[Physics] Mathematical proof Maxwell's equations are incorrect?

[Arend Lammertink]

Dear List members,

I have been studying Tesla for quite some time now and became
convinced longitudinal waves exist and that they propagate faster than
light. For quite some time, I have been working on the theory, which
culminated in the attached draft paper on revision of Maxwell's
equations. During the past week, I had a discussion about this on the
"Theoretical Physics" LinkedIn group, which made me realise how
important the vector Laplace equation is and believe I now have the
mathematical proof that Maxwell's equations are incorrect. This is the
short version of the argument:

-:-
"The Laplace operator is not some sacred physical law of the universe,
it is a mathematical relation".

Yes, it's a relation of which the correctness is pretty much
undisputable, like 1+1=2.

Equate this equation to zero and one obtains the 3D Laplace equation
of which the solutions are the harmonic functions, which (when worked
out) describe all possible (harmonic) wave phenomena in 3D:

 ∇²𝐅= ∇(∇·𝐅) - ∇×(∇×𝐅) = 0.

This can be re-written as:

 -∇²𝐅= - ∇(∇·𝐅) + ∇×(∇×𝐅) = 0.

Then, the terms in this equation can be written out as follows:

 𝐀= ∇×𝐅
 Φ= ∇⋅𝐅
 𝐁= ∇×𝐀= ∇×(∇×𝐅)
 𝗘=−∇Φ= −∇(∇⋅𝐅)

And because of vector identities, one can also write:

 ∇×𝗘= 0
 ∇⋅𝐁= 0

So, any given vector field 𝐅 can be decomposed like this into a
rotation free component 𝗘 and a divergence free component 𝐁.

There is no argument this is mathematically consistent, nor that the
solutions to the equation -∇²𝐅= 0 are the harmonic wave functions in
3D.

Now compare this to Maxwell's:

 𝗘= −∇Φ− ∂𝐀/∂t

Take the rotation at both sides of the equation and we obtain the
Maxwell-Faraday equation:

 ∇×𝗘= - ∂𝐁/∂t

WP: "Faraday's law of induction (briefly, Faraday's law) is a basic
law of electromagnetism predicting how a magnetic field will interact
with an electric circuit to produce an electromotive force (EMF)—a
phenomenon known as electromagnetic induction."

This is a circuit law, which predicts how a magnetic field will
interact with electrons moving trough a wire. Since this involves
moving charge carriers, which are particles, it is illogical to
introduce this law at the medium/field modelling level. Because of the
wave-particle duality principle, it is known that particles are
manifestations of the EM field. So, by including this law in the
medium/field model one introduces circular logic.

Not only that, it breaks the fundamental separation of the fields into
a divergence free component and a rotation free component.

As is well known, this model eventually leads to two mutually
exclusive theories, which cannot both be correct.

In other words: what you are doing by introducing Faraday's law at
this level in the model is you are insisting 1+1 is not 2, but
something else.

And you end up with 150+ years of trying to find additional equations
to straighten things out, but the bottom line is: 1+1=2, NOT something
else

[...]

"How does it break "the fundamental separation of the fields into a
divergence free component and a rotation free component."? "

As shown, the 3D vector Laplace equation defines two components, one
of which is divergence free and one of which is rotation free.

Since the 3D vector Laplace equaton is nothing but a 3D generalization
of the lower dimensional Laplace equation and results in harmonic
solutions, which is all well established undisputable math, it follows
that the decomposition into a divergence free component and a rotation
free component is fundamental and is therefore the only correct way to
derive wave functions in 3D for any given vector field.

There is no argument that with equating the rotation of the rotation
free component 𝗘 to the time derivative of the divergence free (and
therefore rotational) component 𝐁 by Maxwell results in 𝗘 remaining
to be rotation free and therefore such breaks said fundamental
separation of said components.
-:-

I have some rewriting to do of the article, because I now realize it's
perfectly O.K. to have the primary field, which I denoted [V], as the
null vector field, since in the Laplace equation the right side of the
equation is also zero, so we don't have to resort to discrete math.
So, for the time being, I included part of the discussion on LinkedIn,
which I think you'll find interesting.

In short: I believe to have found the foundation for that Theory of
Everything scientists have been looking for for a very long time.

I would love to hear your opinion about this.

Best regards,

Arend.
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