Light Wavelengths in Moving Frames: A Closer Look at the In-Phase Problem

[Adel Makram]

A source in the middle of a moving train emits two light rays in opposite directions. For the train frame of reference (FOR), two light rays reach two ends at the same time and be in phase. For a ground observer, the distance taken by the light going to the near end is larger than the distance taken by the light goring to the rear end. This means the number of wavelengths the near light takes is also larger than the number of wavelengths the rear light takes which means; the two rays could not be necessarily in phase. Is that true?

 

[Ibix]

No. The two light pulses have different wavelengths when viewed from the ground frame - look up the relativistic Doppler effect.

 

[Adel Makram]

But I don't understand that even at the level of classical Doppler effect. Firstly, the number of wave cycle is equal to the distance traveled divided on the wavelength. If that number is the same on both sides relative to the ground observer, the wavelength of the near light wave must be larger than that of the rear, is this true? If so, the near wave must be red shifted and the rear one blue shifted to the ground observer. But that color shift depends on the location of the observer relative to the light. How can this be conceived here in order for both light pulses to be in phase?

 

[Dale]

The phase at any event is a relativistic invariant. So all frames will agree on the phase at any given event.

However, due to the relativity of simultaneity different events may be considered simultaneous, and therefore it is possible to have two sources in phase in one frame and out of phase in another.

No experimental results will change because all experimental results depend on the local phase.

 

[Ibix]

But I don't understand that even at the level of classical Doppler effect. Firstly, the number of wave cycle is equal to the distance traveled divided on the wavelength. If that number is the same on both sides relative to the ground observer, the wavelength of the near light wave must be larger than that of the rear, is this true? If so, the near wave must be red shifted and the rear one blue shifted to the ground observer.

Correct.

But that color shift depends on the location of the observer relative to the light. How can this be conceived here in order for both light pulses to be in phase?

Not sure what you mean here. Are you thinking of a car engine note changing as it passes you? That's a slightly different scenario, in that you are off-axis. The relevant point is that all observers directly infront of the car will agree on the engine note at all times (until they jump out of the way, presumably) and all observers directly behind the the car will agree on the (different) engine note at all times. That's enough to keep the phase relationship constant.

Does that help?

 

[Adel Makram]

Correct.

The relevant point is that all observers directly infront of the car will agree on the engine note at all times (until they jump out of the way, presumably) and all observers directly behind the the car will agree on the (different) engine note at all times. That's enough to keep the phase relationship constant.

Does that help?

I think there are two different things here; namely, Doppler effect and observed wavelengths.
Please have a look at the attached diagram.
It is clear that the ground observer in the front of the train in this diagram will see a blue-shifted light (short upward gray arrow indicates higher frequency). Will that contradict what the same observer comes to know about the wave length of the light in that direction (look the wavelength as measures by the ground observer represented by a pink arrow).

 

[Ibix]

Excellent diagram. The only problem is that the pink/blue arrows aren't measuring wavelength. The wavelength is the distance between each wave crest andthe next one at a particular instant in time. So the wavelength in the ground frame is the horizontal distance between successive blue lines. That does show the expected behaviour. Similarly, if you measure along a line parallel to your x' axis you will find that the spacing is the same in front and behind.

I'm not sure what your horizontal arrows are measuring. It can't be the wavelength because $c=f\lambda$ and your c and your f are behaving as expected.