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When an electron moves at very close at $c$ (speed of light), is it physical to decompose the motion in two other directions (like what we do in classical case). If so, the motion in each direction will not be relativistic any more (by careful selection of the decomposition direction). What about the EM field associated with this moving electrons. Can they be decomposed in the similar way?

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What do you mean by "motion in each direction will not be relativistic anymore"? What you are doing is basically applying a rotation operator to the original lab frame (which is an element of the set of Lorentz transformations). So nothing strange would happen. – Zhengyan Shi Nov 28 '15 at 2:02

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