5. Derivation and interpretation

Vortices cause big problems to every measuring technician. They have the unpleasant

property to whirl around the sensor even if it is as small as possible. Vortices avoid the

smallest disturbance and then can hardly be detected reproducibly.

From the well-known eddy current we know of this problematic. Instead of the vortex, we

are forced to measure and analyse any effects that arise from the vortex. These can be

measurements of the eddy losses or effects back on the stimulating field. But only

provided that the effect actually occurs.

The prerequisite for an increase in temperature by eddy losses is that the vortex falls apart.

In an ideal medium it unfortunately will not do us this pleasure.

As vrtex of the dielectric the potential vortex will find fairly ideal conditions in air or in

water. How should a vortex be detected, if it does not produce any effect? The classical

measuring technique is here at its wits' end.

From the duality to the well-known eddy current and by means of observation in the previous

chapters numerous properties of the potential vortex have been derived. But these

are not all the properties. The mathematical calculation of the electric vortex field, that we

want to turn to now, will reveal still further meaningful and highly interesting properties.

The observation is important, but it can't replace an exact calculation. A strictly mathematical

derived result has occasionally more expressiveness than a whole book full of

explanations. It will be a big help to derive and to discuss the field equation that all

considerations are based on.

We facilitate the mathematical work by vector analysis. Therefore it is useful that we

choose the differential form (equation 5.1 and 5.4) instead of the integral form (equations

3.1 and 3.2 resp. 3.8).

5.1 Fundamental field equation

We'll start from Ampere's law which provides a value for the current density at any point

pace and this value corresponds to the vortex density of the magnetic field strength

The new electric field vortices demand the introduction of a corresponding time constant

tau2 tha t should describe the decay of the potential vortices, as an extension. The extended

Faraday law of induction now provides a potential density, that at any point of space

corresponds to the vortex density of the electric field strength:

which according to the rules of vector analysis can still be further simplified:

= rot rot E = - grad div E , where we should remember that the divergence has to

vanish (div E = O. fig. 3.2, equation 3.7 ), should the corresponding field vortex be

inserted!

Furthermore the following well-known abbreviation can be inserted: = 1/c2 (5.6)

With that the relation with the speed of light c simplifies to the sought-for field equation:

This equation describes the spatial (a) and temporal (b, c, d) distribution of a field vector.

It describes the electromagnetic wave (a, b) with the influences that act damping. As

dumping terms the well-known eddy current (c) and in addition the newly introduced

potential vortex (d) appear.

5.2 Mathematical interpretation of the fundamental field equation

Every specialist will be surprised to find the Poisson equation (a, e) again as a term in the

wave equation. This circumstance forces a completely new interpretation of stationary

fields upon us. The new damping term, that is formed by the potential vortices (d), is

standing in between.

Let us start with a mathematical analysis. We have applied the curl to equation 5.4*, then

inserted equation 5.1* and obtained a determining equation for the electric field strength

E. Of course we could as well have applied the curl to equation 5.1* and inserted equation

5.4*. This would have resulted in the determining equation for the magnetic field strength

H.

If we insert Ohm's law (5.2) and cancel down the specific conductivity, or we put in the

relations of material (3.5) or (3.6) and cancel down by u respectively then the field

equation can likewise be written down for the current density j, for the induction B or for

the dielectric displacement D.

It just is phenomenal that at all events equation 5.7 doesn't change its form at all. The field

vector is thus arbitrarily interchangeable! This circumstance is the foundation for the claim

of this field equation to be called fundamental.

It does make sense to introduce a neutral descriptive vector as a substitute for the

possible field factors E, H, j, B or D.

The fundamental field equation 5.7 consists of three different types of partial differential

equations: a hyperbolic (b), a parabolic (c and d) and an elliptic (e) type. On the left-hand

side each time the Laplace operator (a) is found which describes the spatial distribution of

the field factor.

The potential equation of the elliptic type (e) is known as Poisson equation. It describes

the stationary borderline case of a worn off temporal process ( resp. = O).

With this equation potentials and voltages can be calculated exactly like stationary electric

currents (5.8).

The hyperbolic equation (b). known as wave equation, shows a second derivative to time.

which expresses an invariance with regard to time reversal; or stated otherwise: the direction

of the time axis can be reversed by a change of sign of t, without this having an influence

on the course of frequency. Wave processes hence are reversible. Equation 5.7 makes

clear that a wave without damping by no means can exist in nature. For that both time

constants would have to have an infinite value, which is not realizable in

practice. Seen purely theoretical, undamped waves could withdraw themselves from our

measuring technique (5.9).

Both vortex equations of the parabolic type (c and d) only show a first derivative to time.

With that they are no longer invariant with regard to time reversal. The processes of the

formation and the decay of vortices, the so-called diffusion, are as a consequence irreversible.

Seen this way it is understandable that the process of falling apart of the vortex,

where the vortex releases its stored energy as heat e.g. in form of eddy losses, can not take

place in reverse. This irreversible process of diffusion in the strict fhermodynamic sense

increases the entropy of the system (5.10).

Because it poses an useful simplification for mathematical calculations, often the different

types of equations are treated isolated from each other. But the physical reality looks

different.

5.3 Physical interpretation of the fundamental field equation

In nature the different types of equations always occur in a combined manner.

1. Let's take the concrete case of the particle-free vacuum. Here the specific conductivity

is zero. The relaxation time constant responsible for the decay of vortices tends

towards infinity according to equation 5.3 and the terms (c) and (e) are cancelled from the

field equation 5.7. What remains is the by potential vortices (d) damped wave equation (b)

(equation 5.12).

2. The reversed case (with ) will consequently occur in materials without

resistance, super conducting materials. We now are dealing with the well-known case of

the wave damped by eddy currents (equation 5.12*).

Vi r t u a l l y all in nature existing materials however don't fulfil these boundary conditions,

from which it follows that both damping terms always occur together and in addition the

stationary term (e) becomes active.

It is true that every antenna demonstrates that the electromagnetic wave is convertible in

high-frequency alternating currents and voltages, which then are amplified in the receiver.

But until this fundamental equation was written down it however was not understood that

this transition takes place by means of a vortex. Used are either antennas from well conducting

material, or wave guides and horn radiators, which only have a minimal conductivity,

because they are filled with air. Actually the wave can be converted in two dual

ways; by means of the rolling up to current eddies or to potential vortices (fig. 1.4).

Now we finally are capable to explain, why wave guides make possible a better degree of

effectiveness: Owing to the concentration effect of the potential vortex the HF-power is

bound in the inside and not emitted until the antenna is reached as happens for a wire for

reason of the skin effect.

Therefore, physically, one has to imagine this relation, which describes the transition of an

electromagnetic wave into a vortex, in the way that the wave spontaneously can roll up to

a vortex in case it is disturbed from the outside. The more vortices are generated, the

larger consequently is the damping of the wave (equations 5.12 and 5.12*).

3. The life span of the vortices is limited and is determined by the electric conductivity.

The at first stored vortices decay with their respective time constant This process is

described by the diffusion equation 5.12**. The final stage of the decaying vortices

finally is described by the Poisson equation (a, e: equation 5.8).

If the vortex falls apart, it converts the in the vortex stored energy in heat. These processes

are known from the eddy current. We speak of heating losses, that the stationary currents

cause in the conductor material.

But new is the concept that such vortex phenomena can occur as dielectric losses in

capacitors or in the air. The microwave oven or induction welding are good examples of

this.

5.4 Phenomenological interpretation of the fundamental field equation

How does a damping by vortices take effect in practice? First of all we notice that the

reception of broadcastings gets worse. "The information signal is neglectable regarding

the noise" explains the radio engineer and means, the number of vortices increases at the

expense of the wave intensity.

Why, does the pupil ask, is it so cold in space? There the sun shines day and night and in

addition much more intensely than on earth! The correct answer would have to read that

because of the extremely small conductivity no diffusion process can take place. We owe

the warmth on our earth solely the here taking place decay of vortices. Responsible is the

conductivity of the atmosphere.

In 60 km to 500 km height over the earth's surface, the region which is called the

ionosphere, the gases predominantly exist in ionized form. There a very good conductivity

prevails and eddy current losses are the result. Correspondingly high are the measurable

temperatures. Besides the diffusion process the eddy currents carry out a damping of the

cosmic radiation. We say the sunlight is filtered and reduced to a for nature bearable

intensity.

But not all frequencies are damped in the same way (fig. 2.8). We observe a blue shift, if

we look into the actually black sky. The blue sky doesn't show any spots or clouds. The

reason is to be sought in the skin effect of the eddy currents, which strive outwards. Since

no edge of a conductor is present here, no skin can form. The vortices spread evenly over

the ionosphere.

The potential vortex however is able to structure. It merely needs a bad conductivity and

this it finds in lower heights between 1 km and 10 km. It damps the wave and with that

also the light, for which reason we say it becomes darker, the sun disappears behind

clouds.

The clouds well visibly form the discussed vortex balls and vortex strings. Clouds can

form virtually from the nowhere during intense solar irradiation, i.e. the waves can roll up

to vortices. But as a rule this takes place above the oceans. Here also the phenomenon of

transport has an effect. Because of the high dielectricity the water surface favours the

formation of potential vortices. So the vortices bind individual water molecules and carry

them away. If a diffusion process takes place, in which the vortex decays, then it rains.

This can happen in two different ways:

1. Either the conductivity increases. If for instance during intense solar irradiation air ions

form, the sun is able to break up clouds and fog. Or when the air is raised in higher

layers with better conductivity, because a mountain forces this, then it rains at the

mountain edge.

2. For potcntial vortices the electric field is standing perpendicular to them. If at one point

an exceptionally lot of vortices join together, which let the cloud appear particularly

dark to black, then the danger exists that the ionization field strength for air is reached,

in which case a conductive air channel forms along which the stored up charges

discharge. Also lightning is a diffusion process, in which potential vortices decay and

rain can form.

In connection with the electromagnetic environmental compatibility great importance is

attributed in particular to the storage and the decay of electric vortices. There not only is

an academic-scientific interest in the question, how many potential vortices are generated,

how many are stored and how many decay, if we make a telephone call with a handy, if

we are staying under a high-tension line or if we are eating food, which has been heated

up in a microwave oven. The necessary mathematical description is provided by the

fundamental field equation 5.7.

5.5 Atomistic interpretation of the fundamental field equation

Let's again turn to the smaller, the atomistic dimensions. Here positively charged protons

and negatively charged electrons are found. Both are matter particles and that means that

seen from the outside both have the identical swirl direction. For reason of the unequal

charge conditions they attract each other mutually and according to fig. 4.9 rotate around a

common centre of mass as differently heavy pair. Chemists say: "the light electron orbits

the heavy atomic nucleus". With small balls they try to explain the atomic structure.

But the model is no good: it contradicts causality in the most elementary manner. We are

dealing with the problem that according to the laws of electrodynamics a centripetally

accelerated electron should emit electromagnetic waves and continuously lose energy, to

eventually plunge into the nucleus.

Our experience teaches that this fortunately is not true - and Niels Bohr in order to save

his model of the atom was forced to annul the laws of physics with a postulate founded in

arbitrariness.

Actually this state only exists for a very short time and then something unbelievable

happens: the electron can't be distinguished as an individual particle anymore. "It is

smeared over the electron orbit" do certain people say; "it possesses a dual nature" says

Heisenberg. Besides the corpuscular nature the electron should in case of its "second

nature" form a matter wave, "the position of the electron is to be looked at as a resonance

which is the maximum of a probability density", do explain us de Broglie and

Schrodinger.

These explanations can hardly convince. If the electron loses its particle nature in its

second nature, then it also will lose its typical properties, like for instance its mass and its

charge. but this is not the case.

THE vortex theory provides clear and causal answers: if the electron were a ball it continuosly

would lose energy, therefore another configuration forms that does not know

this problem. Here the phenomenon of transport takes an effect. The electron opens its

vortex centre and takes the tiny protons and neutrons as atomic nucleus up into itself. The

Bohr electron orbit with that is not a path anymore, but is occupied by the whole particle

as spherical shell. This is confirmed by the not understood measurements exactly like the

photos of individual atoms with the scanning electron microscope.

But now an electron does in its inside have the opposite swirl direction as the proton seen

from the outside. As a consequence a force of repulsion will occur, which can be

interpreted as the to the outside directed current eddy, the force of attraction for reason of

the opposite charge works in the opposite direction and can be interpreted as the potential

vortex effect.

If both vortices are equally powerful: (5.13)

or if both forces are balanced, as one usually would say, then the object which we call an

atom is in a stable state.

It probably will be a result of the incompatible swirl direction, why a very big distance

results, if the electron becomes an enveloping electron. On such a shell not too many

electrons have room. Because of the rotation of their own, the electron spin, they form a

magnetic dipole moment, which leads to a magnetic attraction of two electrons if they put

their spin axis antiparallelly.

As a "frictionless" against one another rotating pair they form two half-shells of a sphere

and with that occupy the innermost shell in the hull of an atom. If the positive charge of

the nucleus is still not balanced with that, then other electrons is left only the possibility to

form another shell. Now this next electron takes the whole object up into itself. The new

shell lies further on the outside and naturally offers room to more as only two vortices.

5.6 Derivation of the Klein-Gordon equation

The valid model of the atom today still raises problems of causality, as has been

explained. An improvement was provided by an equation, which was proposed by the

mathematician Schrodinger 1926 as a model description. This equation in this way missed

the physical root, but it nevertheless got international acknowledgment, because it could

be confirmed experimentally. Looking backwards from the result then the physical

interpretation of the probability density of the resonance of the waves could be pushed

afterwards.

(5.14)

The Schrodinger equation is valid for matter fields (of mass m), while the interaction

with a outside force field the energy U indicates. It can be won from a wave equation by

conversion, which possibly is the reason why it usually is called a wave equation,

although in reality it is a diffusion equation, so a vortex equation!

For the derivation Schrodinger gives the approach of a harmonic oscillation for the

complex wave function

(5.15)

if the entire time dependency can be described by one frequency f = W/h

(de-Broglie relation): (5.16)

The high-put goal is: if the structure of the atom is determined by the fundamental field

equation 5.7 then one should be able to derive the experimentally secured Schrodinger

equation and to mathematically describe the discussed special case. Also we select at first

an approach periodic in time:

(5.17)

with . (5.18)

We insert the approach 5.17 and its derivations into the field equation 5.7 and divide by

the damping term e-wt:

If as the next step the angular frequency according to equation 5.18 is inserted, then

summarized the provisional intermediate result results:

(5.20)

The derived equation 5.20 represents formally seen the Klein-Gordon equation, which is

used for the description of matter waves in quantum mechanics and which particularly in

the quantum field theory (e.g. mesons) plays an important role. Even if it often is regarded

as the relativistic invariant generalization of the Schrodinger equation, it at a closer look is

incompatible with this equation and as "genuine" wave equation it is not capable of

treating vortex problems correctly, like e.g. the with the Schrodinger equation calculable

quantization of our microcosm.

5.7 Derivation of the time dependent Schrodinger equation

With the Schrodinger approach 5.15 and its derivations the derivation is continued:

The for a harmonic oscillation won relations according to equation 5.21 and 5.22 are now

inserted into equation 5.20:

This is already the sought-for Schrodinger equation, as we will see in a moment, when

we have analysed the coefficients. Because, besides equation 5.16 for the total energy W,

also the Einstein relation is valid (with the speed of light c):

we can replace the coefficients of the imaginary part by:

To achieve that equation 5.23, as required, follows from the Schrodinger equation 5.14, a

comparison of coefficients is carried out for the real part:

If thc kinetic energy of a particle moving with the speed v is:

then acccording to De Broglie this particle has the wavelength h/mv. The consideration of

the particle as matter wave demands an agreement with the wave length c/f of an electromagnetic

wave (with the phase velocity c). The particle hence has the speed v, which

corresponds with the group velocity of the matter wave:

According to equation 5.24 on the one hand the total energy is W = w • h and on the

other hand the relation 5.28 gives resp.:

Inserted into equation 5.27* the sought-for coefficient reads (according to eq. 5.26):

5.8 Derivation of the time independent Schrodinger equation

The goal is reached if we are capable to fulfil the comparison of coefficients 5.26:

(5.30)

The angular frequency w is given by equation 5.18. Therefore has to be valid:

(5.31)

(5.32)

As is well-known the arithmetic and the geometric average only correspond in case the

variables are identical. In this case, as already required in equation 5.13:

(5.13)

has to hold.

From this we can draw the conclusion that the Schrodinger equation is just applicable to

the described special case (according to eq. 5.13), in which the eddy current, which tries

to inc rea se the particle or its circular path and the potential vortex, which keeps the atoms

together and also is responsible for the stability of the elementary particles, are of

identical order of magnitude.

As a check equation 5.23 is divided by c2 and equations 5.30 and 5.25 are inserted:

(5.14*)

This is the time dependent Schrodinger equation 5.14 resolved for

Next we replace according to equation 5.21 with acc. to equation 5.24:

(5.33)

If we separate the space variables from time by the Schrodinger approach 5.15 we

obtain:

(5.34)

This quation 5.34 for the function of space coordinates is the time independent

Schrodinger equation:

(5.35)

The solutions of this equation which fulfil all the conditions that can be asked of them (of

finiteness, steadiness, uniqueness etc.), are called eigenfunctions. The existence of

corresponding discrete values of the energy W, also called eigenvalues of the Schrodinger

equation, are the mathematical reason for the different quantum postulates.

5.9 Interpretation of the Schrodinger equation

The interpretation of the Schrodinger equation is still disputed among physicists, because

the concept of wave packets contradicts the corpuscular nature of the elementary particles.

Further the difficulty is added that wave packets at a closer look never are connected, run

apart more or less fast, and really nothing can hinder them doing that. But for a particle the

connection represents a physical fact. Then there can be no talk of causality anymore.

The monocausal division into two different levels of reality, in a space-timely localization

and in an energetic description, does not represent a solution but rather the opposite, the

abolition of the so-called dual nature. As has been shown, the potential vortex is able to

achieve this with the help of its concentration effect.

But from the introduction of this new field phenomenon arises the necessity to interpret

the causes for the calculable and with measuring techniques testable solutions of the

Schrodinger equation in a new way. Laws of nature do not know a possibility to choose! If

they have been accepted as correct, they necessarily have to be applied.

Three hundred years ago the scholars had an argument, whether a division of physical

pheomena, like Newton had proposed it, would be allowed to afterwards investigate

them in the laboratory individually and isolated from other influences or if one better

should proceed in an integrated manner, like for instance Descartes with his cartesian

vortex theory. He imagined the celestial bodies floating in ethereal vortices.

One absolutely was aware that the whole had to be more than the sum of every single

realizato n, but the since Demokrit discussed vortex idea had to make room for the

overwhelming successes of the method of Newton. And this idea after 2100 years was

stamped, to in the meantime almost have fallen into oblivion.

Today, where this recipe for success in many areas already hits the limits of the physical

possibilities, we should remember the teachings of the ancients and take up again the

vortex idea It of course is true that only details are calculable mathematically and that

nature, the big whole, stays incalculable, wherein problems can be seen.

If we consider the fundamental field equation 5.7, we find confirmed that actually no

mathematician is capable to give a generally valid solution for this four-dimensional

partial differential equation. Only restrictive special cases for a harmonic excitation or for

certain spatial boundary conditions are calculable. The derived Schrodinger equation is

such a case and for us particularly interesting, because it is an eigenvalue equation. The

eigenvalues describe in a mathematical manner the with measuring techniques testable

structures of the potential vortex .

Other eigenvalue equations are also derivable, like the Klein-Gordon equation or the

Lionville equation, which is applied successfully in chaos theories. So our view opens, if

chaotic systems like turbulences can be calculated as special cases of the same field

equation and should be derivable from this equation.

The in pictures recorded and published structures, which at night should have come into

being in corn fields, often look like the eigenvalues of a corresponding equation. The ripe

ears thereby lie in clean vortex structures flat on the soil. Possibly potential vortices have

charged the ears to such high field strength values that they have been pulled to the soil by

the Coulomb forces.

Consequences resulting from the derivation of the Schrodinger

equation from the fundamental field equation 5.7:

The relation between the energy of oscillation and the mass is

described by the relation named after Albert Einstein

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