Yet another way ( #6) provided by MrMermaid;
Using an absolute frame of reference, the following diagram displays the effects of spectrum shifting.
In this diagram, the top wave is presumed to be the actual, “absolute frame”, frequency of a light source. A train station, S uses that same type of source to produce a light wave, but it is moving fast enough to cause the light headed toward the absolute frame to be at twice the frequency as measured by that frame, it is “blue shifted”. But also the light leaving the station in the other direction is red shifted to one half the frequency.
A railcar on a train is moving twice as fast toward the absolute frame as the station and thus produces a frequency of 4 times what the absolute frame would have produced, “ultra blue-shifted”.
As the railcar views the light coming from the station, it sees that light through 2 types of filtering, Doppler shifting (D) and Time dilation (T). The effect of the Doppler shift due to the train approaching the station is that the frequency of the light is increased to what the absolute frame would have produced. But the effect of the Time shift due to the train approaching the station would also increase the frequency because the train’s time is running slower, thus blue shifted.
From the station’s perspective seeing the light coming from the train, there are also the two filters but the Time shift for the station is zero relative to the train’s motion toward it. This could also be expressed in terms of the absolute frame wherein both were time shifted, the train twice as much due to going at twice the speed.
But even though the Time shifting doesn’t affect the light seen from the train, there is still a Doppler effect due to the motion difference. The railcar is producing a frequency 4 times absolute, but the station is moving away from the source even though not enough to outrun the train. Thus the Doppler effect is to reduce the frequency from its 4 times to merely a 2 times, thus blue shifted.
Thus without any length dilation involved but merely an absolute frame, the station and the train will perceive the same blue shift.
But if we allow the train to experience length dilation effects, we must add that additional filter (L). Due to any length dilation perceived by the railcar, the wave peaks coming toward it must be perceived as shorter, increasing the frequency again.
Thus if the train and the station see the same degree of blue shifting, there cannot be length dilation occurring on the train.