Frame $\mathcal{S}^{\prime}$ is moving with uniform speed $v$ in the $x$-direction relative to a laboratory frame $\mathcal{S}$. The components of the electric and magnetic fields $\mathbf{E}$ and $\mathbf{B}$ in the two frames are related by the Lorentz transformation

where $\gamma=1 / \sqrt{1-v^{2}}$ and units are chosen so that $c=1$. How do the components of the spatial vector $\mathbf{F}=\mathbf{E}+i \mathbf{B}$ (where $i=\sqrt{-1}$ ) transform?

Show that $\mathbf{F}^{\prime}$ is obtained from $\mathbf{F}$ by a rotation through $\theta$ about a spatial axis $\mathbf{n}$, where $\mathbf{n}$ and $\theta$ should be determined. Hence, or otherwise, show that there are precisely two independent scalars associated with $\mathbf{F}$ which are preserved by the Lorentz transformation, and obtain them.

[Hint: since $|v|<1$ there exists a unique real $\psi$ such that $v=\tanh \psi$.]