On Space Time And The Fabric Of Nature Continuity Equation

Notation:

gradient : ∇

divergence: ∇⋅

curl : ∇ x

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

Euler equations

https://en.wikipedia.org/wiki/Euler_equations_(fluid_dynamics)

In fluid dynamics, the Euler equations are a set of quasilinear hyperbolic equations governing adiabatic and inviscid flow. They are named after Leonhard Euler. 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 fact, Euler equations can be obtained by linearization of some more precise continuity equations.

Since Stowe's aether model has zero viscosity as well as zero thermal conductivity, the Euler equations are a prime candidate for re-deriving Maxwell's equations directly from Stowe's compressible aether model.

Let us first consider some of the terms used in this description.

Cauchy momentum equation

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

The Cauchy momentum equation is a vector partial differential equation put forth by Cauchy that describes the non-relativistic momentum transport in any continuum.

In convective (or Lagrangian) form it is written:

{$$ \frac{D \mathbf{v}}{D t} = \frac 1 \rho \nabla \cdot \mathbf{\sigma} + \mathbf{g}$$}

where $\rho$ is the density at the point considered in the continuum (for which the continuity equation holds), $\mathbf{\sigma}$ is the [Cauchy] stress tensor, and $\mathbf{g}$ contains all of the body forces per unit mass (often simply gravitational acceleration). $\mathbf{v}$ is the flow velocity vector field, which depends on time and space.

Notably, it can be written, through an appropriate change of variables, also in conservation (or Eulerian) form:

{$$ \frac {\partial \mathbf j }{\partial t}+ \nabla \cdot \mathbf \Phi = \mathbf s $$}

where $\mathbf j$ is the [mass flux] momentum density at the point considered in the continuum (for which the continuity equation holds), $\mathbf \Phi$ is the flux associated to the momentum density, and $\mathbf{s}$ contains all of the body forces per unit volume.

[...]

Cauchy equations can be put in the [conservative] form [...] simply by defining:

{$$ {\mathbf j}= \rho \mathbf v $$}

{$${\mathbf \Phi}=\rho \mathbf v \otimes \mathbf v + \mathbf \sigma$$}

{$${\mathbf s}= \rho \mathbf g$$}

where [...] $\mathbf v \otimes \mathbf v$ is dyadic product (outer product) of the velocity.

Here j and s have same length N as the flow speed the body acceleration, while $\Phi$ has size N2.

In the Eulerian forms it is apparent that the assumption of no deviatoric stress brings Cauchy equations to the Euler equations.

Since there are no external body forces within our model, we can take g=0, and thus it folows: s=0. Therefore, we can write:

{$$ \frac {\partial \mathbf j }{\partial t}+ \nabla \cdot \mathbf \Phi = 0 $$}

or:

{$$ \frac {\partial \mathbf j }{\partial t} = - \nabla \cdot \mathbf \Phi $$}

Now filling in

{$$ {\mathbf j}= \varepsilon \mathbf E $$}

{$${\mathbf \Phi}=\varepsilon \mathbf E \otimes \mathbf E + \mathbf \sigma $$}

We get:

{$$ \frac {\partial \mathbf \varepsilon \mathbf E }{\partial t} = - \nabla \cdot { \varepsilon \mathbf E \otimes \mathbf E + \mathbf \sigma } $$}

or:

{$$ \frac {\partial \mathbf E}{\partial t} = - \nabla \cdot { \mathbf E \otimes \mathbf E + \frac{\sigma}{\varepsilon} } $$}

Since there are no body forces, we take g=0.

{$$ \frac{D \mathbf{E}}{D t} = \frac 1 \varepsilon \nabla \cdot \mathbf{\sigma} $$}

The material derivative is defined for any tensor field y that is macroscopic, with the sense that it depends only on position and time coordinates, y = y( x, t ):

{$$ \frac{\mathrm{D} y}{\mathrm{D}t} \equiv \frac{\partial y}{\partial t} + \mathbf{u}\cdot\nabla y, $$}

where ∇y is the covariant derivative of the tensor, and u( x, t ) is the flow velocity. Generally the convective derivative of the field u•∇y, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field u•(∇y), or as involving the streamline directional derivative of the field (u•∇) y, leading to the same result.

{$$ \frac{\mathrm{D} \mathbf{E} }{\mathrm{D}t} \equiv \frac{\partial \mathbf{E}}{\partial t} + \mathbf{u}\cdot\nabla \mathbf{E} $$}

TODO: fill in the above definitions for j and $\Phi$, and take the rotation left and right. This should result in some equation involving B = rot(p) = rot(m v) = rot(j).

Euler equations

https://en.wikipedia.org/wiki/Euler_equations_%28fluid_dynamics%29#Euler_equations

The equations below represent conservation of mass, momentum, and energy: the energy equation expressed in the variable internal energy allows to understand the link with the incompressible case, but it is not in the simplest form.

Mass density, flow velocity and pressure are the so-called convective variables (or physical variables, or lagrangian variables), while mass density, momentum density and total energy density are the so-called conserved variables (also called eulerian, or mathematical variables).

{$$ \begin{matrix} {\frac{\partial \rho}{\partial t} + \mathbf{v} \cdot \nabla {\rho} + {\rho} \nabla \cdot \mathbf{v} = 0} & \quad \text{(conservation of mass)} \\ {\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v} + \frac {\nabla p}{\rho} = \mathbf{g}} & \quad \text{ (conservation of momentum)} \\ {\frac{\partial e}{ \partial t}+ \mathbf{v} \cdot \nabla e + \frac p \rho \nabla \cdot \mathbf{v} = 0} & \quad \text{(conservation of energy)} \\ \end{matrix} $$}

where:

  • v is the flow velocity vector, with components in a N-dimensional space $ v_1, v_2, ..., v_N $,
  • $\rho$ is the fluid mass density,
  • $ p $ is the pressure, $p = \rho w$,
  • $ w $ is the specific (with the sense of per unit mass) thermodynamic work, the internal source term.
  • $ \mathbf{g} $ represents body accelerations (per unit mass) acting on the continuum, for example gravity, inertial accelerations, electric field acceleration, and so on.
  • $ e $ is the specific internal energy (internal energy per unit mass).
  • $ \cdot $ denotes the scalar product,
  • $ \nabla $ is the nabla operator, here used to represent the specific thermodynamic work gradient (first equation), and the flow velocity divergence (second equation), and
  • $ \mathbf v \cdot \nabla $ is the convective operator,

Since in our model, all accelerations and/or forces take place within our aether fluid, there are no external forces or "body accelerations" to account for, hence we take {$ \mathbf{g}=0$}. Further,

, and thus we can write:

TODO: Consider the Lamb form and study further:

https://en.wikipedia.org/wiki/Cauchy_momentum_equation#Lamb_form

{$$ \frac{\partial \varepsilon}{\partial t} + \mathbf{E} \cdot \nabla {\varepsilon} + {\varepsilon} \nabla \cdot \mathbf{E} = 0 $$}

{$$ \frac{\partial \mathbf{E}}{\partial t} + \mathbf{E} \cdot \nabla \mathbf{E} + \frac {\nabla p}{\varepsilon} = 0 $$}

{$$ \frac{\partial e}{ \partial t}+ \mathbf{E} \cdot \nabla e + \frac p \varepsilon \nabla \cdot \mathbf{E} = 0 $$}

Stress Tensor

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

In continuum mechanics, the Cauchy stress tensor $\textbf\sigma\,\!$, true stress tensor, or simply called the stress tensor is a second order tensor named after Augustin-Louis Cauchy. The tensor consists of nine components $\sigma_{ij}\,\!$ that completely define the state of stress at a point inside a material in the deformed state, placement, or configuration. The tensor relates a unit-length direction vector n to the stress vector T(n) across an imaginary surface perpendicular to n:

{$$ $\mathbf{T}^{(\mathbf n)}= \mathbf n \cdot\mathbf{\sigma}$ or $ T_j^{ (n) } = \sigma_{ij}n_i. $$}

Gauss law

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