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On the Strong Equivalence of the Elementary Charge and the Gravitational Charge Watch

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    The Nature of the Electron

    There has been for a number of years now, an idea that the electron is in fact a sphere with a very small radius and inside this sphere is distributed the momentum of photon energy. In effect, this particular model would state that the electron is in fact just a form of ''trapped light.'' This was first speculated upon by Dirac and was continued with depth by Schrodinger on the phenomenon known as zitter motion. In this paper, I have taken the theory further, not only to reconcile the original idea's that the classical (or perhaps) semi-classical electron has a mass of electromagnetic nature and that the gravitational charge (the inertial mass) is in fact intrinsically-related to the same electromagnetic features. To do this, I have had to read for many years now, the theories of times past which involved the photon configuration inside of particles - Lloyd Motz was the first to propose a gravitational charge to a particle and even speculated on bound photon particles in a type of orbital motion; the original idea's brought forth in this paper is how to think of the gravitational charge in terms of the electromagnetic field and we will also study the implications of certain equations under the same investigative field. To make short, the electron is taken in this paper, as a fluctuation of either bound or single photons following toroidal or other topological paths in a dense curved spacetime. We will also propose that the idea Motz brought forward, that the gravitational constant takes on extremely large values inside of such particles can actually play the role of a Poincare stress for the electron and to finish off, we will also study what it means to talk about the spin of an electron.

    Part 1
    The Model of our Electron Theory

    The nature of mass is a mystery to some, even with the hailed discovery of the Higgs Boson. Perhaps a Higgs field describes how mass is given to a particle but what if there was more to it? What if mass has a very important meaning outside of the Higgs Field, something associated to a dynamical process inside of say, an electron itself?

    Electromagnetic theories of mass first began in the late 1800's by Sir Joseph J. Thomson in an attempt to explain the origin of the inertial mass of the electron. Since then, famous scientists such as Max Abraham, Hendrik Lorentz, Oliver Heaviside (who contributed greately to the first respectable models) and even Paul A. M. Dirac had pondered the electromagnetic nature of electrons. Basically, (among many puzzling questions concerning the electron) was that ''how much of the electron mass is contributed by an electromagnetic field?''

    As I said, there was actually many questions surrounding the electron, one of the other puzzles was that the electron's self energy should in fact be infinite if it was treated non-classically! Another problem persisted... the electrostatic forces inside of an electron should effectively rip it apart. Some kind of balancing force was expected to exist in the electron to cancel these tremendous forces. These non-electromagnetic forces capable of cancelling out the electrostatic forces began to be called ''Poincare Stresses'' after their originator Henri Poincare.

    The real problem later that was soon ''assumed to be fixed'' was that the electron is considered a pointlike particle up to lengths of about 10^{-18}m... very small indeed but is it really a pointlike system and why is this a problem?

    Well, most scientists are convinced that the electron is a pointlike system and that speaking of it as not is a heresy of science. But there is an unusual problem with thinking of the electron as a pointlike system and that resides in an equation famous to classical physics:

    U = \int_{|r| \leq R} \frac{\epsilon_0}{2} \mathbb{E}^2 d \vec{r} = \int_{R}^{\infty} \frac{e^2}{8 \pi \epsilon_0 r^2} dr = \frac{e^2}{8 \pi \epsilon_0 R}

    The equation basically says, if R = 0 then the energy of the electron U goes to infinity!

    Infinities are strange things, none have ever been observed in nature, so you may take this to mean that the equation is wrong. But that requires on to be biased that non-classical electrodynamics completely runs the show and that renormalization techniques can solve this problem. Or one can argue, that it is actually indicating particles are not truly pointlike and that is the stance I take in this work and that many today still consider.

    The very first intelligible and promising model of an electron based on having a non-zero radius came in the Ref 1. Not only did these scientists find values which where very close and fitting to an electron but they also attempted to solve the problem of why particles appear to be pointlike. They solved it using a scaling technique involving energy of electrons in collision experiments. In a easier way to imagine this, basically electrons appear to be pointlike up to a certain threshold then any attempts to measure a radius for an electron become ineffective. Similar idea's where used for strings in string theory, where particles are actually 1-dimesionally extended objects.

    In their model, albeit, not an original idea per se, was that the electron was really a particle with an internal (instrinsic) momentum of a photon caught up in a toroidal topology... a pathway if you like inside of the electron which gave it the features of charge and mass. They proposed as many had before them, that there was actually experimental evidence supporting this: in the form of a special type of decay process. When an electron comes into contact with an antielectron (positron) they release gamma photon energy

    e^{-}e^{+} \rightarrow \gamma \gamma

    Two photons of energy are released in this decay process. In fact, all matter which interacts with it's antiparticle release photon energy which may extend the theory to mean that all matter is in fact trapped forms of light. This actually has some very important implications if true, not only about discussing the structure of elementary particles but also implications about how the most early stages of the universe came about, which we will cover much later.

    Before this paper (ref 1) models of bound photons had been discussed by scientists at various points in our history, which came to be called ''Preon'' particles. They may even be pointlike in nature which create other subatomic particles. The photon more or less can be considered a pointlike particle, even though it has a wavelength. That's a much more conplicated issue.

    Lloyd Motz, who was an American astronomer and cosmologist with a keen interest in gravity physics, began to describe elementary particles as bound pairs of photons, similar to a cooper pair only that the difference is that photons are in fact massless particles. But bound photons not only would cancel out the electric charge of a system but he speculated at the time that it could give rise to spin itself. He used this model consecutively to talk about the structure of neutrino's. In this work, we are not going to discuss the nature of other particles, instead we are going to focus on the electron.

    One particular part of his model involved describing mass of particles as a charge itself on the system, just like how an electron has an electric charge by moving in an electromagnetic field, the mass of an electron was also a charge on the system; the problem I suffered for quite a number of years was whether there was a direct analogy... ie. Is the mass of a particle then a charge gained by moving in a gravitational field?

    Gravity and the field itself is such a troublesome thing in physics at the moment. Not only does it appear to be significantly smaller by many magnitudes than the other three forces of nature but it also differs from the other forces that the force itself is not carried by a mediator particle. For instance, the electromagnetic force is mediated by photons, so what is the gravitational force mediated by? The speculation was that a field of ''gravitons'' was responsible for transmitting a gravitational signal from one place to another... but alas none have ever been found. In fact, the search for the fundamental mediator of gravity has been pretty much given up for not many scientists today believe there is even one.

    Instead, gravity was a pseudoforce, like the Coriolis force. Something experienced by a system in an inertial frame of reference. Of course, General Relativity which was formulated by Albert Einstein showed us that it was instrinsically related to the fabric of space and time and that objects accelerating in their frames of reference would not be able to tell whether they where falling or elevating. But what is gravity at the fundamental level?

    As I said, for many years I struggled to find an answer for myself why then there was not a complete analogy between the electric charge of a moving particle in an electromagnetic field and that with a mass charge moving in a gravitational field. The answer finally came to me by realizing that if the gravitational force wasn't actually a real force (ie. ficticious force) then it seems that mass had to come about another way. It can be as I found, still valuable to speak of the gravitational field and think of the particles charge of mass as something related to the curvature of spacetime; and this is why...

    If the electron is really just a photon caught up in a very small confined toroidal topology then we must be talking about the photon following geodesics inside of the particle itself; this must mean that the photon is travelling in a very small curved space and time. To talk about this subject requires that we must open our minds and take an imaginary journey into the interior of the electron, with a few added bonus thought-experiments.

    Part 2
    The Gravitational Charge

    Explaining why we can talk about a gravitational charge and the origin of the idea behind it due to Schwinger, comes from a theory concerning a unit of electric and magnetic charge. The equation he first used to describe this was a quantization condition

    e \mu = \frac{1}{2} n \hbar c

    It seems to say that \mu plays the role of a magnetic charge - this basically squares the charge on the left handside, which is important as you will see in the equation describing the charge of the gravitational mass.

    My very first understanding of mass as being treated as a charge as I explained in the previous part, came from Lloyd Motz (ref 2). In his paper, he calls the gravitational charge (squared),

    GM^2 = \hbar c

    This equation has had... huge implications in the study of elementary particle physics. We cannot cover them all of course, but to give you some idea, the expression GM^2 appears all over classical physics and even cosmological physics involving celestial objects. It can even be applied to the very small of course. The other side of the equation, the expression \hbar c appears all over electromagnetic theory of physics... I give no justice to either quantities but you can take my word for it, they are very much used. He get's this charge, or should I say, the charge had been obtained from a Weyl Principle of Gauge Invariance which led to the relationship.

    Motz explains that there is a uniform gravitational charge distribution throughout the electron. He also further adds that

    \hbar  = \frac{GM^2}{c}

    is a quantization condition on the charge \sqrt{G}M. So how will we start to describe our semi-classical electron? First of all, it has a radius, this is related to many equations which I will feature in this paper. It is taken however that it may not be of the same magnitude what is known, as the classical electron radius

    r = \frac{e^2}{Mc^2}

    So our general radius will be a little smaller than this. We will also talk about how the electromagnetic features of this gravitational arise - so in regards to the gravitational charge we will state that it is a fully-electromagnetic feature of photons confined within the electron. But first, we need to get some facts of our model electron straight.

    Part 3
    The Electric and Gravitational Charge Distribution

    The gravitational charge \sqrt{G}M and the electric charge \frac{e} is related by the Heaviside Relationship

    e^2 = \sqrt{4 \pi \epsilon_0 GM^2}

    where \epsilon_0 is the permittivity. Permittivity just describes how much resistence that a dynamic system experiences when there is an applied electric field in a medium. So as you can see, the electric charge doesn't exactly equal the gravitational charge per se (which is why) throughout all of this work you will see a coefficient factor of k = 4 \pi \epsilon_0.

    I found that the energy and the radius of a particle can also be set directly equal to the gravitational charge squared

    Er = GM^2

    The gravitostatic equation which relates the energy to the gravitational charge then might be taken as

    E = \frac{1}{2} \frac{GM^2}{r}

    This is a gravitational charge analogue of the mass contributed by an electromagnetic field

    E_{EM} = \frac{1}{2}  \frac{e^2}{4 \pi \epsilon_0 R}

    (So we assume the same coefficient of 1/2 attached to the equations). The mass of the system at rest is therefore found as

    M = \frac{1}{2} \frac{GM^2}{r c^2}

    As Wein had noticed, if the gravitational mass was an electromagnetic phenomenon (or even in part electromagnetic), then there must exist a proportionality between the electromagnetic energy, the inertial mass and the gravitational mass. Later in our history due to Einstein, it appears that the gravitational mass and the inertial mass has been shown to be pretty much equivalent in all tests made.

    The attraction of the field between two bodies is therefore given as

    G \frac{\frac{1}{2} \frac{GM^2}{r c^2} M}{R}

    So far this is a very basic way of integrating the relationship of the gravitational charge into theories which considered these forms electromagnetic in nature. The only way to interpret how we can change from one system to the next is by recognizing that not only is the charge e proportional to \sqrt{G}M but that also the gravitational charge may be an electromagnetic manifestation as well.

    Since we assume the gravitational and electric charges are uniformly distributed inside of the horizon radius of the electron, we therefore conclude that there is also a distribution of the electric field. The distribution of electric field inside the electron can be seen as related also to the gravitational charge! The average of this field is,

    <E> = \sqrt{\frac{3 GM^2}{\epsilon_0 \lambda^4}}

    Part 4
    The Electron Spin and The Induced Photon Charge

    A much more complicated issue, but over the years I have found ways to better explain the model. The radius in the previous equations has multiple meanings but instrinsically the same... if there ever was an oxymoronic statement.

    In the model electron, we adopt the theory that the electron is really a photon tied up in a toroidal topology, the radius therefore is a degree of freedom inside of the electron. The degree of freedom can be thought of as a wavelength \lambda or even a mean radius of transport \bar{r} for the photon system.
    What might we mean about an internal degree of freedom noted as the radius \bar{r} of curvature? A volume of radius is given as;


    You can relate the Gaussian curvature to the Newtonian Constant G which assuming, takes on the large value of \frac{\hbar c}{M^2} and is distributed evenly throughout the volume. While just recently realizing that Motz had suggested in his paper that there is in fact a cancelation of the forces. The charge \sqrt{GM^2} distributed throughout the volume has a self-energy of order

    \frac{\hbar c}{(\frac{\hbar}{Mc})} = Mc^2

    He notes that the electric charge will cancel out the same magnitude recognized as being distributed as \frac{e^2}{Mc^2}. I independantly came to the idea of the gravitational stresses in the interior of particles as playing the role of a Poincare stress. I am not quite sure whether Motz was implying such a stress, but it is clear that he saw both the gravitational and the electrostatic forces to be equal in size. He doesn't seem to point out a problem of fine tuning, but I have said that if this is the case we are surely talking about something which is oddly finely tuned to be in nature.

    What is interesting is that he proposes an internal degree of freedom for a bound photon particle both traversing a path around a common center. His idea wasn't given in any great mathematical detail, but he did say it could account for the spin of particles \frac{\hbar}{2}.

    The notion of such internal degree's of freedom brings us to the subject of zitter motion, which was a mathematically-predicted (and only just recently) varified motion of the electron. Something about the electron was moving in a zig-zag like fashion as it moved through space and time. Recently, David Hestenes, a wonderful scientist and great mathematician proved that the motion can be attributed to an electron periodic clock as some kind of intrinsic effect. Probably careful not to mention a subatomic particle being mentioned, but it has been noted in many papers over the years that the zitter motion can be attributed to the motion of photons inside of electrons. Zitter motion is therefore in very non-technical terminology, is a slowed down photon. The photon which creates an electron slows down and takes on the appearance of a mass because it is following a curved and yet tight path in spacetime.

    There are numerous ways we can talk about this trajectory; one way is by first knowing that the momentum of a photon is

    p = \frac{h \nu}{c}

    Taking the integral to and from the helicities characterized as the freedom \bar{r} spoke of from the paper in (ref 1) describing the toroidal topology, you can talk about the angular momentum (spin) of electrons in very a very simple mathematical way

    \frac{1}{2} \int \frac{h \nu}{c} d\bar{r} = \frac{\hbar}{2}

    where \bar{r} is the mean radius of transport and \frac{\hbar}{2} is the sum of the total angular momentum for the photons. You can actually obtain this quantity in a similar way by recognizing that the energy of an electron is

    Mc^2 = \frac{e^2}{4 \pi \epsilon_0 r}

    and you reach the same conclusion as my integral equation by combining the mass and radius

    Mv r = \frac{1}{2}\hbar

    (ref 3.)

    In the bound photon model, there is however a fundamental difference to the free-space case photon following a toroidal topology. A bound pair of photons' angular momentum cancel out and actually give no contribution to the over-all system.
    As Motz informs us, that the photon contributes a gravitational mass we should first assume that this is the gravitational charge - a photon in this case will experience a charge if following a curved geodesic. Since this would simply be the gravitational charge divided by the compton wavelength, this is given as


    Motz notes that this is the same thing as:


    he states that since the energy is \hbar \nu then from the expression GM^2\frac{\nu}{c} we find the usual relationship GM^2 = \hbar c and takes this as meaning the gravitational field inside of the particle is responsible for the existence of the photons. If this is true, the gravitational field \Gamma must in some way couple to the photons; in fact, not only do the photons couple to the ''gravitational forces'' but they may even be called a very special kind of particle, called a graviphoton. A graviphoton appears as a photon but is an excitation of the gravitational field. If the gravitational field is responsible for the existence of the photon then we could assume they are indeed a special type of graviphoton. Not only this, but the gravitational force inside of the particle is considered very large (as we have covered before).

    The geodesic spoke of in which our gravitationally-induced photon is moving in, can be thought of a line element (Weyl). Motz recognizes how important it is to talk about a charge if a photon is following a curved path in space http://www.gravityresearchfoundation.../1966/motz.pdf . The line element is

    ds^2 = \frac{dr^2}{(1 - \frac{2GM}{c^2 r}) + \frac{G}{c^4} + \frac{e^2}{r^2}}

    The path of a photon is well-known to be ds = 0 but for a bound system, you would get two values of the inverse radius. In our single photon model, albeit simpler, we must understand then in our former case. In this case, it is also much easier to think of it trapped in a toroidal topology. A toroidal structure is given by a length

    \bar{r} = \frac{\lambda}{4 \pi}

    Here \lambda is the wavelength of the photon. So, as we see, it is possible the photon couples to the gravitational field inside of the electron... How do we describe such a thing?
    A coupling of the system to the gravimagnetic forces are achieved by a cross product (this explanation can be found in ref 2. or by work by Sciama ''On the Origin of Inertia'')

    \sqrt{G}M\frac{v}{c} \times \frac{2 \omega c}{\sqrt{G}} = \sqrt{\frac{e^2}{4 \pi \epsilon_0}}\beta \times \frac{F}{\sqrt{G}M}

    You get the coupling field \frac{2 \omega c}{\sqrt{G}} by obtaining the Coriolis field -2M(\omega \times v) and dividing it by the gravitational charge \sqrt{G}M. This is how a rotating sphere couples to the external fields. Describing the internal framework, you need to talk about the motion of photons inside of electrons.

    I am yet to find an adequate way of describing the system in such a way.

    Part 5
    The Fundamental Equation of the Gravitational Charge

    It is one thing saying 'gravitational charge' is what makes up inertial mass, but what is the gravitational charge reliant on? What is it in nature which creates the charge? We have seen that the photon follow tight spacetime curvature inside of the electron, this creates a charge and the appearance of a mass, but what are the fundamental relationships which make the gravitational charge itself?

    Like all important fundamental dynamical processes in nature, physics often described them in terms of the constants of nature, such as the speed of light, permittivity, permeability, the gravitational constant ect. The CODATA elementary charge does exactly this. The point of the equation about to be shown, is to attempt to describe charge as a ratio of important fundamental constants

    e = \sqrt{\frac{2 \alpha \pi \hbar}{\mu_0 c}} (1)

    We can be really theoretical about this and create a simple equation which will do the same thing for the gravitational charge. We first of all recall the relationship between the elementary charge and the gravitational charge

    e^2 = 4 \pi \epsilon_0 GM^2

    Square equation (1) and set them equal, this gives

    4 \pi \epsilon_0 \mu_0 c GM^2 = 2 \alpha \pi \hbar

    Solving for the gravitational charge and cancelling out some factors we have a fundamental relationship for the gravitational charge

    \sqrt{G}M = \sqrt{\frac{\alpha \hbar}{2 \epsilon_0 \mu_0 c}}

    What is the interepretation of a dubious looking equation like this? I noticed a few things. First of all, in physics it is recognized that the angular momentum component \hbar of a system is conserved through the fine structure constant

    \frac{e^2}{4 \pi \epsilon_0 c} = \pm \alpha \hbar n

    Actually this is a special quantization condition. We should notice the positive and negative eigenvalues of increment n and we may therefore imagine a same condition on our gravitational charge equation

    \sqrt{G}M = \sqrt{\frac{ \pm \alpha \hbar n}{2 \epsilon_0 \mu_0 c}}

    This means that angular momentum is conserved in our definition of the gravitational charge itself. Notice also that our charge is proportionally dependant on the electric and magnetic resistance constants (\epsilon_0, \mu_0), something you might expect if mass itself was an electromagnetic phenomenon. The magnetic field has three components, in (ref 1.) these components are given as

    B_r = \frac{2 \mu_0 \mu_d\ cos \theta}{4 \pi r^3}
    B_{\theta} = \frac{\mu_0 \mu_d\ cos \theta}{4 \pi r^3}
    B_{\phi} = 0

    Perhaps more importantly, but an even odder interpretation lies in Motz' definition of the source of the gravitational field.

    F = \frac{\sqrt{G}M_1 \sqrt{G}M_2}{r^2}

    This is of course analogous to Coulomb's Force \frac{q_1q_2}{r^2}.

    He notes, that the gravitational charge will be the source of the gravitational field


    If anyone has noticed, this is actually analogy of the magnitude of the electric field \mathbb{E} which is created by a single charge

    \mathbb{E} = \frac{1}{4\pi \epsilon} \frac{q}{r^2}

    Another one to note perhaps, would be the appearance of the gravimagnetic field, given by Motz as \frac{-2\omega v}{\sqrt{G}}.

    \frac{F}{\sqrt{G}M} = -\frac{2\omega v}{\sqrt{G}}

    This is actually the gravitational analogue of the equation

    \mathbb{E} = \frac{F}{q}

    which describes the definition of the electric field. So the gravitational case, must be the definition of the gravimagnetic field or something akin to it since

    -\frac{2\omega v}{\sqrt{G}}

    is also

    v \times B

    I know this because I calculated it (ref. 4). Anyway... drifted away slightly from the main theme, if the gravitational charge is indeed the source of the gravitational field then somehow the constants which makes our fundamental equation

    \sqrt{G}M = \sqrt{\frac{ \pm \alpha \hbar n}{2 \epsilon_0 \mu_0 c}}

    Is also part of the electromagnetic ensemble which is the source of the gravitational field. It seems totally erroneous to think of the source of the gravitational field as an electromagnetic phenomenon simply because, on our every day to day lives, gravity is in fact around 10^{40} magnitudes smaller than the electrostatic force. So how can one be the source of the other? The answer lies in the fact that the gravitational force and electric force are of pretty much the same magnitude inside of particles and this is where our next part has led us.
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    Part 6
    The Gravitational Constant taking the role of the Poincare Stress

    As was explained in the beginning of the work, one of the main problems of early electromagnetic theories of classical electrons is that internal electrostatic forces would rip them apart. To achieve some kind of solution to this problem, Henri Poincare suggested there was in fact some kind of internal non-electromagnetic energy which cancelled this out.

    It was unknown for some time what kind of stress this could be and it wasn't until I read the work of Motz (ref 2.) that one of his speculations was that the gravitational constant took on a very large value inside of the particle

    G = \frac{\hbar c}{M^2}

    It occurred to me, even though Motz had never suggested it, that this could play the role of the Poincare stress but it brings into question a matter of fine tuning. Nevertheless, the gravitational field inside the particle could be strong enough to cancel out exactly the electrostatic forces. For this to be true, G must take on a very large value of G \cdot 10^{40}.

    Part 6
    Uncertainty Relationship for the Gravitational Charge

    To rehash our memory quickly on the toroidal structure of the electron, the photon follows a length/curved trajectory inside the particle. The length of a thing can be given as

    \ell = ct

    If t is measures in seconds, then \ell is measures in meters, then c = 3 \times 10^{8}. To measure it we need a clock with an uncertainty of \Delta t can be no larger than t. Time and energy uncertainty says the product of \Delta t and \Delta E can be no less than \frac{\hbar}{2}

    \Delta E t \leq \Delta E \Delta t \leq \frac{\hbar}{2}

    so that

    \Delta E \leq \frac{\hbar}{2t} \leq \frac{\hbar c}{2 \ell}

    which implies a relationship

    \Delta E \leq \frac{GM^2}{2\ell}

    through the equivalence of

    GM^2 = \hbar c

    There is then a relationship to the uncertainty in the energy and the uncertainty in the length of the system. A correction to this inequality must be made then where the length can be considered \frac{\lambda}{2},

    \Delta E \leq \frac{GM^2}{\lambda}

    The physical meaning of this is that the internal energy flux is bound by the maximal uncertainty in the mean transport of radius which may take the role of the wavelength but we have a correction order.

    \lambda = \alpha \lambda_C

    As the paper http://www.cybsoc.org/electron.pdf recites, the limitation of the speed of light means that only paths within the radius can provide a contribution of inertial energy (mass) to the system. The uncertainty relationship then to satisfy the toroidal system is

    \Delta E \leq  \frac{GM^2}{\alpha \lambda_C})

    Shows us that the energy depends on the varying uncertainty of the compton wavelength up to a maximal radius of transport. The energy is also therefore conserved through the fine structure constant if one applied the correction.

    This relationship might be more important than one realizes as even Motz deduced that there was an uncertainty relationship between the mass of a particle and it's radius. The more you attempted to measure the radius of the particle, the more uncertain it's mass became and vice versa. He spoke about this relationship in (ref. 2) but I will quickly cover it.

    He showed that you can equate the Compton wavelength to the Gaussian curvature 8 \pi \rho_0(\frac{G}{c}) where \rho_0 was the proper density and \frac{G}{c} is the Schwarzschild constant. It can obtain the equation

    \frac{K}{6} = (\frac{\hbar}{Mc})^{-2}

    Introducing the radius of curvature, which would equate to our use of the curvature of transport this would be, keeping in mind that the radius of a three dimensional hypersphere is \frac{1}{6}K = (\frac{\hbar}{Mc})^{-2} you obtain

    R^2 = (\frac{\hbar}{Mc})^{-2}

    and from this he states that one gets

    RMc = \hbar

    He says that one can look upon this as an uncertainty relationship where the mass term and the radius anticommute. In a similar fashion, I have shown above there is an uncertainty relationship with the internal radius of transport and it's gravitational charge.

    Part 7
    The Fine Structure Constant and it's Relationships to the Gravitational Charge

    We've actually not only just seen a relationship of the fine structure constant to the gravitational charge, but we also saw one previously where we described the gravitational charge as a ratio of fundamental constants. There are in fact a few more important one's I wish to discuss.

    The fine structure constant, any fine structure constant, determines the field strength of interaction between particles. It is such a mysterious number, it crops up in physics all the time and was made famous to the public by a number of notable physicists but perhaps most notable of all was Feymann who discussed it in a number of his lectures.

    I remember a physicist once saying, ''to unify physics, you not only need to show how the workings of nature in terms of the fine structure but you also need to be able to explain why the fine structure is!''

    Of course, that is an attempt my feeble mind could never grasp, but as I have done throughout the work so far was showing subtle relationships between the fine structure constant and the investigation into the gravitational charge. Now we will get straight into the rest of them I found.

    Let's consider the phase of the photon in the toroidal model (ref. 1).

    In the proper frame, the phase of both the orbital rotation and the internal photon is incidently

    \phi = \omega t_0

    where t_0 is the proper time where

    t_0 = \gamma(t - \frac{vx}{c^2})

    The phase of the internal photon can be written as

    \omega t_0 = \omega \gamma(t - \frac{vx}{c^2})

    The gravitational fine structure constant raises it's beautiful head from this equation in the form

    \alpha_G = \frac{GM^2}{\hbar c} = (\omega t)^2

    thus naturally

    \sqrt{\alpha_G} = \sqrt{\frac{GM^2}{\hbar c}} = (\omega t) = 2 \pi f \gamma(t - \frac{vx}{c^2})

    Therefore the phase of the photon appears to be related to the gravitational fine structure of our system.

    Suppose we can talk about the electric flux and magnetic flux inside a particle of an area enclosed by a sphere (also known as the horizon of the mean radius of transport) we might have

    \Phi_e = \frac{e^2}{\epsilon_0}


    \Phi_m = \frac{\phi_{0}^{2}}{2 \mu_0}

    knowing that

    c = (\sqrt{\epsilon_0 \mu_0})^{-1}

    \hbar = \frac{\phi_0 e}{2 \pi}

    and the relationship describing fine structure

    \alpha = \frac{1}{2} \frac{e}{\phi_0} \sqrt{\frac{\mu_0}{\epsilon_0}}

    One can meld these equations together to reach two interesting relationships

    \alpha = \frac{1}{2 \sqrt{2}} \sqrt{\frac{\phi_e}{\phi_m}}

    and the square of the gravitational charge

    GM^2 = \frac{\sqrt{2}}{2 \pi}\sqrt{\frac{e}{\epsilon_0}\f  rac{\phi_{0}^{2}}{2 \mu_0}} = \frac{\sqrt{2}}{2 \pi} \sqrt{\phi_e \phi_m}

    Let's just quickly explain what the electric \phi_e and magnetic flux \phi_m are. Doing so might make us realize that the gravitational charge is manifestly an electromagnetic property (something we have been hinting at for quite a bit now).

    In short and simple terminology, the electric and magnetic flux is the amount of electromagnetic field which penetrates a given area... in our case, we are assuming a fundamental length of transport within an area.

    The point however of the two equations obtained revealing the fine structure constant and the gravitational charge is that to solve for them in terms of \phi_e and \phi_m that the ratio and the product become independent of unit representation. According to the author in the reference below (ref 5.), he claims that this is a verification that both \phi_e and \phi_m have the same dimensions.

    However, the author may not have sufficiently known the full interpretation of his equations since there is no mention of the gravitational charge through equivelance \hbar c = GM^2

    GM^2 = \frac{\sqrt{2}}{2 \pi}\sqrt{\frac{e}{\epsilon_0}\f  rac{\phi_{0}^{2}}{2 \mu_0}} = \frac{\sqrt{2}}{2 \pi} \sqrt{\phi_e \phi_m}

    Thinking of the gravitational charge as something to do with electromagnetic field is not a new approach in this work, since we have attempted to describe the electric field as related proportionally to the gravitational charge before. But thinking of it in terms of the fluxes is a new approach and certainly an interesting insight if the propositions above are correct.

    And perhaps the most interesting relationship I found takes us back to the zitter motion, that internal motion predicted from quantum mechanics.

    As it was said in the first intelligible paper of a photon model of an electron (ref 1.) http://www.cybsoc.org/electron.pdf states that there are striking analogies of the Dirac solution of equations describing zitter motion (whom later) David Hestenes showed that it could be considered an ''internal dynamic'' clock. The instantaneous velocity eigenvalues \pm c can be pictorially imagined as the two interlooping path which the photon takes in the toroid. They are actually light-like helices of \frac{\lambda}{2 \pi} which defined the maximal degree of freedom for the mean radius of transport. I found a way we can talk about it mathematically.

    The system of coupled equations describing a particle system comes in the form

    \dot{u} = \frac{1}{r} + \frac{q}{M} F \cdot u

    \dot{p} = F \cdot u + \nabla \phi

    \dot{S} = u \wedge p + \frac{q}{M} F \times S

    As Hestenes points out in his paper http://www.fqxi.org/data/essay-conte...1bae814fb4f9e9 the zitter motion radius is given as

    r^{-1} = (\frac{2}{\hbar})^2 p \cdot S

    where the beautiful mathematics is revealed where one notices the potential depends on the charge to mass ratio

    \frac{e}{M} S \cdot F

    (ref 6.)

    Now, the toroid model (ref. 1) required a tweaked fine structure constant as

    \alpha' \equiv \frac{e^2}{4 \pi \epsilon_0 \hbar c} = (\frac{e}{M})^2 \alpha

    This means we can represent Hestenes equation

    \frac{e}{M} S \cdot F

    can be rewritten with a fine structure coefficient

    \alpha' = (\frac{e}{M})^2 \alpha

    we will state

    \alpha^{\dagger} = \sqrt{(\frac{e}{M})^2} \alpha

    which means the potential can be rewritten as

    \Phi = \alpha^{\dagger} S \cdot F

    It appears like an elegant way to integrate the work of the toroidal model into the zitter motion they said had striking similarities. This is but another discovered. But perhaps what is more insightful is that we arrive at a similar conclusion involving the spin as we did in part 5, that the spin S is again conserved through the fine structure in this particular potential equation.

    Of course, you might even ask why is the potential in this form even important? The part of this potential arises as inversely related to the frequency when there is a coefficient of \frac{2}{\hbar} attached to it. In the toroid model, the total energy (external and internal) external being non-rotational, leads to the expression

    \frac{\alpha ' }{2 \pi a} \hbar \omega_C

    The model accounts for the energy as a sum of external and rotational parts and according to the authors of the original toroid model, means that the effective frequency of

    \omega = \frac{U}{\hbar}

    of the confined photon is slightly smaller than the compton frequency. The relationship might be important when we consider that zitter motion decribes the frequency relationship as

    \omega = w_e + \frac{2}{\hbar} (\alpha^{\dagger} S \cdot F)

    What is interesting is that the zitter motion (which can be fully attributed to the dynamical circulating massless charge) is inversely proportional to the frequency (ref 6.) such that

    \lambda \omega = \lambda_e \omega_e = c = 1

    and that the frequencies are related through the equation

    \omega = w_e + \frac{2}{\hbar} (\alpha^{\dagger} S \cdot F)

    where F is the electromagnetic field using Hestenes notation.

    David Hestenes has even showed how the potential \phi = \alpha^{\dagger} S \cdot F (with my inclusion of the fine structure \sqrt{(\frac{e}{M})^2} \alpha for the toroid), is actually the same thing as the zeeman interaction term

    e \mathbf{r} \cdot \mathbf{E} - \frac{e}{M_e} S \cdot \mathbf{B}

    One can even say that the fine structure constant in the equation is actually an important quantity when the equations begins to describe how zitter frequency and the radius vary with interaction (ref 1. below) so that when mass is increased by an interaction term (potential) the frequency increases and the radius decreases. This must happen to maintain the speed of light.

    When an atom is placed in a magnetic field, its fine structure lines splits into a series of equidistant lines with a distance proportional to the magnetic field strength. As it has been noticed, it can be explained by saying that the electron has a magnetic moment. The magnetic moment of course, can only exist if there is a circular motion involved; in our case, the electrons structure accounts for this circular motion by a charged yet massless photon toroidal knot yielding a magnetic moment

    \mu = \frac{e}{2M}( \frac{1}{2} \int \frac{h \nu}{c} d \bar{r})

    The fundamental fine structure inside of the electron is therefore... carried on in the nature outside of the horizon of the radius of transport.

    (you can find reference to the origin of the name ''horizon of transport'' from their new paper)


    The ''zitter clock'' was varified experimentally. This was an idea created by deBroglie and was confirmed in a channelling experiment. The period of the frequency of the electron clock is given as

    \psi(\tau) = e^{i \omega \tau}

    It has been noted by ref 1. in my main paper, that the electron clock can be attributed to the toroidal motion of our charged massless system. In fact, the electron in it's most elegantly explained form, is that it is a lightlike particle which follows in which the spin is determined by the internal helical dynamics where it even has a curvature and a frequency... and as I showed in my own work, may even be attributed to a phase related to the intrinsic gravitational charge.

    If the electron clock is determined by the circular motion at the speed of light, we may attribute this to a invariant proper time operator \frac{\bar{r}}{c} which would be related to the ''rest'' energy of the system as

    (\frac{\bar{r}}{c}) Mc^2 = \hbar

    Now this invariant time operator coefficient term on the energy giving us the quantization condition \hbar can actually be related as Motz explains; the proper time must be treated as an operator which is canonically conjugate to the rest mass of particles, which was discussed in ref 1 below. It actually brings us back to the electron clock, where we are dealing with the proper time of the electrons history. The zitter motion and the invariant proper time operator therefore might be doing the same thing in respects of giving rise to angular momentum component \hbar.

    ref 1. L Motz, phys. Rev., 93, 901 (1954)

    There is a problem which exists though, concerning the modelling of the zitter motion as a proper time invariant operator, that is that the electron is modelled by Hestenes as a lightlike curve

    c^2 \Delta t^2 = \Delta r^2


    s^2 = 0

    and proper time cannot be defined in such ways. Hestene does however mention this and that a physical definition of the time parameter \tau must be determined by other features of the model

    FQXi - Foundational Questions Institute However, as I pointed out, ''The zitter motion and the invariant proper time operator therefore might be doing the same thing in respects of giving rise to angular momentum component'' and this is meant to be taken seriously that the time parameter derives from the assumption that the electron has an intrinsic angular momentum (which is as we have covered), the toroidal topology of our massless charge

    The spin S in this sense, becomes a function of the time parameter S(\tau) which as the mathematical jargon of Hestene points out adequately, ''is a null bivector.''

    This is how he reaches the potential zitter term \Phi, by introducing the relevant equations of motion for the velocity \dot{u}, momentum \dot{p} and the spin \dot{S}. Very ingenuous, so that the potential zitter interaction term becomes a function \Phi(\tau, z). In this respect from this model, we can talk about the proper time experienced between two events - the incremental events which defines the electron clock. And as we have seen, the electron clock may be written as a function or it could actually be represented as a dynamical time invariant operator on the energy term which in return tells us it has an angular momentum. Or we can write it as an integral

    Mc^2 \int_{P_t} d\tau = \hbar

    Where the integral defines a proper time interval. For an invariant time operator case, where \bar{r} is the mean transport of radius for the charged massless circulating system, we would have

    \hat{t} = \frac{\bar{r}}{c}

    \hat{t}E_0 = \hbar

    where E_0 is the rest energy, or in terms of electron energy which is more accurate we would have

    \hat{t} (\frac{e^2}{4 \pi \epsilon_0 \bar{r}}) = \hbar
    • Thread Starter

    Part 8
    The Coupling of the System to External Magnetic Fields

    My model must have a magnetic moment to be consistent with all experimental evidence concerning the electron. For this to be done, the magnetic moment must be generated by a massless (paradoxically) moving charge.

    The most simplest way to describe the magnetic moment in my theory is

    \mu = \frac{e }{2M c}( \frac{1}{2} \int \frac{h \nu}{c} d \bar{r})

    \bar{r} is the radius of transport which can in some way be related to the trajectory or geodesic of a photon moving in a closed curve.

    The equation also takes into account the electric charge, which is proportional to the gravitational charge of the system. The momentum of our system is described through \frac{h \nu}{c}. The speed of light remains invariant.

    A Hamiltonian can be found from the equation also

    H = \frac{e }{2M c}( \frac{1}{2} \int \frac{h \nu}{c} d \bar{r}) \cdot \mathbf{B}

    A circular gauge (possibly fitting for a toroid model) can be given as

    \mathbf{A} = \frac{1}{2}\begin{vmatrix} -B_y \\ +Bx \\ 0 \end{vmatrix} = \frac{1}{2} \mathbf{B} \times r

    switching from cartesian to spherical coordinates, we know the three components of the magnetic field are

    B_r = \frac{2 \mu_0 \mu_d\ cos \theta}{4 \pi r^3}

    B_{\theta} = \frac{\mu_0 \mu_d\ cos \theta}{4 \pi r^3}

    B_{\phi} = 0

    This means the spin coupling of an electron system to an external magnetic field can be given through a Hamiltonian

    \hat{H} = \mu (\mathbf{B}_r, \mathbf{B}_{\theta}, \mathbf{B}_{\phi}) \begin{bmatrix} g_{rr} & g_{r \theta} & g_{r \phi} \\g_{\theta r} & g_{\theta \theta } & g_{\theta \phi} \\g_{\phi r} & g_{\phi \theta } & g_{\phi \phi} \end{bmatrix} \begin{pmatrix} \hat{S}_r \\ \hat{S}_{\theta} \\ \hat{S}_{\phi} \end{pmatrix} = \mu \sum_{(k,q = r \theta \phi)} g_{kq} \mathbf{B}_k \hat{S}_q

    Quickly going back to the equation

    \mu = \frac{e }{2M c}( \frac{1}{2} \int \frac{h \nu}{c} d \bar{r})

    The quantity

    \frac{1}{2} \int \frac{h \nu}{c} d \bar{r}

    amounts to the angular momentum of the system. Therefore, we have the quantity

    \frac{e \hbar}{2 Mc}

    which is also known as the Bohr Magneton. The spin vector is related to angular momentum as

    \vec{S} = \frac{\hbar}{2} \vec{\sigma}

    The correct relationship describing this with the definition of the magnetic moment, is also related to Bohr Magneton

    \mu = \frac{e}{2Mc} \vec{S} = \frac{e \hbar}{4 Mc} \vec{\sigma}

    Thus we also have

    \mu = \frac{e}{2M c}( \frac{1}{2} \int \frac{h \nu}{c} d \bar{r}) \vec{\sigma}

    This is a more proper way of writing the equation, involving the dimensionless spin matrices (there are three of them). There is an even more correct way by involving the Dirac g-factor but you may do that at your leisure.

    There's a peculiar way in which the gravitational charge can arise from the given equations.

    H = \frac{e }{2M c}( \frac{1}{2} \int \frac{h \nu}{c} d \bar{r}) \cdot \mathbf{B}

    \mathbf{A} = \frac{1}{2} \mathbf{B} \times r

    Binding the two together we have

    GM^2 = \frac{e }{2M c}( \frac{1}{2} \int \frac{h \nu}{c}\ d \bar{r}) \cdot \mathbf{B} \times \bar{r} = \mu_B \cdot \mathbf{A}

    We know this true because of the relationship which I discovered describing the gravitational charge

    GM^2 = E \bar{r}

    So apart from investigating a possible framework for the inner intrinsic model of our moving photon, how it is coupled to the magnetic field resulting in the magnetic moment has been given, (a part of the model which was lacking as owned up in the OP) there is an interesting strong gravitational constant which is involved in some grand unified theories of particle interactions. The strong gravitational constant is given as \Gamma and I now wonder how much it can be related to the strong gravitational force arising from a large value G = \frac{\hbar c}{M^2} assumed inside the particle. In fact, the strong gravitational force inside the particle could explain why the strong gravitational constant may exist in nature.
    Values for the strong gravitational constant do arise from exact calculations of \frac{hc}{M^2} for certain particles, here in this article, we see the constant being found from pion masses.


    No doubt you can carry on this for a reduced Planck Constant. The theory adopted from Motz saying that the gravitational constant takes on large values inside an electron, can in fact be carried on to the strong gravitational constant - in fact, we even have a mechanism now for strong gravity. We may assume the interaction of gravity itself is strong at the fundamental level because it takes on large values inside of the particle.


    The electron was at one time considered to be a classical system with a sphere like structure. Later, it was then suggested this internal structure had some dynamics about it... but like the sweat of the brow of a young lad, it had been swiped away by modern non-classical physics, replacing the electron strictly as a pointlike particle. And that seemed to be the end of that.

    But it wasn't... Years later two scientists came together and provided a model of the electron which had an internal structure - one which involved a photon bound in a very tight toroidal topological path in spacetime and also fixing the question of why we still detect them as being pointlike.

    Even with these latest efforts, which was a number of years ago now, the idea that matter is just trapped forms of light, hasn't caught off well in the physics acadamia; though we cannot say it hasn't been heard. One thing scientists are trained in is to not be biased about an experiment but it appears unfortunately the testing of electrons down to scales which ''appear'' pointlike is the cornerstone of the problem.

    If electrons where not actually pointlike, this wouldn't change physics such that we would have to rewrite everything. Electrons still behave pointlike afterall, so we would just need to understand it as a correction order written in the paper alluded to (ref 1.). Perhaps the hardest struggle is that describing particles as pointlike is much easier than trying to describe them in classical terms, but then not all theories are completely classical. It is entirely probable to describe an electron and have your money's worth in classical and non-classical parts.

    Perhaps in time, scientists will come back to this question and start asking them again to find new solutions which seem to make sense in the strange world of quantum mechanics. Certainly, at least my paper shows that scientists have taken it seriously and some nice and maybe interesting tweaks can be found in the equations once describing them as fact.

    Maybe is for another day though.


    Ref 1. http://www.cybsoc.org/electron.pdf
    Ref 2. http://www.gravityresearchfoundation.../1971/motz.pdf

    Ref 3. You can in fact get the classical electron velocity from this as

    v = \frac{\hbar}{Mr}

    Combining that with the electron energy equation we obtain

    v = \frac{4 \pi \epsilon_0 c^2 \hbar}{e^2}


    v = \frac{c}{\alpha}

    then there is a fine structure relationship

    \alpha = \frac{v}{c}

    Ref 4.

    The Lorentz force is of course evB and to refresh our minds, the Coriolis force is -2M\omega v (where we are omitting the cross products). What is interesting is if you set them equal,

    evB = -2M\omega v

    (setting these two quantities equal with each other should not be a surprise, since the Coriolis force is a type of gravimagnetic field)

    cancel the linear velocities and divide the gravitational charge on both sides you get

    B = -\frac{2 \omega}{\sqrt{G}}

    Ref 5. http://arxiv.org/vc/hep-ph/papers/0306/0306230v2.pdf
    Ref 6. To note, F is the chosen notation by Hestenes describing the electromagnetic field which in mathematical jargon, is a bivector. You can see this from his work on spacetime algebra http://geocalc.clas.asu.edu/pdf/ZBWinQM15**.pdf .
    External Ref. http://www.electronspin.org/electron.pdf

    (Note in some equations, Motz used Gaussian units, meaning it might look like we have dropped some 4 \pi \epsilon terms in there) but we haven't, it's just a different unit system.
    • Thread Starter

    I have changed the definition of \mathbf{G} if anyone noticed. Normally the gravielectric field is given as

    \mathbf{G} = \frac{F}{m}

    However, the quantity which we define is

    \mathbf{G} = \frac{F}{\sqrt{G}m}

    As a full analogue of \frac{F}{e} in Gaussian units. This means in Gaussian units, the full analogue allowing e = \sqrt{G}m treats \mathbf{G} not as a gravitational acceleration. Instead, it looks more like a gravimagnetic field than anything else with units of (v \times \mathbf{B}).

    The gravitational four force can be given as

    F_{\mu} = \Gamma^{\lambda}_{\mu \nu} u^{\mu} p^{\nu}

    where \Gamma is playing the role of the Christoffel symbols which play the role of the gravitational field. The quantity on the right

    \Gamma^{\lambda}_{\mu \nu} u^{\mu} p^{\nu}

    has units of energy over a length. The upper limit of the force is calculated as

    F = \frac{Mc^2}{(\frac{Gm}{c^2})} = \frac{c^4}{G}

    I actually speak about this quantity in the next paper, as the origin of the Planck scales and is an important quantity for the special case of the theorem. If the numerator describes the gravitational self-energy, this would have an order

    \frac{\hbar c}{(\frac{\hbar}{Mc})} = Mc^2
    • Thread Starter

    I think I've found a way to unify all my theories now - I think I can fully explain semi-classical dynamics using the strong gravitational constant to explain fully how the Coriolis and the respective intrinsic spin field appears for fundamental particles from a closed loop toroidal trajectory which generates the magnetic field, which couples to the classical Newtonian gravitational field.

    \frac{1}{4} \frac{\hbar e}{Mc} \cdot \mathbf{B} \times \bar{r} = \mu_{B} \cdot \mathbf{A}

    All one has to do to find how electromagnetism, the toroid photon structure in which a topological charge contributes to the mass of the system through e = \sqrt{G}M in Gaussian units was by simplifying the LHS of

    \frac{e }{2M c}( \frac{1}{2} \int \frac{h \nu}{c}\ d \bar{r}) \cdot \mathbf{B} \times \bar{r} = \mu_B \cdot \mathbf{A}

    The gravitational mass of the system, does indeed come from trapped photon energy. The gravitational charge in which Motz showed a quantization of Schwinger in which the gravitational mass is generated by the gravitational charge. The radius of transport must generate the coriolis force if G_s (The strong gravitational constant) contributes to this theory, which it has to.

    \hat{H} = \mu (\mathbf{B}_r, \mathbf{B}_{\theta}, \mathbf{B}_{\phi}) \begin{bmatrix} g_{rr} & g_{r \theta} & g_{r \phi} \\g_{\theta r} & g_{\theta \theta } & g_{\theta \phi} \\g_{\phi r} & g_{\phi \theta } & g_{\phi \phi} \end{bmatrix} \begin{pmatrix} \hat{S}_r \\ \hat{S}_{\theta} \\ \hat{S}_{\phi} \end{pmatrix} = \mu \sum_{(k,q = r \theta \phi)} g_{kq} \mathbf{B}_k \hat{S}_q

    The equation just presented, shows how the magnetic force couples to the external classical upper limit field for gravity \frac{c^4}{G}

    \frac{e }{2M c}( \frac{1}{2} \int \frac{h \nu}{c}\ d \bar{r}) \cdot \mathbf{B} \times \bar{r} = \mu_B \cdot \mathbf{A}

    The gravitational four-force can be given as:

    F_{\mu} = \Gamma^{\lambda}_{\mu \nu} u^{\mu} p^{\nu}

    where \Gamma is playing the role of the Christoffel symbols which play the role of the gravitational field. The quantity on the right

    \Gamma^{\lambda}_{\mu \nu} u^{\mu} p^{\nu}

    has units of energy over a length. The upper limit of the force is calculated as

    F = \frac{Mc^2}{(\frac{Gm}{c^2})} = \frac{c^4}{G}

    This upper limit of self-gravitational force would produce a Planck Particle.
    • Thread Starter

    This prediction of our quadratic term

    \frac{1}{4} \frac{\hbar e}{Mc} \cdot \mathbf{B} \times \bar{r} = \mu_{B} \cdot \mathbf{A}

    Is that for a toroid symmetry in this model, fits exactly for the model in the following reference:



    ...for a photon which has a mass m = \frac{U}{c^2} and the inertial energy

    U_i = \frac{(2 \pi GM^2)}{\lambda} = \frac{\hbar c}{2 \pi \lambda} is bound so has a wavelength proportional to its energy and mass (unifying de Broglies) work on wave functions being applicable to both matter and energy.

    Where the photon mass is equivalent to the rest mass of the electron, known as the Compton wavelength \lambda_{C}.

    But notice... we have interesting thing in the equation

    U = \frac{2 \pi GM^2}{\lambda_C} = \frac{\hbar c}{2 \pi \lambda_C}

    Our gravitational charge term GM^2 has to commute with the constant geometrical number \pi to make this standard equation:

    U = \frac{\lambda_C}{4 \pi}

    completely ruled by quantum activity.

    This Hamiltonian, with all the spin coupling dynamics and geometry, all account to simply a toroid field (or a fundamental Coriolis field I prefer to call it):

    \hat{H} = \mu (\mathbf{B}_r, \mathbf{B}_{\theta}, \mathbf{B}_{\phi}) \begin{bmatrix} g_{rr} & g_{r \theta} & g_{r \phi} \\g_{\theta r} & g_{\theta \theta } & g_{\theta \phi} \\g_{\phi r} & g_{\phi \theta } & g_{\phi \phi} \end{bmatrix} \begin{pmatrix} \hat{S}_r \\ \hat{S}_{\theta} \\ \hat{S}_{\phi} \end{pmatrix} = \mu \sum_{(k,q = r \theta \phi)} g_{kq} \mathbf{B}_k \hat{S}_q

    = \hbar \cdot \Omega

    where \Omega represents 2 \pi \nu, the fundamental dynamics it appears of the toroid field coupled to the angular spin of the system, which is likely classical.

    This part \nu describes the ''ordinary frequency'' - this must have an important role in the very heart of the fundamental particle.
    • Thread Starter

    The difference now is that the classical R has to be smaller - so small in fact, it's a bound photon in a tight gravitational geodesic. The reason for this is because of the gravitational constant and R < \bar{r}. In the ''is the electron a photon caught in a toroidal path'' paper, the radius becomes the Compton wavelength which can have any frequency directly related to that energy which we fully quantized: Changes in the energy of the confined photon become functions of important corrections

    \Delta U = \int_{\bar{r}}^{\bar{r}'} E_p d\bar{r}

    If we go below this rest state of the energy (which no particle is truly at rest) because there is an internal motion (the photon) which generates the matter we perceive to exist. This equation above is saying directly the change in energy of a system will depend on our non-classical radius. The frequency itself might even be related to the chirality of particles, .... at least logically considering the spin dynamics are at work here. If we can find these ''small'' gaps, we might find a semi-classical theory even describing the entanglement.


    This model linked, is a much older model but has many same principles and idea's, a refreshing change from me quoting Motz and Schwinger,

    ''Instead of a single orbiting charge, the toroidal ring was conceived as a collection of infinitesimal charge elements, which orbited or circulated along a common continuous path or "loop". In general, this path of charge could assume any shape, but tended toward a circular form due to internal repulsive electromagnetic forces. In this configuration the charge elements circulated, but the ring as a whole did not radiate due to changes in electric or magnetic fields since it remained stationary. The ring produced an overall magnetic field ("spin") due to the current of the moving charge elements. These elements circulated around the ring at the speed of light c, but at frequency f = \frac{c}{2 \pi R}, which depended inversely on the radius R.

    My more modern model presents itself also depending on the radius of the system

    GM^2 = E\bar{r} = \hbar c

    If the integral is performed in a similar fashion where energy compresses as \bar{r} \rightarrow 0 means we have reached the classical upper limit force of unification. In other words, the system becomes a black hole, or grey hole as Hawking now prefers to call them.
    • Thread Starter

    The difference now is that the classical R has to be smaller - so small in fact, it's a bound photon in a tight gravitational geodesic. The reason for this is because of the gravitational constant and R < \bar{r}. In the ''is the electron a photon caught in a toroidal path'' paper, the radius becomes the Compton wavelength which can have any frequency directly related to that energy which we fully quantized: Changes in the energy of the confined photon become functions of important corrections

    \Delta U = \int_{\bar{r}}^{\bar{r}'} E_p d\bar{r}

    If we go below this rest state of the energy (which no particle is truly at rest) because there is an internal motion (the photon) which generates the matter we perceive to exist. This equation above is saying directly the change in energy of a system will depend on our non-classical radius.

    It's also possible that a bound photon in a strong gravitational coupling G_s to it's mass m = \frac{U_{electro}}{c^2} may be though of a standing wave... it makes sense no? Matter has often been called standing waves.
    • Thread Starter

    This prediction of our quadratic term

    \frac{1}{4} \frac{\hbar e}{Mc} \cdot \mathbf{B} \times \bar{r} = \mu_{B} \cdot \mathbf{A}

    Is that for a toroid symmetry in this model, fits exactly for the model in the following reference:



    To prove this quickly how it could be important, is a little rearranging involved in

    \frac{1}{4} \frac{\hbar}{Mc} \cdot \mathbf{B} \times \bar{r} = \frac{(\mu_{B} \cdot \mathbf{A})}{e}

    The expression remaining on the LHS of particular interest proves itself

    \frac{1}{4} \frac{\hbar}{Mc}

    To find

    \frac{1}{4} \lambda_{reduced}

    as a reduced Compton wavelength as both are equivalent, just as I predicted by stating R < \bar{r} since \lambda = R < \bar{r} = \lambda_{reduced}

    Because of this, for a photon model to work, this has to be an strict inequality.
    • Thread Starter

    Now what about the circular gauge? \mathbf{B} \times \bar{r} which made mass an electromagnetic phenomenon in the equation:

    \frac{1}{4} \frac{\hbar}{Mc} = \frac{\mu_{B}}{e}

    - removing the circular gauge leaves us with the equation above, which says the motion of the toroidal photon is determined by the dynamics of


    (which says the Bohr Magneton is weighed by a factor of the charge e = 2\sqrt{\pi G}M.)

    A quick dimensional analysis will show this:

    \frac{\mu_{B}}{e} = \frac{(\frac{e \hbar}{2m_ec})}{e} = \frac{\hbar}{Mc} = \lambda_{red}
    • Thread Starter

    Our theory of Gauge Fields where \phi is a Gauge Boson, has a structure in the equations which invariance. This has been called, Gauge Invariance and this invariance allows physicists to make certain transformations to fields. The laws of physics always remain the same under gauge invariance, indeed, that is what it's all about! It means there is no such thing as an absolute position in physics - the only thing which does count is relative positions. Another feature of the gauge fields is that they retain a symmetry of the theory.

    A symmetry can be understood from the most simplest langrangian term

    \partial\phi^{*} \partial \phi

    Suppose we had a transformation

    \partial \phi' \rightarrow e^{i\theta}\phi(x)

    In this transformation, the derivatives of \phi' are concerned only with \phi. This transformation will look like

    \partial \phi'(x) = e^{i \theta} \partial \phi(x)

    \partial \phi'^{*}(x) = e^{-i \theta} \partial \phi^{*}(x)

    Therefore, when you multiply \partial \phi ' with \partial \phi ^{*} the e^{i\theta} and e^{-i\theta} will cancel out because that is how you would compute them with their conjugates. Thus

    In other words, our field \theta is constant, and does not require the same derivatives as our boson. In return, we would just get

    \partial \phi^{*}\partial \phi

    and viola! This was the most simplest demonstration of a symmetry conserving field. An even simpler demonstration would be:

    \frac{d(x +C)}{dt}

    is in fact simply the same as


    this means that the equations where symmetry existed and what we have in these symmetries are extra constants always remain the same, just as our Gauge example above.

    But what if \theta also was a function of position \theta (x)? Well let's see shall we.

    \partial \phi' = (\partial \phi + i \phi \frac{\partial \theta}{\partial x})e^{i \theta}

    and our conjugate would be

    \partial \phi'^{*} = (\partial \phi^{*} - i \phi^{*} \frac{\partial \theta}{\partial x})e^{-i \theta}

    Multiplying the two, we need to factorize it

    (\partial \phi + i\phi\frac{\partial \theta}{\partial x})(\partial \phi^{*} - i\phi^{*}\frac{\partial \theta}{\partial x})

    which gives

    =\partial \phi^{*} \partial \phi + i(\phi \partial \phi^{*} - \phi^{*} \partial \phi)\frac{\partial \theta}{\partial x} + \phi^{*} \phi (\frac{\partial \theta}{\partial x})^2

    The reason, again why this equation turned out to be such a mess was because our \theta-field now depended on position, so the derivatives of our massless boson field also included it.

    This motivated scientists to find a symmetry again in the equations, and to do so required the use of the Covariant Derivative which originally came form the work on fibre bundles.

    To restore symmetry, we need to define the Covariant Derivative as

    A_{\mu}' \rightarrow A_{\mu} - \partial_{\mu} \theta

    Here, we can see our four-vector potential again A_{\mu} - you can basically build the electromagnetic fields form this. It's time component in A_0 but that is really not relevant right now. Our Covariant Derivative and the respective conjugate fields are usually denoted as

    D_{\mu} \phi = \partial_{\mu} \phi + iA_{\mu} \phi


    D_{\mu} \phi^{*} = \partial_{\mu} \phi^{*} - iA_{\mu} \phi^{*}

    So calculating it all together, we just define the whole thing again as:

    D\phi' = (\partial \phi + i\phi \frac{\partial \theta}{\partial x})e^{i\theta} + i(A_{\mu} - \partial_{\mu} \theta) \phi e^{i \theta}

    Well, with this, we can see straight away that some terms cancel out. The i\phi terms cancel, and \frac{\partial \theta}{\partial x} is in fact the same as \partial_{\mu} \theta. So what we are really left with is

    D\phi' = D \phi e^{i\theta}

    and so its conjugate is

    D\phi^{*} = D\phi^{*} e^{-i\theta}

    Again, the latter terms cancel out when you multiply these two together and so what you end up with is

    D \phi^{*'} D \phi ' = D \phi^{*} D \phi

    and so by using the Covariant Derivative, we have been able to restore the lost U(1) symmetry where U(1) symmetries deal with rotations.

    But when physicists talk about a mass, we don't want to retain symmetry in these fields. In fact, the very presence of a mass term will imply an explicit symmetry breaking. The process of course, is a little more complicated however. It involves another boson, called a Goldstone Boson, which can be thought of as a ground-state photon which lives in the minimum of a Mexican Hat potential. Something which exists in the minimum is the same as saying our system does not contain a mass term and so \phi=0. In such a potential, we may describe our field as

    \phi = \rho e^{i\alpha}

    Here, \rho is a deviation from the ground state. \rho is in fact our Higgs Boson and \alpha is our Goldstone Boson. If \alpha is a frozen (constant) field, then there are no changes in the equations. But if there is a deviation of the Goldstone Boson from the minimum of our potential, then we are saying that it costs energy to do so, and this energy is what we mean by particles like a photon obtaining a mass. In fact, the Goldstone Boson is gobbled up by the Higgs Boson which gives the system we speak about a mass. Our flucuation from the minimum has the identity f \ne 0 where f plays the role of mass.

    Let's discuss this mass term in terms itself of the electromagnetic field tensor. Such a tensor looks like:

    F_{\mu \nu}F^{\mu \nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}

    here, F_{\mu \nu}F^{\mu \nu} is in fact just F^2 and it makes up the langrangian. F_{\mu \nu} is an antisymmetric object with respect to swapping its indices. It is like a four dimensional curl.

    It is Gauge Invariant, but checking that, remember our transformation

    A' \rightarrow A - \partial \theta

    plugging that into the tensor gives:

    F_{\mu \nu}' = \partial_{\mu}A'_{\nu} - \partial_{\nu}A'_{\mu}

    That is

    \partial_{\nu} \partial_{\mu} A_{\nu} - \partial_{\mu} \partial_{\nu} \theta - \partial_{\mu} \partial_{\nu}A_{\mu} + \partial_{\nu} \partial_{\mu} \theta

    Since the order of partial differentiation doesn't matter, the -\partial{\mu} \partial_{\nu}\theta and \partial_{\nu}\partial_{\mu} \theta cancel and what you are left with is invariance.

    There are some numerical factors left out of this, such as a quarter, but that really isn't all that important in this demonstration. And so, we may have a completely Gauge Invariant term

    D_{\mu}\phi D_{\mu} \phi^{*} - V(\phi^{*}\phi) + F_{\mu \nu} F^{\mu \nu}

    One should keep in mind that the potential term here V(\phi \phi^{*}) is also manifestly invariant. What becomes interesting however, is the question of what cannot be added to this but still can remain Gauge Invariant. What we cannot add to the electromagntic part (F^2)-part, is this:


    This is just the same as \frac{M^2 A^{2}_{\mu}}{2}. The reason why, is because a mass-term for a photon would typically look like this. This actually breaks local invariance. In nature there is only one massless spin-1 boson - the photon - then it seems that if we where to use a local gauge group that is less trivial than U(1), then the symmetry would be broken. It's interesting, that when we speak about symmetry breaking for a photon, we are in fact talking about an electromagnetic term which is squared in the Langrangian.

    Studying Gauge theory, one thing seems quite clear - that being the presence of matter seems to be synonymous with the presence of charge. One can see this clearly when you study the equations which describe massless boson fields which satisfy a charge e = 0. As we have seen, a mass term in an electromagnetic tensor equation would have the appearance of

    \frac{1}{2} M^2 A^{2}_{\mu}

    If there is a symmetry we have

    \delta \mathcal{L} = 0

    The langrangian can be symmetric but the vacuum may not be

    e_i|0> \ne 0

    Where e is the charge. In this case for symmetry breaking, Noethers theorem does not imply a conserved charge. In this terminology we say the ground state is in fact degenerate.

    The only problem with idea of mass being invariant with charge is the theoretical models involving neutrino's. Neutrino's have a very very small mass but is said to contain no charge. This seems to be the only exception in nature!

    But if we took the model Motz proposed, we could have a bound pair photon model in which the charge is totally neutral. The appearance of mass is when you have a goldstone boson fluctuate away from the ground state in a mexican hat potential. Of course, one could also argue that if it has a vanishgly small mass, and knowing that the gravitational charge \sqrt{G}M depends on the electric charge, then may come to think it's electric charge could also be vanishingly small. But physicists don't think of the neutrino in such a way, for many reasons I won't know about but I do know that it's mostly mathematical.

    The squared Langrangian which physicists know created the symmetry breaking is also a quadratic term. This is how it may all implement.
    • Thread Starter

    You can find relationships with spin matrices found in Diracs relativistic energy-momentum equation, using the laws of zitter motion.

    (\frac{\hbar}{M}) = (\frac{1}{\hbar})^{-2} p \cdot \mathbf{S}

    \hbar m = (\frac{h}{2 \pi})E= (\frac{2}{\hbar})^2 (\alpha p^2 + \beta M) \cdot \mathbf{S}

    maybe interesting to note, that m \hbar is the re-scaled spin with addition of ''wedding quantum mechanics to relativity, as we are using their respective spin vector \math{S}.

    The parameter of local dynamics, may as well have a similar role for the model of a toroid photon: interestingly, the photon would not only be an excitation of the field, but rather emergent from a deeper level of momentum dynamics; I've been attempting to find some idea's of why a quadratic term in my specific equations. So far I have noticed one thing:

    (\Delta \prop \frac{1}{4})

    This equation is called the quadratic stark effect. It basically comes from the interaction of an emitter with an electromagnetic field - A ground state boson (Goldstone) is just a photon with it's energy equal to the electrons wavelength. For mass to enter the theory, we could use the logic of the statement I made:

    ''It basically comes from the interaction of an emitter with an electromagnetic field''

    Instead in our model, the ground state photon becomes excited when they collide with high energy photons to make matter as soon as the Planck epoch and the electrostrong epoch had appeared, allowing symmetry breaking to occur (the appearance of mass which makes makes matte a temperature problem and if Tegmark is right including myself, then mobile (living) matter subsystems of a low energy epoch). This was what the early universe was like... It was a very hot ball of a state of matter, in a very unique sense, which when cooled down becomes a photon gas universe (black body radiation).
    • Thread Starter

    Let me just quickly cover something else quickly and take a look into the dynamics of the quantized charge condition \sqrt{\hbar c}= \sqrt{G}M,

    What about a relativistic momentum? We need to consider that \hbar c is related to the mometum. It so happens, I found a whole bunch of relationships using some matrix mechanics.

    The canonical relativistic momentum i\hbar \gamma^0 as a matrix involving the spin \hbar is to be multiplied through by c.

    i \hbar c \gamma^0 = \begin{pmatrix} iGM^2 & 0 & & 0 \\0 & iGM^2 & 0 & 0 \\0 & 0 & -iGM^2 & 0 \\0 & 0 & 0 & -iGM^2 \end{pmatrix}\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}

    (\gamma^0)^2 gives us the unitary matrix. In squaring, we remove the imaginary parts

    -\hbar^2 c^2 \mathbb{I} = \begin{pmatrix} -GM^2 & 0 & & 0 \\0 & -GM^2 & 0 & 0 \\0 & 0 & GM^2 & 0 \\0 & 0 & 0 & GM^2 \end{pmatrix}^2 \cdot \gamma^0

    This is beneficial because it has no imaginary coefficients.

    The matrix for \gamma^0 can also be written as

    -\hbar^2 c^2 \begin{pmatrix} 0 & \sigma^1 \\ \sigma^2 & 0 \end{pmatrix} = \begin{pmatrix} -GM^2 & 0 & & 0 \\0 & -GM^2 & 0 & 0 \\0 & 0 & GM^2 & 0 \\0 & 0 & 0 & GM^2 \end{pmatrix}^2\begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}

    and since \gamma^0 = \beta (the standard beta matrix) then \beta a^k = \gamma^k so we can make

    -\hbar^2 c^2 \gamma^k = \begin{pmatrix} -GM^2 & 0 & & 0 \\0 & -GM^2 & 0 & 0 \\0 & 0 & GM^2 & 0 \\0 & 0 & 0 & GM^2 \end{pmatrix}^2 \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}a^k

    \gamma^k can now be written in it's sub-matrix form

    -\hbar^2 c^2 \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix} = \begin{pmatrix} -GM^2 & 0 & & 0 \\0 & -GM^2 & 0 & 0 \\0 & 0 & GM^2 & 0 \\0 & 0 & 0 & GM^2 \end{pmatrix}^2 \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}a^k

    working with

    \begin{pmatrix} -GM^2 & 0 & & 0 \\0 & -GM^2 & 0 & 0 \\0 & 0 & GM^2 & 0 \\0 & 0 & 0 & GM^2 \end{pmatrix}^2 \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}a^k

    writing out the matrix case for a^k gives us

    a^k = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix}

    using it we have

    \begin{pmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & -1 & 0 \\0 & 0 & 0 & -1 \end{pmatrix}\begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & 1 & 0 \\0 & 0 & 0 & -1 \\0 & -1 & 0 & 0 \\-1 & 0 & 0 & 0 \end{pmatrix}

    To solve the RHS we then have

    -\hbar^2 c^2 \begin{pmatrix} 0 & \sigma^k \\ -\sigma^k & 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 & -GM^2 & 0 \\0 & 0 & 0 & GM^2 \\0 & -GM^2 & 0 & 0 \\-GM^2 & 0 & 0 & 0 \end{pmatrix}^2

    We have used here, Pauli matrices \sigma_1 and \sigma_3. The equation above in a more simplified version:

    \begin{pmatrix} 0_2 & -\hbar^2c^2 \cdot \sigma^1 \\ \hbar^2 c^2 \cdot \sigma^3 & 0_2 \end{pmatrix} = \begin{pmatrix} 0 & 0 & -GM^2 & 0 \\0 & 0 & 0 & GM^2 \\0 & -GM^2 & 0 & 0 \\-GM^2 & 0 & 0 & 0 \end{pmatrix}^2

    where the full form of

    \begin{pmatrix} 0_2 & -\hbar^2c^2 \cdot \sigma^1 \\ \hbar^2 c^2 \cdot \sigma^3 & 0_2 \end{pmatrix}


    \begin{pmatrix} 0 & 0 & 0 & -\hbar^2 c^2 \\0 & 0 & -\hbar^2 c^2 & 0 \\-\hbar^2 c^2 & 0 & 0 & 0 \\0 & \hbar^2c^2 & 0 & 0 \end{pmatrix}

    We can finally equate the two

    \begin{pmatrix} 0 & 0 & 0 & -\hbar^2 c^2 \\0 & 0 & -\hbar^2 c^2 & 0 \\-\hbar^2 c^2 & 0 & 0 & 0 \\0 & \hbar^2c^2 & 0 & 0 \end{pmatrix}=\begin{pmatrix} 0 & 0 & -GM^2 & 0 \\0 & 0 & 0 & GM^2 \\0 & -GM^2 & 0 & 0 \\-GM^2 & 0 & 0 & 0 \end{pmatrix}^2

    If you multiply the RHS with the LHS, you get back real numbers, like this: However they are (so far) has no trace.

    \begin{pmatrix} 0 & 1 & 0 & 0 \\1 & 0 & 0 & 0 \\0 & 0 & 1 & 0 \\0 & 0 & 0 & 1 \end{pmatrix}

    This is in fact a submatrix with entries

    \begin{pmatrix} \sigma^1 & 0_2 \\ 0_2 & \mathbf{1}_2 \end{pmatrix}

    What is interesting about this? I will get to it soon enough.
    • Thread Starter

    It was once said to me that there was no operation that could make this submatrix entry with a trace, albiet even Hermitian. It can be by introducing a new matrix. First of all, you get the null matrix via

    \begin{pmatrix} \sigma^1 & 0_2 \\ 0_2 & \mathbf{1}_2 \end{pmatrix}\cdot\begin{pmatrix  } (1-1) & 1 & 0 & 0 \\1 & (1-1) & 0 & 0 & \\0 & 0 & 1 & (1-1) \\ 0 & 0 & 0 & 1 \end{pmatrix}

    = \begin{bmatrix} 1 & 0 & 0 & 0 \\0 & 1 & 0 & 0 \\0 & 0 & 1 & 0\\0 & 0 & 0 & 1 \end{bmatrix}

    There are in fact infinitely amount of different matrix operations which can make it Hermitian, (another word for ''real.'') This matrix

    \begin{pmatrix} (1-1) & 1 & 0 & 0 \\1 & (1-1) & 0 & 0 & \\0 & 0 & 1 & (1-1) \\ 0 & 0 & 0 & 1 \end{pmatrix}

    can look intimidating, but don't forget we can also simplify this into sub-components again by noticing null matrices 0_2

    Reducing it gives you a 2X2 matrix, easier to mentally cope with where the diagonal elements are unique.
    • Thread Starter

    Now what about the circular gauge? which made mass an electromagnetic phenomenon in the equation:

    - removing the circular gauge leaves us with the equation above, which says the motion of the toroidal photon is determined by the dynamics of

    ....I came across some reading:

    ''15.- The revised Bohr magneton value divided by the elementary charge yields a distinct fractal value for the Bohr magneton in eV T-1.

    9.2743042 / 1.602176487 = 5.788565913''

    http://earthmatrix.com/sciencetoday/pla ... tants.html

    It's funny perhaps to think of a closed flux of the photon's wavelength to something being fractal, however, Bohr intends this in the numerical importance for energy levels.
    • Offline

      This is a study and homework help forum, so I think you need to find somewhere else to post your article.
      I suggest taking a look at physics forums, where people post such things.
      • Thread Starter

      I disagree... for the first time IN A LONG TIME have I come across a forum not run by clique idiots. I really respect the team and their efforts for preserving my work. I have a right to be here as anyone else, and if I wish to talk about subjects, why hold me back?
      • Thread Starter

      I'm a strong believer that if peoples minds come together, instead of clashing, they can achieve more.
      • Thread Starter

      Here's something I was writing on the train today. I was remember the energy-correction equations for the famous ''is the electron a photon trapped in a topology'' paper.

      I was wondering... everything has to be a potential to even be consider the probability of having an existence, which brought me back to some of the classical dynamics we spoke of early on in this investigation.

      The total energy of a non-rotating system is a misnomer for all particles except the Higgs Boson, standard or not - this non-rotating total energy is given with our correction earlier

      \frac{a'}{2 \pi a}\hbar \omega

      This expression is a system which possesses an angular momentum term, but isn't coupled gravitomagnetically - for a full Poincare Group expected naturally within the equations of GR this should happen. For a full energy representation of the energy associated to Coriolis energy fields, we would have to add an extra term, so that the extrinsic field interacts appropriately with the intrinsic field of the system.

      \hat{H} = \frac{\alpha'}{2 \pi \alpha}\hbar \omega + \hbar \Omega

      Not only has this become an operator, but the two terms (\omega, \Omega) apply to the over-all energy of the system: The \omega plays an internal motion role, speculated to be trapped photons in strong gravitationally-warped closed geodesic. The symbol \Omega couples to the external fields as the active Coriolis field.

      This can further be represented as the potential by recognizing that \omega = \frac{U}{\hbar} - plugging it in gives us the potential for the system to allow coupling of external and internal field dynamics in a semi-classical electron:

      \hat{H}(U) = \frac{a'}{2\pi a} U + \hbar \Omega(\bar{r})

      where, do not forget, \bar{r} is the ''mean radius of transport, for our topological charged path in which our photon follows.

      (Original post by Graviphoton)
      I'm a strong believer that if peoples minds come together, instead of clashing, they can achieve more.
      Have you considered approaching university professors with your work?

      Nothing will come out of posting this paper on this forum as the majority of users ( 95%+) lack the knowledge, experience and in most cases the interest, to discuss the work.
      If you are looking for confirmation, you won't find it here.
      Posted from TSR Mobile
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