ABSTRACT: I have reverted to the classical concepts of ether, space and time to develop a pure wave theory of the electron (or fermion) as a simple ROTATING WAVE. The essential postulate is that an electromagnetic wave is brought into classical rotation by a local binding energy. The spin model then yields the required phenomanae of charge, relativity, mass , gravity, and quantum mechanics in a naturally derived and graphic fashion.
Acknowledgements: I wish to thank Ernest Von Rosen M.Sc.Physics for his support and earlier attempt at publication. Also, thanks goes to Steve Belleguelle for his program AmiWeb. I first heard the term "Wavicle" mentioned by Dr.Carl Sagan. I admire H.A.Lorentz for all of his work as I do others, particularly Albert E.Einstein, all of whom I could not envisage the Wavicle without their diligent math and logic. To all of my teachers who inspired me to "always learn to be aware of what we are dealing with" - thank you. W.H.F.Christie
It is well known that in 1887 A.A.Michelson and E.W.Morley could not detect any luminiferous ether by detecting a path length difference between light transmitted in the direction of the earth's travel through the stationary ether and perpendicular to that direction by a fringe shift. In 1889, H.A.Lorentz suggested that the null result of this famous experiment might be due to an actual physical length contraction of the interferometer as measured in the direction of its motion of the earth's orbit. According to Lorentz, the length contraction was merely due to an electrodynamic effect on physical processes within the electron particle make-up. Lorentz also suggested that such physical processes had a cyclical function, or local time, which must slow down or dilate in conjunction with their length contraction. However, the reasons for local time dilation and length contraction as suggested by Lorentz required a complicated and unconvincing model of an electron to be composed of many like charges bound by a nonelectric force infinitely strong at the centre. Ironically, the Lorentz transformation equations of time dilation and length contraction survived to form the basic math of a much more acceptable, although revolutionary, theory put forth by Albert E.Einstein in 1904.
Albert E.Einstein's Special Theory of Relativity was much more acceptable because it was simple and universally consistent. His Special Theory stated that all laws of nature, including mechanical as well as electromagnetic field laws, must be invariant with respect to the Lorentz transformations. He proved the consistency of his theory by applying the principle of equivalence to the Special Theory and created his General Theory of Relativity, which in turn correctly predicted the influence of a gravitational field on light.
Also in 1904, Albert Einstein demonstrated that light behaved as individual particle - like packages of energy called photons' in order to explain the photo-electric effect. Thus, in a quasi-corpuscular theory akin to that of Sir Isaac Newton, Einstein stated that the nature of light must have a wave-particle duality. However, matter was still considered to be only particle-like until twenty years later when Louis de Broglie suggested that matter as well as light might have a dual, wave-particle nature. Both the Lorentz and Einstein interpretations of the Michelson - Morley experiment required that the interferometer be "particle-like" only. I began to wonder how the Michelson - Morley experiment might have been interpreted had the dual, wave-particle nature of matter been established prior to this famous experiment. Furthermore, I considered that matter might simply be a localized wave and that its particle-like nature is simply a phenomenon derived of its localization.
According to Einstein's definition of simultaneity there is no need for a physical length contraction of the Michelson-Morley interferometer. However, if as Lorentz contended, the length of the interferometer did indeed contract in the direction of its motion, then there could not be any fringe shift detected on the interferometer and real time could remain constant. One could disagree with Einstein's assumption that his operational definition of simultaneity is valid in any inertial reference frame and revert back to the nineteenth century classical view of light which accepts such a definition as "valid only in the rest frame of ether". My alternative conclusion about the experiment, coupled with the fact that matter is localized energy and that energy in the form of electromagnetic waves is indeed transmitted as light, compelled me to find a basic form of matter: a particle equivalent to a localized electromagnetic wave with an innate time cycle that would explain the fact of time dilation and a configuration that would require its own physical length contraction. Most other forms of matter that arise out of the electromagnetic ether would only be variations or further developments in the evolution of this principle form of matter which is from hereon referred to as the "Wavicle". The electromagnetic ether field is here stated as a universal reference frame only to a degree of which our perception is capable. The intent of this work is, therefore, to explain by the electromagnetic ether field, the creation of the principle form of matter and the phenomena of relativity, gravity, and quantum mechanics beginning with two postulates which compare with those of Einstein's theory as rephrased by Casper and Noer in "The Evolution of Physics" on p. 330:
I. The Principle of Relativity According to the Wavicle: No physical measurement can distinguish one inertial reference frame from any other inertial reference frame - because - such distinction is obviated by changes in the actual physical time cycle, length, and mass of a particle with respect to that particle's motion relative to the stationary, electromagnetic ether. Such changes are illustrated by the Wavicle.
II. Independent of the motion of the light source, only wave fronts of light which proceed in a straight line with respect to a Euclidean universal reference frame (defined by the stationary electromagnetic ether) always propogate in empty space (the vacuum state of ether) with a definite velocity C relative to that universal reference frame. Other wave fronts which do not proceed in a straight line, propogate with an angular velocity such that all wave fronts remain planar.
From the two postulates above, I can simply state that the null result of the Michelson-Morley experiment is due to an actual physical length contraction of the interferometer in the direction of its motion. The actual physical time cycle of the whole interferometer apparatus slows down in unison and therefore time is not the culprit. The two postulates above also provide an explanation to the outcome of another experiment: the low rate of decay of mesons entering the earth's atmosphere is due to a combination of some degree of actual physical length contraction of the earth's atmosphere and some degree of actual physical slowing of the time cycle decay of mesons.
2. The Electron as a Basic Form of Matter
The electron and its antiparticle, the positron, elegantly fit the requirements of this basic form of matter. The electron wavicle complies with the universal laws of electromagnetism and the previous two postulates. By doing so, it inherently explains relativity, mass, and quantum mechanics. According to the electromagnetic theory of light, the change in the electric field of the photon induces a magnetic field and, conversely, the change in the magnetic field induces an electric field. These changes are made with particular direction at the speed of light through the electromagnetic field, which was once labelled as "ether". In order to exist, the photon must move forward through this electromagnetic field such that the changes in both the electric and magnetic fields induce each other. The photon cannot be at rest with respect to the electromagnetic field and therefore does not have any rest energy or rest mass. However, there are two types of motion: translational (linear) and rotational (angular). We perceive the photon to move forward through space in translational motion. If the two vectors of the electromagnetic wave of a photon could be brought to spin, not as in the case of a circular polarized wave, but as in the case of the electron model shown in Figures 1 and 2, then it might create a magnetic dipole and an electric monopole.
Just as the magnetic field of a photon changes in translational motion, thus inducing an electric field, so does the magnetic field of the electron change in rotational motion thus inducing the electric field of the electron. Likewise, as in the photon, the change in the electric field of the electron induces the magnetic field of the electron. In compliance with the electromagnetic theory of light, the electron wavicle (and likewise the positron) is nothing but one half of a gamma photon brought into rotation by a binding energy equivalent to another half of a gamma photon. This is confirmed by experiment. If a gamma ray is brought under enough localized energy, an electron-positron pair will be created. All of the energy will be conserved as rest mass and kinetic energy of the particles. When the electron and positron are then brought together in annihilation in free space, they produce two photons whose total energy is equal to that of the original photon and the localized energy from which the particles were created. When positive and negative charged particles are oscillated towards each other they will induce an electromagnetic wave which spreads out in every direction except the path of their oscillation. Similarly, two photons are emitted in opposite directions during the annihilation of an electron-positron pair in free space. Note that the counter-clockwise spinning electromagnetic wave which is shown as a dark straight line in Figure 1 is actually a sectional view of a planar wave front with the maximum (or nodal) electromagnetic magnitude. Other lines representing lesser magnitudes are not drawn for the sake of simplicity. Also note that the speed of the wave must be greater at distances further from the centre of spin.
As noted in Figure 2, the singular connection between the poles of the electron and thus the curved magnetic lines might be due to a reversal of the electric and magnetic directions with respect to the direction of rotation at very low speeds and thus in close proximity to the axis of spin. At a greater radius the rotating nodal wave front might be interpreted as a virtual particle blinking into existence with negative charge while at a lesser radius the rotating nodal wave front might be interpreted as a virtual particle of positive charge. The boundary between the reversal might not be so definite. Furthermore, the reversed fields might provide a binding energy which sustains the speculated rotating wave of the Wavicle.
It is not fully apparent how the originating electromagnetic wave of the photon, as illustrated in Figure 3, is brought into rotation by a local binding energy. However, given a nodal planar wave front which extends to infinity in accordance with the wavicle, one can deduce that any curvature in the photon's translational path will require a definite point of rotation for that wave front and thus an associated magnetic dipole along with a symmetrical electric field. Therefore, any curvature in the photon's path will immediately result in fermion type particles, albeit with negligible and short-lived mass. Given sufficient and local binding energy, the curvature will result in the sustained electron and positron as illustrated in Figure 4.
Figure 3. Photon wave existing in translational motion.
Figure 4. In this case the formation of the electron (left) and the positron (right), each with their own self-sustaining and rotating electromagnetic wave, conserves electric field (charge), magnetic moment, and spin angular momentum.
One immediate variation of the foregoing concept of a rotating electromagnetic wave could be that which creates a magnetic monopole and an electric dipole. Or perhaps the rotating wave might show that the electric and magnetic fields of light are merely two different directions of stress, distinguished from each other only upon the creation of the dipole of the electron. Other immediate variations, such as the muon and tau fermions, could simply be created by the rotation of waves which have different frequencies than that of the gamma photon. Virtual photons, gluons, and gravitons causing electrodynamic, chromodynamic, and gravitational interactions amongst particles might simply be the rotating waves of the particles themselves. However, any speculation of immediate or evolved variations of the rotating electromagnetic wave form will not be further discussed in this paper.
3. Relative Time (Dilation) Redefined
According to the Special Theory of Relativity, the spin of a moving electron, as measured by an outside observer, must be slower than the spin of a stationary electron. However, without regard to relativity, the angular velocity of an electron wavicle's spinning electromagnetic wave must slow down as it gains translational motion simply because the speed of light remains constant at the same distance from its axis of spin. This is illustrated in Figure 5.
Cylindrical model of the vectors of the spinning electromagnetic wave of an electron with translational velocity V and spin 1/2.
At a certain distance from the centre of the electron and midway between its magnetic poles, let us say that the electron wavicle's electromagnetic wave is moving at a tangential velocity (here defined as the rotational velocity) about the centre. If the electron is stationary (without translational motion), then the time it takes the electron wave to complete one cycle about the centre will be If the electron is given a translational motion of velocity , then the new time it takes the electron wave to complete one cycle about the centre can be called . The permeability () and the permittivity () of free space will remain constant about the same radius from the axis of the electron's rotation regardless of whether or not the electron is in translational motion. Therefore the resultant rotational velocity () of the electron wave must decrease as the translational velocity () of the electron increases in the direction of the axis of the electron's spin.
From Figure 5:
Without translational motion ():
With Translational motion ():
which agrees with the time dilation formula of the Special Theory of Relativity.
Since this concept of the electron contains only a spinning electromagnetic wave, then it is only the electromagnetic wave that exhibits time dilation. Furthermore, according to Quantum Mechanics, ionic and thus mechanical functions must slow down merely in response to the time dilation of the electron's electromagnetic wave. Therefore, Real Time does not slow down according to the Special Theory of Relativity; rather, the fact of time dilation is due to the electron's (and positron's) spin of its electromagnetic wave coupled with the constant speed of light at a given radial distance from the axis of the electron's rotation.
4. Required Length Contraction
The direction of propagation of an electromagnetic wave must always be at right angles to the direction of both its electric and magnetic fields. Therefore, a wavicle with translational motion will have each rotating planar wave front reoriented such that each respective resultant velocity , as previously shown in Figure 5, will be normal to the wave front. Figure 6 shows a side elevation of an electron moving with a translational velocity and viewed at an observed radius . Since the electric field of an electron must always point to the centre, then the rotating wave of a moving electron wavicle will be inclined from the normal axis of rotation of a stationary electron such that the rotating wave of a moving wavicle will trace a helical path through the electromagnetic ether field.
Side Elevation of moving electron wavicle with translational velocity V at an observed radius
From Figure 6 we can see that for a given electric field at an observed radius , the length of the vavicle contracts in the direction of its motion by a factor of such that:
which is the length contraction formula for the transformation laws of the Special Theory of Relativity.
Since the equilibrium distance between wavicles is assumed to be ultimately determined by their electromagnetic fields, then the distance between wavicles should contract correspondingly in the direction of motion of such wavicles. Therefore, the required length contraction is due to the wavicle's spin of its electromagnetic wave, coupled with the fact that the direction of propagation of an electromagnetic wave must always be at right angles to the direction of both its electric and magnetic fields.
Note that as indicated in Figure 6, the magnetic poles of the electron are displaced from the line of the electron's translational motion, thus inducing the required magnetic field of a moving field of a moving charge. At any one instance this displacement would be characterized by two polar cones: one extending out forward and one extending out backward from the direction of the electron's motion. A stroboscopic detection of this polar cone by an apparatus similar to a cathode ray oscilloscope would prove the physical length contraction of the electron as required by the concept of the wavicle.
Figures 7, 8, and 9 further clarify how the configuration of the wavicle transforms when put in motion. Figure 9 more clearly illustrates the forward polar cone (rear polar cone is hidden) and illustrates the induced, right-handed magnetic field.
Figure 7. Stationary Wavicle Oblique Elevation of stationary electron wavicle with magnetic lines of force B (electric field not shown here for simplicity) and nodal wave front spinning with tangential velocity C.
Figure 8. Moving Wavicle Oblique Elevation of moving electron wavicle with magnetic lines of force B and nodal wave fronts reoriented at an angle as previously illustrated in Figure 6.
Figure 9. Moving Wavicle Creating a Right-handed Magnetic Field - Oblique Elevation of moving electron wavicle with wave fronts of lessor field strengths shown as well as the nodal wave front (highlighted lines with arrows). This figure clearly indicates a right-handed induced magnetic field.
5. A Definition of Rest (Energy) Mass
Rest Mass is localized energy without translational motion; therefore, according to the concept of the electron wavicle, the energy of Rest Mass called Rest Energy () is simply the Rotational Energy () of the half-photon which is rotating in one spot, plus the Binding Energy () which keeps the half-photon in that rotation. The half-photon has a kinematic rotational energy which is equal to the binding energy such that the electron's total Rest Mass Energy . is thus equal to a whole photon's energy () as given by Planck's constant () and the frequency of the photon ().
According to Rotational Kinematics, where is the rotational energy, is an assigned constant for the derivation of rotational inertia ( ), is a non-relative mass of a symmetrical spinning object, and is the velocity of spin at the radius () of the object, then can be derived. Since, according to the wavicle concept, all mass is assumed to be derived from electromagnetic waves, then the rotational energy and non-relative mass of the stationary wavicle can be equated in the same manner. By convention, is assigned the value of unity for the wavicle, such that all objects are thus correctly allowed to have their rotational inertia computed as the summation of point wavicle masses at their respective radii. Therefore, for a stationary electron and a stationary electron will thus have a total rest energy .
The rotational energy of a stationary electron wavicle can also be defined in kinematic terms by its angular frequency and its angular momentum such that , where is the frequency of electron spin. Since we know the values: joule-sec. and joule-sec., then which is as it should be because according to Quantum Mechanics the spin of the electron must be quantized with respect to the originating gamma photon. Finally, since , then . It is interesting to note that if one were to assume that the rotating electromagnetic wave of the wavicle had a consistent momentum and total energy at any radius, then the mass would vary inversely with the radius. Whether or not such consistency is required by the principle of conservation, relative mass with respect to location is not dealt with in this paper.
6. Relative Mass with Respect to Motion
According to the foregoing Figures 5 and 6, an electron with translational motion will have an angular momentum which according to the conservation of momentum is equal to the angular momentum of a stationary electron. A relative mass () is defined with respect to its translational velocity () because the configuration of the mass as indicated in Figure 6 and 9 changes with respect to . Note the increased flux or density of electromagnetic lines as illustrated in Figure 9.
which agrees with the Special Theory of Relativity where is the relative mass with respect to translational motion. The total energy () of an electron must be the sum of its rest energy () plus its kinetic energy () due to its translational motion such that:
while is constant force over a distance to give the electron translational motion,
Furthermore, the translational momentum () and resultant momentum () of a moving electron wavicle can be vectorially related such that:
and when multiplying by :
According to the wavicle concept, the correct relative mass and its associated momentum and energy have been derived with respect to translational motion regardless of the Special Theory of Relativity because real time has been kept constant.
7. An Electromagnetic Wave Theory of Gravity
According to the Wavicle, free space or vacuum is defined as the electromagnetic ether field in which light and matter waves are able to propogate. The basic premise of this theory of gravity is that the binding energy () holds the classical wave of the wavicle in rotation by affecting the permittivity and permeability of free space. Thus any other incidental wave will similarly be affected by the binding energy of the wavicle, although indirectly and to a lesser degree. Figure 10 illustrates a passing test photon with a given Huygen's wavelet at distance from the centre and midway between the magnetic poles of a single stationary electron wavicle. Regardless of the wavicle charge, the wavelet will be permitted to go at the velocity either in the same or opposite direction of the rotating wave.
Figure 10. Vectorial description of wavicle's rotating wave showing resultant gravitational acceleration of wavelet.
While is the tangential velocity of the rotating wave at distance , and is the frequency of the wavicle's rotation, is a variable dependent upon the strength of the vector field generated by the wavicle at radius . However, as illustrated in Figure 10, changes with respect to time and direction and therefore the photon wavelet would also be accelerated by the amount:
In other words the plane of the passing test photon or electromagnetic wave would be bent to some degree by the rotating wave of the wavicle and would account for the phenomenom of gravity. Obviously, according to the wavicle concept, a Huygen's wavelet closer to either of the magnetic poles, but at the same distance from the centre of the electron wavicle, will experience less gravitational acceleration. However, the second premise of the theory of gravity is that uniformity of gravitational acceleration at a constant distance about a spherical mass is simply due to the fact that such mass is composed of many wavicles at various angles of spin and random motion. Therefore, while the electron has a gravitational field severely distorted by its magnetic poles, larger non-elementary particles of matter have more uniform gravitational fields at a given radius. While is the real constant assigned to the gravitational acceleration of the basic particle, a modified can be assigned in a limiting case to the more uniform gravitational fields of larger particles at a given radius. However, as is only an average value of , then it can be renormalized to for simpler calculations that follow.
Since the wavicle must comply with Gaussian law at distances beyond which charge is screened, then the macroscopic laws of gravity must be able to be deduced in the same way. Gauss's Law gives a connection between the flux () for the Gaussian Surfaces () and the net charge () enclosed by the surface: where and is the electric field strength; is the universal permittivity constant. Similarly, a universal constant , which can be proved by deduction, operates on the gravitational field strength () and its surface integral such that:
Since: and the surface integral is a sphere, then:
and from above: and thus: .
However, according to the General Theory of Relativity, our clocks run faster at higher heights and therefore, our acceleration at higher heights is actually greater than it appears to be such that:
where and therefore
where is the apparent acceleration and is the apparent constant mass.
8. The Principle of Equivalence and the Gravitation of Matter
Albert Einstein predicted correctly by the "principle of equivalence" in his General Theory of Relativity that matter would gravitate or bend light and slow down time. He found a deeper significance than mere coincidence that a gravitational reference system could be made equivalent to a uniformly accelerated reference system because it enabled him to extend his Special Theory of Relativity to his all-encompassing General Theory. However, according to the foregoing Electromagnetic Wave Theory of Gravity, light is simply bent by the rotating wave (or matter wave) of the Wavicle. Since the rotating wave of the Wavicle is itself light, then the path of that rotating wave will be bent in the same manner by another Wavicle; thus two Wavicles will attract each other by such gravitation. The gravitation of a Wavicle by a larger mass centred at point is illustrated by Figures 11 and 12.
Figure 11. Sectional plan view of rotating wave front of test wavicle centred at distance from mass centred at .
Figure 12. Sectional Side View of rotating wave front of wavicle centred at distance from mass centred at .
Since the motion of the electron Wavicle is limited to the direction of the axis of its spin, then the acceleration of the wavicle as illustrated in Figures 11 and 12 need only be considered in the direction of the axis of its spin. In figure 11 the rotational velocities of the Wavicle's wave front are indicated as arrows (and ) coming up from the page and the wave front lies in the plane of the page. Note that at different distances along the same radial direction from , the ratio of velocities and, therefore, the instantaneous effects of both the Wavicle and larger mass M on the permittivity and permeability of free space are consistent. However, the lines drawn radially from point reveal different rotational velocities at a constant radius where the permittivity and permeability of free space is affected to some degree by the larger mass . Since the permittivity and permeability of free space is also instantaneously affected by the Wavicle, such that the observed rotational velocity of the Wavicle increases in direct proportion to its observed radius of spin, then the wave front of the Wavicle's rotating wave must remain planar.
However, as illustrated in Figure 12, the Wavicle's planar wave front will be bent instantaneously as it rotates about its axis of spin which is directed towards . Although the wave front of the Wavicle's rotating wave remains planar, its inclination with respect to its axis of spin increases as the wave rotates and the Wavicle is thus accelerated to point . The rotating wave of the Wavicle equally shows by Huygen wavelets how light is gravitated or bent and thus how matter is gravitated.
Also, the Electromagnetic Wave Theory of Gravity shows by Huygen wavelets how light is permitted to go at greater speeds at distances further from the centre of the Wavicle. Again, since the rotating wave of the Wavicle is light, then the velocity of that rotating wave will be affected in the same manner by the proximity of another Wavicle. Hence the spin frequency of the two Wavicles in close proximity will be less than that of two Wavicles separated by a greater distance. Therefore, the Electromagnetic Wave Theory of Gravity enables a reinterpretation of the General Theory of Relativity by predicting, first, the gravitation and slowing down of light and, second, the gravitation and slowing down of the time cycle of matter. The "principle of equivalence" can be completely explained by the equivalence of light and matter according to the concept of the Wavicle.
An interesting corollary can be made at this point. If one were to counter-revert to the conventional physics of relativity and reassert the constant velocity of light in any direction regardless of the motion of its source, the rotating wave theory of the Wavicle might still be able to be constructed. In such a theory the binding energy would affect space and time such that the speed of light remains constant at any radius from the centre of the Wavicle and thus sustain the planar rotating wave fronts of the Wavicle and its definition of mass. The foregoing Electromagnetic Wave Theory of Gravity and the "principle of equivalence" according to the Wavicle might be deduced in the same manner. While such a theory would be more acceptable from the point of view of conventional physics, it would be more difficult to visualize and deduce derivation of the and explain quantum mechanics according to the Wavicle and the following.
9. A Graphical Derivation of the in the General Theory of Relativity
Tensor math forms the mathematical foundation upon which the laws of Einstein's General Theory are defined. The Wavicle also shows how tensor math allows for the calculation of intrinsic curvature of space-time. According to relativity, it must be just as valid in one reference frame to analyze the path of free falling light influenced by the gravitational field of a stationary mass in order to describe the curvature of space-time as it is to postulate in another reference frame the curvature of space-time by the mass energy density tensor of an attracting mass which is in motion. Furthermore, according to the principle of equivalence of the Wavicle, the analysis of the path of free falling light in the the rotating wave of the Wavicle must be equivalent to describing the gravitational field about a stationary attracting mass. The velocities of the rotating wave of the test Wavicle in one reference frame must correspond to those of the mass energy density tensor of an attracting mass in another reference frame.
Whereas in the General Theory of Relativity the attracting mass is defined by the mass energy density tensor and related to the Reimannian curvature of space-time, the following derivation of the is directly deduced by first defining the path of the Wavicle's rotating wave as a geodesic on a generated surface and then second deriving the intrinsic curvature of that surface. Consider the helical path of the wavicle's wavefront as illustrated in Figure 13. The helical path lies within a generated helical surface that can be described in a two dimensional space-time by: or in four dimensions by: . Also, the helical surface extends radially from the centre of axis of spin and is normal to the wave front. Note that: as illustrated before in figures 5 and 6. The generated helical surface can equally be defined at any angle about the Wavicle's axis of spin.
If we are to construct such a helical surface in which the helical path of light is constant (for each radius from the Wavicle's axis of spin) as in the non-accelerating reference frame, then the must be equally gauged from the Wavicle's axis of spin in exactly the same way that and are such that , just as and where as shown in the following Figure 13. If the is not gauged, then the resultant direction of light would not be consistent with at any radius from the Wavicle's axis of spin. The same can be argued for the "uniformly accelerating reference frame". Thus the intrinsic curvature of such a helical surface can only be defined properly if all of the velocities are gauged the same way.
Certainly, the corresponds to the rotational velocity of the Wavicle while corresponds exactly to , so the must as a result correspond to the translational velocity of the Wavicle. While, the velocities of the mass energy density tensor in the current General Theory of Relativity are not gauged in the same way, it could be shown through long arduous derivations that the same basic result (with various, anomolous extra terms) can be derived with the gauging and non-gauging on any of the velocities. In this article the basic result of the derivation is found with all of the velocities gauged equally (thus all on the same footing) and the full comparison between non-gauged and gauged velocities is left for future discussion.
Helical Path of Wave Front which lies in the Helical Surface generated by . The calculation of the of the Helical Surface (by defining the with respect to the pointing vectors) leads to .
While any helical surface is extrinsically curved, it can be shown that the helical surface of the Wavicle does not necessarily have intrinsic curvature. If there is no attracting mass, then the axis of spin will not change and the above generated helical surface will be intrinsically flat. Such a helical surface when unwrapped from its helical path will become extrinsically flat. Even if there is a plane of mass infinitely extended everywhere at right angles to the wavicle's axis of spin (as in the case of the "uniformly accelerated reference frame"), the helical surface will still have no intrinsic curvature. Such a helical surface when unwrapped from its helical path will only have extrinsic curvature like that of a "snow shovel" and no intrinsic curvature like that of a "bowl". However, if there is an attracting mass which is not aligned the Wavicle's axis of spin, then the path of the Wavicle and its axis of spin will change and the helical surface will have intrinsic curvature. Also, likewise, if there is an attracting mass which is symmetrical with and aligns with the Wavicle's axis of spin, then the helical surface will have intrinsic curvature.
The helical path illustrated in Figure 13 can be described also by: such that the curvature invariant with the use of the metric tensor and Einstein's summation convention of summing on repeated indices. According to Minkewski space-time coordinates, . Note that the tensor relationship, which is graphically illustrated by the Wavicle, can be more fundamentally described by the equation: in which is an equally valid metric tensor and now . This enables us to eliminate the imaginary component and construct the from conventional base vectors. From here on the summation convention will be used implicitly and the such that to keep things simple. At the conclusion of the following derivation of , one can then compare the wavicle with that of the Minkewski space-time coordinate system and the mass energy density tensor .
The intrinsic curvature of the helical surface can be calculated by the curvature tensor and sufficiently by the Ricci tensor which is the contraction of the curvature tensor. and
Now let a vector and another vector
Summing on to get the contracted Ricci tensor we find that:
The can now be interpreted by partial derivatives of the pointing vectors which are shown in Figure 13:
Note that there is no gauging on as its base vector is always in the direction of the wavicle's axis of spin while indeed for the derivations that follow. Since the model in Figure 13 is in 3-dimensional space, then we can define with respect to three fixed, euclidean base vectors:
and by the chain rule:
and since we can find one quadrant in 3-dimensional space where indeed holds true, then for the purposes of deriving intrinsic curvature we can state:
Also, the velocities are gauged from the axis of spin such that:
The same chain rule can be applied to the partial derivatives of the velocities such that:
The following underlined and bracketed terms cancel in :
Reversing the indices according to we find:
Or in a more simpler vectorial description with bars above:
from the above equation for we find:
and thus: .
The first group of equations are further resolved:
And the second group of equations are further resolved:
And for the last four equations we find first:
Note that: is at right angles to and
Then gathering the reduced terms above in we have:
and we find:
and according to tensor calculus we find:
which compares some what with the Einstein tensor where the mass energy density tensor has been written.
Also from: we find that:
Thus the curvature scalar and
and such that
10. An Explanation of Quantum Mechanics
The concept of the Wavicle simply explains why Plank's constant () is intimately included and quantized in the characteristics of the electron and all cyclical functions. According to the Definition of Rest (Energy) Mass, automatically enters any equation of matter. Also, since the two nodes of the wavicle rotate, the effect of their electromagnetic and gravitational field strengths on an adjacent particle must vary in a cyclical fashion. Therefore, an orbiting electron wavicle must have a distance from the nucleus which undulates in direct correspondence to the electron's spin. Relativistic effects of the orbiting electron and nuclear particles would of course further define the path of orbit. Furthermore, since the nuclear particles themselves spin and therefore must have cyclical variations in their effective field strengths, then the electron wavicle must make an integral number of spins for each orbit in order to stabilize in that orbit. Thus orbital energies must be quantized with respect to spin or rotational energies ().
Matrix quantum mechanics requires that the energy level of an electron orbiting in an atom be where is the frequency and is the number of typical photons which are absorbed or emitted. According to Rotational Kinematics, the energy level should be equivalent to the total angular energy of the electron wavicle, which is equal to its orbital energy plus its innate rotational energy (). The rotational energy explains why the ground state or "zero point" energy of the electron in an atom must be where for an electron with no orbit.
Also, the concept of the wavicle simply explains why matter has a wave probability distribution. Since the wavicle is purely a rotating electromagnetic wave, then the probability of detecting the wavicle near some point in space should be described by a wave function. The derivative operators, postulated by Erwin Schrodinger, are simply deduced from the basic wave equation of the wavicle and the effects of relativistic quantum mechanics are graphically illustrated with the addition of potentials. The only real wave function of the wavicle is that which describes the amplitude at a certain time and length as measured along the wave's helical path shown in Figure 14.
Figure 14. Cylindrical model of the helical path of real wave of amplitude and measurements of : in the direction of the particle motion and in the direction of the wave's rotation.
Given an arbitrary maximum amplitude , wavelength , and spin angular velocity , the sinusoidal real wave function of the travelling wavicle must be defined as:
Also, as illustrated in Figure 10:
where and are measurements of in their respective directions of motion and rotation of the wavicle. Since we know that , then:
Deriving twice with respect to , , and we find that:
and thus: with , , and as either simple variables or derivative operators. Of course, the energy operator becomes:
In comparison to Schrodinger's postulates, the two new factors: and are deduced along with their respective derivative operators and perhaps heretofore undetected because: at low speeds relativistic effects are not discernible anyway and at high speeds approaches unity while is negligible in comparison to . With regard to these additional factors a review of the Davisson - Germer type of experiments may prove supportive of the Wavicle concept.
The momentum operators may also be related in the vector sum: . Furthermore, if the wave function is converted to a column vector, then the momnta may be related by the sigma matrices of Paul A.M. Dirac such that: which when squared satisfies: . The Hamiltonian () of a particle with charge (), moving in an electromagnetic field of vector potential and scalar potential , may be derived with the use of the Lagrangian function to yield:
According to the wavicle, the same result can be found simply by finding the vector sum of the momenta and potentials as illustrated in Figure 15.
Figure 15. The vector potential is added to the 3 - space vector momentum while the spin momentum is maintained at right angles to the direction of motion. The dotted lines indicate the resultant momenta with the added potentials.
The newly introduced potential has a magnitude and direction such that the spin momentum is always conserved in magnitude and changed only in direction in order to maintain itself at right angles to the direction of motion of the particle. The wavicle is subjected to the same conditions of Dirac by applying the momenta and their respective potentials to the column vector wave function, again with the use of sigma matrices, in the vector sum:
After squaring both sides of the equation, the same extra term of Dirac indicating the magnetic momenta of the particle is derived when the potentials are not considered to be derivative operators themselves and thus non-commutative with the momenta. However, as the wavicle illustrates, the combined momenta and their respective potentials should both be considered as derivative operators and thus commutative, because the addition of potentials refers only to another state of the particle, ie. with added speed and direction at a later point in time.
11. Possible Solutions to Paradoxes of Modern Physics
The foregoing concept of the electron wavicle might resolve well known paradoxes of physics such as the wave-particle duality of both light and matter, the EPR (Einstein, Podolsky and Rosen) paradox, and the age-old Zeno's paradox of motion.
By this concept one would expect the electron to be affected by a double diffraction slit in just the same way that a photon would, simply because the electron is composed of half a spinning photon plus binding energy. Similarily, the photo-electric effect might be explained by the consequent one-on-one relationship of the photon and the electron wavicle.
In 1935 Einstein, Podolsky and Rosen published a paper on a thought experiment which stated that if the position and momentum of only one of two particles is measured after an initial measure of their combined position and momentum, then the final momentum of the two particles cannot be the same unless there is some kind of instantaneous "action at a distance" or "non-local reality" between the two particles. However, modern experiments, which have measured the polarization of twin photons instead of position and momentum of particles, have confirmed the "violation of Bell's inequality" and thus support the "non-local reality" of Neils Bohr's Quantum Theory and the EPR paradox. The apparent "action at a distance" might be simply due to the action of the infinitely extended rotating wave of the wavicle. The apparatus which interferes with one photon while measuring its polarization would, according to the wavicle, also interfere with its twin photon of equal and opposite momentum. The twin photon, although at a greater distance from the centre of the interfering apparatus, would still be affected by the same electromagnetic wave function of that measuring apparatus. The locality of a particle might be thus better interpreted as the centrality of spin of the wavicles's electromagnetic wave.
Zeno's paradox prescribes discrete motion of particles. However, the electron wavicle is transmitted through the electromagnetic ether field as a wave. Ether particles, which may also be wavicle type localized forms of energy predicated upon a substrata energy field, perhaps enable the creation and transmission of electromagnetic waves and move themselves very likely with speeds greater than that of light. Given an infinite amount of real time and space, an infinite number of energy strata could have evolved into something as substantial as light and thus the existence and motion of matter.
The Wavicle model does yield additional insights which might stimulate unification theory as well as provide models for teaching current theory. For example, by applying the Definition of Rest (Energy) Mass according to the Wavicle, a radius , at which a hypothetical wave of the quark is rotating at the speed of light, can be defined as:
where and are the stationary masses of the electron and quark respectively. is, of course, the radius of the electron wavicle at which its rotating wave is travelling at the speed of light. the value of not only complies with the Regge Trajectories but is curiously calculated to be about centimeters or exactly one-half the distance to which the strong force extends. Such speculations might provide some enlightenment upon current QCD theory. Also, any energy expended by the wavicle model in its coupling with other particles or waves would, by its definition of mass, require its gravitational field to decrease as the wavicle's time cycle of spin slows down. The wavicle's inherent connection between gravity, spin and time could thus provide an alternative and elegant explanation to the exact rate and homogenious character of the expansion of the universe.
The fact that the Wavicle allows us to naturally derive the velocity vectors plus the and the extra term: in the in a clear graphic fashion, might provide more insight to the General Theory of Relativity. Although it might be considered a very radical departure from the conventional way that physicists view the cosmos, the Wavicle does provide a consistent explanation of charge, relativity, gravity, and quantum mechanics. Both relativity and quantum mechanics are inseparably innate to the real wave equation of the wavicle. Furthermore, the wavicle function clearly explains why a magnetic field is induced by the motion of a charged particle. Finally, such a theory might not have been so radical if the wave nature of matter had been established prior to the famous Michelson-Morley experiment.
Measures have already been taken to copyright this theory and I do reserve all rights to this series of files on "The Wavicle:". I expressly prohibit any sale of this series of files on "The Wavicle:" in any form without my prior written consent. I do encourage the wide distribution of this theory for education, but I do ask that all documentation and files be kept together under the acknowledgement of my authorship.
Some Personal History
I enjoyed Physics in high school as I was intrigued by the mystery and admired its true logic. However, I despaired half-way through my undergraduate degree that relativity could not be understood intuitively. While turning to the macro social theories in economics, anthropology and politics, I still wondered about the mystery of "something other than nothing or something". In my last year of Architecture school during 1978, while trying to resolve a concept for my graduating project, one of my mentors impressed upon me how important it is to "pull back and be aware of what we are dealing with". At the same time, I was reading a book "The Evolution of Physics" by Albert E. Einstein and his understudy Leopold Infeld and I realized that by "pulling back" from the the Michelson-Morley experiment, one could interpret the null result of fringe shift in a different manner, ie., everyone and the apparatus were doing the same thing that light was: being transmitted through space and all together fringe shifting (so to speak) - so there was no net fringe shift. After trying various concepts and acknowledging the attempt by H.A.Lorentz, I then realized the spin (rotating wave) model of the Wavicle in 1984 and then began to study Physics again in earnest thus successfully interpreting the Special Theory of Relativity by the Wavicle. The Wavicle agreed with my philosophical concept of "something other than nothing or something" as being made up of infinitesmal disturbances which evolved into substantial light and matter. I made failed attempts at publications in 1986 to respected Physics journals, partly due to my limited knowledge, the speculative nature of the theory, and its counter current view. At that time, if I could have been proven wrong, I would have put the Wavicle to rest. However, I still had a lot of confidence in the Wavicle concept and as I already realized what gravity could be due to, I resolved myself to interpret it in any noninertial or accelerating reference frame, perhaps even with the actual derivation of the Reimannian curvature tensor in terms of velocities. I have many notebooks with derivations and finally in November of 1996 I began to zero in on the solution of the curvature for the helical surface. With greatly increased confidence, I decided to publish this speculative theory on the Internet. My goal now is to find out how the Rotating Wave of the Wavicle might explain the 5-dimensional metric tensor of Theodore Kaluza. I am now striving for a thorough understanding of electrodynamics with the Wavicle and a review of the mass energy density tensor. Again, if I could find anything that would prove the rotating wave theory wrong, then I would be content to put the Wavicle to rest. In the least, I will have been able to contribute another angle of insight to the mysterious symmetries of nature. If you have any comments, please write to me or send me e-mail and I will keep it in confidence.