$\mathcal{S}$ and $\mathcal{S}^{\prime}$ are two reference frames with $\mathcal{S}^{\prime}$ moving with constant speed $v$ in the $x$-direction relative to $\mathcal{S}$. The co-ordinates $x^{a}$ and ${x^{\prime}}^{a}$ are related by $d x^{\prime a}=L^{a}{ }_{b} d x^{b}$ where

$$L_{b}^{a}=\left(\begin{array}{cccc} \gamma & -\gamma v & 0 & 0 \\ -\gamma v & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right)$$

and $\gamma=\left(1-v^{2}\right)^{-1 / 2}$. What is the transformation rule for the scalar potential $\varphi$ and vector potential A between the two frames?

As seen in $\mathcal{S}^{\prime}$ there is an infinite uniform stationary distribution of charge along the $x$-axis with uniform line density $\sigma$. Determine the electric and magnetic fields $\mathbf{E}$ and B both in $\mathcal{S}^{\prime}$ and $\mathcal{S}$. Check your answer by verifying explicitly the invariance of the two quadratic Lorentz invariants.

Comment briefly on the limit $|v| \ll 1$.