The key point concerns this line: "electric and magnetic fields don't change their structure even when they are not stationary." I interpret that to mean that you believe that
E and
B are the same as viewed in references frames in motion with respect to each other. However, that's not true. Neither
E nor
B is invariant under change of reference frames. As you change reference frames, each becomes a linear combination of the two fields in the initial frame.
The Special Relativistic transformations were originally designed precisely to assure that if Maxwell's equations worked in one inertial frame they would work in all such frames.
Note that the relativistic Doppler effect is not the same as the classical one. The frequency multiplier is sqrt((1+v/c)/(1-v/c)). If you change the sign of the velocity, you get the inverse effect. The classical factor (1+v/c) is not the inverse of (1-v/c), the value for the opposite velocity. The relativistic factor is thus independent of whether you say the receiver or the sender is stationary. The classical factor is not.
Mike W.
(published on 04/13/2011)