Can Maxwell's Equations be expressed only in terms of an electric field and special relativity?

Question

It is well known that the magnetic field is a consequence of special relativity, due to length contraction.

Because of this, I was wondering if you could express Maxwell's equations only in terms of the Electric field, and depending on the reference frame.

Imagine we have a test particle, moving with velocity $v$, with respect to the $S$ frame. The particle is at rest in the $S'$ frame. Now we will define the following vectors: $$ \mathbf E: \text{Electric field in the $S$ frame} $$ $$ \mathbf B: \text{Magnetic field in the $S$ frame} $$ $$ \mathbf E': \text{Electric field in the $S'$ frame} $$ $$ \mathbf B': \text{Magnetic field in the $S'$ frame} $$ $$ \mathbf r: \text{position coordinates in the $S$ frame} $$ $$ \mathbf r': \text{position coordinates in the $S'$ frame} $$

Would be reasonable to do the following: $$ \mathbf B(\mathbf r) = \mathbf E'(\mathbf r) $$ $$ \mathbf B'(\mathbf r') = \mathbf 0 $$ It is possible to create such theory? Does this theory already exists?

Answer 1

The quantities $\vec{E}\cdot\vec{B}$ and $|\vec{E}|^2-|\vec{B}|^2$ are Lorentz invariants. In particular, if the latter is negative (as it would be around eg a neutral current carrying wire, where $\vec{B}\neq\vec{0}$ but $\vec{E}=\vec{0}$) then there cannot be a Lorentz transformation such that in the new frame $\vec{B}=0$.