Intricate conservation issues in relation to Doppler shifts

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Atre
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Intricate conservation issues in relation to Doppler shifts

Postby Atre » Tue Sep 22, 2009 4:51 pm UTC

When explaining life, the universe and everything scientific to an interested but more arts minded friend I discovered my meandering explanation set up a question that I couldn't answer... So here it comes with some background.

1) Question that is fine: A star whizzing through the universe (relative to you) redshifts some of its emitted photons and blueshifts others. This is okay because overall energy radiated remains constant regardless of speed (redshift energy loss vs. blueshift energy gain).

2) Question that is an intriguing puzzle:If I sit in a very fast rocket and strap a torch the the front + back, does conservation of momentum mean that I slow down?
p=(hbar)k on the photons emitted,impiles more momentum from blue photons at the front than red at the back and yet as far as the torch batteries are concerned there is the same amount of energy being fed into each torch, so there is an inconsistency somewhere.
I remember that my GCSE physics teachers were fundamentally incapable of answering this question, I satisfied myself as to a solution at some point but have forgotten it since then (possibly 'cos it was a BS escape clause about no preferred reference frame in special relativity)


3) Question that is a full on issue: CMB photons are a fossil from when the universe became transparent to EM radiation, meaning that they originally corresponded to Black body spectrum circa 4000K.... Today they correspond to a 2.7K spectrum. The difference in energy between these two spectra is considerable, where the hell did that energy go?
With the expanding "all points accelerate away from all other points" universe, all observers agree that the CMB loses energy over time. How is the energy conserved?


PS.I'm between 2nd and 3rd year physics at an excellent university, don't be afraid to go technical on me - I'll enjoy it :P.
PPS. In case anyone's wondering, the "GCSE"s I mention are the exams sat at age 15/16 in the British school system

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doogly
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Re: Intricate conservation issues in relation to Doppler shifts

Postby doogly » Tue Sep 22, 2009 4:59 pm UTC

No special frame isn't BS, it's the essential truth of relativity. If you prefer to work in an awkward frame rather than the rest frame of the rocket, you just transform the 4-momentum and see if that is conserved. Energy, the zero component of the 4-momentum, may not be conserved in some frame, but it is just a component.

CMB temp goes down because the universe expanded. Same amount of energy, just diluted now.
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schrodingasdawg
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Re: Intricate conservation issues in relation to Doppler shifts

Postby schrodingasdawg » Tue Sep 22, 2009 6:15 pm UTC

Regarding redshifts and blueshifts in curved spacetime: it's a bit complicated.

As long as you have a metric that's time independent, energy is conserved. If, say, you're far away from a (non-rotating) black hole, a photon coming at you from right outside the event horizon will have a constant energy throughout its trajectory. If it has constant energy, why is there a redshift? Well, it has to do with the fact that Planck's law is only true in a locally flat reference frame.

If you have an observer right outside the event horizon, where the photon originates, and the observer is stationary (with respect to you - the faraway observer), he will measure the photon as having a different energy than you would. This is because the highly warped nature of space and time in his vicinity. He will also measure the photon's frequency as obeying (approximately) E=hv (if the black hole is big enough that the metric is approximately flat over small volumes). When the photon eventually reaches you, you will also observe it as obeying E=hv, but for a different E; the v changed.

Or, to put it another way, since frequency is the number of events (e.g. the number of crests in a sinusoid passing through a plane) per unit (proper) time, the warped nature of space and time causes frequency to depend on energy in a different manner than E=hv.

Another approach is to note that the frequency is a scalar quantity. It changes over time, but it transforms as a scalar. That is, frequency is the multiplicative inverse of a change in proper time (such as between two successive crests in a sinusoid). Energy, in contrast, is a component of a four-vector. So the two won't transform the same anyway.

Begin edit:
I stupidly forgot that the Robertson-Walker metric isn't even time-independent. Maybe the energy of cosmologically redshifted photons actually isn't conserved. Anyone with more expertise in cosmology want to confirm or deny this?
/end edit

Now, let's look at your first two questions, regarding special relativity. Let's simplify this to a problem where two photons are emitted at a time: one to the right, one to the left. (We can repeat many times to get the whole two-torch thing.) In the reference frame of the emitter, the photons are emitted with equal energies in opposite directions; that is, they have opposite momenta p=E/c and p=-E/c. You want to find a reference frame in which the emitter is moving at a constant velocity relative to you. So what you need to do is transform the four-vectors of the photons appropriately. Look up the four-vector transformation laws and apply them. (Make sure you get all the signs right; messing up signs is the best way to confuse yourself ... I sure as hell do it all the time.) If you did it right, you'll get that one of the photons is redshifted (less momentum) and one is blueshifted (more momentum). At the momenta together to get the total momentum of the two-photon system.

Now you want to look at a process that could have created the photons. Say a particle with an energy 2E (in the emitter frame) decayed. It has zero momentum. What you want to do is apply the four-vector transformation law that you used in the last problem to the particle. Get, in particular, the momentum. Compare this to the momentum of the two photons derived above; the difference between the initial and final momenta is your impulse, and if non-zero will indicate an accelerating force acting on your emitter. If you did the calculation right, however, you'll find that there was no momentum change between the pre-emission and post-emission stages.

Hopefully that was helpful. I didn't want to do all the work for you, plus I don't know how to post equations on this board yet.

Atre
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Re: Intricate conservation issues in relation to Doppler shifts

Postby Atre » Fri Oct 09, 2009 11:24 am UTC

doogly wrote:CMB temp goes down because the universe expanded. Same amount of energy, just diluted now.

That much is relatively clear but it doesn't offer a mechanism. The CMB is "fossil" photons - yet they have managed to change wavelength, why this has happened was the tricky part.

schrodingasdawg wrote:Regarding redshifts and blueshifts in curved spacetime: it's a bit complicated.
...
Begin edit:
I stupidly forgot that the Robertson-Walker metric isn't even time-independent. Maybe the energy of cosmologically redshifted photons actually isn't conserved. Anyone with more expertise in cosmology want to confirm or deny this?
/end edit

Thanks, really helpful. The nuggets like "plank's law only true in locally flat spacetime" were the tools I needed to clarify the issue (Can I do the calculations for Q3? No. But I can see the consistency in the theoretical framework) - Although if anyone else can jump in to clarify the Robertson-Walker metric that'd be fantastic. Energy must be conserved somehow, it'd be big news if it weren't; and Noether's law wouldn't be generally applicable to general relativity is it didn't apply to changing curvature.

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Re: Intricate conservation issues in relation to Doppler shifts

Postby doogly » Fri Oct 09, 2009 3:30 pm UTC

The mechanism is dilution. It is like if you take a box of photons, and then double the volume of the box, keeping the pressure constant. Temperature goes down.

Even in special relativity, 'energy' is not conserved. Energy is the name for the 00 component of the four-momentum. So in general relativity, it is certainly not going to be conserved! Individual components of conserved tensors may still have names like energy, but they are much less important; it's not big news at all. You need to look at tensors, not individual components. Noether works in curved space, but you have to apply 'comma goes to semicolon,' that is, you have to replace all regular derivatives [imath]\frac{d}{dx^\mu}[/imath] with the covariant derivative. That gets you the necessary changes.
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