# The Mechanism of the Cosmological Redshift

1. Jul 6, 2009

### nutgeb

It is well established that the cosmological redshift causes both the wavelength and the separation of photons to increase in proportion with the increase in the cosmic scale factor a. The traditional explanation for the mechanism of the redshift is that ‘expanding space’ progressively forces apart both the wave crests of individual photons, as well as the separation between photons in a string, as a light beam crosses the expanding space between the emitter and the observer. See, e.g., textbooks by Misner, Peebles, Peacock, etc. This explanation has justifiably been criticized. Here is Alan Whiting’s take in his http://arxiv.org/abs/astro-ph/0404095v1" [Broken] ‘The expansion of space: free particle motion and the cosmological redshift’, published in The Observatory:

“Misner, Thorne & Wheeler, p. 776 and Peebles (1993)12, p. 96-7, use the picture of a standing wave with expanding boundary conditions… It is not clear, for instance, why there should be a standing wave generated between comoving points in the universe, nor why it should maintain itself. More importantly, as pointed out by Cooperstock et al. (1988) (among others), electromagnetic radiation automatically tracking the universal expansion cannot be right. All our test equipment and comparisons are built of or use electromagnetic forces, and they should also expand with the universe; so any cosmological redshift would be undetectable in principle. At the very least, atoms in the Hubble flow would change their characteristic wavelengths with time (and perhaps with the state of the local gravitational field), leading to strange results indeed.”

The ‘tethered galaxy’ exercise definitively teaches that, if Lambda=0, ‘expanding space’ does not act like a force, and cannot cause an increase in the proper distance separation of wave crests, photons, or any other objects, if they didn’t already have a proper momentum away from each other in the initial conditions. It is clear that successive wave crests and photons do not have a positive radial proper velocity away from each other when they are emitted. If the observer is considered to be the coordinate origin, the slowing expansion rate must affect the emitter’s recession velocity at the time of each subsequent emission at least as much as it affects the approach rate of the previously emitted photons.

If ‘expanding space’ does not force wave crests and photons to separate, then what does?

Not the emitter’s recession velocity

The emitter’s velocity away from the observer does not contribute anything to the redshift. It may seem reasonable, as a number of authors claim, that the emitter’s recession velocity contributes an element of Doppler shift. But on closer examination, it turns out that the emitter’s Doppler shift is exactly offset by the gravitational acceleration of the observer toward the approaching photon over the course of the photon’s journey. This gravitational acceleration imparts a blueshift to the photon (in the observer’s frame) which exactly offsets the classical Doppler shift. In other words, on the photon's inbound trip, gravitational acceleration imparts the same proportional velocity increase to the photon as the gravitational deceleration it imparts to the cosmic Hubble rate.

There is no element of SR time dilation because the FRW metric does not include any time dilation between fundamental comovers, which allows them to share the same cosmological proper time. It is easy to see that the time part of the RW line element is linear:

$$ds^{2} = -dt^{2} + a^{2}(t) \left[ dx^{2} + \left\{sin^{2} \chi, \chi^{2}, sinh^{2} \chi \right\} \left( d\theta^{2} + sin^{2 \theta} d\phi ^{2} \right) \right] \\ \left\{ closed, flat, open \right\}$$

The emitter’s progressive recession movement during the interval between the emission of each successive photon in a string causes the photons to initially be separated from each other by an additional distance factor (1 + v/c). This is a classical Doppler shift. If the photons were to arrive at the observer in this configuration, the observer would measure that the reception is time dilated accordingly. However, again this effect is canceled out en route. The Lambda=0 ‘tethered galaxy’ exercise demonstrates that if a string of two (or more) particles are launched from the same comoving coordinate at constant time intervals, cosmic gravitational tidal forces will cause the proper distance between the lead and tail particles to decrease in proportion to their proximity to the observer. If this effect is considered in isolation, each successive photon will arrive at the emitter after exactly the same time and distance interval at which they were originally omitted.

Traversing the Hubble velocity gradient causes the redshift

It is well established that the peculiar velocity of a non-relativistic particle decays in inverse proportion to the growing cosmic scale factor, or 1/a. A comoving observer interprets the decay in a non-relativistic particle’s peculiar velocity as a decrease in the particle’s momentum.

For a particle which is ‘outbound’ from the coordinate origin, this peculiar velocity decay reflects simply that the particle finds itself overtaking successive galaxies that have increasingly large Hubble velocities, since Hubble velocity is proportional to proper distance from the origin. For a particle which is ‘inbound’ toward the coordinate origin, the peculiar velocity decay is also 1/a, but the picture is less intuitive. An inbound particle observes the local Hubble velocities it passes through to be decreasing, all the way to exactly zero, as it approaches the origin. But since the particle’s peculiar velocity is negative (toward the origin), and the Hubble velocity is positive (away from the origin), the decreasing Hubble velocity in fact represents a decreasing peculiar velocity.

Of course the peculiar velocity of relativistic photons does not decay as the cosmic scale factor increases. Photons must retain a peculiar velocity of exactly c in every local frame they pass through. Tamara Davis comments on this behavior in her http://arxiv.org/abs/astro-ph/0402278v1" [Broken] at the page numbered 51:

“It may seem strange that momentum decaying as 1/a means the peculiar velocities of massive objects decay until the objects are comoving, and yet the peculiar velocities of photons always stay at c. It seems that photons are getting some velocity boost that massive particles miss out on.”

The photons effectively are experiencing a local velocity boost: their velocity is boosted each time they pass from one infinitesimal local frame to the next which has a different Hubble velocity. But in this situation there is no external source of incremental energy (such as gravity) driving the velocity boost. Therefore, energy conservation requires the photon’s momentum, as experienced by a comoving observer, to decrease in the same proportion as its velocity increases. Thus in an expanding universe a photon’s comoving momentum decays at 1/a, the same momentum decay as a non-relativistic particle. The decay in the photon’s momentum at 1/a means that the photon’s wavelength is observed by a comover to be redshifted in exact proportion to the expansion of the universe since the photons were emitted.

Note that the photon’s comoving momentum does not decay from the perspective of the emitter’s frame.

It is logical to interpret the successive boosts of the photons’ velocity to also cause their increasing physical separation. The velocity boosts occur as a function of location (passing through the local frame of a galaxy with a particular local Hubble velocity) rather than as a function of time per se. Since the lead photon in the train arrives at each location before the tail photon, the lead photon experiences each velocity boost sooner than the tail photon does. Therefore, the lead photon’s average proper coordinate velocity is always relatively higher than the tail photon’s, despite (or rather, because of) the fact that the local peculiar velocity of each is always exactly c.

My analysis indicates that the cosmological redshift obeys the following simple equation:

$$\frac{ \lambda_{o} }{\lambda_{e} } = \frac{a_{o}}{a_{e}} = (1 + z) = \int_{t_{e}}^{t_{o}} (1 + H_{t}dt)$$

where the subscripts e and o respectively signify the time of emission and observation, H is in units of ly/y/ly, and t is in units of years. Note that in this equation dt is equal to dD, the change in the photon’s proper distance in light travel time. So this equation simply accumulates, in a linear multiplicative manner, the changes in the Hubble velocity (H * D) experienced by the photon over its journey. The formula for each frame-crossing is the same as for the classical Doppler shift. A rough calculation of this integration on my spreadsheet is within 10% of the correct result for z=1023, which suggests it is likely to be correct.

Note again that SR time dilation contributes nothing to this redshift, because the FRW metric has no place for time dilation between comovers. This is consistent with my rough spreadsheet calculations: when I include SR time dilation, the error increases to 40%.

In net, the cosmological redshift does not seem to occur as a discrete event at the observer due to the relative velocity between the emitter and receiver; instead it occurs progressively en route.

Last edited by a moderator: May 4, 2017
2. Jul 7, 2009

### Wallace

In the end everyone agrees on what redshift you get when you apply GR and assume the Cosmological Principle. Any description of 'what causes' that redshift in terms of distances, velocities or expansions of some function a(t) are only going to apply in a particular co-ordinate system, therefore they are not invariant physical effects.

The debate that occurs from time to time is what is the better mental shortcut to thinking about how it all works, which metaphor is the most appropriate? I have my own view, and I know some agree and some disagree with it. The thing that gets lost far too often in threads about this topic here is a confused mixing of physics and pedagogy.

The physics of cosmological redshift is uncontroversial (unless you consider Halton Arp, but lets not do that), so lets not start to debate what 'causes' redshift. The only thing that is reasonably up for discussion is what is the best metaphorical device to explain how to works to someone who will never (general public), or has not yet (students), learned sufficient General Relativity to understand it without resorting to metaphors.

My personal view is that the expanding space metaphor is okay, but limited and problematic because it can only sensibly describe a completely homogenous universe. The expanding space idea can easily lead to horrible misunderstanding when it comes to the question of why galaxies don't expand. Personally I find it much easier to think in Newtonian terms, receeding galaxies are simply moving away from each other and will continue to do so because of inertia unless acted upon by a force, such as gravity. Even on a computation level if you included a gravitational redshift (by using Gauss's law) Newtonian physics gives you a very good quantitative answer that only diverges from the correct GR one for quite large distances (hundreds of megaparsecs). There is no surprise then that galaxies don't expand, they have no reason to do so so this question would never even occur.

Re-enforcing the fact that we can understand the expanding universe in relatively simple terms that have everyday analogues, such as momentum etc, is in my view much better than implying that the universe expands beacuse of some inherently relativistic process, which is what 'expansion of space' can errouneously imply.

3. Jul 7, 2009

### sylas

But... surely it really IS some inherently relativistic process involved!

It's quite true that this a primarily pedagogical, and that you can't hope to explain the full details to everyone. But any incomplete explanation will have its own problems -- yours also.

My own concern here is that reason you give for your own account -- which is presumably pitched at people who understand a little more of the details -- is incorrect. The "expansion" IS an inherently relativistic process. The expansion of the universe involves changes over time of the volumes of regions of space; and this is not simply co-ordinate dependent. The current consensus model, with a dark energy term and accelerating expansion, has horizons, and these can be identified independently of co-ordinates. The volume of space with horizons is expanding in a way that is only explicable in terms of relativity.

I don't mind using simplified accounts to try and give aspects of the situation to novices, or explanations of where misunderstandings can arise when ANY simplified account is used ... including yours.

But your reason here is just wrong. The expansion of the universe is inherently relativistic, in the sense that general relativity is the only theory we have that makes sense of the deeper details behind simpler approximations.

Cheers -- sylas

4. Jul 7, 2009

### Wallace

Yes, because we know that GR is the most correct theory of gravity we have. But we teach Newtonian physics to students before we teach GR because in most cases they are close enough. The same is true in cosmology, but this is acknowledged far less often. Of course you need GR for the details, but the broad concepts can be understood in basic terms far more easily and transparently using Newtonian physics.

The point is that when you do that you are at least understanding the thing in terms of physics rather than learning the canon of a metaphor.

Of course 'it will have problems' if you start doing calculations with it. Of course it is incomplete.

No it isn't, in the sense that it reduces to the same answer in the Newtonian limit. You can just as easily say that dropping a ball is 'inherently relativistic'. While it is true on some level, it is assinine to suggest that teaching the Newtonian explanation is incorrect.

But an expanding cloud of material increases its volume in the Newtonian description as well. Again the details, but not the basic concepts, are different.

If you really understand what an expanding horizon means (I mean really understand it) then you have sufficient knowledge of GR to simply comprehend the FRW solution without the need of comforting metaphors.

There is a difference, to often confused between the expansion of the universe and the acceleration of the expansion. Understanding why the Universe expands is as simple as understanding inertia. Understanding the origin of the various effects that can change that expansion rate is a different question. This is often confused, see question such as 'I heard this thing called dark energy is making the universe expand'. Because people have been told so many times that 'redshift is caused by expanding space' they think that this mysterious physics (which of course is not really physics) of 'expanding space' must be linke to the dark energy thing. This mixes first and second derivatives with respect to time in a horrible horrible way. It is far easier to see how intertia means that no mechanis is required to keep the expansion happening, and then understand that the increase in the expansion rate is something truly strange, hence the need for the very weird dark energy term.

Look at the kind of questions people with no training ask here at PF and elsewhere. You won't find those kinds of things about horizons etc coming up often. Of course you couldn't answer them in terms of Newtonian physics, but you wouldn't answer them using a Balloon either. You would have to explain them in terms of GR, which is fine, in that case you don't need to invoke simpified concepts like 'expanding space', or my alternative method.

Again, you've missed the point. Of course any explanation that is incomplete will be an incomplete explanation. The point is that 'expanding space' is erronously thought to be a technical description of GR, when it is not. I'm happy for any to disagree about which is the better simplified explanation, as long as you realise that both my approach and 'expanding space' are simplistic descriptions, obviously ultimately flawed since you can't use them to do correct calculations.

Agian, you are agreeing with me in order to refute me? We agree that GR is the best theory of gravity we have, the discussion is which is the better simplfied model to use before one can simply use GR. That was the central point of my post, which you seem to have misunderstood.

Look, I do try and keep away from these expanding space threads, since they tend to be long and full of horrible atrocities against science. Correctly explained and used in appriate limits 'expanding space' is a perfectly good way to get the basics of Hubbles law etc. But it is when you hear that 'the gravity of a galaxy overcomes the expansion of space' that you know that this metaphor has been confused with physics. Likewise when the words 'mechanism of redshift' and 'expansion of space' are put in the same sentence. The expansion of space is a descrption of the expanding universe, but it is not causitive effect. a(t) is a convenient function because you can get is straight from the redshift ratio, but too often it is implied that the form of a(t) makes that redshift ratio occur. What makes a particular form of the redshift with time, and hence a(t) is the physics of the constituents of the universe, be it matter, radiation, dark energy or whatever else. It is that physics (calculated with GR) that gives you the particular functional form. When you wrap it all up us 'expansion of space causes redshift' you lose all of the actuall physics going on.

Anyway, despite having just argued a case against 'expanding space' I really don't mind if people want to use it, just so long as you make it clear that it, like other options, is a simplified description, so you shouldn't be surprised it breaks down when you push it too far. We all agree that GR is the theory to use, and when you understand how to use that you don't need any of these simplified devices.

Last edited: Jul 7, 2009
5. Jul 7, 2009

### sylas

To a point, yes; but many aspects of it cannot be explained easily with Newtonian physics. Even a complete novice is going to hear hints of the wonderful stuff involved in gaining a deeper appreciation of the universe; and wants to know something about them at a suitable level. You can use the Newtonian approximations for some aspects... but not others. Coming up with helpful analogies for a complete novice is hard... there's a real skill in trying to share details of physics with interested amateurs.

It's a mistake to insist on one way of explaining things.

Whoa there wallace. YOU were the one who started out with the charge that teaching the relativistic explanation is incorrect! Specifically, you said is it "erroneous" to imply that the expansion of the universe is inherently relativistic. This is the extract of your post that I am disputing.
I'd prefer we avoid words like "asinine". If you think I am wrong, just say that, please.

I am not saying you are "incorrect" to use Newtonian approximations, for all that they are necessarily incomplete. We ALL use incomplete approximations, and IMO it is pedagogically "correct" to teach things that are technically incorrect but a good approximation. When I was lecturing myself (computing), I sometimes used to warn students that I would tell them lies, and then fix them up when they more of the material under their belt. (Credits to Don Knuth for this notion.)

But I am saying you are "incorrect" to insist that expansion of the universe is not inherently relativistic. Sure, relativity is approximately Newtonian... but when you are talking to a novice about the universe, they really aren't as much interested in the smaller scales where Newtonian approximations work as they are in a larger "universal" picture, which most certainly IS inherently relativistic.

I agree that it is good pedagogy to point out that relativistic accounts reduce to Newtonian accounts on smaller scales of a billion light years or so, and definitely on scales of a single galaxy... and hence that whatever expansion might mean it is not something that is "trying" to pull galaxies or galactic clusters apart.

I am not at your level of understanding, to be sure! But I do comprehend the FRW solution without metaphors and I do have a tolerable understanding of the various horizons.

I am writing here as someone who understands a bit of the technical details, and who has a long standing interest in the pedagogy of finding suitable metaphors and approximations that I can use for explaining to others. I am also someone who is still learning a lot of the technical details, and is learning from more expert physicists such as you.

But I am also aware that experts aren't perfect, and sometimes I presume to challenge something that I think wrong, even if it comes from someone with better credentials. In this case, I think that expansion is indeed an inherently relativistic process, and that some aspects of it ... such as the "event horizon" in an accelerating expansion... have no correspondence in Newtonian physics. The event horizon I mean is the point beyond which no information can ever reach us, no matter now long we wait. If the current consensus model is correct, then some of the very distant galaxies we can see will have since crossed this horizon.

Sure. I have seen some of these confusions from novices struggling to get to grips with the matter.

I've seen the questions, and attempted to answer many of them myself. The "horizon" doesn't come up often, but it does come up sometimes. One of the most common things to come up is "superluminal" expansion. It's a common error also to speak of expansion as a velocity, or to think that only inflation is "superluminal". And by the way ... I think you can explain horizons without going into GR in detail. As always, it is an approximation, but that is what a novice needs... along with a caution that they are using approximations!

Skipped a bit there. Look, what I found really odd was the claim that expansion is not inherently relativistic. I think it is... I'm SURE it is. That's really my issue. On the other hand, I agree that it reduces to Newtonian approximations on smaller scales... certainly like a galaxy, and also for larger regions with many galaxies. I'm not saying it's wrong to use Newtonian approximations for explaining aspects to a novice.

I agree with much of the rest of what you have said here; and I do try to be careful to about recognizing when a description is simplified. But I think properly used, the "expanding space" concept often has a place in explanations for a novice, and that even a novice should be able to appreciate that Newtonian approximations don't actually suffice for describing the universe.

Cheers -- sylas

6. Jul 7, 2009

### Wallace

Expansion of the Universe simply means that things are generally moving away from everything else. You can understand how this would work in Newtonian physics, hence there is nothing inherently relativistic about the expansion. You can even describe the FRW solution exactly using Newtonian physics (you only get a difference when you consider light propogation and very large scale fluctuations in density).

The Universe is expanding because something (inflation) got everything moving away from everything else. The expansion continues due to inertia (Newtons first law). The expansion rate is altered due to forces imparted on it (Newtons second law). None of that is 'inherently relativistic' because the concepts are valid in both a Newtonian and GR description and the numbers agree for calculations over 'short' distances, which are in fact pretty big distances by human standards.

The following things can be understood using these concepts:
1) The Hubble law, v=HD
2) how galaxies form and stabilise (no longer expand)
3) Why the Hubble law does not imply we are at a special point
4) Why the matter content of the Universe alters the z(t) (or a(t)) relationship
5) Why the Universe didn't collapse into a black hole immediately after the Big Bang
6) others...

The following things probably would not be suitably described in this framework
1) Horizon problem (why we need inflation)
2) probably others...

If you look at the kinds of questions people new to this ask, most fit into the first list (in some way). It is better to be able to explain them in a way that has a connection with physics, and Newtonian physics is much more intuitive. The problem with invoking GR inspired metaphors is that the theory is much less intuitive, so you are just learning the canon of a metaphor, rather than being able to thinking about in terms relating to physical concepts.

Pedagogically, I would teach Newtonian cosmology first, and then introduce GR when required. This gives a solid grounding of the physics at play and leads naturally into the fully relativistic description. Even at the highest techincal level (e.g. journal articles) Newtonian physics is used more often than GR. This includes cosmological N-body simulations (entirely Newtonian), cosmological perturbation theory (Newtonian after re-ionisation), semi-analytic descriptions of galaxy cluster abundance (e.g. Press-Schechter theory and extensions). Gravitational lensing calculations are also done exclusively in the Newtonian limit. You could do all of the above in fulll GR, but the results would be no different and much much much harder to obtain. The next generation of instruments measuring dark energy properties will have all their data analysed using theoretical calculations entirely derived using Newtonian physics. The only caveat is that the various exotic physical theories to explain the origin of dark energy are written down usually as Lagrangian theories in GR. However, reduced to the usual parametrised forms (usually via the effective equation of state) these can be put in a Newtonian form (it looks like an anti-gravity term) that gives you the exact correct answer (again except for light propogation where the GR solution differs from Newtonian over large distances).

If it's good enough for the pro's I'd say it should be okay to introduce very basic concepts via simple Newtonian devices.

Last edited: Jul 7, 2009
7. Jul 7, 2009

### sylas

"The" FRW solution? I don't know what you can mean by "exact" description here. Some aspects... yes. Others; no.

It seems by "not inherently relativistic" you are meaning that you can have expansion corresponding to the Hubble law with Newtonian physics, and hence the concept of expansion shows up also in Newtonian physics. Fair enough. Your original remark with which I took issue seemed to me to be saying that the expansion of our universe is not inherently relativistic; which was going too far, I think. Maybe I just took you to be saying more than what you actually intended.

Very nice. I like it.

The Horizon problem relates to the "particle horizon", or how regions seem in opposite directions of the sky can be so similar, as if they have been in contact in the past, when the current expansion rate and development indicates this is not possible. An inflationary epoch resolves this.

The horizon I mentioned previously is the converse "event horizon", which refers to a limit beyond which no signal can reach us into the future. Accelerating expansion has such a horizon, and it becomes particularly relevant when thinking about likely conditions in the distant future, and the "heat death" of the universe.

A common question where Newtonian accounts become inadequate is whether the universe is finite or infinite. In the FRW solutions, this turns upon the curvature term.

There's one exception when it comes to common novice questions, IMO... and I realize this may be contentious. That is the notion of "superluminal" recession velocities. You can give co-ordinates in which recession velocities are not superluminal, with things like the Milne universe, but I personally think it is better at this point to allow that there really is a difference between expansion of the universe and Newtonian expansion of a cloud of gas. The most intuitive notion of "distance" for a novice (IMHO) corresponds to the "proper distance" co-ordinate.

I have no problem with that.

Cheers -- sylas

8. Jul 7, 2009

### Wallace

I agree that this is a tricky question to give a satisfactory answer to, and not have that answer blow up somewhere else. Yes, I agree that Newtonian physics is not going to help here, since the point at which you encounter implied superluminal velocities is well beyond that in which the Newtonian solution agrees well with GR. I do have problems with an explanation to this (not that I'm accusing you of this) that treats it as 'expansion of space isn't like real motion' and hence it can do whatever it wants.

You can't really explain this question properly without addressing the issue that both the distance and time that make up recession velocity are non-trivial to define over such scales, which means dealing with relativity on some level.

So yes, all simplistic explanations have their limits, and the above question should be added to my second list in my previous post. Note that you can't really explain this using the idea of expanding space either, at least not very well.

9. Jul 7, 2009

### sylas

Over and out from me again, I think. Thanks very much for the exchange... I appreciate the opportunity to cross swords with people who do have more expertise than myself, and over time my own understanding develops as a result.

For what it is worth, I do usually use expanding space to explain superluminal recession, but not just by saying it can do "whatever it wants", and this has helped for some people, I think.

Cheers -- sylas

10. Jul 7, 2009

### Wallace

Do you have a link to a previous thread where you have given such an explanation? I would be interested to have a look, since it is a tricky question to get across clearly.

I would caution that just because someone feels they understand something well, and an explanation has helped, doesn't mean that they have actually improved their understanding. I'm sure some people find wack-job explanations of Intelijent Dezine helpful to their understanding of Evolution...

Of course I'm not accusing you of this, since I haven't seen your explanation, but I would be interested to see the details.

11. Jul 7, 2009

### sylas

Sure. I'll give you quite an old one, on a different forum, from 2005. I was still quite new to it all at that point. I might say a few things a bit differently now, but the guts of it remains.

The evcforum where this explanation appears is mainly about evolution/creationism, and so there is less general expertise in physics around, and it was pretty crucial to keep it simple. The context is a couple of genuine nutbars (buzsaw) speaking nonsense, and with little capacity to learn anything new, being corrected by someone else (Percy) who was a novice in cosmology and getting a few matters wrong, though with a good capacity for improving his understanding.

If you can point out things that are actually wrong, go right ahead! But note that I am aware then and now that it is an approximation, and that any explanation at this level is incomplete.

See: this post at the evcforum.

For example... I might be more cautious about now is the account of "stretching" a photon, but it does at least give the result, since the wavelength of a photon for co-moving observers is proportional to scale factor.

Cheers -- sylas

12. Jul 7, 2009

### Rymer

After reading all this I'm even more confused.

1) I don't see any role for General Relativity in the mechanism of the cosmological redshift
2) I do see a geometry one. Simply due to expansion -- for whatever reason.

Note, the scale factor is geometry -- not actually part of GR. The introduction of GR and related arguments seems to be the confusing point.

Also, note the relativistic (Special Relativity) Doppler relation has nothing to do with the emitter of the photon. It only addresses the relative velocity difference of source and observer. I do not see why some form of Doppler doesn't apply IF there is actually a velocity difference.

So is there a velocity difference? Seems that needs to be clarified.

13. Jul 7, 2009

### nutgeb

Wallace,

After setting aside all of the off-topic dialog between you and Sylas, the following quote seems to comprise your entire substantive critique of my analysis:

What 'uncontroversial cause' of the redshift are you referring to?

The redshift is proportional to the scale factor a(t), but it cannot be inferred that one thing 'causes' another simply because they occur together. Causation should be demonstrated through some compelling logic and preferably by ruling out reasonable alternatives.

The traditional explanation for the 'expanding space' paradigm is that an expanding hypersphere of geometry forces photons and wavecrests to separate. A simple analysis of the 'tethered galaxy' exercise flatly demonstrates that an expanding hypersphere model is incapable of forcing photons, wave crests or any other objects to physically separate unless they already were separating in the initial conditions. The expanding hypersphere does not act as a 'force'. So 'geometry' alone cannot be the cause of the cosmological redshift in any metric or coordinate system.

Last edited: Jul 7, 2009
14. Jul 7, 2009

### marcus

That's a thoughtful comment. I'll try to respond and hope the main parties to the discussion will too. GR is our model of dynamic geometry. It is our basic theory of geometry (how it interacts with matter, how it can be nearly Euclidean when there is not much matter and not Euclidean where there is higher density...)

So anything that involves dynamically changing geometry in in that sense "relativistic".

Furthermore the Friedman eqns describe a particular solution of the main GR equation. The Friedman et al metric (which contains the scalefactor) is a solution. The metric (which is what is used in cosmology) and the scale factor are part of GR, therefore.

About "velocity". Sylas has pointed out that it can be confusing to call the recession rate, the rate that a distance is increasing, by the word "velocity". Distance increase as given by Hubble law is different from ordinary motion in the sense that normal motion gets you somewhere. It brings you closer to objects in the direction of motion, as to a destination.
Uniform distance expansion is merely dynamic geometry, it does not get anybody anywhere.
======================

Sylas explained that you can, if you like, apply the Special Rel doppler formula, but IMO it is rather elaborate and the long way round. You posit an infinite sequence of fictitious observers between emitter and receiver. And for convenience let's imagine everybody is at rest relative to Background. (the earlier word for that is "comoving with the Hubble flow" but the simpler criterion is you see no CMB hotspot).
then there is a rigamarole of giving each observer a reference frame and doing an approximate special rel doppler calculation at each infinitesimal step along the way.

That application of doppler is mathematically possible but has never seemed very helpful to me. A straight application without the infinite chain of observers doesn't work.

On the other hand I have never liked thinking of space as a material, some kind of rubber, and pretending that gives a physical explanation. Thinking that way is not something one does (in my experience) it is something one occasionally gets condescendingly accused of doing by people who don't listen.

The way I cope with the Cosmo Redshift issue is I just accept that geometry is dynamic. That we have no right to assume distances won't change. If you take two observers both at rest relative to the microwave Background (or if you like relative to the ancient matter, the ancient fog that emitted the background ) you have no right to assume that the distance between them will not change. It will change. If they are far apart it will most likely increase. And that isn't motion, neither is going anywhere towards some destination. It is merely that geometry changes.

And then I say to myself, well Maxwell's equations are constructed within the context of geometry. Distances between events like two wavetroughs can change. Minkowski space is fictional, the real geometry that waves live in constantly evolving. Maxwell's eqns don't guarantee that the wavelength won't increase as the wave propagates. They only say that when the equations are set up in a rigid geometry. So the Redshift is no big deal for me. It is just how I gauge the changing geometry of the universe. This is a private attitude and just how I think---it would be folly to try to tell everybody to think the same way. Wavelengths are part of geometry. Wavelengths are distances.

I never say "space expanding" because I don't like to imagine space as a substance. I focus on distances, on geometry. If something has to be expanding then I guess it could be the CMB. Being at rest with respect to CMB means no doppler hotspot in your microwave sky. The distances which Hubble law says are increasing at a certain rate (like 1/130 or 1/140 of a percent per million years) are distances between observers which are at CMB rest. I guess at a semantic stretch you could think of those observers at rest as "points" in the CMB, figuratively speaking.

Last edited: Jul 7, 2009
15. Jul 7, 2009

### nutgeb

I think this is a simple terminology problem. I used the word 'emitter' to mean the same thing you mean by 'source'.

If you read my OP, you'll see that I said a classical Doppler effect does occur, due to the fact that the distance between the emitter and observer does increase in the interval between each photon emission. That's true whether or not comovers are considered to be 'stationary' or to have a true recession motion. The occurrence of classical Doppler effect is perfectly consistent with the 'expanding space' paradigm.

But my point was that this classical Doppler effect subsequently is exactly offset by the linear contraction of the photon train during its journey. In 'expanding space' terminology, this linear contraction occurs (within the context of this effect in isoloation) because the tail photon(s) have a higher proper velocity than the lead photons at any point in time. This result is predicted by the 'tethered galaxy' exercise.

The complete picture is more complex however, because there is another effect (frame-crossing in the Hubble flow) which tends to cause a linear expansion of the photon train during its journey. The cosmological redshift results from the combination of these countervailing effects.

I think the mechanism is pretty straightforward, but there are several moving parts.

Last edited: Jul 7, 2009
16. Jul 7, 2009

### Rymer

Frankly, I don't see the difference when talking about light going from some A to B -- as far as converting a redshift into an apparent velocity of recession.

You seem to be saying that the perceived velocity isn't 'real'. It would seem to me that it is. Everything I have read indicates it DOES take the light longer to reach us -- so that the space between is actually getting larger. So in our 'super-duper' spaceship would we have to travel farther to get there?

What am I missing?

17. Jul 7, 2009

### marcus

Well I can't tell you how to think about evolving geometry. I can however ask you to come into my head and look at it my way, briefly. You don't have to accept it, just understand how I visualize things.

Check out the balloon sticky thread if you want. The analogy is the galaxies aren't moving because they don't change their latitude longitude. The photons (I assume you've watched the movie) are the wiggly things which travel from galaxy to galaxy always at a fixed rate like 1 mm per second.

If you havent watched the movie, google "wright balloon model". It takes about 3 minutes to watch. Then you will know pretty well how I think.

The galaxies do not have a velocity. None of them is going anywhere. In the simple model only the photons are moving.

So I would urge you (and if you see how I think you understand why) never to speak of a
"recession velocity". A velocity has a direction and at least provisionally a destination, none of the galaxies in the simple model have a velocity in that sense. None have a destination they are approaching. All you see there is a change in geometry. Separation between them increasing at a certain percentage rate. Do not attribute a velocity to a galaxy (I urge) because they don't have them. Each of its neighbors might think the galaxy is moving in a different direction, but none of them are moving the distances are just increasing. Watch the movie and you will see what I mean.

And the model does not suggest that space is made of rubber or some other material. It is simply a vehicle for picturing the geometry of a 2D surface. No physical explanation, just picture the surface of the balloon and visualize how distance relations are changing. Then carry that analogy up to 3D (where there is no balloon to help )

18. Jul 7, 2009

### sylas

What perceived velocity do you mean?

Suppose we observe a galaxy, with a redshift of z=2. You can use the Doppler formula to convert that to a velocity... but it is not the same as the rate of change of distance for most of the useful notions of distance we use in cosmology. You can, of course, define your co-ordinates in such a way that you get a velocity with whatever value you like... including the velocity with a Doppler shift of z=2. (which happens to be 80% light speed). Is that what you mean by "perceived velocity"?

It's not, however, the rate of change of "proper separation" distance, and it is certainly not the velocity used in the hubble relation. A redshift of z=2 arises if the photon was emitted at a time when the "scale factor" of the universe was 1/3; that is, everything was three times more dense than it is at present. The recession velocity of the galaxy as rate of change of proper distance with time, will depend on how the expansion rate has varied over time. In the current consensus model (dark energy plus dark matter), the recession velocity of that galaxy is "now" a little bit greater than the speed of light. At the time of emission (a bit over 10 billion years ago) the recession of that galaxy would have been less (due to acceleration) but still greater than the speed of light.

The recession velocity is really a rate of change of a chosen distance co-ordinate between particles that are so widely separated that you cannot really treat it like the velocity of a particle measured by a single observer.

The numbers I have given, by the way, are calculated using ΩM = 0.27 and ΩV = 0.73, with a flat universe, and a Hubble constant H0 of 71 km/s/Mparsec. In this model, proper recession velocity between emitter and receiver was about 5.5% more than c at emission, and about 12.4% more than c at reception, unless I stuffed up the calculations.

I suspect there is a bit of a problem thinking in terms of a "mechanism" for redshift. I'm with Marcus; it follows from geometry. There's nothing happening to the photon to force it to be redshifted, any more than anything special happens to photons to give a Doppler effect. A photon from deep space has a redshift because it is emitted from a galaxy that is not in your reference frame, where your reference frame is the nice flat Euclidean approximation that works locally for you. Whether you explain it by the increasing space between wavefronts, or by a recession velocity, or whatever, this is not really "mechanism"; it's just the transformation from the frame of the emitter to that of the observer.

Cheers -- sylas

PS. I wrote this before seeing Marcus' caution about "recession velocity". I can't help it... I do use the term. But it's not the same as a velocity of a particle that can be measured by a local observer. It is the rate of change of separation between two particles, using nominated co-ordinates. I use "proper time" and "proper distance" co-ordinates, which corresponds most closely (IMO) to what people think of as distance, and which gives the v term used in the Hubble relation for redshift.

Last edited: Jul 7, 2009
19. Jul 7, 2009

### Rymer

OK. Too much above to quote, so I'll just start.

1) I was NOT advocating using Doppler per se. I believe Doppler only works for local 'lab' geometry -- not cosmological. What is needed is a properly corrected version for cosmology.

2) However, I do believe this velocity is 'real' in a sense.

Simply saying -- for example that the universe expanded by some amount during the 'flight' of the photon and that this not real due to everything being expanded proportionally (which is what I'm getting for the explanations) is incorrect. If it were then we would be calculating the same size for the universe now that it was 13 billion years ago.

Separation is increasing. An those of us made of matter -- while expanding -- are not expanding as fast as 'space' itself. So at least some of the APPARENT recession velocity is real. ( I did use apparent). It is the real 'part' that causes the redshift our measurements 'see'.

20. Jul 7, 2009

### marcus

Thanks, amigo.

21. Jul 8, 2009

### sylas

I don't quite get what you intend by "real" or "not real", still less what you might mean by having a "part" of it being real.

The recession is certainly a real feature of the universe. Over large scales, the notions of distance and time become a little more ambiguous than what is defined for a nice simple Euclidean space... though the Euclidean approximation works perfectly well on small enough scales. Hence you can describe the same real phenomenon in different ways, or using different co-ordinates, and they are mutually consistent, with means to transform between different ways of giving co-ordinates.

None of this says it isn't "real", and it certainly does give scope for lots of errors by people who aren't sure of what they are doing. That includes me. I try to be cautious about straying into more detail than I can manage, and I don't always succeed.

I have no problem with the term "recession velocity", myself, but because it is not something that can be measured locally, you can trip up if you think about it in the same way as a velocity of a particle measured by a local observer. Recession velocities for galaxies we see in deep space, with redshifts of z >= 2 are greater than the speed of light... and that is really real. The "proper distance" between us and those galaxies really is increasing at more than c.

(The z value for a proper recession velocity of c is model dependent, but 2 is a nice round value with proper recession safely above c.)

Cheers -- sylas

22. Jul 9, 2009

### Rymer

Just used to emphasize that it can be more than 'apparent'. The degree to which this may or may not be true is dependent on the model used.

23. Jul 22, 2009

### nutgeb

Wallace, I've thought about what you said, and I think your critisism is misplaced. I think that the analytical components I described do combine to cause the cosmological redshift, and this can be demonstrated in any reasonable coordinate system.

As I said in the OP, the three components are:

1. An accumulation of classical Doppler redshifts as a photon crosses from each infinitesimal local frame in the Hubble flow to the next, all along its worldline. The photon must accelerate its proper velocity (relative to the observer) at each such frame-crossing in order to maintain a peculiar velocity of exactly c in each local frame. Integrating these accumulated Doppler shifts yields a result which happens to be equal to the expansion of the scale factor between emission and observation. My formula for this Doppler shift is equivalent to Peacock's equation 3.27.

2. A one-time classical Doppler redshift caused by the emitter's proper velocity away from the observer at the time of emission.

3. An accumulation of gravitational blueshifts as the photon is accelerated by the gravity of the sphere of cosmic matter centered on the observer (Gauss' law), in each successive local frame along the photon's worldline. This gravitational blueshift happens to exactly offset the classical Doppler redshift of component #2. (Interestingly, this gravitational blueshift is not accompanied by gravitational time dilation. I'll be happy to explain.)

If the emitter and observer both are fundamental comovers (each at rest in its local Hubble flow), then component #1 is the only cause of the cosmological redshift because components #2 and #3 exactly offset each other.

If either the emitter or observer has a nonzero peculiar velocity, then there is a 4th component:

4. A one-time SR Doppler redshift equal to the sum of the peculiar velocities of the emitter and observer. This component is then multiplied with component #1 to yield the final redshift.

These analytical components combine to cause the cosmological redshift. This can be demonstrated using either FRW comoving coordinates or proper distance coordinates.

Of course, when using proper distance coordinates, one cannot assume that the photon's proper velocity toward the observer is c; that assumption necessarily would lead to incorrect results. That is why neither the "standard" SR Doppler shift nor the gravitational shift, alone or in combination, will correctly predict the observables. At each point on its worldline, the photon's geodesic must be such that its proper velocity relative to the local Hubble flow (H * D as calculated from the observer's frame) is always exactly c. This is a very simple requirement.

One must also keep in mind that there is no "net" time dilation as between fundamental comovers in the FRW metric.
There are a variety of approximations that are close at z << 1. But they are all wrong at higher z's, because they are not the correct components of the cosmological redshift. Gauss' law alone or in combination with SR Doppler shift will not yield a calculation of the cosmological redshift that is correct at all z's. But of course you already agree with that.

Last edited: Jul 22, 2009
24. Jul 22, 2009

### Rymer

Generally I disagree with most the statement. Not because I doubt what you are saying about the interpretation given to cosmological redshift by GR. But your blanket statement about redshift and Doppler.

You are correct is saying that Doppler by itself does not perform well except in low z.

However, if a proper geometric correction is applied to the relation between the Doppler
recession velocity and scaled luminosity distance, a linear relation is found that so far extends through the recent supernovae data and on though the gamma ray burst data.

My point in mentioning this is not necessarily this model, but to warn that there are many
assumptions and outstanding issues still in question. Not a decided issue.

25. Jul 23, 2009

### Chronos

If we stick to what is observed, galaxies in the observable universe appear to have a doppler-like redshift. These distances also correspond nicely with those derived from Sn1a, Cepheid, Tully-Fisher and Faber-Jackson relationships, to name a few. If anyone is suggesting these distance indicators are also in dispute, I can only say you have a lot of work left to do before I will seriously consider any alternative theory. You cant decry redshift without explaining away all the other good observational evidence that favors it.

Last edited: Jul 23, 2009