 Solution

For a boost of speed $v$ in the $x$ direction, withthe Lorentz transformation of electromagnetic fields isExpanding $E_y^{\prime }{B}_y^{\prime }+E_z^{\prime }{B}_z^{\prime }$, the mixed terms cancel and the remaining transverse terms acquire the factor ${\gamma }^2(1-v^2/c^2)=1$. Together with the unchanged longitudinal product, this proves the Lorentz invariantAn independent invariant isIf only $E_y=E$ and $B_y=B$ are nonzero, thenand direct subtraction gives

Now put $cB=\lambda E$ and $\beta =v/c$. Invariance of the dot product gives ${\mathbf{E}}^{\prime }\cdot {\mathbf{B}}^{\prime }=\lambda E^2/c$, while the displayed components giveHenceAt $\beta =0$ this equals $\mathrm{sgn}\lambda$, and as $|\beta |\uparrow 1$ it tends continuously to zero with the same sign. Thus boosts realize every $0\le \theta <\pi /2$ when $\lambda >0$, and every $\pi /2<\theta \le \pi$ when $\lambda <0$.