Components of electromagnetic tensor in a moving frame

Question

I need to find the electric and magnetic components of electromagnetic tensor in an inertial frame S' moving in the +x direction with a speed $\beta$ relative to frame S. Electromagnetic tensor in S frame $F_{\mu\nu}$ is given below. $$F_{\mu\nu}=\begin{pmatrix} 0 & -E_x & -E_y & -E_z \\ E_x & 0 & B_z & -B_y \\ E_y & -B_z & 0 & B_x \\ E_z & B_y & -B_x & 0\end{pmatrix}$$ Since$$F_{\mu'\nu'}=\Lambda_{\mu'}^{\mu}\Lambda_{\nu'}^{\nu}F_{\mu\nu}$$ Is it correct to use lorentz transformation matrix $$\Lambda_{\mu'}^{\mu}=\begin{pmatrix} \gamma & -\gamma\beta & 0 & 0 \\ -\gamma\beta & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{pmatrix}$$in the equation.

Answer 1

Yes, it is correct to use the given Lorentz transformation, but you need to be careful about how you interpret the expression $F'_{\mu'\nu'}=\Lambda^\mu{}_{\mu'}\Lambda^\nu{}_{\nu'}F_{\mu\nu}$ if you are to express it in terms of a matrix equation rather than summing over the dummy indices. When applying a Lorentz transformation on both indices of a rank-2 tensor, the equivalent matrix expression is $F'=\Lambda F\Lambda^\top$.