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Consider an inertial system $\mathcal{O}$ and a Lorentz boosted system $\mathcal{O}'$, moving with a velocity $\vec{v}$ with respect to $\mathcal{O}$. Then we have expressions for the electromagnetic fields as follows: $$\vec{B}=\gamma\vec{B'}+\frac{\vec{v}}{v^2}(\vec{v}\cdot\vec{B'})(1-\gamma)+\gamma\frac{\vec{v}}{c}\times\vec{E'}$$ $$\vec{E}=\gamma\vec{E'}+\frac{\vec{v}}{v^2}(\vec{v}\cdot\vec{E'})(1-\gamma)-\gamma\frac{\vec{v}}{c}\times\vec{B'} $$ Now, I want to find the condition that $\vec{E'}$ and $\vec{B'}$ have to satisfy such that there exists a $\vec{v}$ such that $\vec{E}=0$. I reckoned that the third term $\vec{v}\times\vec{B'}$ is perpendicular to the second term $\vec{v}$, so those two cannot cancel each other. However, how can I formulate these conditions in terms of $\vec{E'}$ and $\vec{B'}$?

James
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    If you are intended to follow @Chiral Anomaly's instruction then take a look in the ADDENDUM of my answer there :Is it a typo in David Tong's derivation of spin-orbit interaction?. Start with equation (08): $$ \mathbf{E}\boldsymbol{=0}\quad\boldsymbol{\Longrightarrow}\quad\mathbf{E}' \boldsymbol{=}\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{B}' \tag{08}\label{08} $$ – Frobenius Sep 13 '20 at 13:18
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    After finding from (08) the necessary conditions on $\mathbf{E}',\mathbf{B}'$ for the existence of velocities $\boldsymbol{\upsilon}$ you must try to prove the sufficiency, that is the validity of the inverse of equation (08) : $$ \mathbf{E}' \boldsymbol{=}\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{B}'\quad\boldsymbol{\Longrightarrow}\quad\mathbf{E}\boldsymbol{=0} \tag{08'}\label{08'} $$ – Frobenius Sep 13 '20 at 22:58
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    To accomplish this start from my equations (04a),(04b) and prove that $$ \mathbf{E}' \boldsymbol{-}\left(\boldsymbol{\upsilon}\boldsymbol{\times}\mathbf{B}'\right)\boldsymbol{=}\gamma^{\boldsymbol{-}1}\mathbf{E}{\boldsymbol{\perp}} \boldsymbol{+}\mathbf{E}{\boldsymbol{\parallel}} \tag{08''}\label{08''} $$ where $\mathbf{E}{\boldsymbol{\parallel}},\mathbf{E}{\boldsymbol{\perp}}$ the components of $\mathbf{E}$ parallel and normal to the velocity vector $\boldsymbol{\upsilon}$ respectively. – Frobenius Sep 13 '20 at 23:14
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    Thank you, I thinkn I can figure it out from myself right now! I just needed the hint to seperate parallel and perpendicular part – James Sep 14 '20 at 10:17

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