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Is the magnetic field of a moving electron caused by lenghtlength contraction in the direction of motion?

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Is the magnetic field of a moving electron caused by lenght contraction in the direction of motion?

Consider an electron moving relative to us. Because the space in the electron's rest frame is contracted relative to us in the direction of the electron's velocity, the electric field lines are squeezed in the same direction, so the electric field "density" is bigger perpendicular to the electron's motion (but smaller (zero?) in the direction parallel to its motion). Is this the qualitative source of the magnetic field?