How can classical electrodynamics be a complete theory of electromagnetism when it's based only on special relativity?

Question

When charges are not accelerating Lorenz transformations suffice but dealing with accelerations goes formally beyond the scope of special revativity

Answer 1

Speical relativity can describe acceleratong objects if they are described from an inertial frame of referenfe, i.e. the observer is not accelerating. Also, Maxwell's equations can be formulated in curved spacetime, so are compatible with gravity and accelerated frames.