The scalar and vector potentials satisfy the homogeneous wave equation in free space. Often, we choose ϕ = 0 and require that the polarization vector e of the vector potential be transverse to its wave vector: e · k = 0. This is not a Lorentz invariant requirement. However, consider a plane wave solution Aν = eν exp(ik_σ r_σ) of the wave equation, ∂_μ∂_μAν = 0 which satisfies the Lorenz gauge constraint, ∂_μA_μ = 0.
(a) Use covariant notation and show that a change of gauge does not affect the field tensor F_μν.
(b) Suppose that a Lorentz boost destroys the transverse property of the polarization. Show that it is always possible to choose a gauge function Λ (consistent with the Lorenz gauge constraint) which restores the transverse property.