electromagnetism - Why are magnetic fields a consequence of special relativity?

I'm very confused about why it is a consequence of special relativity.

Answer

Let's forget about anything quantitative at all.

Special relativity gives you length contraction -- so, when you're moving at a certain speed, distances along your direction of motion are compressed. Amongst many other things, this means that volumes will shrink, which also means that densities will increase.

Now, electromagnetism tells us that the electric force is proportional to the charge density. So, naïvely, we'd expect the electric force on a test particle external to the charge distribution to be higher in a boosted reference frame. This, however, contradicts the central assumption of special relativity that the net force on an object doesn't depend on the speed of the reference frame.

So, you need some new force that isn't present in the stationary reference frame. Well, in the boosted frame, the compressed charges are moving, so there is a current, so you could perhaps cancel out your excess force with some force that depends on the current distribution of spacetime. If you work this out, it turns out that magentism exactly does the trick, and if you factor in both electricity and magnetism, then the net force on the particle does not depend on whether you are in a stationary or moving reference frame.