Vacuum dynamics

[quiet]

Current physics explains that the state of a portion of space depends on the phenomena, local and / or distant. It also explains that states can and do evolve, change depending on events.

The mathematics that expresses some states is similar to the mathematics that describes the curvature in geometry. In short, in scientific jargon it is said that space can be curved. This abbreviated language also includes contraction, elongation, torsion ...

Does it include excision? That is, the possibility of something whose mathematics is similar to the mathematics of a crack?

I mean the vacuum. In a region where there is no break initially, could one happen? If yes, does something appear as a product of that break? If yes, is that product something known for a long time?

Edited August 26, 2018 by quiet
bad writing

[MigL]

No,  scientific jargon says that the model ( GR ), involving curvature of the co-ordinate system, matches observation/experiment to a very high degree.
That is the model, whether space or space-time is curved ( or can be curved ) is immaterial.

[quiet]

27 minutes ago, MigL said:

No,  scientific jargon says that the model ( GR ), involving curvature of the co-ordinate system, matches observation/experiment to a very high degree.
That is the model, whether space or space-time is curved ( or can be curved ) is immaterial.

Thank you MigL for providing a response. I need now to think a little.

[Strange]

10 hours ago, quiet said:

That is, the possibility of something whose mathematics is similar to the mathematics of a crack?

One of the requirements of the mathematics used to model the curvature of spacetime is that it is continuous. Therefore discontinuities, such as cracks, are not supported. 

[quiet]

30 minutes ago, Strange said:

One of the requirements of the mathematics used to model the curvature of spacetime is that it is continuous. Therefore discontinuities, such as cracks, are not supported. 

Thanks for pointing that out. Let me now replace the idea of splitting by the idea of topological change that happens inside a finite three-dimensional region. I think of a change that produces some kind of symmetry, limited to a finite region. Please allow a rude analogy. Take putty, of that used to put glasses. Without breaking it, in a place of putty you mold something topologically different from the rest of the putty piece. There will be a boundary between both topological conditions, so that a Gaussian surface can wrap around the region where the topology has changed and symmetry has been created. That boundary does not delimit a break, but it delimits a clear topological difference.

[Markus Hanke]

5 hours ago, quiet said:

Thanks for pointing that out. Let me now replace the idea of splitting by the idea of topological change that happens inside a finite three-dimensional region. I think of a change that produces some kind of symmetry, limited to a finite region. Please allow a rude analogy. Take putty, of that used to put glasses. Without breaking it, in a place of putty you mold something topologically different from the rest of the putty piece. There will be a boundary between both topological conditions, so that a Gaussian surface can wrap around the region where the topology has changed and symmetry has been created. That boundary does not delimit a break, but it delimits a clear topological difference.

What exactly do you mean by “topological difference”? If you smoothly deform the region in question (no breaks, holes etc), then all topological invariants remain the same, and you end up with something that is topologically equivalent to what you started with. This is just a diffeomorphism, and GR is diffeomorphism invariant, so you will get the same physics in that region.

As a practical example - you start off with a region of spacetime that has the topology of a torus. You then transform the “shape” of that region into that of a teacup - they look different at first glance, but all topological invariants remain the same, so this transformation is just a diffeomorphism, which leaves the physics untouched. You do the same physics on a teacup as you do on a torus.

[quiet]

14 hours ago, Markus Hanke said:

What exactly do you mean by “topological difference”?

I tried to look for something that I had seen before and, instead, I found something surprising. It is a study of the topological properties of electrodynamics, done in two Spanish universities. It is written in English. It comes to establish the quantization of the charge and relates that to the polarization of the vacuum. It is not based on the breaking of spacetime. It is based on knots. On page 85 you can read a synthesis of the result. It is at the following address.

http://cdn.intechopen.com/pdfs/33435/InTech-Topological_electromagnetism_knots_and_quantization_rules.pdf

[Markus Hanke]

9 hours ago, quiet said:

http://cdn.intechopen.com/pdfs/33435/InTech-Topological_electromagnetism_knots_and_quantization_rules.pdf

Interesting article, thanks for the reference!

[quiet]

My surprise was great, because you get to the same elemental charge of vacuum polarization developing the consequences of the displacement wave. I wrote the beginning of that topic in the thread initiated by Achilles, entitled: Is there anything left to discover in electromagnetism?
The full development is interesting. Did you know, for example, that this elementary charge is a universal constant, property demonstrated in development? Did you know that the fine structure constant is given only by the quotient between the charge of the electron and the polarization elememtal charge? And that this quotient is an intrinsic property of the system of 4 Maxwell-Heavside equations, so that the quotient is calculated in a purely theoretical way, without using a single empirical data? Thus a purely theoretical alpha value is obtained. The development contains much more, as you imagine. That's why the Spanish publication caused me great surprise.

[Markus Hanke]

 

13 hours ago, quiet said:

so that the quotient is calculated in a purely theoretical way, without using a single empirical data

Actually, the elemental charge itself depends on both vacuum permittivity and Planck’s (barred) constant, the numerical values of which have to be determined experimentally. The fine structure constant can also be obtained in a number of other ways, so while this article is interesting, it does not make any unique predictions as such.

[quiet]

I put in this note what I really tried to express before.

- Classical electrodynamics in a vacuum.

- Propagation in a vacuum includes the waves of the fields [math]\vec{E}[/math] and [math]\vec{H}[/math]. It also includes waves from other fields, such as the vector potential [math]\vec{A}[/math] and the electric displacement [math]\vec{D}[/math].

- In the simplest case, flat wave without circular or elliptical polarization, [math]\vec{D}[/math] has a property that all other fields do not have. The wave equation of [math]\vec{D}[/math] in the vacuum admits a complex solution of exponential type.

[math]D = \hat{D} \ e^{i \left( \omega t - kx  \right)}[/math]

 [math]\hat{D} \ \ \ \rightarrow[/math] module of [math] \vec{D} [/math]

Let's write the identity of De Moivre for the case that concerns us.

[math]e^{i \left(\omega t - kx \right)} = cos\left(\omega t - kx \right) + i \ sin\left(\omega t - kx \right)[/math]

Applying that identity we have the following.

[math]D = \hat{D} \ \left[ cos\left(\omega t - kx \right) + i \ sin\left(\omega t - kx \right) \right] [/math]

8. The vector field [math] \vec{D} [/math], which has two components and module [math] \hat{D} [/math], corresponds to a plane electromagnetic wave propagating in the direction of the axis [math] x [/math]. If those components were not mutually perpendicular, they could not correspond to a complex number. They are mutually perpendicular and correspond to different axes of the coordinate system. Which axes?

9. Let's write the vector expression of the electric displacement.

[math]\vec{D}= \vec{P} + \varepsilon \vec{E} [/math]

In the vacuum is [math] \varepsilon = \varepsilon_o [/math]. We apply it.

[math]\vec{D}= \vec{P} + \varepsilon_o \vec{E} [/math]

The components [math] \vec{P} [/math] and [math] \varepsilon_o \vec{E} [/math] are mutually perpendicular. Can they be both cross-sectional? Let's reason. In terms of local results, polarization is a field with colinear symmetry that does not alter the electrical neutrality. That means that, within a finite length segment, there is a pair of equal and opposite vectors, resulting from all local contributions. In the case we are dealing with, could polarization be transversal? Impossible, because two transverse vectors that correspond to different values of [math] x [/math] are not collinear. Two longitudinal vectors corresponding to different [math] x [/math] values are collinear, because both vectors have the [math] x [/math] axis  direction.

10. What does the vacuum do when a wave propagates? Is it inert or participate in any way? The velocity of propagation is determined only by two properties of the vacuum, which are the permeability [math] \mu_o [/math] and the permitivity [math] \varepsilon_o [/math]. That leaves no doubt. The vacuum participates. How do it participate? The expression of displacement leaves no doubt. Participate polarizing. In that way it set the speed [math] C [/math] of propagation. That means that the displacement has a transversal component [math] \varepsilon_o \vec{E} [/math] and a longitudinal component [math] \vec{P} [/math].

[math]\vec{D}= \vec{x} P + \vec{y} \varepsilon_o E[/math]

In terms of the wave function we have the following.

[math] \vec{D} = \vec{x} \hat{D} \ cos\left(\omega t - kx \right) + \vec{y} \hat{D} \ sin\left(\omega t - kx \right) [/math]

[math] \vec{D} [/math] has finite divergence, corresponding to the charge density of the polarization. That divergence has the form of a wave function.

[math]\nabla \cdot \vec{D} = \hat{D} \ k \ sin\left( \omega t - kx \right)[/math]

Does that mean that some charge travels in a vacuum when the wave propagates? No charge needs to travel to produce that divergence. In the cities there are giant illuminated signs, which are panels populated by thousands of luminous cells, controlled by a programmable device. A program can achieve that the brightness of each cell varies sinusoidally, in the form corresponding to a wave function. You can program two colors, say blue and red. The first cell is initially dark. Then the blue light grows sinusoidally, reaches the maximum and decreases sinusoidally, until the cell becomes dark. Follow the sinusoidal stage of the red light, which does the same. All the cells are immobile on the board, but the program manages to see alternate blue and red areas traveling along the board. The effect is equivalent to colors in movement. At each point of the vacuum, the charge density varies sinusoidally. The signs of the charge do the same as the colors. The effect is equivalent to alternating zones with opposite charges traveling in the direction of propagation, although no infinitesimal or finite charge is actually moving.

- Although the charge does not really move, the effect is equivalent to moving charge, as happens with colors that look like moving sections with [math]\tfrac{1}{2} \ \lambda[/math] in blue and [math]\tfrac{1}{2} \ \lambda[/math] in red.

- The effects of the virtual movement of the charge density are equivalent to a virtual density of current in the direction of propagation, equal and opposite to the longitudinal component of [math]\dfrac{\partial \vec{D}}{\partial t}[/math] . This derivative is part of the Ampere-Maxwell law.

Ampere-Maxwell law [math]\rightarrow \ \ \ \vec{\nabla} \times \vec{H}=\vec{j}+ \dfrac{\partial \vec{D}}{\partial t}[/math] .

Finally [math]\vec{\nabla} \times \vec{H}[/math] it remains, in a net form, identical to the result obtained for the real solutions of the wave equation, which lack a longitudinal component.

- The law of Ampere-Maxwell allows to calculate the inductance of the propagation in a vacuum. Then the energy of the magnetic field is calculable by two methods. One is to integrate the energy density, as we usually do. Another is based on the inductance and the currents involved. Equating the results of both methods all the geometrical measurements and the shape of the autonomous configuration that has the inductance and capacity calculated. It is a cylinder whose length is equal to [math]\lambda [/math] and its diameter is

[math]\dfrac{\lambda}{2 \pi}[/math]

The inductance and the capacitance have the following values.

[math]\mathcal{L}= \mu_o \ \dfrac{\lambda}{2 \pi}[/math]

[math]\mathcal{C}= \varepsilon_o \ \dfrac{\lambda}{2 \pi}[/math]

So it turns out that the total energy of the stunned configuration is

[math]E=2 \ \pi \mu_o \ C \ Q_o^2 \ \ \nu[/math]

[math]Q_o \ \ \ \rightarrow[/math]   elemental charge of vacuum polarization

The development shows that [math]Q_o[/math] It is a universal constant. Then the energy of the autonomous configuration is directly proportional to the frequency and the Planck law of quantum energy is proved. There is also the Planck constant expressed in electrodynamic constants melting.

[math]h=2 \ \pi \mu_o \ C \ Q_o^2[/math]

If in the formal definition of the fine structure constant you express h in that way, you get

[math]\alpha=\dfrac{1}{4 \ \pi} \ \left( \dfrac{e}{Q_o} \right)^2[/math]

That is, alpha is given by the quotient between the charge of the electron and the elementary charge of the polarization of the vacuum, that same charge that appears in the Spanish article of the electromagnetic knots.

The next stage of the development is to analyze the mutual frontal collision of two autonomous configurations, that is, of two photons having the cylindrical shape and the calculated measurements. Photons do not pass through one another like ghosts. They interpenetrate, but not totally. Mutual penetration only reaches a certain depth, because the impulse gradually decreases. The partially interpenetrated photons form a larger object for a moment, which ends up breaking in one of the two sections that delimit the penetration zone. The large object is fragmented. One of the fragments takes the fraction of charge that the other contributed to the penetration zone. That fraction is equal to [math] e [/math]. One of the fragments, the electron, remains with charges [math]-(Q_o+e),+Q_o [/math] and its net charge is [math]-e[/math]. The other fragment, positron, remains with [math]-(Q_o-e),+Q_o[/math] and its net charge is [math]+e[/math].
By proposing the wave functions and the energy balance, the theoretical value of the quotient between the net charge [math]e[/math] and the elemental charge of vacuum polarization [math]Q_O[/math] is obtained as a result, without using a single empirical data, that is, without involving physical constants. The result is

[math]\dfrac{e}{Q_o}=\dfrac{-3+\sqrt{13}}{2}[/math]

In decimal numeration is

[math]\dfrac{e}{Q_o}=0,30277563773199...[/math]

The inverse is the factor that appears in the Spanish article of electromagnetic knots.

[math]\dfrac{Q_o}{e}=3,30277563773199...[/math]

As that article points out, [math]Q_o[/math] It is about 3.3 times greater than the electron charge.

- The force between a pair of charges [math]Q_o[/math] It has the same properties as the force between the plates of a capacitor, that is, it is a force independent of the distance between charges. And in relative terms it's [math](3,30277563773199...)^2[/math] , about 11 times greater than the force between two charges [math]e[/math]. Do those characteristics remind us of something? Strong force has the same characteristics.

- Let's formulate [math]\alpha[/math]

[math]\alpha=\dfrac{1}{4 \ \pi} \ \left( \dfrac{e}{Q_o} \right)^2[/math]

[math]\alpha=\dfrac{1}{4 \ \pi} \ \left( \dfrac{-3+\sqrt{13}}{2} \right)^2[/math]

In decimal numeration is

[math]\alpha=0,0072951124566757786721625768237...[/math]

It differs 0.03% with respect to the empirical value published by CODATA. As far as I can analyze, the calculation of [math]\alpha[/math] in this context, it does not allow retouching that equals the value given by CODATA. If the empirical value is fully reliable, the development based on classical electrodynamics should be understood as a first approximation.

- That development includes more details. For example, it shows that the electron and the positron are constituted by cylindrical rotating waves, electromagnetic waves obviously. And determined what proportion of the constituent energy of each particle is in the electric field and what proportion in the magnetic field of the wave. 65.1% of the energy [math]m_o \ C^2[/math] of the electron is in the electric field. And 34.9% in the magnetic field. That means that the magnetic field can not overcome the potential barrier of the electric field and, for that reason, the electron does not decompose. In the positron the percentages are identical, but the greatest proposal corresponds to the magnetic field. That is why the positron decomposes easily. Many more issues appear in the development, but exposing them to all coherently demands 130 pages of writing, a space that we do not have here. We do not even have space to adequately expose the details shown in this note.
 

Edited August 30, 2018 by quiet