Static magnetic grating in a moving frame ($K^{\prime 2}=-k_u^2$)

Boosting a static spatially periodic magnetic field produces temporal oscillations together with an electric field. Its spatial-grating four-wavevector remains spacelike, rather than the null four-wavevector of a vacuum plane wave. For a boost along the grating axis, $|E^{\prime }|/|B^{\prime }|=v<c$. Ultrarelativistic motion can make the field approximately wave-like along a prescribed trajectory without making it an exact source-free radiation field.