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Electroscalar waves

Phytomech Industries

Abstract

So basically we derive an equation for electrostatic waves in vacuum under the assumption vacuum polarization means we can’t neglect ρ\rho or J\bold{J}.

Scalar Electrostatic Waves

I was taught these things were impossible. Something about something not being conserved. I don’t remember the exact verbage that was used, but longitudinal polarziation of light was out. Didn’t fit with the theory. That’s of course a sophmoric understanding of physics, but that’s what I had to work with.

I went full crank. Eventually I was saved by the grace of nuclear energy and communism pulling me back from the brink, but that’s to be expected when your trained SFT on sophmore level physics slop by scholars disconnected from the science.

Anway I hunted everywhere for this equation. Thought it was the missing link. Where we zigged when we should have zagged. Oh well! I build AI agents now. That’s pretty freaking cool you have to admit. So yeah without further ado, the derivation of scalar waves under the assumption the vacuum is essentially an unexcited plasma (ωp=0\omega_p=0).

Maxwell’s Equations

Eqn. (1.1) describes Gauss’ Law for electricity, which is basically a mathematical way of saying that the flux of E\bold{E} fields across some surface ∂V\partial V can be used to find the charge density inside the volume VV. This is a fancy way of saying that static E\bold{E} fields have no curl and are created entirely by static charges/charge densities ρ\rho.

∇⋅E=ρϵ0\nabla \cdot \bold{E} = \frac{\rho}{\epsilon_0}
(1.1)

Eqn. (1.2) is Gauss’ Law for magnetism, which says that magnetic fields B\bold{B} are solenoidal/divergenceless and have no monpole sources (B\bold{B} fields are created by current densities J\bold{J}). Do not let the next equations fool you. The source of E\bold{E} fields are static charges and the source of B\bold{B} fields are current densities J\bold{J} AKA moving charges.

∇⋅B=0\nabla \cdot \bold{B} = 0
(1.2)

Eqn. (1.3) is known as Ampere’s Law and relates changing magnetic fields with respect to time with the curl of the electric field E\bold{E}, which is of course a differential relationship between the electric field and space. That describes the torsions the electrical field is capable of generating through accelerating electrical densities/potentials.

∇×E=−∂B∂t\nabla \times \bold{E} = - \frac{\partial \bold{B}}{\partial t}
(1.3)

Eqn. (1.4) relates the torsions in the magnetic field (∇×B\nabla \times \bold{B}) due to changing electrical fields with respect to time (∂E∂t\frac{\partial \bold{E}}{\partial t}) and current density J\bold{J}. This equation allowed for the unification of electricity and magnetism into a single electromagnetic field and they both describe the forces and potentials generated by the motion of charged objects through space.

∇×B=μ0J+μ0ϵ0∂E∂t\nabla \times \bold{B} = \mu_0 \bold{J} + \mu_0 \epsilon_0 \frac{\partial \bold{E}}{\partial t}
(1.4)

An equally important equation in our derivation of scalar light is Eqn. (1.5)

∇⋅J=−∂ρ∂t\nabla \cdot \bold{J} = -\frac{\partial \rho }{\partial t}
(1.5)

And the mathematical identity for the curl of the curl, Eqn. (1.6)

∇×∇×A=∇(∇⋅A)−∇2A\nabla \times \nabla \times \bold{A} = \nabla (\nabla \cdot \bold{A}) - \nabla^2 \bold{A}
(1.6)

Electroscalar Light

Starting with Eqn. (1.6) and plugging in our electric field E\bold{E}, we get

∇×∇×E=∇(∇⋅E)−∇2E=1ϵ0∇ρ−∇2E\begin{align} \nabla \times \nabla \times \bold{E} &= \nabla(\nabla \cdot \bold{E}) - \nabla^2 \bold{E} \\ &= \frac{1}{\epsilon_0} \nabla \rho - \nabla^2 \bold{E} \end{align}
(1.7)

Through application of Ampere’s Law Eqn. (1.3)

∇×∇×E=∇×−(∂B∂t)=−∂∂t[∇×B]=−∂∂t[μ0J+μ0ϵ0∂E∂t]\begin{align} \nabla \times \nabla \times \bold{E} &= \nabla \times - (\frac{\partial \bold{B}}{\partial t}) \\ &= - \frac{\partial}{\partial t} \left[ \nabla \times \bold{B}\right] \\ &= - \frac{\partial}{\partial t} \left[ \mu_0 \bold{J} + \mu_0 \epsilon_0 \frac{\partial \bold{E}}{\partial t} \right] \end{align}
(1.8)

Adding together Eqn. (1.8) and Eqn. (1.7) we arrive at

−μ0∂J∂t−μ0ϵ0∂2E∂t2=1ϵ0∇ρ−∇2E∇2E−μ0ϵ0∂2E∂t2=1ϵ0∇ρ+μ0∂J∂t\begin{align} - \mu_0 \frac{\partial \bold{J}}{\partial t} - \mu_0 \epsilon_0 \frac{\partial^2 \bold{E}}{\partial t^2} &= \frac{1}{\epsilon_0} \nabla \rho - \nabla^2 \bold{E} \\ \nabla^2 \bold{E} - \mu_0 \epsilon_0 \frac{\partial^2 \bold{E}}{\partial t^2} &= \frac{1}{\epsilon_0} \nabla \rho + \mu_0 \frac{\partial \bold{J}}{\partial t} \end{align}
(1.9)

The left hand side of Eqn. (1.9) should look familiar because it’s the second order wave operator.

The right hand side probably doesn’t look as familiar but it’s actually one half of a wave operator itself. Let’s see what happens to the right hand side of Eqn. (1.9) when □2E=λ\Box^2 \bold{E} = \bold{\lambda} where λ\bold{\lambda} is a constant vector (λx,λy,λz)(\lambda_x, \lambda_y, \lambda_z).

1ϵ0∇ρ+μ0∂J∂t=λ∇ρ+μ0ϵ0∂J∂t=ϵ0λ∇⋅[∇ρ+μ0ϵ0∂J∂t]=∇⋅ϵ0λ∇2ρ+μ0ϵ0∂∂t(∇⋅J)=0\begin{align} \frac{1}{\epsilon_0} \nabla \rho + \mu_0 \frac{\partial \bold{J}}{\partial t} &= \bold{\lambda} \\ \nabla \rho + \mu_0 \epsilon_0 \frac{\partial \bold{J}}{\partial t} &= \epsilon_0 \bold{\lambda} \\ \nabla \cdot \left[ \nabla \rho + \mu_0 \epsilon_0 \frac{\partial \bold{J}}{\partial t} \right] &= \nabla \cdot \epsilon_0 \bold{\lambda} \\ \nabla^2 \rho + \mu_0 \epsilon_0 \frac{\partial}{\partial t}(\nabla \cdot \bold{J}) &= 0 \end{align}
(1.10)

Applying the continuity equation Eqn. (1.5) we arrive at a scalar wave equation due to the presence of current and charge densities in every section of space that is capable of vacuum polarization

Spoiler Alert

All of space is capable of vacuum polarzation. We’ve known lasers have longitudinal E\bold{E} fields in hard vacuum since 1974

∇2ρ+μ0ϵ0∂∂t(∇⋅J)=0∇2ρ−μ0ϵ0∂2ρ∂t2=0\begin{align} \nabla^2 \rho + \mu_0 \epsilon_0 \frac{\partial}{\partial t}(\nabla \cdot \bold{J}) &= 0 \\ \nabla^2 \rho - \mu_0 \epsilon_0 \frac{\partial^2 \rho}{\partial t^2} &= 0 \end{align}
(1.11)

Which is of course □2ρ=0\Box^2 \rho = 0, a scalar wave equation for charge density. Sorry Charlie, the vaccum is a plasma and light is an alternating current, not unlike sound at all.

Scientific Computing with Python
Journal
June
N-balls in high dimensions