3. Velocity
The first problem with the particle is how does it propagate through the aether? Normally electromagnetic radiation propagates at the speed of light and spreads out in all directions. The particle on the other hand must propagate at some speed less than light and furthermore, it cannot spread out but maintain the identical field intensity and density throughout its journey. Do Maxwell's equations even allow for such a process? A hint to the answer already exists from the last section, where an electric field, which should have propagated away at the speed of light, just stayed put with zero velocity and no change in field intensity. In this section the symmetrical field described in section one is slightly altered by increasing the intensity of the field in a single plane, maintaining cyllindrical symmetry but having an "oblate" field intensity. The derivation shows that there is a propagation about the perimeter of the soliton (a coherent non-spreading wave pulse) at a speed less than the speed of light. This brings energy from the back to the front of the soliton so ultimately transports the whole soliton to a new position at a speed less than light and also less than the wave velocity of the perimeter. This has not been fully demonstrated, but only suggested by this derivation. Another problem is that when the energy is transported to the front of soliton, there is a sort of dead-end since there is no route by which to continue the process. This means, at the least, that the simple spherical symmetrical model of a particle is not sufficient for motion within the aether. This problem will, in fact, be readdressed and solved in Section 23. However, it is still not known whether Maxwell's equations can support this peculiar type of wave and some subtle alterations might be in order. Despite this, what is shown in this section is a fairly strong plausability.
Imagine a spherical symmetric field that is stationary. However, instead of being perfectly symmetrical, the field in a two dimensions is stronger than the field in the other (e.g. Ex=Ey=kEz,k£1). How would this field propagate? The presence of the electric intensity differential allows for the field to propagate since the magnetic field due to the decay of the electric field is no longer perfectly canceled1. This can be represented mathematically by multiplying the field in the two dimensions by a constant such that
(1)
where the unprimed electric field components represent those of a perfectly symmetrical field. k is a constant factor by which we will deform the field. Using these values in the wave equation (1-6) (using cylindrical coordinates and assuming that there is no field in q ) yields
(2)
(3)
As in Eq.(2-3), ¶Er /¶z=¶Ez /¶r,
so substituting this into both equations (in reverse for the second equation) gives
(4)
(5)
Note that this decouple the equations and shows that the transverse component from Eq.(4) is propagating at a velocity slower than light. This velocity is given by:
(6)
Note that the energy must be propagated perpendicular to both the electric and magnetic fields, so it appears that there is an angular propagation around the center of the particle. This would ultimately result in the field in front increasing and the field in back decreasing. This would make the whole particle move at some velocity less than the wave velocity shown here, but related to how much energy is ultimately transfered in the process. While this has not yet been solved precisely, it should be noted that from the equipartition of energy, given a small velocity, approximately 2/3 of the total energy is being transported in the direction of the particle's motion, the other 1/3 being sort of a dead weight which delays the transport of the particle. 2.
Note how in Eq.(6) with no deformation (k=1), the field has zero wave velocity as it should and it could only propagate with the speed of light at k=¥ , that is, it can never be propagated at the speed of light since the transverse field would have to be infinite3.
1. One can imagine an electric vector field that is oriented in only one direction, having a maximum at some point then falling off rapidly to zero in each direction on a surface (as in the following diagram)
Fig.1: Propagation of a uniformly directed electric field.
As the field decayed a magnetic field would be created angular about the center (as shown). This would then decay creating an electric field again, but displaced outward. There would thus be a wave motion radially outward from the initial electric field. If this surface were shaped into a sphere, each region of electric field would try to decay and the angular magnetic field created would always conflict with the adjacent regions. This causes a destructive interference which prevents the magnetic field from being created. Thus the normal propagation is prohibited. (This is a physical explanation of what is being expressed by Eq.(2-1) and Eq.(2-2).)
2. The conditions for particle existence are supported by the postulates when the particle is at rest. I have not yet been able to solve the problem explicitly with velocity and there are some qualitative issues, one being that as the field transfers to the front, it is sort of a dead end, since there is nothing to create a shifted field on the back of the particle. One could imagine that the field could also propagated backwards to restore the symmetry of the particle and by the delay end up shifting the particle in the proper direction. But with this model there is no avenue by which this could occur, therefore, there may be required some subtle alteration of the field equations to allow this or this particle model itself is simple not geometrically correct. This later may be the case as we will see in Section 23. However, while being accepted here to give formal cause to the existence of particle like solitons, this process is not applied in the development of the following derivations, which are valid with the traditional view of charge as well. It is an appealing approach, though, in that it provides a plausible physical model with few parameters.
3. This solves two problems often associated with ether theories. We see here a natural cause for why the velocity of mass cannot exceed the speed of light. Also, in traditional ether theories, mass was still considered a separate independent object, which would have to push the ether out of the way when it moved, or conversely, be affected by the ether in some way. Thus, there were proposed many explanations dependent on ether drag. Here, however, the ether remains at absolute rest, but does not hinder the progress of mass moving "through" it.
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