Mass and gravitation

The model of the elementary particles presented in the first post makes it tempting to philosophize over the nature of mass and gravitation. Is there a logical concept of these matters that smoothly fits in? I have made an attempt to find one...

Obviously, a clue must be that the mass of at particle is proportional to its total energy. Assuming that electromagnetic waves travel in a background medium¹, one should expect that the waves displace some of it. In fact, it is this displacement that defines the wave. In the enclosed waves making the elementary particles, the net displacement of medium is radially outwards because of the 180° twist for each half wavelength. This will hold true for simple circular waves as well as for folded ones. The more energy contained in an enclosed wave, the more medium it displaces. The mass is a measure of the total medium displacement, and is therefore a function of the total energy of the wave. Generally, outside the enclosed wave the medium will be compressed, and inside the wave it will be diluted². This assumes an elastic medium. If the background medium is perfectly elastic, the gravitational field will never reach zero.

Because the electromagnetic wave compresses the medium in two dimensions (and not in three), the degree of the compression is inversely proportional to the square (and not to the cube) of the distance from the wave-particle. Gravitation is therefore an inverse-square law.

This interpretation can explain the gravitational attraction observed between two objects with mass: When the compression of medium created by one particle partly coincides with the dilution created by another, their total displacement of medium is reduced. Their total mass and energy are reduced accordingly. As the objects approach each other, medium displacement (mass) is reduced even more, and energy is lost in radiation.

Because the area of diluted medium inside a particle usually is very small compared to the squared distance to a neighboring particle, their common reduction in mass is negligible compared to their total mass. A notable exception is of course the case in the atomic nucleus.

_________________
¹ This medium was thought to be a “luminiferous aether" in the late 19th century, but this idea was rejected by the famous Michelson–Morley experiment in 1887, and also by following experiments. Today, the medium that carries electromagnetic waves is generally thought to be spacetime itself.

² Charged particles will also produce an area of dilution outside the particle, enabling electrostatics. This will be discussed in a later post.

A simple model of electrons, protons, neutrons and their antiparticles

© 2008

Abstract
A strictly classical model of electrons, protons, neutrons and their antiparticles is presented. The model explains mass, spin and charge for each particle. An approximation of the value of the elementary charge can be calculated, and is found to be 0.956 times that of the measured value. Further, the model strongly suggests why neutrons can be unstable. A structure of the atomic nuclei is proposed, which explains why the ratio N:Z for stable nuclides follows a limited range as Z increases. Only the electromagnetic force is needed to support the model.

General hypothesis
It is hypothesized that an elementary particle is simply a standing enclosed electromagnetic wave with a half or whole number of wavelengths (λ). For each half number of λ the wave will twist 180° around its travel path, thereby giving rise to chirality. As for photons, the Planck constant (h) can be applied to determine the total energy (E): E = nhc/λ, where n = 1/2, 1, 3/2, 2, etc., and c is the speed of light in vacuum. The mass (m) can be expressed as a function of λ, since E = mc² gives m = nh/cλ from the formula above.

If the electric field of a wave is completely cancelled out by the wave itself, the particle will possess no charge. But if electric field vectors escape, the particle will be charged. Negative and positive values of charge arise from opposite chiralities.

Like electric currents flowing in the same direction attract each other, also loops of electromagnetic waves attract. They align with the waves moving in the same direction and with their magnetic moments merging.

The electron and positron
The electron represents an enclosed electromagnetic wave of 1/2 λ, analogous to a Möbius band because of the 180° twist. The mass is determined by the wavelength. The incomplete wave period causes an asymmetry in the standing wave and allows a given fraction of the electric field to escape.

The wave can release potential energy by coiling upon itself once, making a double loop and switching chirality in the process. An attempt to compute the value of the elementary charge suggests that the coiled structure is the one occurring naturally.

By convention, the charge of the electron is negative and the electric field points toward it. This implies that the electromagnetic wave processes in a left hand mode through the internal twist. One full twist will take two loops, explaining the 1/2-spin attribute of electrons.

The positron has the same half wavelength, structure, mass and spin as the electron, but opposite charge because of right hand twisting.

The proton and antiproton
The proton and antiproton are built like the positron and electron, respectively. But they have a much higher total energy, giving them a shorter wavelength and more mass.

The neutron and antineutron
The simplest elementary particle with no charge consists of 1 λ, and the neutron apparently does. The measured mass of the neutron is just slightly larger than that of the proton, implying a λ nearly twice that of the proton. It has like the electron a measured negative magnetic moment, suggesting left hand twisting.

The observed 1/2-spin of free neutrons seems to contradict the whole λ, but can be explained by folding. Because of the internal 360° twist, tension is released in the neutron if it folds into two lobes (like converting the number 0 to an 8). In a common plane, the formed two lobes have opposite spins and magnetic moments. By folding again, perpendicular to the intersection, the neutron aligns upon itself and releases potential energy in the process. It now resembles two parallel waves with a 180° twist each, and therefore appears with a 1/2 spin. Locally, at the “hinge” region, a strain is created which makes the neutron unstable. Eventually it breaks down in beta decay, forming new waves.

It is likely that the two lobes of the neutron go through coiling, analogous to that of the 1/2-λ particles. In that case the neutron will consist of two antisymmetric double-loops, with opposite twist compared to the uncoiled neutron.

The antineutron should materialize and behave like a neutron, but with an opposite twist.

Approximate calculation of the elementary charge
The measured elementary charge (e) of 1.602176487 × 10⁻¹⁹ C can be approximated theoretically fairly well by assuming that the charged particle appears in a coiled configuration and that the 180° twist is effectively concentrated near the node of the enclosed wave.

In the electromagnetic wave making an elementary particle, half the total energy is stored in the electric field (the sum of vectors E) and the other half in the magnetic field (B). Using radians, one can let |E| be represented by the definite integral

, where a is a constant. Letting the integral curve twice around the circumference of a circle, the 1/2-λ wave encloses a sphere of radius (r) λ/8π and volume (V) λ³/384π².

Only a fraction of E is not cancelled by the wave itself, and contributes to the charge. This fraction can be approximated by assuming a perfect double circular path and neglecting the effect of the internal 180° twist. Because of symmetry around the line crossing the node + peak (0 and π/2, respectively) and π/4 + 3π/4, the vector sum of E will be parallel to this line. The net electric field will point in the direction of π/4 + 3π/4, and its fraction can be expressed as:

/
= (−a/30)(5 cos(3π) − 3 cos(5π) − 5 cos(0) + 3 cos(0)) / a(− cos(π) + cos(0))
= (−a/30)(− 5 + 3 − 5 + 3) / 2a = 15⁻¹

The effect of not taking into account the 180° twist concentrated near the node will be to underestimate the fraction of escaping field vectors. This is because some of the vectors subtracted from the sum in the computation, actually are perpendicular to the net sum.

Gauss’ law for electric field gives:

hc / λV	=	ε0|E|2
|E|2	=	384hcπ2 / λ4ε0
|E|	=	(384hc)1/2π / λ2ε01/2

, where ε₀ is the electric constant. Also, Coulomb’s law applied on a point charge gives the electric field associated with it:

|E|

=

Q / 4πε0r2

, where Q is the charge of the particle.

An approximation of the absolute value of the charge (Q_est) carried by the particle can now be found:

15−1(384hc)1/2π / λ2ε01/2	=	Qest / 4πε0r2
16(3hc)1/2π / 15λ2ε01/2	=	64π2Qest / 4πε0λ2
(3hc)1/2 / 15λ2ε01/2	=	Qest / ε0λ2
Qest	=	(3hcε0)1/2 / 15
Qest	=	1.531376790 × 10−19 C

So, Q_est/e ≈ 0.956. Apart from the already mentioned cause of underestimation, deviations from a perfect double circular path and possibly other factors as well, can make the estimated elementary charge at variance with the measured value.

As is shown, the absolute value of the charge does not depend on λ in a double circular wave. It will not for wave particles of other shapes either.

The atomic nucleus
For an element above hydrogen in the periodic table, the nucleus consists of a defining number of protons (Z) in a framework of neutrons (N). The “8-configuration” of the neutron is apparently stabilized if a proton aligns with one of the lobes. This may be due to the electric field topography of the proton. The opposite lobe is free to align with another proton and/or other neutron lobes. Thus, several neutrons can build a structure holding the protons. The positions of a proton in a neutron lobe may be restricted by the electric fields from other protons in the nucleus. Protons may therefore share different binding energies with their neutrons.

In a stable nuclide, there must be sufficient framework (N) to prevent overlapping of the electric field vectors from the charged units (Z). On the other hand, if the ratio N:Z is too high there will not be enough protons to stabilize all the neutrons. A neutron will eventually fold onto itself and go through beta decay. A too large Z will destabilize the structure as the separate electric fields more frequently conflict each other, and thereby repel the protons.

By accounting for nuclear spin and nuclear magnetic moment, and also fitting binding energy into the model, it should be possible to find likely configurations for any nuclide. Different configurations may correspond to alternative excitement states of the nucleus. The alpha particle (⁴He²⁺), for example, could be assumed to be in its lowest state when the two protons are sandwiched between the two neutrons.

Conclusions
The presented model, or a modification of it, should shed light on the structure of other elementary particles as well.

The model suggests that both the electromagnetic force and gravitation can only originate from electromagnetic waves. This may give insight into understanding the intrinsic structure of these only two forces of nature.

Several testable predictions can be made from the model, making it falsifiable.

Acknowledgements
HyperPhysics (http://hyperphysics.phy-astr.gsu.edu/hbase/HFrame.html; Department of Physics and Astronomy, Georgia State University, Atlanta, GA, USA) was an excellent source when exploring basic concepts in physics, and Wolfram Mathematica Online Integrator (http://integrals.wolfram.com/; Wolfram Research, Inc.) was a helpful tool when calculating the integrals.