= Field-removing boosts for perpendicular electric and magnetic fields
For perpendicular <electric field> $\mathbf E$ and <magnetic field> $\mathbf B$, the sign of $E^2-c^2B^2$ determines which field can be removed by a <Lorentz boost>. If positive, $\mathbf w=c^2\mathbf E\times\mathbf B/E^2$ gives $\mathbf B'=0$; if negative, $\mathbf w=\mathbf E\times\mathbf B/B^2$ gives $\mathbf E'=0$. Both results follow from $\mathbf E'_\perp=\gamma(\mathbf E+\mathbf w\times\mathbf B)$ and $\mathbf B'_\perp=\gamma(\mathbf B-\mathbf w\times\mathbf E/c^2)$; 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.
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