Does Redshift Change Over Time for a Constant Recession Speed?

[JohnnyGui]

[Moderator's note: post spun off from previous thread.]

I'd really like some verification on the following.

After thinking about light redshift, I came to the following conclusion:

Is it true that λObs / λEmit for 1 particular star doesn't stay constant over time even if its recession speed is constant?

The reason I'm thinking this is because, unlike doppler shift, cosmological redshift is dependent on the time for the light to travel and not only the recession speed. As the star recesses over time, light of that star would increasingly need more time to reach us and thus it would be longer and longer subject to the expansion which is continuously occurring and responsible for the redshift. λObs for that star would therefore increase over time and thus the ratio λObs / λEmit would get larger.
So even is the recession speed is constant, λObs would still get larger since the expansion has more time to redshift its light because of the increasing distance from us.

So, in case of a constant linear expansion rate, would this statement be true for (maybe one of the many) the above mentioned reason?

 

[andrewkirk]

This is by no means obvious, but the question is actually incomplete. The redshift will depend on what happens in between emission and reception, so we need to know what the recession rate is at all the places between emitter and observer.

I can think of no natural model that delivers a constant recession speed for the emitter, as opposed to the recession speed proportional to distance that arises under the standard cosmological model. If you assume that all points are receding at 10 km/s then a point only one cm away from the observer will be receding at 10km/s, which would be very weird, not to mention probably a discontinuity of the spacetime manifold. It would make it very difficult to eat one's breakfast.

Did you have a particular model in mind for the pattern of recession/expansion that would deliver a constant recession speed for the emitter?

 

[Chronos]

Cosmological redshift only occurs at cosmological distances. Individual stars at such distances are not resolvable: save in the case of supernova - which explains why they are so important in cosmology.

 

[andrewkirk]

Cosmological redshift only occurs at cosmological distances.

According to my calculations, it occurs at all distances, provided the emitter and the observer are both stationary wrt the CMBR. It's just that the effect is only large enough to be measured at cosmological distances.

 

[JohnnyGui]

This is by no means obvious, but the question is actually incomplete. The redshift will depend on what happens in between emission and reception, so we need to know what the recession rate is at all the places between emitter and observer.

I can think of no natural model that delivers a constant recession speed for the emitter, as opposed to the recession speed proportional to distance that arises under the standard cosmological model. If you assume that all points are receding at 10 km/s then a point only one cm away from the observer will be receding at 10km/s, which would be very weird, not to mention probably a discontinuity of the spacetime manifold. It would make it very difficult to eat one's breakfast.

Did you have a particular model in mind for the pattern of recession/expansion that would deliver a constant recession speed for the emitter?

I don't really have a specific model in my mind but what I was trying to do here is understand the difference between dopplershift and redshift regarding the formula λObs / λEmit. I chose a constant recession speed for one particular star/sound source to see how that formula would behave because for sound, λObs / λEmit would stay constant. I'd think that for redshift however, λObs / λEmit would get larger since light of a star would take increasingly more time to reach us and thus have more time to be redshifted, even at a constant recession speed of that star.

 

[PeterDonis]

what I was trying to do here is understand the difference between dopplershift and redshift regarding the formula λObs / λEmit.

It's the difference between flat and curved spacetime. See below.

I'd think that for redshift however, λObs / λEmit would get larger since light of a star would take increasingly more time to reach us and thus have more time to be redshifted, even at a constant recession speed of that star.

Step back for a moment, and think of the star moving away from us in flat spacetime. By analogy with your argument for sound,
λObs / λEmit should be constant for this case, even though the light from the star takes increasingly more time to reach us. (And this is correct, in the sense that it is what SR predicts.) So whatever the difference is between this case (ordinary SR Doppler shift) and cosmological redshift, it can't be just a matter of the light taking more time to reach us.

The real difference between the two cases is that in the SR Doppler shift case, spacetime is flat, while in the cosmological case, spacetime is curved. One way of seeing this is to observe that, in the cosmological case, the universe looks homogeneous and isotropic to both the receiver and the emitter (we'll ignore the fact that we on Earth are not actually "comoving" observers in the cosmological sense--we'll assume that the receiver and the emitter are both "comoving", which is what cosmologists usually do when discussing this kind of scenario). In the SR Doppler shift case, if the receiver sees everything else in the "universe" as homogeneous and isotropic, the emitter won't (unless the universe is entirely empty except for the two of them, in which case the question of homogeneity and isotropy is meaningless anyway). So the reason the SR model doesn't give the right answer in the cosmological case is that it's the wrong model: it assumes a flat spacetime, when the spacetime of our universe is actually curved.

 

[JohnnyGui]

It's the difference between flat and curved spacetime. See below.
Step back for a moment, and think of the star moving away from us in flat spacetime. By analogy with your argument for sound,
λObs / λEmit should be constant for this case, even though the light from the star takes increasingly more time to reach us. (And this is correct, in the sense that it is what SR predicts.) So whatever the difference is between this case (ordinary SR Doppler shift) and cosmological redshift, it can't be just a matter of the light taking more time to reach us.

The real difference between the two cases is that in the SR Doppler shift case, spacetime is flat, while in the cosmological case, spacetime is curved. One way of seeing this is to observe that, in the cosmological case, the universe looks homogeneous and isotropic to both the receiver and the emitter (we'll ignore the fact that we on Earth are not actually "comoving" observers in the cosmological sense--we'll assume that the receiver and the emitter are both "comoving", which is what cosmologists usually do when discussing this kind of scenario). In the SR Doppler shift case, if the receiver sees everything else in the "universe" as homogeneous and isotropic, the emitter won't (unless the universe is entirely empty except for the two of them, in which case the question of homogeneity and isotropy is meaningless anyway). So the reason the SR model doesn't give the right answer in the cosmological case is that it's the wrong model: it assumes a flat spacetime, when the spacetime of our universe is actually curved.

Thanks for the extensive explanation!

Sorry for being so stubborn but would you please explain to me how in a flat spacetime, SR would predict a constant λObs / λEmit for a traveling light even when it takes increasingly more time to reach us? Since the recession of the star and thus the expansion of the universie, which is the cause for the redshift, is constantly happening while at the same time, light takes increasingly more time to arrive to us, wouldn't light then be longer subject to the expansion and thus the redshift?

 

[PeterDonis]

Sorry for being so stubborn but would you please explain to me how in a flat spacetime, SR would predict a constant λObs / λEmit for a traveling light even when it takes increasingly more time to reach us?

How did you arrive at the same prediction for sound? If a sound source is moving away from you, the sound waves take increasingly more time to reach you, yet you're saying the ratio λObs / λEmit is constant in that case. Why?

the expansion of the universie, which is the cause for the redshift

Not in flat spacetime. Remember that when I talk about a flat spacetime SR model, I'm not talking about the model we actually use in cosmology. I'm talking about a different model, which would make predictions different from what we actually observe in the universe as a whole--but whose predictions, such as the ordinary Doppler effect, are correct in other scenarios, not cosmological ones.

 

[JohnnyGui]

How did you arrive at the same prediction for sound? If a sound source is moving away from you, the sound waves take increasingly more time to reach you, yet you're saying the ratio λObs / λEmit is constant in that case. Why?
Not in flat spacetime. Remember that when I talk about a flat spacetime SR model, I'm not talking about the model we actually use in cosmology. I'm talking about a different model, which would make predictions different from what we actually observe in the universe as a whole--but whose predictions, such as the ordinary Doppler effect, are correct in other scenarios, not cosmological ones.

Ah, I think my problem was that I didn't think of light as independent waves as soon as it leaves the emitter, just like sound. I was thinking of it as a sinusoidal elastic rope which is tied to the emitter and therefore will be kept being stretched by the recessing emitter during its travel.I must say that I need to learn more about flat and curved spacetime since I didn't know expansion causing redshift couldn't be applied to a flat model. I'd appreciate a few beginner's links on the explanation and differences between these two models. Do you happen to know some?

Your way of explaining definitely helped me notice my misunderstanding, thank you for that!

 

[PeterDonis]

I didn't know expansion causing redshift couldn't be applied to a flat model.

The reason why is that a flat model is not expanding. (There are some technicalities here that I won't go into; I'll just note that you may find people who will claim they have an "expanding" model in which spacetime is flat. The Milne universe is an early example, from the 1930's IIRC. These models are not "expanding" in the sense we're using the term here.)

I was thinking of it as a sinusoidal elastic rope which is tied to the emitter and therefore will be kept being stretched by the recessing emitter during its travel.

This isn't a valid model even in the cosmological case, where the universe is expanding (in the relevant sense for this discussion). If you want to think of the light as being "stretched" by the expansion (which might not be the best way to think of it in any case), it's not because the light is tied to the emitter.

I'd appreciate a few beginner's links on the explanation and differences between these two models.

I don't know if it is quite a "beginner's" link, but Ned Wright's Cosmology FAQ and tutorial is a resource I've found very helpful in understanding this subject. This particular entry should be a good place to start:

http://www.astro.ucla.edu/~wright/cosmology_faq.html#MX

One thing to be very careful of: when I talked about "flat spacetime" above and in earlier posts, that is not the same as a "flat universe" as cosmologists usually use the term (and as Wright does). The latter refers to a universe with flat spatial slices (in standard cosmological coordinates); spacetime in such a universe is curved. Many people get themselves confused by not keeping careful track of this distinction.

Your way of explaining definitely helped me notice my misunderstanding, thank you for that!

You're welcome!

 

[andrewkirk]

I think the reason confusion sometimes arises over whether a flat model can be expanding is that in discussion of this it is not always clear whether the adjective 'flat' is applied to the spacetime or to the constant-time hypersurfaces (CTHs) of that spacetime.

An expanding spacetime is necessarily curved, but its CTHs can be flat, in which case it is someties described as 'spatially flat'. Indeed, the current expectation appears to be that our universe, while expanding and hence curved as a 4D spacetime is either spatially flat or very close thereto, at cosmological scales.

I think the de Sitter universe is the simplest example of a spacetime that is expanding (hence curved) but spatially flat.

 

[PeterDonis]

I think the de Sitter universe is the simplest example of a spacetime that is expanding (hence curved) but spatially flat.

Even this depends on which slicing you adopt. dS spacetime is unusual in that it has a flat slicing, an open slicing, and a closed slicing, all of which have the form of FRW solutions with inertial "comoving" observers at rest in the spatial coordinates of the slicing. (It also has a static slicing, but observers at rest in the spatial coordinates of that slicing are not inertial.)