It is a mistake to think that special relativity (SR) causes magnetism. The article you're reading is arguing that
- (A) "If electricity exists, and SR is true, then magnetism has to exist".
This statement is true, I admit it. But it's equally true that
- (B) "If magnetism exists, and SR is true, then electricity has to exist".
In reality, electricity and magnetism are equally fundamental parts of physics. Special relativity unites electricity and magnetism into electromagnetism, in exactly the same way that it unites space and time into spacetime. Time does not cause space, space does not cause time, and SR causes neither space nor time. SR merely reveals the relatedness of space and time. Similarly, electricity does not cause magnetism, magnetism does not cause electricity, and SR causes neither electricity nor magnetism. SR merely reveals the relatedness of electricity and magnetism.
A lot of people come across (A) in their high school or intro college physics classes, and wind up misunderstanding it as the fundamental reason that magnetism exists. Why is that? And why do textbooks almost never point out (B)?
The reason is simply that nobody questions why electricity exists--electrical attraction and repulsion seem perfectly natural--whereas magnetism seems more mysterious. In other words, this pedagogical asymmetry between electricity and magnetism has nothing to do with physics, and everything to do with our inborn preconceptions and intuitions.
Suppose you start with a linear charge density $\lambda^+$ of positive charges and $-\lambda^-$ of negative charges in the wire, everything at rest.
Case 1: No current, test charge stationary
You assume you have a neutral wire with no current. Therefore $\lambda^- = \lambda^+$. There's no other frame worth considering, since nothing is in motion anyway.
Even if you did go into another frame, any change in charge density will affect electrons and nuclei equally. Thus the wire is neutral in all frames, and test charges are entirely unaffected by it.
Case 2: Nonzero current, test charge moving with electrons
Now suppose you have a wire with a current. Again, the wire is neutral in the lab frame $S$, where the bulk of it is not moving. In this frame, we still must have $\lambda_S^- = \lambda_S^+$, even though the electrons are moving and the nuclei aren't.
If we slip into the rest frame $S'$ of the bulk electron motion, then the spacing between electrons must be different, and in fact it must be larger. Since charge doesn't change when changing frames, we know $\lambda_{S'}^- < \lambda_S^-$. Similarly, the nuclei spacing will be length-contracted, so $\lambda_{S'}^+ > \lambda_S^+$. In this frame, then, $\lambda_{S'}^+ > \lambda_{S'}^-$, so the wire looks positively charged, and any (positive) test charge at rest in this frame $S'$ will be repelled.
As you can check, this is exactly what the Lorentz force law tells you. If the electron bulk motion is in the $-z$-direction, then the current is in the $+z$-direction, and the magnetic field along the $+x$-axis (assuming the wire coincides with the $z$-axis) is in the $+y$-direction. A positive charge with velocity in the $-z$-direction in a magnetic field in the $+y$-direction will experience a force in the direction of $(-\hat{z}) \times (+\hat{y}) = +\hat{x}$, away from the wire.
Case 3: Nonzero current, test charge stationary
Now consider the setup as follows. In $S$, the nuclei and test charge are stationary, but the electrons are moving in the $-z$-direction. Just as before, we can transform into the electrons' rest frame, where we will find that the wire is positively charged. However, we also have that the test charge is moving in the $+z$-direction in $S'$, and that there is a current of positive charges in the $+z$-direction (which we could neglect earlier). Here the full Lorentz force law tells us there is a $qE$ repulsion, and also a $q \vec{v} \times \vec{B}$ attraction, and in fact they perfectly balance in this frame, so there is still no net force.
Summary
The space between electrons expands only if you keep yourself in their rest frame as you accelerate them. The spacing measured by an observer who doesn't accelerate is unchanged, in keeping with the assumption that the wire stays neutral in the lab frame. You can only use the electrostatic Coulomb's law if you are in the frame where the test charge of interest is stationary. If you are in a frame where the charge is still moving, you need the full Lorentz law, using whatever electric and magnetic fields are present in that frame.
Best Answer
From the point of view of Special Relativity, there are no two separate electric and magnetic fields, and charge and current densities are not independent when passing from one reference frame to another.
$E^2-B^2$ is a relativistic invariant (I use units such as $c=1$). Its value must be the same in every inertial reference frame, although the intensity of the electric and magnetic fields may vary separately. This fact puts an essential constraint on the possible changes of electric and magnetic fields from one reference frame to another. In particular, if only an electric field exists in one reference frame, a magnetic field may appear in other frames. Still, in no frame, the electric component of the electromagnetic field may vanish (that would reverse the sign of the relativistic invariant).
The invariance of $E^2-B^2$ helps to understand that the argument showing the magnetic field as an effect of a change of reference frame, when staring with a reference where only the electric field is present, can be reversed. It is equally possible to show that an electric field must appear in another reference frame if only a magnetic field is present in the original. In any case, Vweritasium's video is not showing that the electric field is more fundamental than the magnetic field. It just shows how the unitary description of the electromagnetic fields introduced by SR allows us to understand the deep link existing between electric and magnetic forces. Without a hierarchy among them.
In the case of an electromagnetic wave, the invariant is zero, and then it remains zero in all inertial frames. Said in another way, there is no possibility of converting an electromagnetic wave into a purely electric or magnetic wave.