- #1
Roberto Pavani
- 8
- 3
- TL;DR
- 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 .
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 .