Electromagnetic field invariants ($F_{\mu \nu }{F}^{\mu \nu },F_{\mu \nu }{\tilde{F}}^{\mu \nu }$)

For signature $(-,+,+,+)$ and a consistent orientation,Overall signs depend on tensor and orientation conventions, while their vanishing does not.

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.

For perpendicular electric field $\mathbf{E}$ and magnetic field $\mathbf{B}$, the sign of $E^2-c^2{B}^2$ determines which field can be removed by a Lorentz boost. If positive, $\mathbf{w}=c^2\mathbf{E}\times \mathbf{B}/E^2$ gives ${\mathbf{B}}^{\prime }=0$; if negative, $\mathbf{w}=\mathbf{E}\times \mathbf{B}/B^2$ gives ${\mathbf{E}}^{\prime }=0$. Both results follow from ${\mathbf{E}}_{\perp }^{\prime }=\gamma (\mathbf{E}+\mathbf{w}\times \mathbf{B})$ and ${\mathbf{B}}_{\perp }^{\prime }=\gamma (\mathbf{B}-\mathbf{w}\times \mathbf{E}/c^2)$; the relevant strict inequality ensures $|\mathbf{w}|<c$. A nonzero null crossed field cannot be reduced to a single field by a finite Lorentz boost. A particle initially at rest in the laboratory has initial velocity $-\mathbf{w}$ in the chosen inertial frame, which must be retained when solving its trajectory.

A null field has both electromagnetic invariants zero:For a nonzero constant null field, the mixed tensor is nilpotent of index three,