Electromagnetic field under different perspective

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The discussion presents a document analyzing the electromagnetic field from a new perspective, introducing a scalar field A and its spatial gradient F, along with a modified Laplacian operator. It establishes a relationship between this modified Laplacian and the d'Alembert equation, which governs electromagnetic potentials in vacuum. By setting the constant k to -1, the equation simplifies to match the d'Alembert equation, demonstrating mathematical equivalence to classical electromagnetism. The findings suggest that the proposed model can effectively describe electromagnetic phenomena using the parameter k. This approach emphasizes the significance of the speed of light in the propagation of events within differential equations.
Roberto Pavani
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TL;DR Summary
A different "interpretation" of the electromagnetic field
google doc document link is: https://drive.google.com/file/d/1b5TgfDGFbx3Y1BcfEV3PRF9maq1hY3Zr/view
Here a little document I'd like to have comments on it

Electromagnetic Field from another perspective

Analogy between the Electromagnetic Field and a Field Defined by the Constant k

Definition of the Field:
We have defined a scalar field A and its spatial gradient F:
F = (∂x A, ∂y A, ∂z A)
and its time derivative:
q = ∂t A.
We introduced a modified Laplacian, defined as:
∇̃2 A = A - k ∂t2 A.
If we impose that A satisfies a modified Laplace-type equation:
∇̃2 A = 0,
we obtain:

A - k ∂t2 A = 0.
Comparison with Electromagnetism
In classical electromagnetism, the electromagnetic potentials ∇̃2 = (φ, Ax, Ay, Az) satisfy the
d'Alembert equation:
□ = 0,
where the d'Alembertian □ is defined as:
□ = - ∂t2. (assuming c = 1)
This equation governs the propagation of electromagnetic potentials in vacuum.
Special Case with k = -1
If we choose k = -1, our equation becomes:

A - (-1) ∂t2 A = 0,

which simplifies to:
A + ∂t2 A = 0.

This is exactly the d'Alembert equation:
□ A = 0.

Since the electromagnetic field satisfies this same equation for the potentials , with k = -1, our
theory becomes mathematically identical to the relativistic description of electromagnetic potentials in
vacuum.
Conclusion
- Our model, parameterized by the constant k, describes a scalar field A with propagation defined by a
modified Laplacian operator.
- When k = -1, the resulting equation is identical to that of the electromagnetic potentials in
electrodynamics.
- This shows that our formalism with k = -1 exactly matches the relativistic electromagnetic theory for the
potentials .
 
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Short explanation:
The reason why the metric is ( 1, 1, 1, -1) is just because events are propagated by the speed of light and this must be taken into account also in differentials equations.
k reflect a coefficent to take this into account
 
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