent" and essential part of field, as physical reality along with E and H 2) The expression for the energy flux W was changed, with an additional member ε0 · (E · V) · E and I hope someone will conduct experiments to confirm or reverse that assumption 3) There are non-trivial suggestions, what "forces" may affect the velocity derivative over time 4) Assumptions about the internal structure of electron-like particles, that inside the main part with a predominant charge is a zone with a charge of the opposite sign, and due to what factors it arises

I want to present my hypothesis for your consideration ehveng.htm

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