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Matter as a Solution to Maxwell's Equations
Towards a Theory Of Everything: Part 2
By Ray Tomes, 28-06-95
Concentric Standing Electromagnetic Waves as Particles
There is a stable solution to Maxwell's equations which is equivalent to
a continuous standing electromagnetic wave arranged concentrically about
a point. It is the Author's hypothesis that such stable waves are in fact
the basis of fundamental particles and also exist on many scales from galaxies,
stars and planets down to atoms and sub-atomic particles. Standing waves
of intermediate sizes explain the Rydberg constant and the fine and superfine
structures of spectral lines. Some particles and all atoms are expected
to be composites of different sized waves within each other.
Thought experiment
To give a description of these waves which I call consons, here is a thought
experiment. Imagine a perfect spherical mirror with a light or other EM
source at the centre. Let the source have perfectly coherent output of
a wavelength which divides exactly into the radius of the sphere. Imagine
the source to be turned on long enough for the light to reach the mirror
and return to the centre at which point the source vanishes. This description,
with the size of the mirror taken to infinity or the size of the universe
at which point the mirror is no longer required is the description of a
conson.
Energy distribution
The wave is essentially stable as long as its energy is low. We will come
back to the stability later. The energy density of the wave is inversely
proportional to the square of the distance from the centre. Therefore the
typical amplitude is inversely proportional to the distance from the centre.
If the wavelength is lambda and r=lambda/2 then at multiples of r from
the centre the electric field will be zero. In effect the electric field
oscillates with an amplitude proportional to sin(2pi.r)/r which has nodes
at all multiples of r except r=0 where the function has its maximum value.
Diagrams
The amplitude of the electric field looks like this (O is the centre):
_-_
/ \
/ \
__ _-_ / | \ _-_ ___
\ / \ / | \ / \ / \
---+-----+-----+-----+-----O-----+-----+-----+-----+-----+-----+
\___/ \ / | \ / \___/ `---'
\_/ | \_/
Note that the central part of the wave has twice the length that the other
half waves do. That is, there is a change of phase at the centre by 1/2
cycle. This is due to r changing sign.
A cross-section through the wave looks like this.
__________
/ \ Arranged about a central point are equally
/ ) \ spaced spheres. These are in fact nodes,
/ ____ \ or places where there is no motion.
/ / \ \ In between are regions where shear waves
| / ( \ | are oscillating sideways with maximum
| | . | | amplitude midway between the nodes.
| | | | This sideways motion should be seen as
| \ ) / | The correct physical interpretation of
\ \____/ / Maxwell's equations. Alternate half waves
\ / are moving in opposite directions at any
\ ( / one time. They reverse direction every half
\__________/ cycle of the wave.
Here is a slice through a conson frozen at a moment in time.
Multiple solutions
Actually there are a whole set of solutions to Maxwell's equations which
take this basic form. They all have the same nodal structure and property
that the energy is distributed as the inverse square of the distance from
the centre, but the differences are due to the possible different polarisation
schemes for the light in the wave. It is not possible for all of the light
in the wave to be unpolarised. This is the same situation as the ball of
fur which has to have at least two places where there is a crown. To put
it another way, if the wave is considered to be a displacement of space
(as an alternative explanation to Maxwell's equations) then any rotation
of a spherical shell must leave two points unmoved.
There are additional modes of oscillation possible also, and between
these it seems possible that the explanation for charge, spin and all the
other quantum variables can be found.
Hot matter does not tend to uniformity
One very important thing about this solution is that it does not have an
even distribution of energy throughout space, and does not tend towards
an even distribution. As far as I know all existing models of the universe
make the erroneous assumption that hot matter (EM waves included) will
always tend towards an even distribution of energy over time.
The edges
I have left a bit of a loose end at the edge of the universe which is quite
likely to disturb some readers. It would be possible to close such a wave
in a universe which was a hypersphere, in which case it would have an antiwave
exactly opposite it. However the wave can also be stable in configurations
in which multiple waves are present. These configurations can be exactly
like the arrangement of atoms in a crystal, including a square matrix and
various other forms. In that case each wave may be seen as occupying a
cell which is a cube or some other shape which is space filling. Within
that shape will be a whole number of wavelengths, and even though the distance
from the centre to the edge is unequal in different directions the wave
will still be stable and have a high energy concentration at the centre.
Interaction of waves
It is to be understood that electromagnetic waves do actually have a shear
motion. This means that when two waves pass through each other at angles
they do interact. For typical waves in our normal experience the interaction
is negligible. However for very high energy concentrations there are mensurable
effects. Effects are expected to be of the order of G m / (c^2 r) where
r is the half wavelength. That is why a black hole is a limit, as it has
extreme interactions with everything.
All sizes
The waves can form at any size, including galactic, stellar and planetary
scales. The forces that then apply can explain the observed quantisation
of galactic, stellar and planetary distances. Smaller waves can sit inside
larger waves as baryons do inside electrons and atoms. Much more will be
said about the sizes in a later article.
Stretching of Space
If the vector potential, A, in the alternate form of Maxwell's equations
is taken to be a measure of the actual physical displacement of space then
some interesting things happen. One is that the rapidly oscillating rotation
near the centre of the conson causes the space there to expand. This may
be looked at as a centrifugal force due to the rotation. When I built a
computer model of this I was expecting to have to add some extra terms
to Maxwell's equations to achieve this, but found that it happens automatically
with the equations as they are. This leads to the further interesting consequence
that no matter what scale an observer chooses for the original space metric
that the co-ordinates can still map (by stretching and rotation) on to
any other set, giving a sort of relativity of scale.
Further work
I have attempted to work out the possible equations for various consons,
however I am sure that there are people who can do a much neater job of
it than I have, so I have left them out of here. I am aware that the above
is incomplete. There clearly is such a solution to Maxwell's equations
however. It is important that those more capable than me at this area of
maths should complete this work. In the following articles it will be demonstrated
that this conson solution not only provides a probable model for fundamental
particles but also explains why all the interesting things in the universe
happen at the scales that they do relative to each other.
Non-linearities
Because of the stretching of space there are non-linearities. As mentioned
above the effect is very small for most consons, being of order 1 only
for black holes. Given that a particle such as a proton has G m / (c^2
r) of only about 10^-39 then the non-linearities beyond one radius are
negligible. Even so, with the proton oscillating some 2.27x10^23 times
per second it can be seen that a change of only 10^-41 per oscillation
would result in the proton evolving at the Hubble rate. The nature of this
evolution is the subject of the next instalment.