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

[Arend Lammertink]

Hi Maurice and General,

Let me write out the symbols in text format, which makes things a lot
easier to read and add some explanation. I had quite some trouble with
the symbols myself when I started working on this like 15 years after
my education.

We begin with the Laplace operator, which is elementary math. It is
the vector equivalent of the second order derivative, denoted in one
dimension by ∂²/∂x² , i.e., the curvature, also called "jerk":

https://en.wikipedia.org/wiki/Jerk_(physics)
"In physics, jerk or jolt is the rate at which an object's
acceleration changes with respect to time. "

In vector theory, this concept is generalized in three dimensions and
represented by the symbol ∇² or 𝞓. In vector theory the _definition_
for the second order derivative, or jerk, is given by:

∇²𝐅= ∇(∇·𝐅) - ∇×(∇×𝐅)

This is also a so called "vector identity" that is mathematically
proven to be correct.

This can be written out in text format as follows:

∇²𝐅= grad (div 𝐅) - rot (rot 𝐅)

All three operators, grad div and rot, are spatial derivatives which
all have a unit of measurement in per meter [/m].

The rotation, rot or curl operator (all three the same thing) can be
described as follows:

https://en.wikipedia.org/wiki/Curl_(mathematics)#Intuitive_interpretation
"Suppose the vector field describes the velocity field of a fluid flow
(such as a large tank of liquid or gas) and a small ball is located
within the fluid or gas (the centre of the ball being fixed at a
certain point). If the ball has a rough surface, the fluid flowing
past it will make it rotate. The rotation axis (oriented according to
the right hand rule) points in the direction of the curl of the field
at the centre of the ball, and the angular speed of the rotation is
half the magnitude of the curl at this point."

The gradient or grad:

https://en.wikipedia.org/wiki/Gradient
"The gradient can be interpreted as the "direction and rate of fastest
increase"."

And the divergence:

https://en.wikipedia.org/wiki/Divergence
"In physical terms, the divergence of a vector field is the extent to
which the vector field flux behaves like a source at a given point. It
is a local measure of its "outgoingness" – the extent to which there
is more of the field vectors exiting an infinitesimal region of space
than entering it."

One can also think of divergence as a measurement of compessibility,
for instance of a fluid or gas.

So, what we have here is this elementary definition for how to
calculate "jerk" or "curvature" in 3D vector calculus:

∇²𝐅= grad (div 𝐅) - rot (rot 𝐅)

When we equate this equation to zero, ∇²𝐅=0, we obtain Laplace's
equation, of which the solutions are the harmonic functions.

By negating this equation, this can be re-written as:

-∇²𝐅= - grad (div 𝐅) + rot (rot 𝐅)

What we can then do, is to take the terms in this equation, write them
out and label them by equating them to a symbol:

 𝐀 = rot 𝐅
 Φ = div 𝐅
 𝐁 = rot 𝐀= rot ( rot 𝐅)
 𝗘 = − grad Φ= −grad ( div 𝐅 )

And because of other vector identities, we can also write:

 rot 𝗘 = 0
 div 𝐁 = 0

So, what is rather interesting is that this mathematical second order
derivative establishes a separation into a irrotational component 𝗘
and an incompressible component 𝐁.  So, from a physics point of view,
we are looking at a fundamental distinction between rotational
movements (represented by 𝐁) and translational movements (represented
by 𝗘). One can also compare mass-spring movements with gyroscopic
movements to illustrate that this separation is rather fundamental
within physics.

Now when we look at Maxwell's equations, we find a/o the following definition:

 rot 𝗘= -∂𝐁/∂t,

which is obviously not equal to zero, so we can establish that what we
are looking at is an obvious violation of elementary math, which
messes up this fundamental distinction between translational and
rotational movements within physics.

What this equation describes within Maxwell is the Maxwell-Faraday law
of induction:

https://en.wikipedia.org/wiki/Faraday%27s_law_of_induction

"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 currents trough a wire, which are electrons moving
trough a wire. Since this involves moving charge carriers, which are
particles, it is illogical to introduce this law at the medium
modelling level, the level which should be the basis for our theory
describing a fluid-like medium called aether.

Because of the wave-particle duality principle, it is known that
particles are manifestations of the EM field. So, by including
Faraday's law in the medium/field model one introduces circular logic.

The wave-particle duality principle is very important in modern
physics, which forms the basis for Quantum Mechanics:

https://en.wikipedia.org/wiki/Wave–particle_duality
"Wave–particle duality is the concept in quantum mechanics that every
particle or quantum entity may be described as either a particle or a
wave. It expresses the inability of the classical concepts "particle"
or "wave" to fully describe the behaviour of quantum-scale objects.

[...]

Wave–particle duality is an ongoing conundrum in modern physics. Most
physicists accept wave-particle duality as the best explanation for a
broad range of observed phenomena; however, it is not without
controversy. Alternative views are also presented here. These views
are not generally accepted by mainstream physics, but serve as a basis
for valuable discussion within the community."

It is clearly very important to resolve this conundrum.

IMHO, the logical approach would be to start with a model for the
medium,  which is characterized by a permittivity 𝞮of 8.854 pF/m, a
permeability 𝞵 of 4𝞹 x 10^-7 H/m and a characteristic impedance of
377 𝞨. This matches to the characteristics of a fluid, hence the idea
of the existence of a physical aether.

Since in my view, there can only be one aether and we can define
fields using the Laplace equation in order to describe it's
*dynamics*, we can conclude that ALL physical phenomena are
manifestations of the thus described fields, which can be detailed by
the harmonic wave functions that are the solutions for this equation.

Because a fluid cannot sustain transverse waves and there are
significant differences between the near and far fields, the logical
conclusion is that the far field must be some kind of vortex or vortex
ring.

It is not hard to see that vortex rings can be combined into complex
structures and that these show both attributes of particles as well as
waves:

 http://www.tuks.nl/img/dualtorus.gif

Given the fact that charge is a property of certain particles, the
nature of what charge is should follow from the particle model and
should not be taken as a fundamental quantity.

According to Paul Stowe, one can indeed compute the value for e,
elemental charge, from a ring vortex topology, hence the idea that the
electron should be modelled as a vortex ring.

So, IMHO, the concept of charge should fundamentally follow from the
particle model, which should be described on top of the medium model.

Otherwise, one irrevocably runs into circular logic problems, as also
illustrated earlier around the introduction of Faraday's law into the
model at a place where it just does not fit.

In a nutshell:

*) Either particles are caused by the EM fields OR the EM fields are
caused by particles, but NOT both at the same time!

*) The introduction of Faraday's law into the model at the wrong place
introduces a heck of a lot of problems.




On Thu, Apr 23, 2020 at 5:16 PM Maurice Daniel <5D at earthlink.net> wrote:
>
> Arend Lammertink,
>
> If you are not already aware of the works of Professor Oleg D. Jefimenko, (1922 to 2009)  physicis and Professor Emeritus at West Virginia University; author of such works as: “Causality Electromagnetic Induction and Gravitation” then I suggest you read some of his books.  He has discussion of these topics, and like you, he has dispute with two of Maxwell’s equations.  He was also able to combine electromagnetism and gravity.
>
> I studied vector math at one time, but now I have forgotten most of it so I cannot follow your arguments.  It would be interesting to know if you have reached the same conclusions as Prof. Jefimenko.  If so, this would lend powerful support to your arguments.
>
> Let me know if this is helpful to you.
>
> - - - Maurice Daniel - - -
>
>
>
>
> Maurice Daniel
> 5D at earthlink.net
>
>
> On Apr 23, 2020, at 2:09 AM, Arend Lammertink <lamare at gmail.com> wrote:
>
> 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.
> <Revision of Maxwell equations DRAFT.pdf>_______________________________________________
> Physics mailing list
> Physics at tuks.nl
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>
>
>
>
> Maurice Daniel
> 5D at earthlink.net
>
>
>
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