General Relativity: Gravitational-Red Shift Confused with Doppler Effect?

Great analogy. I haven't run into that one, but it seems to work quite well.

In the simpler model you set up, you basically have an emission point and reception point, with a gravity well in the middle.

In your case, there would be no frequency shift, because the frequency would have been compressed and stretched by the same amount, as it passed the low pressure area. Great analogy, once again.

In my case, however, there would be a NET frequency shift (observed by any reference frame), as modelled in my first post.

I think maybe you didn't understand my scenario. The low pressure area isn't in between
the two sources. One of the sources is in the middle of it and the other source is in high pressure. One signal goes from low pressure to high and the other goes from high to low.
Comparable to sending signals between a large mass and a higher altitude.
How is this essentially different from your problem other than the lack of time dilation????
The critical factor in both cases is the difference in potential/pressure at the locations.
As you just pointed out localized areas that are passed through in between aren't important.
Do you see there would be no frequency shift in this situation???

Again, great analogy.

Yes, more accurately, a signal emitted in a LARGE high pressure area and then received in a SMALLER, lower pressure area, would be the same scenario.

1)Wouldn't there be a frequency shift in the above scenario? In this case, "red-shifted"?Also, aren't small amounts of energy lost into the medium as waves travel from one medium in to another, and from traveling from dense to denser areas of the same type of medium?

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2)The mechanics of how this scenario would operate might be different for photons traveling through curved spacetime, and for molecules vibrating in air, though. Quantum mechanics is needed to describe what happens to photons as they travel through curved spacetime.

Hi
1) no change . Look at yuiop"s last post for a detailed description of why.
or a quickie: frequency is a function of time. As long as the clocks at each end are running at the same rate and the source and receptor are at rest wrt each other there will be no change in frequency.
AS Jonathon described; as long as each wave peak, marble or whatever takes the same amount of time to make the whole trip , the intervening conditions , no matter what manner or how extreme make no difference. The time interval between successive transmissions and the time interval between successive receptions will be exactly equal..
And equal in both directions in this setup.
Energy loss would have no effect on frequency

2)In this regard there is no difference between photons and sound propagation in air.
QM may be needed to adequately describe the intervening conditions wrt photons but unless our observations of gravitational shift are in error , those conditions have no measurable effect

Thanks for the corrections in regards to your sound wave scenario.

Photons traveling through curved spacetime do undergo gravitational time dilation, though, which effects the clock rate, and so the frequency as well, I'm pretty sure.

In the scenario of the first post: as the photon is gravitationally red-shifted for a long time and then gravitatinally blue-shifted for a shorter time, its reference frame's clock-rate would have been changing as well. The frequency would have been changing dynamically the whole time.

Finally, when it is abruptly absorbed, there would be a net change in the frequency, as would be observed by any reference frame. This net change would be directly the result of gravitational time dilation.

AS has been explained; photons themselves are not gravitationaly shifted, it is the frequencies of the emitters and receptors that are effected by dilation.
The effects of curved spacetime can effect light speed , wavelength and perhaps other things but in the current structure of GR do not effect frequency.
No change enroute in the photons frequency. In the scenario weve been discussing, the same frequency would be received at any arbitrary point you cared to measure along the way.
Photons cannot rationally have a reference frame and time itself does not apply to them.

Here is an even better analogy, because you can actually do this at home and demonstrate to yourself what really happens. Get a piece of gutter or flexible curtain track or even a track for toy matchbox cars. Set up the track so that start is higher than the finish. Get some marbles and roll them down the track starting them off at one minute intervals. Notice that whatever the incline of the track they always arrive at the far end in one minute intervals. The conclusion you should reach is that the receiving frequency is always the same as the transmitting frequency. Now vary the track so that it slopes down at the beginning and then goes upwards on the last half. The marbles initially speed up on the first half of the track and then slow down on the last half of the track and yet they still arrive at the far end of the track at the same frequency. What does change is the speed and the distance between the marbles. You can think of individual marbles as peaks in a wave and the length of the gap between the peaks is the wavelength. Therefore a wave passing through various media at different speeds always maintains its frequency, but the speed and wavelength can change. If you have any doubts about the truth of that then actually do the experiment for real. It will only cost you some loose change. If that is still too much do a computer simulation with two sections of track with points travelling at one speed one first section of track and different speed on the second section of track and demonstrate that at any point along the track, the points always pass with the same frequency. If after all that you still not convinced yourself that frequency stays constant then I can only assume you lost your marbles.

The same is true for gravitational redshift. Using Schwarzschild coordinates, the wavelength and speed of a photon climbing out of a gravity well increases, but the frequency remains constant. It is only because the clocks of different observers at different heights run at different rates, that the frequency appears to slow down from the point of view of local observers.

In cosmology the redshift of light from distant galaxies is put down to stretching of space between galaxies as the universe expands which in turn stretches the wavelength of the light in transit and slows it down to a certain extent. (There are corrections for gravitational redshift due to the mass of the galaxies but this is a minor effect.) This is a difficult concept because it is difficult to imagine how a vacuum can stretch. How do we know the redshift is not simply due to the galaxies receding away from us in static space? The main clue is the cosmic microwave background (CMB) radiation. The frequency of the CMB is consistent with a extreme high frequencies during the big bang followed by billions of years of expanding space stretching the wavelength to the values we observe today.

Quite helpful and great analogy indeed.

Focusing on the photon's wavelength:

Photons that are gravitationally red-shifted for a long time and then gravitationally blue-shifted for a shorter time, would have a net stretching of their wavelength (relative to all reference frames).

If there is a net stretching of the wavelength, wouldn't there be a net loss of energy?

There is no long time short time. The shift occurs as a result of the difference in emitting and receiving frequencies of the electrons. I.e. All relatively instantaneously at the beginning and end. The time in transit is irrelevant. As for possible stretching of the wavelength during transit I think photons are pretty elastic ;-) and so any effects would be purely transitory according to local conditions so once again length of time in any local condition would not have any effect as far as the end reception.
AFAIK the energy is only dependant on frequency and once again it does not gain or lose energy in transit. If it is emitted at a higher potential it intrinsically has more energy than a comparable photon emitted at a lower potential.

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What happens to the photon dynamically during transit is important, because the photons are changing, and probably those changes are relevant here.

If the photon's wavelength changes, so does its energy, as per E=hc/λ.

According to local observers, the frequency of falling light is getting greater and the wavelength is getting shorter, so whether he uses E=hf or E=hc/λ they always consider the energy of the falling photon to be increasing.

Now for a Schwarzschild coordinate observer, the coordinate frequency (f') remains constant but the coordinate wavelength (λ') is getting shorter by gravitational gamma factor squared (1/γ²) and importantly the coordinate speed of light (c') is also getting slower by a factor of 1/γ², so the energy by his calculations is either:

E' = hf' = hf

or

E' = hc'/λ' = h(c/γ²)/(λ/γ²) = hc/λ

so whether the coordinate observer considers energy to be a function of frequency or wavelength he always comes to the conclusion that the coordinate energy of a falling photon is unchanging.

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... If it is emitted at a higher potential it intrinsically has more energy than a comparable photon emitted at a lower potential.