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Why do we need to apply Lorentz transformation twice when transforming the electromagnetic f...

We are not applying it twice, what happens is that this is how rank 2 tensors transform. A rank n tensor will transform like $$ F^{'}_{a_1,a_2,\dots,a_n} = \Lambda_{a_1}^{b_1} \Lambda_{a_2}^{b_2} \dot …
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