How to transform Electromagnetic fields for Lorentz Boost in arbitrary direction?

[Lusypher]

Homework Statement

In an inertial frame $\boldsymbol{S}$, components of electric and magnetic fields are $\vec{E}=2 \hat{i}+3 \hat{j}+5 \hat{k}$ and $\vec{B}=\overrightarrow{0}$ respectively, in some chosen system of unit. In another inertial frame $\boldsymbol{S}^{\prime}$, which is moving with uniform speed with respect to the former, components of magnetic field are $\overrightarrow{B^{\prime}}=2 \hat{i}+1 \hat{j}+3 \hat{k}$. What may be the components of the electric field $\overrightarrow{E^{\prime}}$ in the second frame of reference?

Relevant Equations

The Field Strength tensor is goven by
$$
F^{\alpha \beta}=\partial^\alpha A^\beta-\partial^\beta A^\alpha=\left[\begin{array}{cccc}
0 & -E_x & -E_y & -E_z \\
E_x & 0 & -B_z & B_y \\
E_y & B_z & 0 & -B_x \\
E_z & -B_y & B_x & 0
\end{array}\right]
$$

I know the Lorentz matrix for a boost along any of the Cartesian axes. But, since all the components of the magnetic field has changed in the frames, the boost is along an arbitrary direction, rather tahn a particular axis. How do I solve the problem in this case?

 

[anuttarasammyak]

Say 4-vector A is Lorentz transformed as
$$A'^{\mu}=\Lambda^\mu_\nu A^{\nu}$$
which is product of matrix and vector, 4-tensor B with form of matrix is Lorentz transformaed as
$$B'^{\mu\nu}=\Lambda^\mu_\alpha B^{\alpha \beta }(\Lambda^{-1})^\nu_\beta$$
which is product of matrices. In order that we can chose arbitrary boost direction
$$\Lambda^\mu_\nu=R^\mu_\alpha \mathbf{\Lambda}^\alpha_\beta (R^{-1})^{\beta}_\nu$$
where $\mathbf{\Lambda}^\alpha_\beta$ is e.g, z boost and $R^\mu_\alpha$ is matrix for a rotation so that the boost direction be z axis.

 

[dextercioby]

Wikipedia https://en.wikipedia.org/wiki/Lorentz_transformation is your friend. Check the section called "Vector transformations" towards the middle of the page and you should get your answer.