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Green Destiny

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## Homework Statement

I am not an expertize at algebra, and recently I was considering derivisions of equations based on postulates already known to physics. I just wanted a little guidence if any of the equations are inconsistent, which I am sure probably quite a few are, knowing my own personal track record :)

So if any mistakes are pointed out, or that I have missed something, I'd appreciate it if you recognize it aloud for me, so to say.

## Homework Equations

The relevant equations are progressive derivations, so nothing is set in stone at first hand.

## The Attempt at a Solution

**Preliminaries**

First deriving the property of our electric field [tex]\mathbb{E}[/tex] takes the form of:

[tex]\mu_0 D= c^2 \mathbb{E}[/tex]

Rearranging gives:

[tex]\frac{\mu_0 D}{c^2}= \mathbb{E}[/tex]

Squaring both sides, and then multiplying by the permitivvity gives:

[tex]\epsilon_0 (\frac{\mu_0 D}{c^2})^2= \epsilon \mathbb{E}^2[/tex]

We will now employ our rules, having a conservative electric field irrespective of any external magnetic changes. Now I consider the equation:

[tex]\mathbb{E}^2(qt^2)^2 \int v dt= I^2[/tex]

Again we take a similar route and do the following:

[tex]\epsilon_0 \mathbb{E}^2(qt^2)^2 \int v dt= \epsilon_0 I^2[/tex] [*]

If we take one half of the quantity [tex]\epsilon_0 I^2[/tex] where [tex]I[/tex] is the inertial moment, gives us on the right hand side a relation to the field density,

[tex]\frac{1}{2}\epsilon I^2=u_eq^2t^4 \int v dt[/tex]

where the left hand side has been simplified to give one whole value of the inertial moment. This equation as I interpret it, describes the electric field strength as being inversely related to the inertial moment of the particle, and so this would be related to a certain type of electromagnetic inertia.

The equation with the star [*] is like most of the equations, describing a charge in an electric field in a state of motion.

We will soon be considering the dynamical effects of the electric field [tex]\mathbb{E}[/tex] and it's associated density [tex]u_e[/tex] in energy per unit volume of spacetime - because one can minimize the integral action, this immediately turns into a lowest frequency, or ground state solution of the vacuum. In other words, it would have exactly [tex]E=\frac{1}{2}\hbar \omega[/tex] as fluctuations in the bubbling vacuum of spacetime.

The electric field is something which describes spacetime surrounding electrically-charged particles or even a time-varying magnetic field. However, the above is seen in light of a conservative electric field with no magnetic field present. Thus the electric field in this case can be seen as exerting a force on particles.

If a field can exert a force on a particle, why may it not give rise to a force associated similar to that of an inertial force?

The force resisting acceleration could very well be related to the field densities acting as a type of electromagnetic inertia.

**Inertia as Innert Energy**

Einstein never ruled out completely the cause of inertia, but in a series of work, he did show it was possible that the inertial energy of a system could cause inertia, which is slightly different to the above, but shares a fascet in which that the field density can be verbally replaced but not mathematically equivalent to an energy desnity itself, since the electric field is not a field which has the dimensions of energy.

The conservative field equations I have been using, are themselves a type of equation of motion, where the integral must govern a change in position, with a velocity which by choice of the observer, can change. It's linear however, but this can still invoke a more complicated set of dervations I would have presumed which we could allow a system to experience a varying magnetic field.

Einstein, as I said, involved the genious idea that perhaps the rest energy of a material system is what causes the inertia of matter, but in a sense, energy and matter are but fascets themselves of the same manifestation under Einsteins mass-energy relationship - but, with a little thought, I question the methodologies of using energy to explain inertia alone. Afterall, saying energy is responsible for inertia in a material system is just trivial as saying matter is what causes the inertia of trapped energy - neither really are enlightening to the cause of inertia itself, other than something pointing to the inherent structure of the particle (intrinsic structures).

In response to Einsteins idea, it was then a consideration to see if we could see the above equations, in a slightly different light, this time concerning energy.

I came to the equation:

[tex]\epsilon_0 \frac{(\frac{\mu_0 D}{c^2})q^2t^4}{t^2} \delta \int v dt= \epsilon_0 E_0 \ell[/tex]

where [tex]E_0[/tex] is for inertial energy and [tex]\ell[/tex] is basically a distance travelled, with obviously dimensions of L. Here I have used a minilizing integral on the change of position, which in the sense is meant to indicate it takes the least energy required to move that distance. This means that it satisfies a ground state of energy, so I am assuming this eq. can be altered to the following form;

[tex]\epsilon_0 \frac{(\frac{\mu_0 D}{c^2})q^2t^4}{t^2} \delta \int v dt= \epsilon_0 (\hbar \omega) \ell[/tex]

Where [tex]\hbar \omega[/tex] is the lowest frequency, or ground state of energy per unit volume of spacetime.

**The Four Equations of Motion Relating the Electric Field, The Inertial Mass-Moment, Field density and the Energy**

So far through the derivisions made and the restraints of what the theory so far permits for the respective variables we are using, there are four equations which justify each other and unify the sense of inertia being related to the density of the electric field, the rest energy of the inertia (which will be a concept challenged later), the electric field which acts like a grid not in spacetime as you would with operating with Cartesian Coordinates, but one which has itself it's own abstract field, and this will measure the density in relation to field projected by the particle in question.

The first equation is

[tex]\epsilon_0 \mathbb{E}^2q^2t^4 \delta \int v dt = \epsilon_0 I^2[/tex]

This unifies the electric field with the inertial moment with a relation to the electric permitivvity.

[tex]\frac{1}{2}\epsilon_0 I^2=u_e q^2 t^4 \delta \int v dt[/tex]

The second equation here unifies that half the quantity of inertia goes to the electric field strength. [A]

The Third equation now, unified the energy with the electric field density, again in quite a familiar pattern:

[tex]\frac{1}{2}\epsilon_0 E^{2}_{0}t^4=u_e q^2 t^4 \delta \int v dt[/tex]

The lower script in the energy-term on the right hand side is for two useful reasons. The first is to destinguish it notationally from the electric field notation which is similar, but also to demonstrate we are talking about an inertial energy, which some define as a rest energy.

However, because it has a minililizing on the integral, the true way to express the third equation is really by this form:

[tex]\frac{1}{2}\epsilon_0 (\hbar \omega)^{2}_{0}t^4=u_e q^2 t^4 \delta \int v dt[/tex]

The fourth equation comes in the form of relating the energy to the electric field itself, and it's not always completely obvious, but wrapped up in the signs [tex]\epsilon \mathbb{E}^2[/tex] does have meaning as being closely related to the displacement if it were not for the extra multitude of electric field. This being related to the energy in its cryptic-sense should not seem to bizarre, as displacement does involve the use of a kinetic energy.

[tex]\frac{1}{2}\epsilon_0 E^{2}_{0}t^4=u_eq^2t^4 \int v dt[/tex]

What is interesting, is that the charge has been the one unit which has been pretty much invariant throughout each derivation of the four equations, all logical from each other, and relating their different properties in unique ways.

[A] - I have considered possible definitions of what this could mean. Because mathematics is an abstractual representation of cold rigor, sometimes it's difficult to come to a conclusive resuly in your heart what it might mean. I have come to interpret this however to mean that inertia arises as an emergent property of a

**charged**particle

**moving**in an electric field. This would basically confirm my original hypothesis that there is somehow a description of an electro-inertia condition. It may be even possible that the field strength has an opposing force on the charge, and I will explore that possibility in another post using coupling theory [tex]g(x)[/tex] and try and allow it to work alongside some description of a Higgs Influence.

**The Coupling of the Invariant Charge and a Gravitational Mass**

Since charge [tex]q^2[/tex] has been invariant throughout the equations, it might be noted that the quantum mechanical definition of a charge density is [tex]q(\int |\psi|^2)[/tex]. The density of a charge will remain constant in a thermally-abundant environment. Since the electric field strength was shown to play a

*possibly*large role in the understanding of it's connection to electric field, there maybe some indication here that perhaps this is what the thermally-abundant environment could be. The electric field istelf - this would require making a coupling constant making charge it's coefficient of reference [tex]g(q^2)[/tex], since it will be the charge in this hypothesis which couples to the mass of the particle. The equations above will lead to some description of this, but it will require more derivations.

The coupling between the mass term [tex]M[/tex] and the charge [tex]q[/tex] will first be identified as:

[tex]M=q \phi[/tex]

that should be [tex]M=q \phi[/tex]

This equation is actually well known, but not illuminating. From here, my goal will be to derive the mass-term relationship to the electric field equations. But this will be no good if I have a lot of mistakes. I am very keen to learn as much as I can on calculus. I have got quite old and the memories of college just gave me a boost this year to get right stuck into all this stuff I read many years ago. It certainly does escape your memory fast.

I will appreciate anyone who has the patience to study the equations I have written so far, and if they notice any mistakes, I am open-ears so that I may fix them. I am very very rusty at calculus, hence why I am keen for someone to have the patience with me and take the time out to have a wee look over them.

Thank you in advance!

Green destiny

ps. This site looks good. I think I am going to enjoy it here, and maybe contribute what I can :)

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