The Mystery of Expanding Space: Uncovering the Truth Behind Dark Energy

[Peter Watkins]

Why "expanding space"?

When the "faster with distance" view was first discovered the natural assumption was that this movement was ballistic. As far as I am aware, this remained the case for most of the rest of the 20th century. Nowadays this movement is said to be due to the expansion of space, (which, as an aside, begs the question, "where is the need for dark energy"?). So what aspect of the the original "expansion from a single point of origin, via the energy imparted by the big bang," cannot be explained without the need for either/or expanding space and dark energy?

 

[Dmitry67]

Dark energy is needed to explain why expansion is accelerating, while it should be slowed down by gravity

 

[Chronos]

What makes you think the universe arose from a 'single point of origin'? It was not a 'point' in the manner you suggest.

 

[Peter Watkins]

Logic. If a missile passes by and heads away from your right, then even if you didn't see it coming, it would seem logical that if you could reverse time, that it would come back and head away from your left. Similarly, it would seem logical that the galaxies that Hubble saw moving away from us and each other would come together at some point if time, and hence their movement, were reversed. As for the apparent acceleration, the "raisin bread" analogy can explain that.
And as Thaddeus says in post# 64 of "all on one page", if the big bang were an everywhere at once event, wouldn't galaxies be headed every which-way?

 

[marcus]

When the "faster with distance" view was first discovered the natural assumption was that this movement was ballistic...
Nowadays this movement is said to be due to the expansion of space...

You are wildly misinformed, Peter. Ballistic was never the view of the scientific mainstream.

There were a small number on the fringe who tried to promote the "ballistic" interpretation. One name that comes to mind is Eddie Milne in 1935, a kind of contrarian or renegade. His ballistic view was not taken seriously by the professional majority.

The ballistic view gives a bad mathematical fit, for many reasons, one being that so much of the distance expansion we see goes on at rates far exceeding the speed of light. Verbally it may sound good to you but quantitatively the explosion picture works only in a vague kinda sort of way.

From it's beginning in 1915, General Relativity has been understood as a theory of dynamic geometry---of distances that change in other words.

As early as 1923 it was shown that GR applied to cosmology called for either a pattern of increasing distances, or the reverse: a pattern of shrinking distances.

No ballistics, Peter. That is only what you get in popularized accounts for the general public--and the journalists who put things in oversimplified terms like that do the public a disservice.

GR is our theory of gravity and its predictions have been shown to be exquisitely accurate. However it says you have no right to expect distances to remain constant unless locked to some physical system like a piece of metal, or rock, or a planetary orbits. Unless you have a model of gravity which can rival the precision of GR, and which does not have expanding largescale distance as corollary, you have to accept dynamic geometry, as scientists of our grandparents' and great-grandparents' generation already did back in the 1920s.

============

Btw I only respond to where you misrepresent the scientific consensus all thru the 20th century. We shouldn't mislead people about what the astronomers back then thought.

But if you personally want to believe in an explosion model, that's fine.

 

[marcus]

...
And as Thaddeus says in post# 64 of "all on one page", if the big bang were an everywhere at once event, wouldn't galaxies be headed every which-way?

In case anyone wants to respond, here is Thaddeus post #64 from the other thread:

If it is to be asserted that the Big-Bang was not of "point" origin then how is it justified in terms of -everything- expanding away from other items .. as though it were simply an outward expansion .

IF as positioned earlier the bigbang is not to be seen as a point radiation but as a whole universe instantaneous? radiation then stuff should be flying in all directions equally .. yes or no ?

And just because the claim is that there is no point origin of the big bang .. how can it be asserted logically that this means there is no center point to the universe ?

Maybe it would make more sense as a hypothesis that matter is shrinking lol .. sometimes feels that way mumble mumble .. .. :)

I don't think we've heard from Thaddeus for a while.

 

[Peter Watkins]

Hello Marcus. I'm sure you already know this, but Hubble went to his grave not believing Einstein's notions on space curvature etc.. Re Chronos on #2; the einstein-online. info that you yourself regularly promote, states that a reversal of time would see a gathering of all matter at a single point. With regard to a "bounce", it will not happen. This little "altered state" universe is a one off, single use, throwaway universe.

 

[marcus]

Hello Marcus. I'm sure you already know this, but Hubble went to his grave not believing Einstein's notions on space curvature etc...

No indeed! I didn't already know this! You have taught me an interesting bit of history, if true. I wish you had a link to an online source for that Hubble story.

I'd rather not have to scroll thru pages of stuff searching so if you can give me a link and say how many paragraphs down the page I'd like to see where you think Einstein Online says everything in the universe was collected into one dense blob. They may have actually said everything in the currently observable portion of the universe. Or they may have qualified it in some other way. I'd need to see what they actually said, in context.

 

[Ich]

Hi marcus,

The ballistic view gives a bad mathematical fit, for many reasons, one being that so much of the distance expansion we see goes on at rates far exceeding the speed of light. Verbally it may sound good to you but quantitatively the explosion picture works only in a vague kinda sort of way.

I tend to disagree. Since https://www.physicsforums.com/showpost.php?p=1366649&postcount=24" set me on the right track (you witnessed), I learned to distrust the "Davis/Lineweaver-approach".
The Milne Model is just one coordinate transformation away from an empty expanding universe, and so are superluminal speeds from subluminal. Expanding space means increasing distances - nothing else -, and there is always a region of space, or a local coordinate system, where it is sensible to speak of increasing distances as velocity. It's just a matter of coordinates.
Of course, our universe is not empty, and the Milne model is an artificial setup like the ether, but it presents a different point of view - at least locally - that helps to demystify some coordinate-dependent statements that tend to mislead struggling students like me. One of these is the notion of motion through space as opposed to motion of space. Motion through space is not a well defined concept, it contradicts the principle of relativity, and in fact makes sense only in an (admittedly somehow "preferred") coordinate basis.
It's my point of view that we can break down "cosmological mechanics" to the familiar concepts of ballistic motion and gravitational influence on a local scale, and that GR allows the extrapolation that this picture is valid at every point in space, mostly because it gets rid of the concept of gravitational (proper) acceleration.
Sorry, this post seems to be somewhat incoherent, but as promised I will try to explain my thoughts in the "Balloon Analogy" thread.

 

[russ_watters]

Logic. If a missile passes by and heads away from your right, then even if you didn't see it coming, it would seem logical that if you could reverse time, that it would come back and head away from your left. Similarly, it would seem logical that the galaxies that Hubble saw moving away from us and each other would come together at some point if time, and hence their movement, were reversed. As for the apparent acceleration, the "raisin bread" analogy can explain that.
And as Thaddeus says in post# 64 of "all on one page", if the big bang were an everywhere at once event, wouldn't galaxies be headed every which-way?

Everything in that explanation is fine, but it doesn't necessarily imply what you claimed it does. What you are missing is just that since every point in the universe was at that single point of origin, after the big bang, every point remains a usable origin for the expansion. The rasin bread analogy is just an analogy and is not meant to show an edge or center. You are misusing it.

Your problem here is your understanding of the geometry...in addition to a simple refusal to consider that your view could be wrong: You keep asking the same questions over and over. You aren't going to get different answers.

 

[marcus]

Hi marcus,
I tend to disagree. Since https://www.physicsforums.com/showpost.php?p=1366649&postcount=24" set me on the right track (you witnessed), I learned to distrust the "Davis/Lineweaver-approach".

Thanks for the link, Ich. I'll go back and have a look at your conversation with Old Smuggler.

For me, what anchors me whenever we have a controversy about this at PF is the CMB.
The bath of radiation from the evenly dispersed matter in the early universe provides a criterion of rest.

If some galaxy really were moving at 0.999 c then its people would be roasting from a Doppler hotspot in the CMB sky.

Coordinate transformations that depart from universal rest seem contrived and unintuitive to me.

That and the fact that the Hubble law and Friedman model use my kind of coordinates and my kind of distance.

I'll have to see what Old Smuggler could have said to so Milnerize you. I know you're smart so there must be something persuasive in it.

 

[Ich]

I'll have to see what Old Smuggler could have said to so Milnerize you.

No, it's not that bad. I'm fully aware of the advantages of a global coordinate system, and I'd never say that "private space"-coordinates are the only ones. Nor do I think of the Milne model as more than a useful toy model.
But I believe that the description in comoving coordinates somehow got reified among many cosmologists and especially in public outreach, giving rise to some misconceptions and inappropriate mysticism.
Now, I start sounding like a crank. I better show what I mean in a separate post.

 

[marcus]

No, it's not that bad. I'm fully aware of the advantages of a global coordinate system, and I'd never say that "private space"-coordinates are the only ones. Nor do I think of the Milne model as more than a useful toy model.
But I believe that the description in comoving coordinates somehow got reified among many cosmologists and especially in public outreach, giving rise to some misconceptions and inappropriate mysticism.
Now, I start sounding like a crank. I better show what I mean in a separate post.

No! you are certainly not sounding like a crank.
The best paper I know in line with what you say is Bunn and Hogg (2008).
I will get the link.
They show that cosmo redshift can (and they think for mathematical rigor's sake ought to) be considered as the cumulative limit of a very large number of small Dopplershift steps.
I think there is no question about this, and that the paper is quite solid and well-intentioned.

They see the main danger that people will re-ify space and think of it as a rubber sheet and believe that the wavelengths are actually stretched by being in this physical material which is being stretched. So they defend against what they see as the main danger.

What I see as the worst obstacle to understanding that people come here to PF with is something different. Newcomers think of the cosmo redshift naively as a one-time (or perhaps two-time) Dopplershift involving one or two definite velocities----and they picture a cosmology as an explosion. In my opinion this misconception is more primitive and deep-rooted than the rubbersheet reification (imaginary materialization) and more of an obstacle.

In fact the excellent Bunn Hogg paper shows that cosmo redshift is not any simple kind of Doppler. It can be analyzed mathematically as an infinite series of small "epsilon" Dopplers. The effect depends on the whole expansion history while the light is in transit.

And if one should make a mental picture of that long series of small Dopplers, it comes to much the same as the stretching rubber, except that there is no rubber.

So everybody has their own different pedagogical problems, and needs different tracts, pamphlets, and sermons
Bunn Hogg is a sermon preached from the other side, but it is a good one which actually supports the conventional view as well, that the way to treat redshift is simply as

1 + z = a(now)/a(then)

the ratio of the Friedmann model scale-factor.

http://arxiv.org/abs/0808.1081
The kinematic origin of the cosmological redshift
Emory F. Bunn, David W. Hogg
14 pages; Am. J. Phys.
(Submitted on 7 Aug 2008)
"A common belief among cosmologists is that the cosmological redshift cannot be properly viewed as a Doppler shift (that is, as evidence for a recession velocity), but must instead be viewed in terms of the stretching of space. We argue that the most natural interpretation of the redshift is in fact as a Doppler shift, or rather as the accumulation of many infinitesimal Doppler shifts..."

I haven't followed all the discussion about this topic. It's possible you have already seen this paper, but if not I think you would like it.

 

[tickle_monste]

If the universe is expanding, what is it expanding relative to? Couldn't one say there's an equally accurate perspective from which one would see something other than an expanding universe? If the universe had a boundary that increased in size over time, THEN one could say that the universe is actually expanding. But if the universe had a boundary that was expanding, it would be expanding relative to its own center, and if you were to be floating right along next to the boundary as it expanded, you would be able to look at the center and see everything getting smaller. As I understand it, the universe has no boundary, and no center, so this hypothetical scenario doesn't work, and all motion can still stay relative. But I just don't see how the universe could be called "expanding" without having an expanding boundary and a center relative to which it can be called "expanding". As an aside, the way I see it, cooling is generally linked to contraction in size. A cool galaxy should seemingly occupy much less space than it would were all its energy just "free".

 

[Wallace]

The problem with this whole discussion is that both in the public domain (such as these forums) and the scienitific literature, much of the noise is made by the least informed. People can convince each other of all kinds of non-sense but unfortunately it doesn't make it any more right.

Bunn and Hogg seems like a reasonable paper, I think anyone who reads and understands that should be satisified and no longer feel the need to endlessly debate this uninteresting topic. The problem is that every few years someone comes along and re-mangles the ideas all over again (see for instance several papers by Ambramowicz et al in recent years) that requires someone else coming along to set the story straight again.

The only thing I would tell anyone about the expansion of space is that it is a metaphor. Not physics. If you like, you can debate the usefulness of the metaphor, but the problem that comes up over and over again is that people start debating and motion over the surface of balloons and through rising bread and think they are discussing physics. They are not. It is simply not possible to assess the truth or otherwise of the Big Bang by thinking and talking about expanding space. Unfortunately this is what Peter Watkins and others are trying to do. It simply won't work because the expansion of space is a flawed metaphor, as all metaphors must be at some level. You can break the metaphor very easily but that does not mean you have broken the physics. You need to understand the physics to do that, and to that there is no shortcut.

The science behind modern cosmology makes no reference to the ideas that are most frequently discussed in forums such as these as if they were fundamental pillars. I'm not sure what the solution is, but it's a sorry state of affairs.

 

[marcus]

Bunn and Hogg seems like a reasonable paper,..

I am glad you think so, Wallace. I spotted it when it first came out and thought it was excellent. That is, mathematically. The interpretive part where they preach a little sermon subsequent to their mathematics----well if you find the controversy over different people's spins tiresome, I would heartily agree!

 

[Peter Watkins]

Marcus, einstein-online.info cosmology/spotlight on relativity/tale of two bangs, para.3.

 

[marcus]

Marcus, einstein-online.info cosmology/spotlight on relativity/tale of two bangs, para.3.

Thanks Peter! It's so nice to have a pinpoint reference and not have to scan stuff.
(Old eyes.)

Yes. They are describing not the whole universe being compressed down to a small volume (in the classical 1915 theory a zero volume but we think the classical theory does not apply).
They are talking about only the part we now see being compressed to a small volume.

In conventional mainstream cosmology one does not assume that a singularity existed, only that is what you get if you push the 1915 theory beyond its limits. So people are busy replacing the old theory with a model that doesn't break down.

And in conventional cosmology one does not assume the singularity is small volume, it can be infinite volume. (This is the commonest case to use for purposes of analysis.)

Your earlier language suggested you were thinking of the whole universe---all matter and all space---compressed to a very small volume.

We can't say that, the whole universe might be infinite and therefore the imagined singularity would be.

Einstein Online does not say that the whole of space and the matter it contains was compressed down into a point in our space. That would be contradictory. It would mean there was a place in space that you could point your finger at which was where everything came from. That would be crazy

Here is what the paragraph actually says. We need to read it carefully. If they use any confusing or misleading language we should write E-O and suggest a correction!

"If we simply follow the predictions of Einstein's theory of general relativity for the evolution of a simple expanding, homogeneous universe filled with matter and radiation, then our journey into the past will eventually come to an end - a point in time where we cannot go back any further. At this moment, all the galaxies that we see around us today were compressed into a region of zero volume - to a single point in space. Since density is defined as mass divided by volume, the density was infinite. In Einstein's theory, matter influences the way that the geometry of space and time is distorted, and at this moment of infinite matter density, the curvature of spacetime was infinite, as well. Within the simple cosmological models based on general relativity, there is no possibility to go to any earlier times than this. Such a boundary of time (or, more generally, of spacetime) is called a singularity."

It doesn't say compressed to a zero volume point in OUR space. But all this zero volume infinite density stuff is nonsense anyway. What the paragraph mainly does is lay out what is obviously wrong with the vintage 1915 theory.

The important stuff is what they say later:
"Did the big bang really happen? If you are talkinng about...the hot early universe as described by well-known physical theories ... then there is good evidence that,...

[BUT!]
Whether or not there really was a big bang singularity is a totally different question. Most cosmologists would be very surprised if it turned out that our universe really did have an infinitely dense, infinitely hot, infinitely curved beginning. Commonly, the fact that a model predicts infinite values for some physical quantity indicates that the model is too simple and fails to include some crucial aspect of the real world..."

 

[Peter Watkins]

Re old eyes; me too. By "ballistic" what I mean is an initial shove by whatever means and thereafter the matter is prey to any gravitational influence that it encounters, or carries with itself. As you well know gravity can pull from the front causing acceleration, which in the case of a long line of matter, would cause spreading. Drag from the rear would also cause spreading and, when viewed from the rear, would give the impression of acceleration in the direction away from the viewer and the gravity source.
Unlike the apparent majority of you contributors, I don't think that the universe is infinite, far from it. So the faster with distance view would never produce a faster than light problem, so the need for expanding space does not exist.
Also; Heat. Is it always transmitted via radiant motion, does this produce a wavelength that increases or decreases with cooling? And can I assume that straight, non wave motion cannot transmit heat?

 

[Peter Watkins]

At last, there is something on which we can agree, your statement that "all this zero volume infinite density stuff is nonsense". Thank you for your time. I would describe a gravitational/ballistic model, but I'm not allowed to on this forum, plus it's more than a few sentences.

 

[nutgeb]

Bunn and Hogg seems like a reasonable paper, I think anyone who reads and understands that should be satisified and no longer feel the need to endlessly debate this uninteresting topic.

I understand and sympathize with Bunn & Hogg's assertion that cosmological redshift is in some manner (which the article does not reduce to a mathematical formula) related to Doppler redshift. I also agree that the rubber sheet model of expanding space is fundamentally limited and cannot yield a meaningful explanation of cosmological redshift.

But I think Bunn & Hogg miss a crucial point. In a homogeneous, isotropic gravitational universe, the Friedmann equations dictate that the clocks of fiducial observers who are privileged to exactly experience the Hubble flow themselves (i.e., with 0 peculiar velocity with respect to each other) are all perfectly in synch with each other. The clocks of nearby privileged observers are synchronized, and so are the clocks of very distant privileged observers. Since the SR redshift formula incorporates a clock differential between the emitter and the observer (SR time dilation), it follows axiomatically that SR time dilation cannot be an element of the cosmological redshift observed by privileged observers. Nor can cosmological redshift be an accumulation of a large number of tiny SR time dilations along the light path.

SR redshift without any SR time dilation is simply classical Doppler redshift. Yet we know that cosmological redshift is not equal to (nor does it approximate) classical Doppler redshift at cosmological distances, even when computed as Bunn & Hogg prescribe, using the velocity of the galaxy at the time of light emission relative to the observer at the present time.

 

[marcus]

(which the article does not reduce to a mathematical formula) related to Doppler redshift...

That is just the point, nutgeb. It is not possible to reduce redshift to a simple Doppler formula, or any simple formula based on Doppler. That is because the redshift is the cumulative result of the entire expansion history.

The only simple formula is the conventional one you learn in class
1+z = a(now)/a(then)

This is exact, and intuitive, and simple. 1+z is just the ratio of the Friedman metric scalefactor now to what it was when the light started out.

If you took Bunn Hogg's infinite chain of observers with overlap patches and calculated all the relative speeds and the Dopplers and all that, then in the end what you would get is
1+z = a(now)/a(then)

This is why students are taught not to treat cosmo redshift as a Doppler shift and to treat it as exactly paralleling the percentage change in distances.

 

[nutgeb]

This is why students are taught not to treat cosmo redshift as a Doppler shift and to treat it as exactly paralleling the percentage change in distances.

I agree with your statement, but maybe I'm missing your point. Earlier in this thread you seemed to endorse the explanation given in the Bunn & Hogg paper. But the central point of that paper is that cosmological redshift is an accumulation of SR Doppler redshifts along the light path. I'm not arguing with you, just trying to understand whether Bunn & Hogg have missed a crucial point.

As Bunn & Hogg says:

"Since the Doppler Family is the by far the most natural family to work with, it is natural to interpret the cosmological redshift as a Doppler shift, and it is curious, to say the least, that this interpretation is generally frowned on."

"The common belief that the cosmological redshift can 'only' be explained in terms of the stretching of space is based on conflating the properties of a specific coordinate system with properties of space itself. This is precisely the opposite of the correct frame of mind in which to understand relativity."

 

[marcus]

I agree with your statement, but maybe I'm missing your point. Earlier in this thread you seemed to endorse the explanation given in the Bunn & Hogg paper. But the central point of that paper is that cosmological redshift is an accumulation of SR Doppler redshifts along the light path. I'm not arguing with you, just trying to understand whether Bunn & Hogg have missed a crucial point.

I know I must sound inconsistent.
1. I think Bunn Hogg are good mathematicians and they do a correct analysis.

2. What their analysis shows is that there is no simple doppler formula for redshift, and the conventional handle (not the doppler) is the practical way to get a grip on it.

3. Therefore they put the wrong pedagogical interpretation on their result. Their math is correct but they put the wrong spin.

It is a strawman to say people think of space as rubber. Of course it's not substance. We only use substance as a crutch to aid in imagining changing geometry.
Expansion is not a material expansion, it is a pattern of increasing largescale distances-- changing geometry, not rubber or bread-dough.
But after their analysis, Bunn and Hogg preach a sermon to the already converted: that people should not re-ify space and think of it as rubber.
OK, we know this already.

The important thing is that if newbies come to PF and start off thinking of redshift as doppler they will get into endless misconceptions, because they will think of it as the doppler of the recession speed. So then you have the explosion picture all over, and the paradox of distances increasing faster than light (which the Hubble law says distances to most galaxies). It can lead to a big mess. I've seen it happen

Instead what I find works better is to give beginners the usual formula you get in class.
1+z = a(now)/a(then)
which is simple and correct.
And get them started using Ned Wright's calculator, if they're willing.
Wright's calculator essentially embodies the LCDM model. So you get hands-on experience with the standard cosmological model.

 

[nutgeb]

I know I must sound inconsistent.
1. I think Bunn Hogg are good mathematicians and they do a correct analysis.

2. What their analysis shows is that there is no simple doppler formula for redshift, and the conventional handle (not the doppler) is the practical way to get a grip on it.

3. Therefore they put the wrong pedagogical interpretation on their result. Their math is correct but they put the wrong spin.

I understand your broader message about pedagogicality and not causing confusion. I agree there is no simple Doppler formula for cosmological redshift, but I read Bunn & Hogg to say that there is a specific mathematical approach to solving it, which involves parallel transport of the velocity four-vector along the light path by integrating a large series of tiny SR Doppler redshifts. That's what their Fig. 3 illustrates. They say:

"Imagine many comoving observers stationed along the line from the observed galaxy to the observer. Each observer has a local reference frame in which special relativity can be taken to apply, and the observers are close enough together that each one lies in within the local frame of his neighbor. Observer number 1, who is located near the original galaxy, measures its speed v1 relative to him and gives this information to observer 2. Observer 2 measures the speed u of observer 1 relative to him, adds this to the speed of the galaxy relative to observer 1 using the usual special-relativistic formula, [equation 5] and interprets the result as the speed of the galaxy relative to him. He passes this information on to the next observer, who follows the same procedure, as does each subsequent observer. At each stage, the velocity of the original galaxy relative to the observer will match the redshift of the galaxy in accordance with equation (4)." [Equation 4 is the SR Doppler redshift formula.]

I think their mathematical approach (above) is wrong. As I explained in my first post, relativistic Doppler redshift incorporates an element of SR time dilation. But SR time dilation is not possible as between privileged emitters and observers who exactly comove with the local Hubble flow. Without SR time dilation, SR Doppler redshift is nothing but classical Doppler redshift. We know definitely that the latter by itself is not the solution to cosmological redshift.

Do you agree with my assessment of their approach?

 

[marcus]

"Imagine many comoving observers stationed along the line from the observed galaxy to the observer. Each observer has a local reference frame in which special relativity can be taken to apply, and the observers are close enough together that each one lies in within the local frame of his neighbor. Observer number 1, who is located near the original galaxy, measures its speed v1 relative to him and gives this information to observer 2. Observer 2 measures the speed u of observer 1 relative to him, adds this to the speed of the galaxy relative to observer 1 using the usual special-relativistic formula, [equation 5] and interprets the result as the speed of the galaxy relative to him. He passes this information on to the next observer, who follows the same procedure, as does each subsequent observer. At each stage, the velocity of the original galaxy relative to the observer will match the redshift of the galaxy in accordance with equation (4)." [Equation 4 is the SR Doppler redshift formula.]

I think their mathematical approach (above) is wrong. As I explained in my first post...

Do you agree with my assessment of their approach?

Nutgeb, you are obviously sophisticated and I think you would benefit from discussing this sort of thing in the General Rel forum if you can get a thread started there. Challenge the experts there, put it as a mathematical or theory question.

I'll give you my two bits. I don't agree with your assessment. And I think Bunn Hogg analysis is correct. But that's just my view.

Since adjacent observers are comoving, they see the distance between them to be increasing. So they each measure a definite nonzero velocity that the other has.
So just like Bunn Hogg says, they keep adding on the speed of their neighbor as they go along the chain. It makes totally good sense to me, so much that I hardly see the point of discussing it. But you may get more discussion from other people, which I hope you do.

 

[nutgeb]

I'll give you my two bits. I don't agree with your assessment. And I think Bunn Hogg analysis is correct. But that's just my view.

Since adjacent observers are comoving, they see the distance between them to be increasing. So they each measure a definite nonzero velocity that the other has.
So just like Bunn Hogg says, they keep adding on the speed of their neighbor as they go along the chain.

I want to make sure we're not talking past each other. I agree with you that each observer measures a definite nonzero velocity that the other has, and they keep adding the speed of their neighbor as they go. That describes an accumulation of a series of Doppler shifts.

Where I think Bunn & Hogg go wrong is in stating that SR Doppler redshift occurs as between each successive observer. Since as I said the Friedmann equations dictate that the clocks of all such comoving observers are synchronized together at all times, it is impossible for any SR time dilation to occur as between any of them, individually or in aggregate. Therefore any redshift resulting from velocity accumulation must be limited to classical Doppler redshift alone.

I don't disagree that a Doppler accumulation occurs - I disagree only that it is SR Doppler.

 

[marcus]

... That describes an accumulation of a series of Doppler shifts...

I don't think so. It describes a cumulative series of velocities. I don't think they need bother to calculate the galaxy's Doppler shift along the way. So talking about it as you do seems irrelevant. I may be missing something but I think the whole point is to come up with a figure for the galaxy's velocity, that we back here at Earth can use. It's an SR velocity addition game. This endresult velocity is somewhat artificial and will have contributions from the entire history of expansion. It is not anybody's instantaneous.

But we don't have to worry about this multitude of observers calculating Doppler shifts. I think. Or?

I have to go out. Hopefully you will get responses from other folks.

BTW anybody please correct me if I am wrong, but I picture the Bunn Hogg velocity information passing along the chain of observers with approximately the speed of light.

So that the artificially constructed speed of the galaxy (cumulative result of a huge number of calculations) arrives as information at the same time as the redshifted light from the galaxy.

This is beautiful in a way because it means that the speed construct that we get, at the end of the chain, takes into account the entire expansion history of the universe that occurred while the light was in transit.

So it is only fitting that this complicated and subtle Bunn Hogg process would in the end give the same answer as the simple non-Doppler formula
1+z = a(now)/a(then)

Because the ratio of size now to size then is precisely what summarizes the expansion history that occurred while the light was in transit.

Please correct me if I am missing some essential feature, but I like the Bunn Hogg exposition because it shows why it is pedagogically such a depressing idea to present redshift as a Doppler effect. Elaborate rigamarole is necessary and in the end one gets the simple non-Doppler result.

Not manifestly or explicitly Doppler, tho as they show it can be analyzed by a Doppler chain.

Anyway, nothing new in Bunn Hogg. A kind of sexy alternative re-working of what we already know.

 

[nutgeb]

I may be missing something but I think the whole point is to come up with a figure for the galaxy's velocity, that we back here at Earth can use. It's an SR velocity addition game.

I agree with your description, except that I think it is wrong to add SR velocities in this scenario. I think the velocities must be added non-relativistically because comoving observers by definition cannot experience SR time dilation as between each other.

Thank you for your lengthy responses Marcus. I'll wait to see if others join the discussion.

 

[chronon]

I think it is wrong to add SR velocities in this scenario. I think the velocities must be added non-relativistically because comoving observers by definition cannot experience SR time dilation as between each other.

Why not? Central to GR is that it is locally like SR. Hence two sufficiently close comoving observers can also be thought of as two observers moving apart in minkowski space.

 

[Wallace]

Bunn & Hogg get the right answer numerically. If you work through there method you will see that as Chronon suggests, they use the equivalence principle to join an infinite series of Minkowski frames together that are space at infinitesimal distances. They in effect re-derive the effects that nutgeb is insisting need to be tacked on again. It's clever but not original. Bunn and Hogg can clearly write well but are not such good readers of the literature.

 

[Ich]

Hi nutgeb,

comoving observer's times are definitely not synchronized via the standard method. Cosmological time is nothing but the proper time of each comoving observer.
But if you regard only nearby observers, you can ignore relativistic corrections and use the classical doppler effect instead.

SR redshift without any SR time dilation is simply classical Doppler redshift. Yet we know that cosmological redshift is not equal to (nor does it approximate) classical Doppler redshift at cosmological distances, even when computed as Bunn & Hogg prescribe, using the velocity of the galaxy at the time of light emission relative to the observer at the present time.

I think you misunderstand the method. Total redshift is calculated as the sum of infinitely many small classical redshifts. The velocity of the galaxy is not used.
If you're interested in the galaxie's velocity, bear in mind that in this procedure you sum over infinitely many small Lorentz boosts. The result is a single boost not with argument \int dv, but with argument \tanh \int dv: you can add rapidities, not velocities.
That is a crucial point, the difference between a coordinate velocity and an "actual" one.

marcus, I disagree with you when you say:

This endresult velocity is somewhat artificial and will have contributions from the entire history of expansion. It is not anybody's instantaneous.

It's true that this velocity is not anybody's instantaneous, but that's a stregth of the definition, not a weakness. As simultaneity is relative, a coordinate dependent concept, any definition relying on it is bound to be artificial. The velocity defined through parallel transport along the connecting path (or, as Bunn and Hogg advertise, through addition of infinitesimal redshifts) is the only one that is not dependent on the coordinate system you use, therefore I'd say it's the natural one.

The important thing is that if newbies come to PF and start off thinking of redshift as doppler they will get into endless misconceptions, because they will think of it as the doppler of the recession speed.

that is true, but it's not a problem of the explanation as a doppler shift. The problem arises when one speaks (and thinks) of d (a \chi )/dt, without further specification, as a velocity. It's a coordinate velocity, and these things do whatever they want and mustn't be thought of as "the velocity". I've seen misconceptions concerning this point even in standard papers.

Where I agree with you: Bunn and Hogg's method is as usless as it is correct for all practical purposes. They hide everything interesting in the details of a fully relativistic calculation (which they explicitly do not perform), so the pedacogical value is strongly limited to showing the possibility of different coordinate approaches.

 

[nutgeb]

comoving observer's times are definitely not synchronized via the standard method. Cosmological time is nothing but the proper time of each comoving observer.

Ich, I'm not sure what your point is with that statement. "Cosmological time" is the name for the single universal clock rate which by the Friedmann definition is the proper time of every privileged comoving observer and is identical for all of them. Let's not get diverted by the question of whether or not those comoving observers have physically synchronized their local (arbitrary) timekeeping formats. Regardless, they must agree in principle that all observed comoving events in the universe have the same duration and occur simultaneously, when corrected for light travel distance from the comoving emitter to each comoving observer (and correcting for any intervening inhomogeneities).

I think you misunderstand the method. Total redshift is calculated as the sum of infinitely many small classical redshifts.

But that's just what I said: Any methodology for integrating an infinite series of infinitesimal local redshifts should use only the classical Doppler redshift, not SR Doppler redshift. Bunn & Hogg are wrong to suggest the latter. It's not just a matter of tiny discrepencies which are insignificant locally: when an infinite number of those tiny discrepencies are integrated over the full light path, the SR Doppler formula will yield significant errors. The amount of error derives entirely from the inclusion of the element of SR time dilation, the occurence of which is fundamentally inconsistent with the fact that the comoving emitters and observers all share the same cosmological time.

 

[nutgeb]

Bunn & Hogg get the right answer numerically. If you work through there method you will see that as Chronon suggests, they use the equivalence principle to join an infinite series of Minkowski frames together that are space at infinitesimal distances. They in effect re-derive the effects that nutgeb is insisting need to be tacked on again.

Are you saying that their mathematical approach first derives an (incorrect) interim solution by integrating SR Doppler redshifts (which includes within it an integration of the local SR time dilations), and then corrects for that error in a subsequent 2nd calculation step which removes the SR time dilation component? I don't see that in their math, but even if that's what they are doing, it would be simpler to integrate only the classical Doppler redshifts in the first place. Then it is a 1-step approach, avoiding the need to tack on a 2nd step.

 

[nutgeb]

Central to GR is that it is locally like SR. Hence two sufficiently close comoving observers can also be thought of as two observers moving apart in minkowski space.

I agree broadly with that statement Chronon, but taken to the extreme it is circular. Nearby comoving observers are "sufficiently close" to apply SR accurately only when they are so close that the SR time dilation effect is so small that it can't be measured by available instruments. If the resolution of the instruments is improved, introducing an SR time dilation element would result in detectible error in the measurement. Of course this error can be eliminated by selecting a different set of comoving observers who are even closer together, ad infinitum.

At nearly infinitely small distances, the erroneous SR time dilation component is nearly infinitely small, which is acceptable for any single measurement. But if the individual observations of a nearly infinitely large number of adjacent observers are integrated mathematically, those nearly infinitely small errors will accumulate into one large, significant error.

I think this point is clear if you focus on this specific question: How can there be any element of accumulated SR time dilation as between a comoving emitter and a distant comoving observer, if both of them experience exactly the same duration of elapsed cosmological time for the event?

Also, keep in mind that there is no SR time dilation effect built into the Friedmann equations. For example, the instantaneous Hubble rate is defined to be exactly proportional to proper distance, without applying any time dilation adjustment in the calculation.

 

[Old Smuggler]

Why not? Central to GR is that it is locally like SR.

True, but..

Hence two sufficiently close comoving observers can also be thought of as two observers moving apart in minkowski space.

This does not follow. Recall that the set of comoving observers taken at some instant of cosmic time makes up a 3D spatial hypersurface. Two neighbouring comoving observers can be thought of as two observers moving apart in Minkowski space-time only if this 3D hypersurface coincides with the counterpart 3D hypersurface in the tangent space-time in the limit when the distance between the neighbours goes to zero. This works well for open FRW models but not for closed (or flat) FRW models.

This means that for closed (or flat) FRW models, the cosmic redshift should be thought of as an effect coming from the curvature of space-time, with no "kinematic" contribution (somewhat similarly to the observed spectral shift observed between two hovering observers at different heights in Schwarzschild space-time). For open FRW models the cosmic redshift should be thought of as an effect coming from both curvature and "kinematic" contributions, but such that for small enough distances, the "kinematic" effect will always dominate. A discussion of this including calculations can be found in the newsgroup sci.astro.research about November 2004.

 

[Ich]

Hi again, Old Smuggler.

This works well for open FRW models but not for closed (or flat) FRW models.

This works well in all FRW spacetimes, as long as you look at first order effects only. Locally, spacetime is flat and expansion is the same as motion.

This means that for closed (or flat) FRW models, the cosmic redshift should be thought of as an effect coming from the curvature of space-time, with no "kinematic" contribution (somewhat similarly to the observed spectral shift observed between two hovering observers at different heights in Schwarzschild space-time).

Gravitation is second order in distance. Bunn and Hogg explicitly exclude second order effects.

Hi nutgeb,

At nearly infinitely small distances, the erroneous SR time dilation component is nearly infinitely small, which is acceptable for any single measurement. But if the individual observations of a nearly infinitely large number of adjacent observers are integrated mathematically, those nearly infinitely small errors will accumulate into one large, significant error.

No, the error vanishes in this limit. But I have a correction to my earlier post: It's not a sum of classical redshifts, but a combination, i.e. a multiplication. If we denote rapidity with w, the effect is not
f=f_0 \, (1+w), but
f=f_0 \, e^w.

Regardless, they must agree in principle that all observed comoving events in the universe have the same duration and occur simultaneously, when corrected for light travel distance from the comoving emitter to each comoving observer (and correcting for any intervening inhomogeneities).

No. Events don't have a duration, and when they correct for light travel distance, they use the standard method and thus will not agree. That is the point I am talking about: cosmological coordinates are not Minkowski coordinates and if you treat them as Minkowski coordinates, you will lose the way.

 

[nutgeb]

No, the error vanishes in this limit.

I agree that at the extreme limit, the accumulated error of including SR time dilation in the computation goes to zero. Zero X infinity = zero.

But that just brings us full circle on the analysis of Bunn & Hogg. When the difference between using SR Doppler and classic Doppler goes to zero as accumulated SR time dilation goes to zero, then Bunn & Hogg's approach becomes the same as if we had just used an accumulation of classical Doppler redshifts in the first place. And since we know that the accumulation of classical redshifts alone over the full light path does not yield a correct number for the cosmological redshift, Bunn & Hogg's mathematical approach must be fundamentally invalid.

Added by edit: I expect that at the extreme limit, the accumulated classical Doppler shift would also go to zero, because the discrete Doppler shift measured by each adjacent observer becomes infintesimal and approaches zero. This doesn't seem helpful to generating a useful outcome. If the accumulated SR and classical Doppler shift both approach zero regardless of how large the total proper distance is, the redshift calculation always has a 0 result which clearly is invalid.

No. Events don't have a duration, and when they correct for light travel distance, they use the standard method and thus will not agree.

I used the term "duration" to mean the duration of a discrete process, such as the duration of a supernova as viewed by two distantly separated observers. I don't understand your statement that because the two comoving observers will use the "standard method" to correct for light travel distance they will disagree on the duration of a comoving supernova process. Both comoving observers' clocks are running identical cosmological time. And they each will measure light travel distance from the single distant emitter to themselves by using the standard cosmological redshift formula based on the expansion of the scale factor. Why will that cause them to disagree about the light travel distance and/or the duration of the supernova process?

 

[Chronos]

What 'cosmological time' are you referring to nutgeb?

 

[nutgeb]

What 'cosmological time' are you referring to nutgeb?

Here's the quick and dirty Wikipedia definition. I'll look around for a more explanatory reference.

"Cosmic time (also known as "time since the big bang") is the time coordinate commonly used in the Big Bang models of physical cosmology. It is defined for homogeneous, expanding universes as follows: Choose a time coordinate so that the universe has the same density everywhere at each moment in time (the fact that this is possible means that the universe is, by definition, homogeneous). Measure the passage of time using clocks moving with the Hubble flow. Choose the big bang singularity as the origin of the time coordinate.

Cosmic time is the standard time coordinate for specifying the Friedmann-Lemaître-Robertson-Walker solutions of Einstein's equations."

 

[marcus]

Yes I think Wikipedia is basically right. It is a common accessory to doing cosmology. The Friedman equations model is based on it. Hubble law assumes it. When you say the comoving distance now is proportional to the rate it is increasing now, all that is meaningful because you have a standard way to slice, a standard concept of simultaneity that everybody uses.

I wouldn't compete with Wiki but I think of cosmological time as the time told by the temperature of the CMB. Two observers who both measure the same CMB temp (as long as one is not significantly farther down in some gravity well, as long as they are roughly on the same level so to speak) are contemporaries. Pun.

 

[nutgeb]

I agree (despite the blushworthy pun) that any privileged comoving observers will measure the same CMB temp. Without any dipole.

Here's another description of cosmological time, from Cosmology Physics by John Peacock (1999) p.67:

"COSMOLOGICAL TIME The first point to note is that something suspiciously like a universal time exists in an isotropic universe. Consider a set of observers in different locations, all of whom are at rest with respect to the matter in their vicinity (these characters are usually termed "fundamental observers"). We can envisage them as each sitting on a different galaxy, and so receding from each other with the general expansion (although real galaxies have in addition random velocities of order 100 km/s and so are not strictly fundamental observers). We can define a global time coordinate t, which is the time measured by the clocks of these observers - i.e. t is the proper time measured by an observer at rest with respect to the local matter distribution. The coordinate is useful globally rather than locally because the clocks can be synchronized by the exchange of light signals between observers, who agree to set their clocks to a standard time when e.g. the universal homogeneous density reaches some given value. Using this time coordinate plus isotropy, we already have enough information to conclude that the metric must take the following form [omitted]. Here, we have used the equivalence principle to say that the proper time interval between two distant events would look locally like special relativity to a fundamental observer on the spot: for them c²dt² = c²dt² - dx² - dy² - dz² . Since we use the same time coordinate as they do, our only difficulty is in the spatial part of the metric: relating their dx etc. to spatial coordinates centred on us."

 

[Chronos]

That is exactly right, nutgeb. I think there may be some constants lurking in the background, but, I agree with you [and Peacock] in principle.

 

[Old Smuggler]

This works well in all FRW spacetimes, as long as you look at first order effects only. Locally, spacetime is flat and expansion is the same as motion.

You have been seriously mislead by a really bad paper (see below).

There is no way to foliate Minkowski space-time into flat hypersurfaces such that their evolution
describes an isotropic expansion. The only way to foliate (a subset of) Minkowski space-time to describe an isotropic expansion is via hypersurfaces with negative curvature; this is the empty FRW (Milne) model. (Minkowski space-time cannot be foliated into hypersurfaces with positive curvature.)

By a suitable (constant) scaling of the time coordinate in the Milne model, it is possible to set up an
initial-value problem such that the metric of space and the first derivative of the spatial metric
with respect to time (essentially the extrinsic curvature) are identical (at some initial hypersurface)
for the Milne model and any open FRW model. The differences in evolution then appear only via
the second derivative of the spatial metric with respect to time (essentially space-time curvature).
This shows that the expansion for open FRW models can be said to be "kinematic" for small
enough times/distances.

For a flat FRW model no such correspondence exists (since flat hypersurfaces are geodesically
embedded into Minkowski space-time, their extrinsic curvature vanishes, i.e., the first derivative of the spatial geometry with respect to time vanishes. No scaling of the time coordinate can change that). This means that the expansion described in a flat FRW model has none resemblance whatsoever to any counterpart in Minkowski space-time. Thus for a flat (and closed) FRW model, the expansion is purely a curvature effect.

Gravitation is second order in distance. Bunn and Hogg explicitly exclude second order effects.

That paper is built on a simple misunderstanding. They notice that the components of the Riemann curvature tensor is small enough to be neglected. This is true, but the curvature of space-time do not influence spectral shift calculations via the Riemann tensor, but via its effect on the connection
coefficients. This effect is of first order in general, and cannot be neglected.

Since the Bunn & Hogg paper is built on a false premise, their conclusions are wrong and misleading. IMO this is a crackpot paper of the worst kind; it contains a lot of bogus claims and very little sound science. Hopefully it will not pass peer review. However, it would seem that merely by
posting this paper on the preprint server, Bunn & Hogg have made laughing stock of themselves.

 

[Ich]

And since we know that the accumulation of classical redshifts alone over the full light path does not yield a correct number

Why do you think so? Of course it yields a correct number, it couldn't be otherwise. Why should the result be different if you check it at infinitely many points?

And they each will measure light travel distance from the single distant emitter to themselves by using the standard cosmological redshift formula based on the expansion of the scale factor.

Ah, I see where we came apart. The "standard method" to get the light travel distance is to measure it, not to infer it from redshift. Whatever, the result is that you divide the redshifted duration by the classical doppler effect, which leaves time dilation.

There is no way to foliate Minkowski space-time into flat hypersurfaces such that their evolution describes an isotropic expansion.

I don't need to find a global foliation - or the other way round, the standard foliation will do because it deviates from the local "tangent space" (flat spacetime approximation) only in second order. IOW, if I take any comoving observers 4-velocity as time and create an orthonormal basis, expansion looks like motion.
I don't know why we disagree here, basically Bunn and Hogg say that spacetime looks locally flat, and that is simply true. The deviations are of second order, one higher than the velocity - expansion equivalence.

They notice that the components of the Riemann curvature tensor is small enough to be neglected. This is true, but the curvature of space-time do not influence spectral shift calculations via the Riemann tensor, but via its effect on the connection
coefficients. This effect is of first order in general, and cannot be neglected.

Their procedure is essentially a parallel transport of the wave vector. They don't neglect the effect of the nontrivial connection, it is implicitly incorporated in the shift from one flat frame to the next - which they don't specify.
That's one of the point I'd criticize, if you want to get quantitative results from their approach, you're thrown back on "general spacetime geometry", which means that you have to calculate everything the tedious way where this explanation as doppler shift doesn't helt at all.
Another point is that they deny the importance of gravitational redshift. If you include second-order terms, you get a quantitatively useful approximation, and gravitational redshift comes into play. Of course it was there all the time, it's just that "spacetime curvature which we don't have to care about" now gets its name and can be included as a perturbation.
But that paper is far from crackpottery, Bunn and Hogg just have an onesided agenda which they try to push.

 

[Old Smuggler]

I don't need to find a global foliation - or the other way round, the standard foliation will do because it deviates from the local "tangent space" (flat spacetime approximation) only in second order. IOW, if I take any comoving observers 4-velocity as time and create an orthonormal basis, expansion looks like motion.

You are confused. When discussing space expansion in context of the FRW models, these models do not come as featureless 4D manifolds, but as manifolds foliated into a specific set of hypersurfaces. The evolution of these hypersurfaces in terms of a parameter (time) describes the expansion. Equivalently, one may view space-time as threaded by a specific family of observers moving orthogonally to the hyper-surfaces. Other foliations of the FRW models or the evolution of other families of observers with time are simply irrelevant when discussing space expansion in context of the FRW models. In particular, any model involving Minkowski space-time foliated into flat hyper-surfaces and
describing the "expansion" by means of receding test particles (such that their velocities mimic the Hubble Law as seen by a particular observer), has nothing to do with the FRW models whatsoever.

This means that no part of space expansion (for small distances/times) in a closed or flat FRW model is captured in each tangent space-time - in other words, that for these cases, the expansion cannot reasonably be interpreted as motion in flat space-time, not even locally.

I don't know why we disagree here, basically Bunn and Hogg say that spacetime looks locally flat, and that is simply true. The deviations are of second order, one higher than the velocity - expansion equivalence.

The important deviations are in the connection coefficients - not negligible in general.

Their procedure is essentially a parallel transport of the wave vector. They don't neglect the effect of the nontrivial connection, it is implicitly incorporated in the shift from one flat frame to the next - which they don't specify.

Consider any flat FRW model (use cartesian space coordinates). The non-zero connection coefficients all contain the first derivative of the scale factor with respect to time. Now take the corresponding connection coefficients of the tangent space-time (same foliation). Here all connection coefficients vanish. It should be clear that for this case, by parallel transporting vectors along a null curve, no contribution at all to the spectral shift comes from each flat frame. Trying to avoid this fact by claiming that "the effect of the non-trivial connection coefficients are incorporated in the shift from one flat frame to the next" is just mumbo-jumbo when it is not shown how this yields the same quantitative result as parallel transport with the full set of connection coefficients coming from curved space-time. Anyway, in absence of space-time curvature, there would be no need to shift between flat frames.

That's one of the point I'd criticize, if you want to get quantitative results from their approach, you're thrown back on "general spacetime geometry", which means that you have to calculate everything the tedious way where this explanation as doppler shift doesn't helt at all.
Another point is that they deny the importance of gravitational redshift. If you include second-order terms, you get a quantitatively useful approximation, and gravitational redshift comes into play. Of course it was there all the time, it's just that "spacetime curvature which we don't have to care about" now gets its name and can be included as a perturbation.
But that paper is far from crackpottery, Bunn and Hogg just have an onesided agenda which they try to push.

This paper has succeeded in utterly confusing the issue and fooling (at least one of) its readers.
That is the most dangerous form of crackpottery.

 

[Peter Watkins]

Marcus, and others, thank you for your time. Post #18 seems to have resolved the multiverse question! It states that the "einstein online" reference to all being compressed refers only to that part of the universe that we can see. The clear implication being that there is more that we can't see. If there were other expansions like ours, whether hundreds, thousands or billions, ie. the big bang occurring everywhere, presumably we would never see them due to the expansion of the space between us and them? This in turn means that we only have to think about our own little universe and forget questions about infinity. I assume that our omni-directional expansion would have produced a more or less spherical universlet. Whilst we are in the centre of our sphere of vision, it is doubtful that we are at the centre of our universe. This means that we could only be positioned between the centre and the outer "edge". Is it known approximately where, along this radial line, we are positioned?

 

[nutgeb]

Why do you think so? Of course it yields a correct number, it couldn't be otherwise. Why should the result be different if you check it at infinitely many points?

Ich, I'm not sure your comment relates to what I said, which is that equation for calculating classical Doppler shift does not even remotely yield a correct approximation of cosmological redshift, whether or not one uses the Bunn & Hogg approach of comparing the emitter's recession velocity at emission time with the observer's recession velocity now.

For example, here are a few selected cosmological redshifts, compared with the corresponding emitter velocity at emission and observer velocity now in units of c (as calculated with the Wright and Morgan cosmic calculators):

============================================

Cosmological Redshift (z+1): ...4...32...256

Emitter V @ emission: ...26.58c...8.13c...1.62c

Observer V now:.....3.26c...2.76c...1.53c

Calculated Doppler shift: ...7.7...13.32...23.97

============================================

Using the classical Doppler equation:

\frac{\lambda_{now}}{\lambda_{emit}} = \frac{c + V_{em}}{c-V_{now}}

the calculated classical Doppler shift is larger than the cosmological redshift at low z values and many times smaller than it at high z values. There's just no way to manipulate recession velocities to yield a classical Doppler shift even remotely as large as the cosmological redshift at (z+1) = 256.

The example of calculated Doppler shift in the table above divides both recession velocities by 2, assuming the rough approximation that the emitter and source respectively would be moving away from an imaginary midpoint between them at half of their relative total velocities.

 

[Chronos]

I agree with Old Smuggler on that point, nutgeb. Your math is inconsistent.

 

[Ich]

Other foliations of the FRW models or the evolution of other families of observers with time are simply irrelevant when discussing space expansion in context of the FRW models.

Sorry, I don't follow. You may use whatver coordinates you like, and if the authors choose to use local standard inertial frames, that's perfectly legitimate. And since we're discussing this paper, this approach is anything but irrelevant.

In particular, any model involving Minkowski space-time foliated into flat hyper-surfaces and describing the "expansion" by means of receding test particles (such that their velocities mimic the Hubble Law as seen by a particular observer), has nothing to do with the FRW models whatsoever.

This specific "model" is nothing but a differet coordinate representation of a specific (the empty) FRW solution. Nothing wrong with it.

This means that no part of space expansion (for small distances/times) in a closed or flat FRW model is captured in each tangent space-time - in other words, that for these cases, the expansion cannot reasonably be interpreted as motion in flat space-time, not even locally.

I don't know how you come to this conclusion. If we ignore second order effect, any spacetime can locally (and for a short time) be described as flat minkowski space with moving particles in it. That has nothing to do with space curvature of the original foliation, that's second order and irrelevant.
Hey, for 70 years, nobody knew wheter space is flat or positively or negatively curved. This is irrelevant for nearby redshift observations, we see galaxies moving away from us, and that's it. It's irritating that you seem to deny this fact, maybe I misunderstood you. When you say "locally", don't you mean also "for a short time"?

Trying to avoid this fact by claiming that "the effect of the non-trivial connection coefficients are incorporated in the shift from one flat frame to the next" is just mumbo-jumbo when it is not shown how this yields the same quantitative result as parallel transport with the full set of connection coefficients coming from curved space-time.

We both agree that parallel transporting the emitter velocity to a nearby absorber along a null curve gives the correct SR doppler shift. Actually, you teached me that.
We both agree that on small scales, for short time, there is a standard inertial frame that covers any smooth spacetime and is accurate to firat order.
We both agree that parallel transport along arbitrary paths leaves a vector unchanged (again, to first order).
Which means that, in this frame, the emitter has some definite velocity relative to the observer, and that this velocity gives the correct SR doppler shift. The classical doppler will do also, because we're ignoring second order effects.

Anyway, in absence of space-time curvature, there would be no need to shift between flat frames.

Of course you have to boost from one frame to the next, if you use Bunn and Hogg's procedure, where the local observers are at rest in the respective inertial frame. Those small dv 's add up to the accurate rapidity.

This paper has succeeded in utterly confusing the issue and fooling (at least one of) its readers.

Agreed, but until now you haven't convinced me that I am this reader.