On Space Time And The Fabric Of Nature Charge

The Nature of Charge

"The partial differential equation came into theoretical physics as a servant, but little by little it took on the role of master." - Albert Einstein (1931)

Perhaps one of the most fundamental yet unanswered questions in Physics is: What is (electric) charge? On WikiPedia, the following description is given:

Electric charge is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charges: positive and negative. Like charges repel and unlike attract. An object is negatively charged if it has an excess of electrons, and is otherwise positively charged or uncharged. The SI derived unit of electric charge is the coulomb (C). In electrical engineering, it is also common to use the ampere-hour (Ah), and, in chemistry, it is common to use the elementary charge (e) as a unit. The symbol Q often denotes charge. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that don't require consideration of quantum effects.
The electric charge is a fundamental conserved property of some subatomic particles, which determines their electromagnetic interaction. Electrically charged matter is influenced by, and produces, electromagnetic fields. The interaction between a moving charge and an electromagnetic field is the source of the electromagnetic force, which is one of the four fundamental forces.
Twentieth-century experiments demonstrated that electric charge is quantized; that is, it comes in integer multiples of individual small units called the elementary charge, e, approximately equal to 1.602×10-19 coulombs (except for particles called quarks, which have charges that are integer multiples of 1/3e). The proton has a charge of +e, and the electron has a charge of -e. The study of charged particles, and how their interactions are mediated by photons, is called quantum electrodynamics.

Let us note the following:

  • Charge is considered to be a property of matter;
  • in this sense, it is quantized i.e. charged particles are found to exhibit a charge which is always an either positive or negative integer multiple of (an integer division of) the elementary charge, denoted by e.

Let us now introduce the (mathematical) concept of divergence:

In vector calculus, divergence is a vector operator that produces a signed scalar field giving the quantity of a vector field's source at each point. More technically, the divergence represents the volume density of the outward flux of a vector field from an infinitesimal volume around a given point.
As an example, consider air as it is heated or cooled. The velocity of the air at each point defines a vector field. While air is heated in a region, it expands in all directions, and thus the velocity field points outward from that region. The divergence of the velocity field in that region would thus have a positive value. While the air is cooled and thus contracting, the divergence of the velocity has a negative value.

As an intuitive explanation, one can say that the divergence describes something like the rate at which a gas, fluid or solid "thing" is expanding or contracting. When we have expansion, we have an out-going flow, while with contraction we have an inward flow.

Let us note that:

  • the resulting value is calculated over an infinitesimal volume, which means something like "an infinitely small volume". This is called "taking the limit to 0", denoted by {$\lim_{\to 0}$};
  • since the resulting value is divided by the volume over which the divergence is calculated, we obtain a result in the form of a density;
  • since the resulting value is defined for all points in space, it essentially expresses the variation or distribution of the density value in space. That is why this notation/calculation method is called differential form;
  • the differential form in vector calculus is a multi dimensional notation, wherein multiple partial differential equations are taken together in a compact notation.

For example, in normal 3 dimensional space, with vector calculus notation we don't have to write out all the partial derivatives for all 3 dimensions and combinations thereof in 3 rather long equations, but we can take them together in one vector/matrix notation, which significantly improves readability, while at the same time significantly reducing complexity and thus the chance of making errors.

This notation is specifically useful for the description of fields and it can be used to derive wave equations. The equations can also be written out and can then be numerically solved with great accuracy using a computer, which is called a simulation. This involves the implementation of the finite-difference time-domain or Yee's method (FDTD), whereby the core "bare bone" algorithm can be written down in just a few lines of computer code.

For manual mathematical analysis, however, it is often useful to use the so-called integral form of the same equations, which are essentially the same as the differential form, with the essential difference that in the integral form, we are calculating with actual volumes, surfaces, etc. without taking the limit to 0. In that case, we are not calculating with the density of a certain parameter [/m, /m2, or /m3] , but with the actual parameter. So, the differential form can always be converted to integral form, as for example with Maxwell's equations.

As an intuitive explanation, one could say that the differential form gives you a detailed view (when simulated on a computer), while the integral form gives you sort of a helicopter view of the same phenomenon.

With this in mind, it is clear that for the analytical consideration of a "toroidal topology", we would use the integral form for the calculation of parameters associated with such a topology. Such a consideration leads to rather remarkable results, as Paul Stowe has shown. Let us now review some of his work.

Stowe's aether model

The basis of Stowe's theory is the definition of a simple model for describing the aether as if it were a compressible, adiabatic and inviscid fluid. Such a fluid can be described with Euler's equations:

The equations represent Cauchy equations of conservation of mass (continuity), and balance of momentum and energy, and can be seen as particular Navier–Stokes equations with zero viscosity and zero thermal conductivity.

In other words, with such an aether model, we can describe the conservation of mass, momentum and energy and if the hypothesis of the existence of such a kind of aether holds, these are the only three quantities that are (fundamentally) conserved.

The definition of his aether model is straightforward and can be found in his "A Foundation for the Unification of Physics" (1996) (*):

We will start by defining a single vector entity (a basic quantum [not a photon, neutrino, graviton]). The fundamental properties of this quantum entity is; it has momentum P, occupies space consisting of volume s, obeys Newton laws of motion, exerts no force, and no external forces are exerted on it. These quanta therefore move through four dimensional space (x,y,z,t) at velocity V and have an apparent mass m, equal to (P/V).
Next, a population of n of these quanta, having random orientation, occupying volume s', [...] results in a system described by basic kinetic theory (without friction or interacting forces {a superfluid state}). Since each quantum, by definition, has an intrinsic momentum P, the system momentum p_s, becomes simply n[p].

With this definition, all kinds of considerations can be made, for example about the question of whether or not an aether model should be compressible or not. In a Usenet posting dated 4/26/97 he wrote(*):

A little history of Maxwell's work. Maxwell fully acknowledges that his Treatise's were, of necessity, incomplete (or as he phrased it: "in our current state of ignorance"). He take the classical simplification of assuming an incompressible medium. This is done because it significantly simplifies the resulting derivations, and unless the media departs significantly from its equilibrium density, such compressibility has very little (negligible) impact on the results under consideration. But compressibility does affect the basic properties. Assumption of incompressibility mathematically defines the divergence of field velocity v as:

{$$ div \, \mathbf{v} = 0 $$}

where v is the media's particulate velocity. A direct consequence of this definition is that waves cannot be created or propagated in such a system (wave speed is infinite). But, as we all know, even though we assume incompressibility, every media (even liquids and solids) are not incompressible. The consequence of this is, for field velocity v:

{$$ div \, \mathbf{v} > 0 $$}

Thus the momentum field property (p = mv) is

{$$ div \, \mathbf{p} > 0 $$}

This has measurable physical consequences, and IS A FUNDAMENTAL UNIQUE PROPERTY of the field! Given that divergence is defined as:

{$$ div = \lim_{V \to 0} \oint \frac{\delta A}{\delta V} \qquad \qquad \text{(A is area)}$$}

and has physical units of inverse distance (meters), Div v become the measure of an oscillation in the velocity field at any point in the continuum. The resulting momentum fluctuation is ... elemental charge, a unique property that is a consequence of the field's compressibility.

This statement illustrates the reasoning which led to Stowe's interpretation of the concept of charge, which he interprets as being a property of the field c.q. the medium. In "A Foundation for the Unification of Physics" (1996) he explained that in order to calculate the value of e, one needs to consider a torroidal topology, whereby both the enclosed volume as the surface area can be expressed in terms of the large toroidal radius, R, and the poloidal axis,r (*):

We can define the systemic fluctuations in the momentum content of a limiting volume element. This is known as the divergence. Divergence is defined as:

{$$ div \, S = \lim_{S \to 0} \frac{\int \delta A}{\delta S} $$}

Where A is the surface area of volume S.
Taken for momentum we get:

{$$ div \, \mathbf{P} = \lim_{S \to 0} \frac{\mathbf{P} \int \delta A}{\delta S} $$}

(Image from later paper)

This term e, becomes $ \pm 2 \mathbf{P} / r$ in a torroidal topology (predominantly consisting of vortex rings {this is an assumption based on the spinor topology of superstring theories and consistent with the earlier atomic vortex theories}), $A = 4 \pi^2 R r$ and $S = 2 \pi^2 R r^2$ {R is the large toroidal radius and r the poloidal axis} and represents an intrinsic fluctuation of the quantized particulate momentum in the limiting volume element.

This is a truly remarkable finding, which finally gives us a basis for understanding what charge is and how the effect it causes, the electric field, propagates trough the medium. However, as we will see, we will have to refine Stowe's interpretation in order to come to a complete understanding of the phenomena of charge and the electric field.

Nonetheless, it is the particular idea of considering a toroidal topology within the context of the postulated existence of a physical aether which enables us to "adapt the theoretical foundation of physics to this new type of knowledge (Quantum Theory)", as Einstein once put it.

(*) Slightly edited for clarity, replaced ascii formulas with math symbols, added epmhasis, etc.

Decomposing Stowe's vector field

Now that we are aware that the consideration of a toroidal topology yields remarkable results - which can even explain some anomalies - we can apply vector calculus in order to come to a formal derivation and verification of the results acquired by Stowe, which we shall do by decomposing the field proposed by Stowe into two components. We will begin by following Stowe and define a vector field analogue to fluid dynamics, using the continuum hypothesis:

At a microscopic scale, fluid comprises individual molecules and its physical properties (density, velocity, etc.) are violently non-uniform. However, the phenomena studied in fluid dynamics are macroscopic, so we do not usually take this molecular detail into account. Instead, we treat the fluid as a continuum by viewing it at a coarse enough scale that any “small” fluid element actually still contains very many molecules. One can then assign a local bulk flow velocity v(x,t) to the element at point x, by averaging over the much faster, violently fluctuating Brownian molecular velocities. Similarly one defines a locally averaged density ρ(x,t), etc. These locally averaged quantities then vary smoothly with x on the macroscopic scale of the flow.

We define this vector field P as:

{$$ \mathbf{P}(\mathbf{x},t) = \rho(\mathbf{x},t) \mathbf{v}(\mathbf{x},t), $$}

where x is a point in space, $\rho(\mathbf{x},t)$ is the averaged aether density at x and $\mathbf{v}(\mathbf{x},t)$ is the local bulk flow velocity at x. Since in practice, this averaging process is usually implied, we consider the following notations to be roughly equivalent:

{$$ \mathbf{P} = \rho \mathbf{v}, $$} {$$ \mathbf{p} = m \mathbf{v}, $$} {$$ \mathbf{P} = m \mathbf{v}. $$}

We will attempt to use P for denoting the "bulk" field and p to refer to an individual "quantum", but that may not always be the case.

Helmholtz decomposition

Let us now introduce the Helmholtz decomposition:

In physics and mathematics, in the area of vector calculus, Helmholtz's theorem, also known as the fundamental theorem of vector calculus, states that any sufficiently smooth, rapidly decaying vector field in three dimensions can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field; this is known as the Helmholtz decomposition.

The physical interpretation of this decomposition, is that the a given vector field can be decomposed into a longitudinal and a transverse field component:

A terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component. This terminology comes from the following construction: Compute the three-dimensional Fourier transform of the vector field Fv. Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k.

It can be shown that performing a decomposition this way, indeed results in the Helmholtz decomposition. Also, a vector field can be uniquely specified by a prescribed divergence and curl:

The term "Helmholtz Theorem" can also refer to the following. Let {$\mathbf{C}$} be a solenoidal vector field and {$\mathbf{d}$} a scalar field on {$ \mathbf{R}_3$} which are sufficiently smooth and which vanish faster than {$ 1/r^2 $} at infinity. Then there exists a vector field {$\mathbf{F_v}$} such that

{$$ \nabla \cdot \mathbf{F_v} = d \text{ and } \nabla \times \mathbf{F_v} = \mathbf{C} $$}

if additionally the vector field {$\mathbf{F_v}$} vanishes as {$r \to \infty $}, then {$\mathbf{F_v}$} is unique.
In other words, a vector field can be constructed with both a specified divergence and a specified curl, and if it also vanishes at infinity, it is uniquely specified by its divergence and curl. This theorem is of great importance in electrostatics, since Maxwell's equations for the electric and magnetic fields in the static case are of exactly this type.

So, let us define a vector field {$\mathbf{A}_T$} for the magnetic potential, a scalar field {$\Phi_L$} for the electric potential, a vector field {$\mathbf{B}$} for the magnetic field, a vector field {$\mathbf{E}$} for the electric field and a vector field {$\mathbf{G}$} for the gravitational field by:

{$$ \mathbf{A}_T = \nabla \times \mathbf{v} $$} {$$ \Phi_L = \nabla \cdot \mathbf{v} $$}

{$$ \mathbf{B} = \nabla \times \mathbf{A}_T = \nabla \times (\nabla \times \mathbf{v}) $$} {$$ \mathbf{E} = - \nabla \Phi_L = - \nabla (\nabla \cdot \mathbf{v}) $$}

{$$ \mathbf{G} = \nabla \mathbf{E} = - \nabla (\nabla (\nabla \cdot \mathbf{v})) $$}

According to the above theorem, {$ \mathbf{v} $} is uniquely specified by {$\Phi_L$} and {$\mathbf{A}_T$}. And, since the Curl of the gradient of any twice-differentiable scalar field {$ \Phi $} is always the zero vector, {$\nabla \times ( \nabla \Phi ) = \mathbf{0}$}, and the divergence of the curl of any vector field P is always zero, {$\nabla \cdot ( \nabla \times \mathbf{v} ) = 0 $}, we can establish that {$\Phi_L$} is indeed curl-free and {$\mathbf{A}_T$} is indeed divergence-free.

For the summation of {$ \mathbf{E} $} and {$ \mathbf{B} $}, we get:

{$$ \mathbf{E} + \mathbf{B} = - \nabla (\nabla \cdot \mathbf{v}) + \nabla \times (\nabla \times \mathbf{v}) = - \nabla^2 \mathbf{v}, $$}

which is the negated vector Laplacian for {$ \mathbf{v} $}.

Since {$\mathbf{v}$} is uniquely specified by {$\Phi_L$} and {$\mathbf{A}_T$}, and vice versa, we can establish that with this definition, we have eliminated "gauge freedom". This clearly differentiates our definition from the usual definition of the magnetic vector potential, about which it is stated:

[The usual] definition does not define the magnetic vector potential uniquely because, by definition, we can arbitrarily add curl-free components to the magnetic potential without changing the observed magnetic field. Thus, there is a degree of freedom available when choosing A.

With our definition, we cannot add curl-free components to {$ \mathbf{v} $}, not only because {$ \mathbf{v} $} is well defined, but also because such additions would essentially be added to {$ \Phi_L $}, which encompasses the curl-free component of our decomposition.

https://en.wikipedia.org/wiki/Newton%27s_laws_of_motion#Newton.27s_second_law

The second law states that the rate of change of momentum of a body, is directly proportional to the force applied and this change in momentum takes place in the direction of the applied force.

{$$ \mathbf{F_N} = \frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t} = \frac{\mathrm{d}(m\mathbf v)}{\mathrm{d}t} $$}

The second law can also be stated in terms of an object's acceleration. Since Newton's second law is only valid for constant-mass systems, it can be taken outside the differentiation operator by the constant factor rule in differentiation. Thus,

{$$ \mathbf{F_N} = m\,\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t} = m\mathbf{a}, $$}

where FN is the net force applied, m is the mass of the body, and a is the body's acceleration. Thus, the net force applied to a body produces a proportional acceleration. In other words, if a body is accelerating, then there is a force on it.

Voor Phi kunnen we div P nemen, welke Stowe associeert met "q", maar het definieert eigenlijk de source/sinks van een flow, een flow van momenta p = m v. Dat heeft weer zo zijn gevolgen voor het beschrijven/afleiden van behoudswetten - q is immers een flow en geen "quantity" van het een of ander.

Maar goed, onder aan de WP pagina trof ik een zeeer interessante note aan:

Poloidal–toroidal decomposition for a further decomposition of the divergence-free component \nabla \times \mathbf {A}.

https://en.wikipedia.org/wiki/Poloidal%E2%80%93toroidal_decomposition

For a three-dimensional F, such that

{$$ \nabla \cdot \mathbf {F} =0, $$}

can be expressed as the sum of a toroidal and poloidal vector fields:

{$$ \mathbf {F} =\mathbf {T} +\mathbf {P} =\nabla \times \Psi \mathbf {r} +\nabla \times (\nabla \times \Phi \mathbf {r} ), $$}

where \mathbf {r} is a radial vector in spherical coordinates {\displaystyle (r,\theta ,\phi )}, and where {\mathbf {T}} is a toroidal field

{$$ \mathbf {T} =\nabla \times \Psi \mathbf {r} $$}

for scalar field {\displaystyle \Psi (r,\theta ,\phi )},[2] and where {\mathbf {P}} is a poloidal field

{$$ \mathbf {P} =\nabla \times \nabla \times \Phi \mathbf {r} $$}

for scalar field {$ \Phi (r,\theta ,\phi ) $}. This decomposition is symmetric in that the curl of a toroidal field is poloidal, and the curl of a poloidal field is toroidal"

Dit geldt dus voor een solenoidal vector veld, een divergentie-vrij vector veld:

https://en.wikipedia.org/wiki/Solenoidal_vector_field

In vector calculus a solenoidal vector field (also known as an incompressible vector field or a divergence free vector field ) is a vector field v with divergence zero at all points in the field:

{$$ \nabla \cdot \mathbf {v} =0.\, $$}

Meer over toroidal / poloidal (met een plaatje er bij): https://en.wikipedia.org/wiki/Toroidal_and_poloidal

De crux is dan dat je zowel oscillaties in de toroidale richting als in de poloidale richting kunt hebben, waarmee je een wiskundige basis hebt voor twee verschillende divergentie-vrije golf verschijnselen aka "near" en "far" fields...

Ik ga hier eens mee aan de slag....

Second derivatives

Curl of the gradient

The Curl of the gradient of any twice-differentiable scalar field {$ \phi $} is always the zero vector:

{$$\nabla \times ( \nabla \phi ) = \mathbf{0}$$}

Divergence of the curl

The divergence of the curl of any vector field A is always zero: {$$\nabla \cdot ( \nabla \times \mathbf{A} ) = 0 $$}

Divergence of the gradient

The Laplacian of a scalar field is defined as the divergence of the gradient: {$$ \nabla^2 \psi = \nabla \cdot (\nabla \psi) $$} Note that the result is a scalar quantity.

Curl of the curl

{$$ \nabla \times \left( \nabla \times \mathbf{A} \right) = \nabla(\nabla \cdot \mathbf{A}) - \nabla^{2}\mathbf{A}$$} Here,∇2 is the vector Laplacian operating on the vector field A.

Taking Gauss' law to the limit

All these fifty years of conscious brooding have brought me no nearer to the answer to the question, ‘What are light quanta?’ Nowadays every Tom, Dick and Harry [jeder Lump] thinks he knows it, but he is mistaken. - Albert Einstein (1951)

Gauss's law may be expressed as:

{$$ \Phi_E = \frac{Q}{\varepsilon_0} $$}

where ΦE is the electric flux through a closed surface {$ S(\Omega) $} enclosing any volume {$ \Omega $}, Q is the total charge enclosed within {$ \Omega $}, and ε0 is the electric constant. The electric flux ΦE is defined as a surface integral of the electric field:

{$$ \Phi_E = \iint_{S(\Omega)} \mathbf{E} \cdot \mathrm{d}\mathbf{S} $$}

where E is the electric field, dS is a vector representing an infinitesimal element of area of the surface, and · represents the dot product of two vectors. The electric flux ΦE can also be defined as a volume integral of the charge density, which we denote by {$\rho_E$}:

{$$ \Phi_E = \frac{1}{\varepsilon_0} \iiint_\Omega \rho_E \,\mathrm{d}V $$}

Of course, the volume integral and the surface integral give the same result and thus can be equated, which is the form Gauss' law is generally included in Maxwell's equations:

{$$ \iint_{S(\Omega)} \mathbf{E}\cdot\mathrm{d}\mathbf{S} = \frac{1}{\varepsilon_0} \iiint_\Omega \rho_E \,\mathrm{d}V $$}

The above equation is written in integral form and can also be written in differential form:

{$$ div \, \mathbf{E} = \frac{\rho_E}{\varepsilon_0} \, $$}

While the integral and differential forms are mathematically equivalent by the divergence theorem, in practical application this only holds within the limitations of continuum mechanics. Also, the equations using {$\rho_E$}, the charge density, assume the charge to be uniformly distributed within the volume under consideration, which is not necessarily true under all circumstances, especially when considering a charge distribution at a very small scale, like when computing the field of a handful of particles like electrons, protons or atom nuclei at the nano-scale.

However, since within Stowe's aether model, we equate electric permittivity {$\varepsilon$} to mass density {$\rho_M$}, we can write:

{$$ \Phi_E = \iint_{S(\Omega)} \mathbf{E}\cdot\mathrm{d}\mathbf{S} = \iiint_\Omega \frac{\rho_E}{\rho_M} \,\mathrm{d}V = \frac{\rho_E}{\rho_M} |V| = \frac{Q}{M} , $$}

or:

{$$ div \, \mathbf{E} = \frac{\rho_E}{\rho_M} $$}

in the case we are considering a situation whereby there are only charge carriers within the volume under consideration.

Let us compare this with WikiPedia's definition of divergence:

The divergence of a vector field F at a point p can be defined as the limit of the net flow of F across the smooth boundary of a three-dimensional region V divided by the volume of V as V shrinks to p. Formally,

{$$ div \mathbf{F}(p) = \lim_{V \rightarrow \{p\}} \iint_{S(V)} \frac{\mathbf{F} \cdot \mathbf{n}}{|V|} \; \mathrm{d}\mathbf{S} $$}

where |V| is the volume of V, S(V) is the boundary of V, and the integral is a surface integral with n being the outward unit normal to that surface. The result, div F, is a function of p. From this definition it also becomes explicitly visible that div F can be seen as the source density of the flux of F.
In light of the physical interpretation, a vector field with zero divergence everywhere is called incompressible or solenoidal – in this case, no net flow can occur across any closed surface.
The intuition that the sum of all sources minus the sum of all sinks should give the net flow outwards of a region is made precise by the divergence theorem.

Stowe's Aether model

Now that we have formulated a basic conceptual model of electromagnetism, which requires a model of a compressible aether, we can consider how to describe such a model. Paul Stowe already proposed such a model, which he outlined as follows:

I have determined that in my opinion all of physical processes can be defined in terms of the aether populational momenta (p). Such that, 
          Force     (F) -> Grad p
          Charge    (q) ->  Div p
          Magnetism (B) -> Curl p
Gravity for example is Grad E where E is the electric potential at x. This resolves to Le Sagian type process as outlined in the Pushing Gravity models. The electric potential E in turn is created by charge which is Div p...
My model is a direct extension of Maxwell's vortex model of interacting rings (the smoke ring model). I have been able to define all fundamental constants in terms of basic parameters, including the gravitational constant G. Further, G is, within this system, seamlessly integrated to all others, fitting into a unified system.
The key to this system's definition is the realization that charge is fundamentally a result AND the measure of the compressibility of Maxwell's aether.

[...]

Quantity SI Conversion Factor to Maxwell's Ether Based Units 

Length meter   (m)                  meter(m)
Mass Kilogram  (kg)                 Kilogram (kg)
Time Second    (sec)                second (sec)
Force Newton   (Nt)                 kg-m/sec^2
Energy Joules  (J)                  kg-m^2/sec^2
Power Watts                         kg-m^2/sec^3
Action         [h] (Planck's Const) kg-m^2/sec
Permitivitty   [z] (Q^2/kg-m^3)     kg/m^3 {1}
Permeability   [u] (kg-m-sec^2/Q^2) m-sec^2/kg {2}
Charge         [q] (Coulomb)        kg/sec
Boltzmann's    [k] (J/°K)           m-sec
Current        [I] (Amp)            kg/sec^2
Electric Field [E]                  m/sec
Potential      [V] (Voltage)        m^2/sec {3}
Displacement   [D]                  kg/m^2-sec
Resistance     [R] (Ohms)           m^2-sec/kg
Capacitance    [C]                  kg/m^2
Magnetic Field [H] (Henries)        m^2
Magnetic Flux  [B] (Gauss)          (dimensionless)
Inductance     [L]                  m^2-sec^2/kg
Temperature   [°K] (Kelvin)         kg-m/sec^3

{1} - density
{2} - modulus
{3} - Kinematic Viscosity

The basic physical quantities in this system are the medium properties identified by Maxwell in his 1860-61 "On Physical Lines of Force". We quantify the mean momentum (quanta) [ß], characteristic mean interaction length (quanta) [L], the root mean speed [c], and a mass attenuation coefficient [¿].

Their values are, 

ß = 5.154664E-27 kg-m/sec
L = 6.430917E-08 m
¿ = 3.144609E-06 m^2/kg
c = 2.997925E+08 m/sec

In other words, all of the major observed and measured constants of physics can be derived from the above terms.

Note that he directly associates the concept of electric elasticity with the compressibility of the aether itself, as we proposed would be necessary. He worked this out in his artlcle "The nature of Charge"(1999).

Gauss' law

Notation:

gradient : ∇

divergence: ∇⋅

curl : ∇

Note: on WP, $\rho$ is used for charge density, while in our model this is already used for mass density. In order to avoid confusion, we will use $\theta$ instead

https://en.wikipedia.org/wiki/Electric_potential#Electrostatics

The electric potential at a point r in a static electric field E is given by the line integral

{$$ V_\mathbf{E} = - \int_C \mathbf{E} \cdot \mathrm{d} \mathbf{\ell} \; , $$}

where C is an arbitrary path connecting the point with zero potential to r. When the curl ∇ × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. In this case, the electric field is conservative and determined by the gradient of the potential:

{$$ \mathbf{E} = - \mathbf{\nabla} V_\mathbf{E}. \, $$}

Then, by Gauss's law, the potential satisfies Poisson's equation:

{$$ \mathbf{\nabla} \cdot \mathbf{E} = \mathbf{\nabla} \cdot \left (- \mathbf{\nabla} V_\mathbf{E} \right ) = -\nabla^2 V_\mathbf{E} = \theta / \varepsilon_0, \, $$}

where $\theta$ is the total charge density (including bound charge) and ∇· denotes the divergence.

This holds in the case the field is conservative, which means that $ \theta $ is considered to be constant over the volume wherein path C can be chosen. This also holds in infinitesimal consideration when the volume goes to zero, provided we also include the partial derivative of $ \theta $ with respect to time ($ \frac{\partial \theta}{\partial t} $).

Gauss' law in differential vorm is given as:

{$$ \mathbf{\nabla} \cdot \mathbf{E} = \frac{\theta}{\varepsilon_0} \, $$}

where ∇ · E is the divergence of the electric field, ε0 is the electric constant, and $\theta$ is the total electric charge density (charge per unit volume).

In Stowe's "Atomic Vortex Hypothesis", we find the following (pg. 2,3):

"In any compressible media there exist cyclic fluctuations in the physical contents of every point. The scalar magnitude of the point momentum variance is Divergence. [...] The textbook dimensions of charge ( in terms of mass/length/time) remains undefined and is assigned arbitrary names ( Coulombs in SI, ESU in cgs, ... etc .) in different unit systems. In this model this is not the case and for the SI unit system it is defined as one (1) Coulomb equals 1 kg/sec".

Thus, we can write:

{$$ \theta = \frac{\partial \rho}{\partial t} $$}

Substituting this into Gauss' equation, we get:

{$$ \mathbf{\nabla} \cdot \mathbf{E} = \frac{1}{\varepsilon_0} \frac{\partial \rho}{\partial t} $$}

This is a rather remarkable find. It fundamentally not only couples the concept of charge to the compressibility of the aether itself, it also couples it fundamentally to time variations of the mass density of the aether itself, which is quite logical given the model used for describing the aether as a super fluid, consisting of "particles" defined in terms of average momentum, or the distribution of mass times velocity.

This pretty much implies that longitudinal waves, involving the movements of aether mass, are the fundamental phenomenon that creates the electric field. This suggests that at the atomic scale, the concept of a "charge carrier" is akin to a "Helmholtz resonator" in the acoustic domain, which is capable of producing "jet streams", which can create a (propulsive) force:

http://www.youtube.com/watch?v=PoEyIJx3uM0

Notes and cut/paste stuff

http://www.tuks.nl/wiki/index.php/Main/StoweCollectedPosts

OK, let's look at "Continuum Mechanics", T. J. Chung, Prentice Hall 1988. On page 1&2 we find: 

        "To distinguish the continuum or macroscopic model from a
        microscopic one, we may list a number of criteria. ... A
        concept of fundamental importance here is that of mean free
        path, which can be defined as the average distance that a
        molecule travels between successive collisions with other
        molecules.  The ratio of the mean free path L to the
        characteristic length S of the physical boundaries of interest,
        called the Knudsen number Kn, may be used to determine the
        dividing line between macroscopic and microscopic models."

Bottom line, the limit of validity of the continuum model is when L/S < 1 period. If our boxes become smaller that L we simply can't use the continuum mathematics.

http://www.tuks.nl/wiki/index.php/Main/StowePersonalEMail

The basic physical quantities in this system are the medium properties identified by Maxwell in his 1860-61 "On Physical Lines of Force". We quantify the mean momentum (quanta) [ß], characteristic mean interaction length (quanta) [L], the root mean speed [c], and a mass attenuation coefficient [¿].

Their values are, 

ß = 5.154664E-27 kg-m/sec
L = 6.430917E-08 m
¿ = 3.144609E-06 m^2/kg
c = 2.997925E+08 m/sec

In other words, all of the major observed and measured constants of physics can be derived from the above terms.

https://en.wikipedia.org/wiki/Compton_wavelength

"The CODATA 2010 value for the Compton wavelength of the electron is 2.4263102389(16)×10−12 m."

So, when considering properties of the electron, we get an L/S of:

5.154664E-27 / 2.4263102389e-12 = 2.12448676899e-15,

which means we can safely use continuity mechanics at sub-atomic scales.

For Planck's length however, we get an L/S of:

https://en.wikipedia.org/wiki/Planck_length

In physics, the Planck length, denoted ℓP, is a unit of length, equal to 1.616199(97)×10−35 metres.

5.154664E-27 / 1.616199e-35 = 318937457.578

So, we certainly cannot use continuity mechanics at the Planck scale....

On uniqueness of Helmholtz decomposition:

http://www.bem.fi/book/ab/ab.htm :

When the curl of a vector field is zero, that vector field can be derived as the (negative) gradient of an arbitrary scalar potential field (which we designate with the symbol Φ and which denotes the electric scalar potential). This assignment is valid because the curl of the gradient of any scalar field equals zero. Thus Equation B.13 further simplifies to
(B.14)
According to the Helmholtz theorem, a vector field is uniquely specified by both its divergence and curl (Plonsey and Collin, 1961). Since only the curl of the vector field has been specified so far (in Equation B.12), we may now choose
(B.15)
This particular choice eliminates Φ from the differential equation for (Equation B.17). That is, it has the desirable effect of uncoupling the magnetic vector potential from the electric scalar potential Φ. Such a consideration was originally suggested by Lorentz when dealing with the free-space form of Maxwell's equations.

http://arxiv.org/abs/quant-ph/0311124 :

The conventional decomposition of a vector field into longitudinal (potential) and transverse (vortex) components (Helmholtz's theorem) is claimed in [1] to be inapplicable to the time-dependent vector fields and, in particular, to the retarded solutions of Maxwell's equations. Because of this, according to [1], a number of conclusions drawn in [2] on the basis of the Helmholtz theorem turns out to be erroneous. The Helmholtz theorem is proved in this letter to hold for arbitrary vector field, both static and time-dependent. Therefore, the conclusions of the paper [2] questioned in [1] are true. The validity of Helmholtz's theorem in the general case is due to the fact that the decomposition above of vector field does not influence the field time coordinate, which plays, thus, a passive role in the decomposition procedure. An analysis is given of the mistakes made in [1]. It is noted that for point particle the longitudinal and transverse components of electric field, taken separately, are characterized by the infinitely great velocity of propagation. However, superluminal contributions to the expression for the total electric field cancel each other.

In a Usenet posting dated 9/6/03, Stowe wrote(*):

The basic continuity equation of Continuum Mechanics is given as: 
                           d(rho)/dt  + (rho)Div v = 0    [Eq.  1]

{$$ \frac{\partial \rho}{\partial t} + \rho \, div \, \mathbf{v} = 0 $$}

Where {$ \rho $} is the field density, and v is the mean velocity. If the field is incompressible this simplifies to: 
                           (rho)Div v = 0           [Eq.  2]

{$$ \rho \, div \, \mathbf{v} = 0 $$}

Since with the incompressible assumption, there can be no change in density. We can further simplify the equation by removing density (dividing it from both sides) we then get: 
                           Div v = 0             [Eq.  3]

{$$ div \mathbf{v} = 0 $$}

This is a equation of state, given an incompressible medium the above MUST BE SATISFIED!
There is a problem here however, if we have equation 3, then we have for, 
                           c^2 = 1/uz            [Eq.  4]

{$$ c^2 = \frac{1}{\mu z} $$}

Where {$\mu$} is the coefficient of compressibility and z density, 
                              u = 0              [Eq.  5]

{$$ \mu = 0 $$}

In other words, this definition requires infinite propagation speeds of any perturbations in such incompressible systems, eliminating any possibility of wave activity!
Thus, for c to be finite, {$\mu$} cannot be zero, Eq. 3 is ruled out...
Conversely, in compressible mediums we see that {$\rho \, div \, \mathbf{v}$} equals the time rate of change in the density ({$\frac{\partial \rho}{\partial t}$}). For the limit, as a volume element s go to zero, we get: 
                    s(rho)Div v = s(d(rho)/dt)   [Eq.  6]

{$$ s \rho \, div \, \mathbf{v} = - s \frac{\partial \rho}{\partial t}$$}

This is based on the observation that for the two terms to sum to zero, and therefore must have opposite signs.

Note that Stowe made an error here. In his posting, there was no - sign, which should be there.

This leads directly to: 
                         mDiv v = dm/dt          [Eq.  7]

{$$ m div \mathbf{v} = - \frac{\partial m}{\partial t} $$}

And cannot be zero. This is an important finding, it describes a unique characteristic of all compressible systems. The result of this is a fixed finite propagation speed for any perturbations in the resulting continuum, leading to standard acoustic behavior.
So, what is the above equation saying? It appears to be saying that compressible medium will have a basic oscillation of density fluctuation occurring continuously. Moreover, given a generally uniform density and velocity, this fluctuation will have a distinctive frequency associated with this activity. This is clearly demonstrated by the relationship: 
                          Div v = d/dt           [Eq.  8]

{$$ div \mathbf{v} = - \frac{\partial}{\partial t} $$}

Obviously, d/dt is a frequency... So, what is this???

As we shall see, this is the oscillation frequency of a specific "charge carrier"....

When applied to the Continuum Mechanics of Electromagnetism where is this? There is a fundamental property that has remained undefined (and given arbitrary units), this is charge [q]. So, if we assign to charge the units [kg/sec] and let's now assume it is a result of the definition above, what is the result?
In Coulomb's law, the force resulting from the interaction of two charges is given to be: 
                   F = [1/4pi(eps)][qq/r^2]      [Eq.  9]
Following that assumption we find that permittivity {$\epsilon$} must have units of density to get a result in units of force.
If we can associate permittivity with density, we find that standard acoustic equation matches that given for light propagation exactly. In standard acoustics wave speed c is given by the relationship: 
                          c^2 = Y/z              [Eq. 10]
Where Y is proportional to pressure and the specific heats in a gas, the bulk modulus of a liquid, or Young's modulus in a solid. For a solid we have the further complication of whether we are evaluating the compression <p wave> or shear <Sv, Sh waves>. The relationship between these two in a perfect elastic medium is that the shear wave travel at a speed Sqrt(3) time slower than the compression wave.
We can of course write the above equation in terms of inverse Y [ {$\mu$} ] (in the standard literature this is known as the coefficient of compressibility see Eq. 4 above), and as can be seen: 
                     c^2 = 1/u(eps) = 1/uz      [Eq. 11]
This provides us with confirmation that this definition is, at least, internally consistent for Coulomb's law and the Maxwell/Heavyside relationship to wave speed.
We can now look elsewhere for other possible correlations.
As shown above, Div v = nu (a characteristic frequency in Hertz). With our definition, the charge to mass ratio would suggest that the mass, seen in matter, could be some sort of resulting stable manifestation of this harmonic oscillation in the field.
Exploring this idea, lets look at the thermal (as in black body) frequency which, given the above definitions, results from this relationship. Given: 
                        E = h(nu) = 3kT        [Eq. 12]
and, as defined, 
                           nu = q/m.           [Eq. 13]
We have: 
                        E = hq/m = 3kT         [Eq. 14]
And the resulting temperature T for this relationship is 
                         T = hq/3km            [Eq. 15]
For the smallest stable elemental particle, the electron, this calculates to be 2.8 degrees Kelvin.

In a Usenet posting dated 5/16/97, he wrote(*):

Given Maxwell definition of the point form of Gauss's Law, namely: 
                        rho = DIV D
By its definition, this must be a physical constant. My question is, has it ever been quantified or measured? If so, then this value when divided by elemental charge (e), must yield the definitive volume of the electron and proton. Since I can find no reference to this value I am assuming that it has not. But wouldn't logic dictate that given this definition, and a finite value for e, there can't exist point charges? This would, in turn, make such an assumption of point charges mute?

In a Usenet posting dated 5/27/97, he wrote(*):

Div D yield a charge density, nothing more. We can then apply Gauss's law to "enclosed" the specific region of field (i.e. multiply by a selected volume) where we have defined this density to obtain an enclosed "total charge". This then can be input into 
                    E = Q/4pi epsion R^2 a->
(your 1/R^2 relationship) where a-> is the vector designator, to get the "electric field" at a point R distance from where Div D was derived. However, I fail to see how this defines Div D as Coulomb's Law.
However, I think at this point we have immersed ourselves so much into the forest that we see only trees, not the forest anymore. The point of the original post was to ask if this value, Div D (for an elemental charge e) had been defined. Since this is a well understood mathematical operator, its value should be a fixed constant value (given that e is a fixed constant value). This is a logical conclusion based on the assumed validity of the mathematical definition of Divergence.

While this is a logical conclusion indeed, one must take the limitations of the used mathematics into account. In the case of Continuum Mechanics, these are well known:

Continuum mechanics is a branch of mechanics that deals with the analysis of the kinematics and the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. Research in the area continues today.

[...]

Modelling an object as a continuum assumes that the substance of the object completely fills the space it occupies. Modelling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. Fundamental physical laws such as the conservation of mass, the conservation of momentum, and the conservation of energy may be applied to such models to derive differential equations describing the behaviour of such objects, and some information about the particular material studied is added through constitutive relations.

In other words: when one does not take these limitations into account when using the "partial differential equations" derived this way, your equations take "on the role of master" "little by little" indeed...

Conservation laws

https://en.wikipedia.org/wiki/Fluid_dynamics#Conservation_laws

TODO

Laplace

https://en.wikipedia.org/wiki/Laplace%27s_equation

In mathematics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace who first studied its properties. This is often written as:

{$$ \nabla^{2}\varphi =0 \qquad {\mbox{or}}\qquad \Delta \varphi =0 $$}

where ∆ = ∇2 is the Laplace operator and {$ \varphi $} is a scalar function.

Laplace's equation and Poisson's equation are the simplest examples of elliptic partial differential equations. The general theory of solutions to Laplace's equation is known as potential theory. The solutions of Laplace's equation are the harmonic functions, which are important in many fields of science, notably the fields of electromagnetism, astronomy, and fluid dynamics, because they can be used to accurately describe the behavior of electric, gravitational, and fluid potentials.

[...]

The Laplace equation is also a special case of the Helmholtz equation.

https://en.wikipedia.org/wiki/Vector_Laplacian

In mathematics and physics, the vector Laplace operator, denoted by {$ \nabla ^{2} $}, named after Pierre-Simon Laplace, is a differential operator defined over a vector field. The vector Laplacian is similar to the scalar Laplacian. Whereas the scalar Laplacian applies to scalar field and returns a scalar quantity, the vector Laplacian applies to the vector fields and returns a vector quantity. When computed in rectangular cartesian coordinates, the returned vector field is equal to the vector field of the scalar Laplacian applied on the individual elements.

The vector Laplacian of a vector field {$ \mathbf{A} $} is defined as

{$$ \nabla^2 \mathbf{A} = \nabla(\nabla \cdot \mathbf{A}) - \nabla \times (\nabla \times \mathbf{A}). $$}

In Cartesian coordinates, this reduces to the much simpler form:

{$$\nabla^2 \mathbf{A} = (\nabla^2 A_x, \nabla^2 A_y, \nabla^2 A_z), $$}

where {$A_x$}, {$A_y$}, and {$A_z$} are the components of {$\mathbf{A}$}. This can be seen to be a special case of Lagrange's formula; see Vector triple product.

https://en.wikipedia.org/wiki/Helmholtz_equation

In mathematics, the Helmholtz equation, named for Hermann von Helmholtz, is the partial differential equation

{$$ \nabla^2 A + k^2 A = 0 $$}

where {$ \nabla^2 $} is the Laplacian, k is the wavenumber, and A is the amplitude.

https://en.wikipedia.org/wiki/Gauss%27s_law_for_gravity

The gravitational field g (also called gravitational acceleration) is a vector field – a vector at each point of space (and time). It is defined so that the gravitational force experienced by a particle is equal to the mass of the particle multiplied by the gravitational field at that point.

Gravitational flux is a surface integral of the gravitational field over a closed surface, analogous to how magnetic flux is a surface integral of the magnetic field.

Gauss's law for gravity states:

The gravitational flux through any closed surface is proportional to the enclosed mass.

[...]

The differential form of Gauss's law for gravity states

{$$ \nabla\cdot \mathbf{g} = -4\pi G\rho, $$}

where {$\nabla\cdot$} denotes divergence, G is the universal gravitational constant, and ρ is the mass density at each point.

https://en.wikipedia.org/wiki/Divergence

It can be shown that any stationary flux v(r) that is at least twice continuously differentiable in R 3 {\displaystyle {\mathbb {R} }^{3}} {\mathbb {R} }^{3} and vanishes sufficiently fast for | r | → ∞ can be decomposed into an irrotational part E(r) and a source-free part B(r). Moreover, these parts are explicitly determined by the respective source densities (see above) and circulation densities (see the article Curl):