6 Electromagnetic potential

Finally, the potential 4-vector $A^{\mu}=\left(\phi,c\mathbf{A}\right)$ for a uniformly moving charge could be rewritten in Euclidean form using the same method as used so far:

$$\phi^2=K^2+c^2\left(A_x^2+A_y^2+A_z^2\right)$$ (22)

with $K$ still to be determined as the temporal component in the Euclidean form. The magnitude of the vector potential $\mathbf{A}$ is

$$\sqrt{A_x^2+A_y^2+A_z^2}=\frac{v}{c^2}\phi$$ (23)

while $\phi$ for a moving charge can be written in terms of the retarded potential $\phi_r$ :

$$\phi=\gamma\phi_r.$$ (24)

Using these identities, $K$ can be determined as:

$$K^2=\gamma^2\phi_r^2\left(1-\frac{v^2}{c^2}\right)=\phi_r^2$$ (25)

The Euclidean form thus reads:

$$\phi^2=\phi_r^2+c^2\left(A_x^2+A_y^2+A_z^2\right)$$ (26)

The variable $\phi$ is however not invariant. In order to make the 4-vector invariant under rotations in SO(4) (the Euclidean equivalent of Lorentz transformations, see also [3]) the expression would have to be multiplied with $1/\gamma$ but this is inconsistent with the approach in the other 4-vectors where the aim was to get rid of the gamma's in the Euclidean expressions. This raises the question whether the classical potential 4-vector could be the Euclidean form already, although at first sight this seems in conflict with the traditional $+-$ pattern in its components, necessary to yield an invariant.

Various operations on the potential 4-vector, like e.g. the derivation of the fields of $\mathbf{E}$ and $\mathbf{B}$ in $F_{\mu\nu}=\nabla_{\mu}A_{\nu}-\nabla_{\nu}A_{\mu}$ , involve the operator $\nabla_{\mu}$ which is defined as:

$$\nabla_{\mu}=\left(\frac{\partial}{\partial t},-\boldsymbol{\nabla}\right)$$ (27)

The derivative with respect to $t$ in $\partial/\partial t$ is inconsistent with the derivation with respect to $\tau$ in Minkowski 4-vectors but consistent with the Euclidean 4-vectors. Furthermore, the components $\phi$ and $\mathbf{A}$ both already have intuitive physical meanings, showing in particular from operations like $\mathbf{E}=-\nabla \phi - \partial \mathbf{A}/\partial t$ and $\mathbf{B}=\mathbf{\nabla}\times \mathbf{A}$ . The pattern $+-$ therefor seems an intrinsic property of electromagnetic potentials rather than being related to Minkowski geometry.