Let us consider the Lorentz transformation of the fields. Clearly just transforms like a vector. We could derive the transformed and fields using the derivatives of but it is interesting to see how the electric and magnetic fields transform.

We know that Maxwell's equations indicate that if we transform a static electric field to a moving frame, a magnetic field will be generated, because there is a current in that frame. Think of the case of a point charge in its rest frame and in a frame in which it is moving. It was clear from E&M that and were not simply parts of 4-vectors.

The Electric and Magnetic fields are part of a rank 2 tensor and so they transform accordingly.

Since we could choose any direction for the axis that we boosted along, these results for the field transformation are correct for all boosts.

Jim Branson 2012-10-21