Lorentz transformation of fields

The Lorentz transformation of electromagnetic fields gives the closed-form rule by which the three components of E and three components of B transform under a Lorentz boost between two inertial frames. For a boost along +x at velocity v = βc, with γ = 1/√(1−β²) the Lorentz factor: the parallel components E_x and B_x are unchanged; the perpendicular components mix linearly with the perpendicular components of the other field. Explicitly: E'_y = γ(E_y − v B_z), E'_z = γ(E_z + v B_y), B'_y = γ(B_y + v E_z/c²), B'_z = γ(B_z − v E_y/c²). The result was derived by Hendrik Antoon Lorentz in 1895, ten years before Einstein's special-relativity paper gave the geometric meaning, and is the basis for understanding how a configuration that looks "magnetic" in one frame can look "electric" (or partly electric) in another.

The transformation makes immediate sense once F^{μν} is recognised as a rank-2 antisymmetric tensor: under a Lorentz boost, a tensor transforms as F'^{μν} = Λ^μ_α Λ^ν_β F^{αβ}, and the closed-form result above is just this contraction written component-by-component. The pedagogical consequence is that the electric and magnetic fields are not independent physical entities but two faces of one geometric object F^{μν}, and the apparent "type" of a field depends on the observer's state of motion. A pure electric field of a static point charge in the lab frame becomes an electric field plus a magnetic field in any boosted frame; a pure magnetic field of a current-carrying solenoid in the lab frame becomes a magnetic field plus an electric field in any frame moving past. Two scalar invariants (E·B and |E|² − c²|B|²) are constructed from F^{μν} and pinned down regardless of frame, and these constrain which transformations are physically achievable.