The following aspects of Maxwell's equations make no sense without special relativity:

- the Lorentz force would be different observers have different speeds, see e.g.: charged particle moving at the same speed of electrons thought experiment
- Maxwell's equations imply that the speed of light is the same for all inertial reference frames

When charged particle though experiment are seen from the point of view of special relativity, it becomes clear that magnetism is just a direct side effect of charges being viewed in special relativity. One is philosophically reminded of how spin is the consequence of quantum mechanics + special relativity.

It appears that Maxwell's equations can be derived directly from Coulomb's law + special relativity:

This idea is suggested by the charged particle moving at the same speed of electrons thought experiment, which indicates that magnetism is just a consenquence of special relativity.

This is a well known though experiment, which Richard Feynman used to emphasize

- infinite wire with balanced positive and negative charges, so no net charge, but a net magnetic field
- a single charge moves parallel to wire at the same speed as the electrons

In the above experiment:

- from the wire frame, the charge feels electromagnetic force, because it is moving and there is a magnetic field
- from the single charge frame, there is still magnetic field (positive charges are moving), but the body itself is not moving, so there is no force!

The solution to this problem is length contraction: the positive charges are length contracted and the moving electrons aren't, and therefore they are denser and therefore there is an effective charge from that frame.

This is also mentioned at David Tong www.damtp.cam.ac.uk/user/tong/em/el4.pdf (archive) "David Tong: Lectures on Electromagnetism - 5. Electromagnetism and Relativity" "5.2.1 Magnetism and Relativity".

See also: covariance.

Subtle is the Lord by Abraham Pais (1982) chapter III "Relativity, the special theory" mentions that this fact and its importance (we want the laws of physics to look the same on all inertial frames, AKA Lorentz covariance) was first fully relized by poincaré in 1905.

And at that same time poincaré also immediately started to think about the other fundamental force then known: gravity, and off the bat realized that gravitational waves must exist. general relativities is probably just "the simplest way to make gravity Lorentz covariant".

- www.youtube.com/watch?v=nrBiDRZRK5g Maxwell Lagrangian Derivation by Dietterich Labs (2019)
- www.youtube.com/watch?v=yo-Z3RO-eeY Deriving the Maxwell Lagrangian by Pretty Much Physics (2019)