Question about doppler shift in relativity

[skate_nerd]

I've got a somewhat simple question, and I think its from me overlooking something stupid...but how does using doppler shift work if in any moving rest frame the speed of light is supposed to stay at 3(10^8) m/s? I know that E=hc/λ, so for the wavelength of the light λ to change, the only thing that could change is the energy right? So does moving towards or away from a light source change the energy of the photons coming at you, but only in your frame of reference?

 

[Legaldose]

Well, the wavelength is also related to the frequency, independently of the energy.

 

[skate_nerd]

Okay so yeah I was overlooking something stupid haha. Well I'm just a little confused about this homework problem, its about pair annihilation. A positron-electron pair annihilate into two photons, and I need to find the speed an observer would have to move to observe one of the photons wavelengths as twice that of the other photon's wavelength. So is this saying that if the observer moves at a certain speed, only the frequency of the photons change?

 

[phinds]

This is an excellent question and one that I had a problem with at first because it seems to violate the principle of conservation of energy.

As space expands, or as some prefer to say, as things get farther apart (on cosmological scales), light loses energy and thus changes frequency. The energy just disappears. BUT WAIT, I said ... how can that be. Energy can't just disappear.

Well, turns out that conservation of energy is not a valid concept on cosmological scales and the energy DOES just disappear. Weird, huh?

Now, keep in mind, this happens over BILLIONS of years. The cosmic microwave background, for example, is radiation that started out as visible light (see "surface of last scattering") and is now in the microwave part of the spectrum and is VERY weak but then it has been traveling towards us for almost 15 billion years.

 

[Dale]

Okay so yeah I was overlooking something stupid haha. Well I'm just a little confused about this homework problem, its about pair annihilation. A positron-electron pair annihilate into two photons, and I need to find the speed an observer would have to move to observe one of the photons wavelengths as twice that of the other photon's wavelength. So is this saying that if the observer moves at a certain speed, only the frequency of the photons change?

Have you been taught about four-vectors?

 

[phinds]

Okay so yeah I was overlooking something stupid haha. Well I'm just a little confused about this homework problem, its about pair annihilation. A positron-electron pair annihilate into two photons, and I need to find the speed an observer would have to move to observe one of the photons wavelengths as twice that of the other photon's wavelength. So is this saying that if the observer moves at a certain speed, only the frequency of the photons change?

Ah, that's a different problem than what my previous post addressed. I thought you were asking about cosmological Doppler shift.

 

[skate_nerd]

@phinds, I was actually trying to think of this problem by relating it to cosmological doppler shift, because I'm a little more familiar with that than doppler shift on a quantum scale...@DaleSpam I just recently started learning about four-vectors, yes.

 

[skate_nerd]

Actually, I just found a page in my text containing the derivation for the Longitudinal Doppler Shift Formula of Relativity Theory. Pretty cool stuff. So apparently one photon will have a higher frequency than the "rest frequency", and the other will have a lower frequency than the "rest frequency".

 

[Dale]

@DaleSpam I just recently started learning about four-vectors, yes.

OK, so this problem is fairly easy with four-vectors. Start with the four-momentum of the photons in the COM frame, $P=(E/c,\mathbf p )$. Then use the deBroglie relationship $P=\hbar K$ to get the four-wavevector $K=(\omega/c,\mathbf k )$. Then you can just transform that to the frame where ω of one is twice that of the other.

EDIT: you can make this really easy by using units where c=1 and where ω=1.