The Primary Essentials - Round 2 - The elementary inertial mass particle

A photon which goes into/out-of an optical prism along some direction which is not perpendicular to the prism surface, changes its propagation direction.

[tex]\epsilon[/tex] and [tex]\mu[/tex] values of the spacetime within an optical prism are different from [tex]\epsilon[/tex] and [tex]\mu[/tex] values of the spacetime outside the prism. It is quite logically and physically justified to interpret that fact in the following way:

increased values of [tex]\epsilon[/tex] and [tex]\mu[/tex] properties of the spacetime within the prism are consequences of superpositioned influences of the huge number of elementary particles which constitute the prism – that is how the spacetime reacts to the prism constituents.

The prism surface and its nearest vicinity form a thin transitional spacetime domain (transitional [tex]\epsilon[/tex]–[tex]\mu[/tex]–domain). „Thin“ means that its length in the direction which is perpendicular on the prism surface is several atom/molecule diameters long. „Transitional“ means that within that domain [tex]\epsilon[/tex] and [tex]\mu[/tex] change their values. In the into-the-prism direction, [tex]\epsilon[/tex] and [tex]\mu[/tex] increase. In the out-of-the-prism direction,[tex]\epsilon[/tex] and [tex]\mu[/tex] decrease.

The change of photon’s propagation direction occurs within that thin transitional spacetime domain, that is, the change of photon’s propagation direction occurs there where the change (increase/decrease) of [tex]\epsilon[/tex] and [tex]\mu[/tex] occurs along the photon’s propagation direction.

And here is the explanation why photon’s path bends:

Along the left-to-right direction on the picture, values of [tex]\epsilon[/tex] and [tex]\mu[/tex] increase. Therefore, along the left-to-right direction on the picture, the allowed EM-propagation velocity decreases.

Therefore, the only thing that the photon can do is to move along a path which bends to the right. The photon will continue to turn right for as long as its right side is imposed (by [tex]\epsilon[/tex] and [tex]\mu[/tex] distributions) to move slower than its left side.

So, if a photon enters a prism along the direction which is not perpendicular to the prism surface, it will turn until allowed velocities become equal around its propagation direction.

On the picture, the photon's shape - that is, the shape of the tiny spatial domain which contains the photon's energy - is an assumption, a sketch. Namely, what we know is just the photon's length [tex]\Delta s[/tex]. The photon's width we do not know. We can assume that it has some value which is in the range which starts with some infinitesimal value, and up to the size of the photon's length, or several photon's lengths. The photon's width is not important for the presented explanation of the photon's path banding. In other words, the presented explanation for the photon's path bending is valid, whichever the width may be.

That is also the reason why photons change their propagation direction while they pass very near the edge of some solid body, or through hole(s), or through slits (single slit, double slits, slits of refraction gratings) – very near the edges of solid bodies the values of [tex]\epsilon[/tex] and [tex]\mu[/tex] change (which, in most cases, means that they increase).

And, not only near the solid bodies, composed of many particles – the same happens in the very near vicinity of single elementary particles, too.

The interference patterns, which emerge as the result of two mutually confronting/overlapping coherent EM-waves, are explainable in the following way:

the interference patterns are consequence of changes of [tex]\epsilon[/tex] and [tex]\mu[/tex] , caused by energy density increase when two coherent EM-waves confront each other, and then the change of [tex]\epsilon[/tex] and [tex]\mu[/tex] causes the change of propagation directions of photons which constitute those confronted EM-waves. On some places there would be very small number of photons, or no photons at all, and on some places there would be many photons. And such patterns would repeat all over the domain of confrontation.

Now, all this will be logically and physically essentialized:

The change of energy amount within some constrained spacetime domain, which occurs during confrontation of EM-energy-flows, is the most rapid change possible. Such rapid change causes the change of [tex]\epsilon[/tex] and [tex]\mu[/tex] of the spacetime domain affected with that rapid change.

Also, the phenomenon of reactive power in electric circuits implies this conclusion. That opens the way for a more general mathematical treatment, which includes the complex numbers domain. Namely, the increase of [tex]\epsilon[/tex] and [tex]\mu[/tex] is a way of energy accumulation in the imaginary-number "format". But, that would be an option for further development of this theory, which is the basic, primary, elementary EM theory, and in which it will be demonstrated how the part of physics known as *the mechanics* is to be derived. And, for that, we do not need to delve into complex-numbers calculus. The photons are elementary energy carriers. So, the most elementary situation, in which elementary energy amount changes within some elementary-size space-domain most rapidly, would occur when two photons confront each other, that is, when two photons pass one through the other, and also when they pass very near to each other.

Why would two confronting photons cause [tex]\epsilon[/tex] and [tex]\mu[/tex] to increase? We do not know. But that mystery is reduced to the elementary level circumstances - to the confrontation of two most fast and most fundamental entities in nature. It is quite in accordance with intuition, as well as with the common sense, that *specific circumstances produce specific effects*. And we came up with this conclusion through logical and reasonable deduction, on the basis of the known and detected physical phenomena. We cannot delve deeper into that, so we take it as an axiom, as "that's the way it is". That axiom is the result of reasonable, common sense thinking. What we can do next about it, is to check if we can derive consequences which would be reasonable, too, and which comply to reality. To see if that can explain reality well.Hence, when two photons pass sufficiently and appropriately near to each other, and if these two photons have also the sufficiently and appropriately high energies (short lengths), and if we assume that such occurrence causes the [tex]\epsilon[/tex] and [tex]\mu[/tex] to increase, forming some bumpy [tex]\epsilon[/tex]-[tex]\mu[/tex]-distribution which has maximal value in the center of the distance among the photons, then there are three logically/physically possible outcomes/manifestations:

1) our photons just alter their path directions (which, i.e., is the cause for interference paterns of two confronting coherent EM-waves),

2) our photons form a temporary whirl (that is, temporary particles, short-life particles)

3) our photons form a stable whirl . The conditions for forming of stable photons-whirls are more stringent than for temporary whirls, and therefore there are just several elementary types of such whirls, i.e. electron, positron, quarks, anti-quarks.

The photons-whirl is the most elementary whirling in nature, the tiniest energy whirling possible. And it can be interpreted (as, indeed, any other whirl/vortex) as the continuous (ongoing, lasting) non-direct confrontation, as the continuously almost-directly confronting flow. The conditions induced by two photons which meet each other in the previously described way, make these photons to bend their paths. If the bending is such that the photons remain in the same position with respect to each other, then these conditions remain the same, too.

So, when our two photons have

*appropriate energies (masses)* and meet at some

*appropriate distance*, they will induce the appropriate reaction of spacetime, that is, the appropriate spatial distribution of [tex]\epsilon[/tex] and [tex]\mu[/tex] in the spatial domain occupied with these photons.

The previous underlined words should highlight the fact that both [tex]\epsilon[/tex] and [tex]\mu[/tex] distirbutions obviously have to be the functions of the photons’ energies (mases) and their distance, namely, the functions of the ratio mass/distance (or energy/distance):

- the greater the mass, the greater the influence on [tex]\epsilon[/tex] and [tex]\mu[/tex] ;

- the greater the distance, the smaller the influence on [tex]\epsilon[/tex] and [tex]\mu[/tex].

So, the ratio [tex]\frac{m}{r}[/tex] (or [tex]\frac{E}{r}[/tex]) is the quite logical way to start modelling the influence of some entity. In the case of confrontation of two photons, the ratio [tex]\frac{m}{r}[/tex] would be the measure of influence of the both photons on [tex]\epsilon[/tex] and [tex]\mu[/tex] of the space which contains both of our photons. That means that the mass [tex]m[/tex] is the sum of masses of our two photons. The denominator should be the distance between our photons. We want to model the distribution of [tex]\epsilon[/tex] and [tex]\mu[/tex]. The two-photons-whirl would definitely have the circular symmetry, and that means that the distributions of [tex]\epsilon[/tex] and [tex]\mu[/tex] must have circular symmetry, too. Therefore, we will consider that the total mass [tex]m[/tex] is the mass-center, and that [tex]r[/tex] is the radial distance from the center.

We will assume that the spacetime is continuum, and we will start mathematical modeling of influence of two confronting photons on spacetime with the following general mathematical model:

[tex]\frac{\epsilon}{\epsilon_0}=f_\epsilon(\frac{m}{r})[/tex] [tex]\frac{\mu}{\mu_0}=f_\mu(\frac{m}{r})[/tex]

That what we know for sure, is that if there is no confrontation, then [tex]\frac{\epsilon}{\epsilon_0}=1[/tex] and [tex]\frac{\mu}{\mu_0}=1[/tex], that is, [tex]\epsilon = \epsilon_0[/tex], and [tex]\mu = \mu_0[/tex], hence, [tex]f(no-confrontation) = f(0) = 1[/tex]

The continual analytic functions can be represented with Taylor series. Since we deal here with elementary level of existence, we will start with only first two elements of the Taylor series.

[tex]\frac{\epsilon}{\epsilon_0}=1+k_\epsilon \cdot \frac{m}{r}[/tex] [tex]\frac{\mu}{\mu_0}=1+k_\mu \cdot \frac{m}{r}[/tex]

[tex]v = \frac{1}{\sqrt(\epsilon \cdot \mu)} = \frac{1}{\sqrt(\epsilon_0 \cdot (1+k_\epsilon \cdot \frac{m}{r}) \cdot \mu_0 \cdot (1+k_\mu \cdot \frac{m}{r}))} = \frac{c}{\sqrt(1+k_\epsilon \cdot \frac{m}{r}) \cdot (1+k_\mu \cdot \frac{m}{r})}[/tex]

Next, we could assume that [tex]k_\epsilon = k_\mu = k[/tex], because, for now, there is no reason to assume that they would/should be different. Hence, the distribution of allowed velocities is:

[tex]\frac{v}{c} = \frac{1}{1+k \cdot \frac{m}{r}} = \frac{1}{1+ \frac{R_0}{r}}[/tex]

(graph)

So, in the case of the two-photons-whirl, the velocity distribution is not linear function of the circling-radius. However, it does not have to be linear – it is enough to be sufficiently linear along the radial line segment within the photons, at the moment when photons come into the position of their minimal passing-by distance. However, the circling would occur even if the previous condition is not met.

The further elaboration will demonstrate the two-photons-whirl concept's supreme reality-explanation-power.

Quantitatively, the mass of the two-photons-whirl is equal to the sum of the non-inertial masses of the photons which form that whirl. That what is new, is that the mass of the two-photons-whirl

**whirls**,

**spins**. Another new property is that the velocity [tex]V[/tex] of the

*two-photons-whirl as a whole* may have any value from the range [tex]V \in [0,c)[/tex].

What would be the bahavior of the two-photons-whirl if some outer force affects it (i.e. if some other entity – a photon, or another whirl, or some composite entity – colides with the two-photons-whirl)?

Well, the influence of that outer force would manifest itself as any combination of the following changes:

1) the change of individual masses of the photons which constitute the whirl

2) the change of the velocity distribution within the whirl, that is the change of [tex]\epsilon[/tex] and [tex]\mu[/tex] distributions of the space which contains the two-photons-whirl

3) the change of the spatial position of the two-photons-whirl as a whole, which means, the change of the velocity [tex]V[/tex] of the

*two-photons-whirl as a whole*.

It may happen that the final result is only the change of [tex]V[/tex]. That would mean that the whirl was disturbed by the outer force, but nevertheless, survived that disturbance, in the way that its mass remained the same, and that it whirls further in the stable way. And that is possible because it has the possibility to change its spatial position, that is, its velocity [tex]V[/tex]. So, we could say that the two-photons-whirl is able to keep its mass

*better** than a single photon. And that is the origin of the property which is generally known as

**inertia**.

* we used the word

*better * because there is the case when even the single photon's mass remains the same as it was before interaction. That case is known as

*elastic scattering*.

Comparing to single photons, the two-photons-whirl is the new, higher mode of existence, the new way of energy packaging, whose inevitable consequence is the enhancement of the mass stability, because, at least a part of an outer force influence would change the [tex]V[/tex]. The two-photons-whirl has the intrinsical, inherent, ontological capability to avoid or to reduce the influence of some outer force on its mass. And that capability is the origin of the phenomenon known as

**inertia**.

Hence, inertia is inevitable consequence of energy-whirling, that is, of the non-inertial-mass whirling, that is, of the proto-mass whirling, that is, of the two-photons whirling. In other words: the most elementary inertial mass,

**the most fundamental inertial mass, is the mass of the two-photons-whirl**.

Hence,

**the two-photons-whirl is the most elementary appearance of that what we call matter**.

The state of a two-photons-whirl is [tex]p=m \cdot V[/tex]. We should recall that we have

**derived **the momentum in the previous section. The momentum of the photon, that is, the momentum of non-inertial mass, is [tex]\Delta p= \Delta m \cdot c[/tex]. Before that, we have

**derived ** the force, from the change of the fundamental state [tex]\Delta E \cdot \Delta t = h[/tex], and we have derived that the force can be presented either as the change of energy over length, or as the change of momentum in time. Before and after interaction with some other entity, the velocity of a photon is [tex]c[/tex], and therefore, that what changes has to be and is the [tex]\Delta m[/tex]. The two-photons-whirl may have any velocity [tex]V \in [0,c)[/tex]. If only the [tex]V[/tex] changes during interaction, we have:

[tex]\frac{dp}{dt}=\frac{d(m \cdot V)}{dt}=m \cdot \frac{dV}{dt}=m \cdot a[/tex]

So, we have fundamentaly derived and explained the Newton’s force definition, too.

Starting with the two-photons-whirl and up - in the sense of the matter composite structures - we have the intrinsic energy (energy of the photons which form the whirls, that is, which form the matter), and the kinetic energy, that is, the energy of the movement of matter. In the case of a single photon, these two energy "types" are indistinguishable, that is, the intrinsic and the kinetic energy of a single free photon is the one and the same thing.

**What would be the radial energy distribution of some energy which whirls?**In the radial direction towards the center of the whirling, energy rises, that is, in any radial direction away from center, the energy decreases. At each point [tex]r[/tex], energy has a certain value [tex]E(r)[/tex]. Along the radius, the energy does change: [tex]\frac{dE}{dr}[/tex]. This is force, by definition.

At some given radius [tex]r[/tex], that force has a certain value [tex]\frac{dE(r)}{dr}[/tex]. Both force and energy do not change along the circle of that given radius [tex]r[/tex]. Hence:

[tex]\frac{dE(r)}{dr} = - a(r) \cdot m(r) = - \frac{v^2}{r} \cdot \frac{E(r)}{c^2} \Rightarrow \frac{dE(r)}{E(r)} = - \frac{v^2}{c^2} \cdot \frac{dr}{r}[/tex]

[tex]v = \omega \cdot r, dv = \omega \cdot dr \Rightarrow \frac{dr}{r} = \frac{dv}{v}[/tex]

[tex]\Rightarrow \frac{dE(r)}{E(r)} = - \frac{v \cdot dv}{c^2} \Rightarrow \frac{E(v)}{E_0} = e^{ - \frac{1}{2} \cdot \frac{v^2}{c^2}}[/tex]

Hence, we got the distribution which is the synonym of stability -

**the Gaussian**.

In other words, the Gaussian is intrinsically related to the canonical whirling.

**This is an unprecedented explanation and derivation of the Gaussian, its physical origin, its essential ontology.**

The Gaussian form is the most stable form. It is mostly known as the final form that each distribution tends to, the form which envelopes each type of the so called *chaos*. All of the existing derivations and interpretations of the Gaussian form are based on and derived from statisctics, probability, oodles, chaos. (http://en.wikipedia.org/wiki/Central_limit_theorem, section History)

However, it is not only the finall, the end form – it is, primarily, the archetype of the continual form, the archetype for the continual forming – it is one of the essential principles upon which the world is built, starting from elementary particles. It is the fundamental continuous form determinator, the synonym for order. It is the essential soliton-form, the basic building block for any other form – any localized physical form can be composed by the set of Gaussians. In other words, there is no ground for any doubts about the possibility to have a stable two-photons-whirl. And it could be even more stable - the two-photons-whirl could form some type of electromagnetic self-adjusting phase-locked-loop (PLL).

The photons-whirl concept definitely requires deeper/broader analysis and investigation, but that analysis and investigation simply cannot compromise this concept. Not only that this concept – that the photons are the basic building blocks for inertial-mass-particles, namely, that the elementary inertial-mass-particle is the photons-whirl – is both logically and physically completely justified, but there does not exist any logical or physical argument against this concept. If someone would argue that the „annihilation“ of a slow electron and a slow positron results with just two [tex]\gamma[/tex]-photons, that would be easy to explain:

- It is the resonant, perfect (precise, accurate) match, whose result is: one photon of the electron and one photon of the positron merge into one [tex]\gamma[/tex]-photon, and the rest two photons merge into the other [tex]\gamma[/tex]-photon. These new photons have higher energy (bigger mass), which affects their lengths. Also, the allowed-velocities-distribution is affected. Also, the whirl-radius is affected. All of these changes together disturb the geometry and balance of whirling enough to cause that these new photons – two of them – fly away.

So far, the new definition for the space given at the beginning of this work, which logically lead to the two-photons-whirl concept, enabled the unprecedented understanding of reality. And that is just the beginning.