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

[Ich]

I suspect you are mixing up definitions of H. In conventional GR based cosmology, H can be defined as (da/dt)/a where a is the scale factor (wrt to present) and t is a proper time co-ordinate. H~1/t is the nice simple solution you get for an empty universe using GR.

With all respect, I suspect that you're missing a crucial point: The empty expanding universe is a valid FRW solution, with \Omega_{\Lambda} = \Omega_M = 0, and it's only 3 sigma away from LCDM. Empty spacetime is flat. SR can handle a flat spacetime, you simply have to use a different set of coordinates. Predicted observations, such as redshift of test particles, are independent of the choice of coordinates.
Maybe you want to read what http://www.astro.ucla.edu/~wright/cosmo_02.htm#MD" has to say, or you want to convince yourself.
Start with FRW coordinates (a(T)=T, T0=age of the universe)
ds^2=dT^2-T^2dr^2
and apply the transformations
<br /> T = \sqrt{t^2 - x^2}<br />
<br /> r = T_0 \tanh^{-1}(x/t)<br />
You'll get
ds^2=dt^2-dx^2
and you can perform the necessary calculations (redshift, luminositiy distance, angular size distance...) purely in SR.

It's perfectly normal to think they've done something wrong; and this is precisely because they tackle popular and entrenched misconceptions. If you think that they have made a mistake, you are probably in a good position to be learning something.

Ha, that's what I'm telling crackpots all along.
Pease understand that I'm not trying to sell a pet theory of mine. Davis&Lineweavers' analysis contradicts http://books.google.de/books?id=e-w...universe"&as_brr=3&ei=ta7USYHbHYGuzATAmuTeAg", you can convince yourself if you're familiar with th idea of a metric, you can read what other authorities in the field have to say. Or you can take the fact that even Old Smuggler, who disagrees generally with everything I say, agrees with me as evidence with the status of a mathematical proof.
Really, I'm not doing original research here, that chapter is simply wrong.

 

[sylas]

With all respect, I suspect that you're missing a crucial point: The empty expanding universe is a valid FRW solution, with \Omega_{\Lambda} = \Omega_M = 0, and it's only 3 sigma away from LCDM. Empty spacetime is flat. SR can handle a flat spacetime, you simply have to use a different set of coordinates. Predicted observations, such as redshift of test particles, are independent of the choice of coordinates.

That is not what is meant by an SR model. I do know about the FRW solutions.

The SR model described in Davis and Lineweaver is the model obtained by taking redshift as due to motions in a simple non-expanding space, and calculated as Doppler shift.

That's DIFFERENT from the FRW solution with an empty universe.

There's no error in the Davis and Lineweaver paper on this point, because they are quite clear on what they mean by SR model. It's not just taking an FRW solution and applying SR. It's taking redshift as being a Doppler shift in non-expanding space.

The luminosity distance with z arising from Doppler shifts for particles receding with at uniform velocity from a common origin event is different from that in the empty FRW model.

Cheers -- Sylas

 

[Wallace]

I hope I can contribute here. I think you (sylas and ich) are both basically right.

The empty FRW universe is indeed only 'ruled out' at 3 sigma, but as sylas suggests this is not the model D&L mean by saying 'SR model', they are referring to a particular assumption, valid at low redshift, that gives a bogus result at high redshift.

The point that leads to disagreement is actually a bit subtle. In post #61 ich makes a conformal tranformation between the empty FRW metric and a Minkowski like metric. This is all well and good, however this is only valid radially. If you put the angular terms back into the first line you will see that your transformation does not return a fully conformally Minkowski metric. This means that you cannot use this to determine either the angular diameter or luminosities distances. You need to do a more complex fully conformal transformation in order to do this.

Some technical details of this can be found http://adsabs.harvard.edu/abs/2007MNRAS.381L..50L".

I*think* that the error in the SR model the D&L discuss is that if you work through the details, you can see that that way we define distance in the SR model violates simultaneity, which is why it is okay for small distances but gets worse and worse the further you go.

So yes, a *correct* SR model is identical to an empty FRW universe and to work out the relationship between the FRW co-ordinates and the co-ordinates of this model you need to do the fully conformal transfomation, but D&L are talking about a model that, due to the misidentification of the meaning of co-ordinates, is only a low redshift approximation.

I hope that helps!

 

[Ich]

The SR model described in Davis and Lineweaver is the model obtained by taking redshift as due to motions in a simple non-expanding space, and calculated as Doppler shift.

No, the SR model they use is, frankly, BS. Read this:

However, since SR does not provide a technique for incorporating
acceleration into our calculations for the expansion of the Universe, the best we
can do is assume that the recession velocity, and thus Hubble’s constant, are approximately
the same at the time of emission as they are now6.

A non accelerating universe has \dot a = const., thus H=1/t. Constant H is the de Sitter universe, nothing to do with SR, and the extreme case of an accelerating universe.

There's no error in the Davis and Lineweaver paper on this point, because they are quite clear on what they mean by SR model. It's not just taking an FRW solution and applying SR. It's taking redshift as being a Doppler shift in non-expanding space.

Please try to understand my point: in the empty universe, the only difference between "expanding space" and "constant velocity in non-expanding space" is a coordinate transformation. That's nothing more than just taking an FRW solution and applying SR.

The luminosity distance with z arising from Doppler shifts for particles receding with at uniform velocity from a common origin event is different from that in the empty FRW model.

No, if "uniform velocity" means what it should for particles receding from a common origin.
Now this is your claim, please back it up with calculations. You are probably in a good position to be learning something.

 

[Wallace]

A non accelerating universe has \dot a = const., thus H=1/t. Constant H is the de Sitter universe, nothing to do with SR, and the extreme case of an accelerating universe.

Hmm, good point. I guess the best we can say then is that D&L introduce a very bad model and then demonstrate that it doesn't fit the data. I'm not sure that they intended it to be a 'correct' model in the sense of it correctly using relativity (SR and GR are of course identical if the universe is empty), I think they were trying to show that a mugs 'SR' model doesn't work, but maybe it was a bit too muggy.

I really think the points of agreement are much more than those of disagreement here, stemming from maybe some loose terminology. I think we can all agree that the 23 sigma model from D&L is not a 'correct' SR model. The disagreement appears to be just how incorrect it is, yes?

 

[Ich]

In post #61 ich makes a conformal tranformation between the empty FRW metric and a Minkowski like metric.

Stop, no! I just match coordinates locally to first order, and drop all the higher order terms. It's neither valid radially nor in the transverse directions, if you're looking at higher orders.
What I'm doing here is an exact coordinate transformation. The angular directions (hyperbolic to flat space) transform correctly, no need to bend the laws of physics. We're talking about a flat spacetime in both cases.

This means that you cannot use this to determine either the angular diameter or luminosities distances.

Of course you can. The hyperbolic space in FRW coordinates stems solely from the definition of the radial coordinate as being measured by comoving observers. If you "fix" that, everything is ok again.

but D&L are talking about a model that, due to the misidentification of the meaning of co-ordinates, is only a low redshift approximation.

Yes, they talk about the wrong model and therefore come to wrong conclusions. I think this is most clearly seen in the passage I quoted before, where they identify "not accelerated" with "constant H", which is bogus.

 

[Wallace]

Alright, I don't want to introduce additional disagreement. As you say, minkowski space and an empty FRW metric are both flat space-times (they have a vanishing Ricci scalar). You can transform between these two co-ordinate systems, without being forced to be vaild only to a given order, via a fully conformal transformation.

Of course you can. The hyperbolic space in FRW coordinates stems solely from the definition of the radial coordinate as being measured by comoving observers. If you "fix" that, everything is ok again.

Right, this 'fixing' is exactly what the transformation does.

I get what you are saying, any co-ordinate transformation is exact, so if your original space-time is flat the transformed one is as well. Just pointing out that the one you suggest doesn't work, on it's own to relate FRW co-moving co-ordinates to their Minkowski counterparts. Clearly you agree with this point, it just wasn't clear to me what you were demonstrating with it original, but now I see.

 

[Ich]

I guess the best we can say then is that D&L introduce a very bad model and then demonstrate that it doesn't fit the data.

Yes, it's a strawman.

I'm not sure that they intended it to be a 'correct' model

They say "the best we can do", so I'd say that they simply didn't know better. I'm convinced that this section would look quite different if they'd write it today.

The disagreement appears to be just how incorrect it is, yes?

Of course. D&L claim incorrectly that the "Doppler/SR interpretation" is ruled out by 23 sigma by SNIa observations alone, I (we) say it's ruled out by ~3 sigma. Taking other observations into account, I think we're rather getting back to 23 sigma.

 

[Old Smuggler]

For the benefit of readers who may possibly fall for the misunderstandings Ich seems to be promoting,
I will contribute with one last post in this discussion.

No. I approximated the scalar function a(t) by its tangent at the point of interest, therefore there are no first order deviations. That's not an assumption, that's basic calculus. All deviations are of second order in cosmological time, therefore at most second order also in private time and private space.

Your "linearisation procedure" is, with a suitable choice of constants, equivalent to expanding a(t) as a truncated Taylor series around some arbitrary time t_0. That is, you set a(t) = a(t_0) + \dot a(t_0)(t-t_0) and neglect higher order terms. But then you assume that no space-time curvature effects are included since the series is terminated after the linear term. This is not necessarily true since curvature effects
may be included into \dot a(t_0) (as is the case, in general). This simple misunderstanding may be appropriately called "the Bunn/Hogg fallacy", and you have endorsed it.

So what? It can be done locally, and that's what we are talking about. More to the point, I actually showed how it is done locally, so unless you're objecting to specific points in the transformation, there's no use telling me that curved space is not globally flat. I know this.

The problem is that you do not do what you think you do. Take again the FRW model with flat space
sections. The non-zero connection coefficients are proportional to \dot a, as usual. But here, since
we cannot perform any relevant coordinate transformation in order to change the connection coefficients
(the coordinates already have the standard form), the correct flat space-time approximation is to neglect \dot a altogether. On the other hand, for a non-empty, open FRW model where
the line element is expressed in comoving coordinates, a coordinate transformation to standard
coordinates will change the connection coefficients, but not get rid of them altogether. What is left
should be due to curvature and must be neglected in the correct flat space-time approximation. It
is only for the empty FRW model a coordinate transformation from comoving to standard coordinates
can completely get rid of all the connection coefficients.

On the other hand, approximating a(t) as a Taylor series to first order the way you do, is effectively to include all the crucial effects of the connection coefficients (expressed in comoving coordinates) at the time t_0, since the relevant connection coefficients expressed in comoving coordinates are always proportional to \dot a. After making a local transformation to standard coordinates, the resulting non-zero velocity field is then just an expression of the fact that the connection coefficients (expressed in comoving coordinates) at the time t_0, are proportional to \dot a(t_0). You have absolutely no guarantee that these connection coefficients do not include some effects of curvature so that this procedure yields the correct flat space-time approximation for the issue we were discussing. In fact, it fails.

Wow, the line element is different, but the coordinates are the same. Now that's interesting.

You think so? Of course you can keep the coordinate system and change the metric as long as
the coordinate system covers the relevant part of the manifold. That is basic differential geometry.
You should try to learn it some time.

And what does "flat FRW model" mean? The empty one? One with flat space?

I have consistently used "flat FRW model' to mean the FRW model with flat space sections.

That concludes all I have to say in this discussion. You are on your own now. Good luck.

 

[sylas]

No, if "uniform velocity" means what it should for particles receding from a common origin.
Now this is your claim, please back it up with calculations. You are probably in a good position to be learning something.

There's no question about that! I'm sure most of you guys here know more than I do about GR, and metrics and tensors. I learn a lot by trying to work through these kinds of problems.

In any case, I'll go away and try my own analysis, and report back.

Cheers -- Sylas

 

[sylas]

In any case, I'll go away and try my own analysis, and report back.

Cheers -- Sylas

OK; I have now done this more thoroughly for myself as you suggest. You're right; and I was wrong. In fact, the luminosity distance in the SR case is the same as obtained in the FRW model with an empty universe, and the SR model used in section 4.2 of Davis and Lineweaver has no sensible correspondence to anything. It is, as you point out, nonsense.

I'm not an expert in GR; I can solve the differential equations for scale factor and energy density which are used in the FRW models; but I can't derive the equations themselves. In any case, I didn't need any of that, because the issue is simply the SR model.

The SR model corresponds to a realistic situation that could, in principle, be set up and tested right now, and SR is the appropriate way to analyze it.

Take a large collection of particles, and at a point in time, have them all start moving at constant velocity from a common point. (An explosion in space.) After elapsed time t, an observer on one of the particles makes observations of all the others.

Consider a signal received by one exploding particle from another, and compare with the signal from another equivalent particle at the same distance, but with no velocity difference. The signal received from the moving particle is weaker by a characteristic amount. The factors to consider are

• Redshift. Each photon arrives with less energy, by a factor (1+z).
• Time between photons. The time between successive photons is increased by precisely the same factor as the distance between wave crests. Think of a radiator sending out pulses of radiation, according to an onboard clock. The individual photons are redshifted. The frequency at which pulses of radiation arrives is reduced also, by the same factor. This reduces the energy by another (1+z).
• Angular size of the radiating surface. This is unchanged. There is no Lorentz contraction perpendicular to the direction of motion, so the stationary particle and the moving particle subtend the same angle at the same distance.

Hence, the signal received from the moving particle is weaker than a signal from the stationary particle at the same distance, by a factor (1+z)^2. Equivalently, the angular distance is less than the luminosity distance by this factor.

But that is precisely the relation for all the FRW models, empty or otherwise. Davis and Lineweaver, in their section 4.2, used a factor of (1+z) for the so-called SR model, which can only be seen as an error. There are still differences in comparing z with the apparent magnitude across the different FRW solutions, but the ratio of angular distance and luminosity distance is the same for everything.

Using Ned's formulae for the empty universe, I get the angular distance as follows:
D_A = \frac{c}{H_0}(1-(1+z)^{-2})/2

Using Lorentz transformations for the SR model I have described here, and using H₀ as the inverse of time since the explosion, which makes sense, I get the same thing. Hence the SR model gives the same relation between z and luminosity distance as the empty FRW solution.

Thanks very much. I have learned something indeed.

Cheers -- Sylas

 

[Ich]

Take again the FRW model with flat space
sections. The non-zero connection coefficients are proportional to LaTeX Code: \\dot a , as usual. But here, since
we cannot perform any relevant coordinate transformation in order to change the connection coefficients
(the coordinates already have the standard form), the correct flat space-time approximation is to neglect LaTeX Code: \\dot a altogether.

Sorry, that's nonsense. I need a coordinate transform that is accurate to first order only, and this is always possible. You start with coordinates where ds²=t'²-a²dr² -where parallel transport changes coordinate velocity - and transform to coordinates where ds²=t²-dx², where there is a definite notion of velocity. You simply have to make sure that the transformation is exact to first order, and you get the exact velocity field to first order. It simply does not matter whether space was flat before and is treated as flat (but is actually curved) after the transformation. That's second order.
In the next paragraph, you seem to concede this point, but then write:

You have absolutely no guarantee that these connection coefficients do not include some effects of curvature so that this procedure yields the correct flat space-time approximation for the issue we were discussing. In fact, it fails.

Now, this gets kind of boring - for the umptieth time you make assertions, without a single line of maths. Especially as the case is quite clear here, curvature is by definition second order, so it can't change the first order accuracy of a result.
I showed you how to get the first order result, and you've done nothing to show where, explicitly, the procedure fails in you view. I appreciate your general, well-meaning, and repeatedly uttered advice that I better learn basic principles of mathematics and physics, and I will certainly continue to do so with the help of this forum, but this discussion seems to lead nowhere.

You didnt'd really believe that you'd have the last word, did you?

 

[Ich]

Hi sylas,

OK; I have now done this more thoroughly for myself as you suggest.

Hey, that's great. Not many people would take the time to get wound up in a specific problem, but that's the most rewarding thing you can do in physics.
I see that you're quite skilled in the art, so I'm looking forward to learning from you. in the future.

 

[Dmitry67]

Just wanted to confirm: even in the particular case where space is flat, spacetime is not flat as it is expanding, right?

 

[Ich]

Just wanted to confirm: even in the particular case where space is flat, spacetime is not flat as it is expanding, right?

Right. If spacetime were flat, space would have negative curvature in expanding coordinates. Energy density gives positive curvature, and at a certain density space is flat even in expanding coordinates. But now time "runs in a different direction" at each point, and spacetime must be curved to make this combination possible.

 

[nutgeb]

OK, I've read some more and thought some more about this.

I think we all agree that cosmological redshift includes no accumulation of SR time dilation, when considered in cosmological time coordinates. And I see no explanatory benefit in translating to global SR time coordinates in a hypothetical "empty" universe, as an alternative coordinate system, because isotropy and homogeneity require a distinctly hyperbolic (negative) spatial curvature in SR coordinates, which is inconsistent with actual observations.

So I next want to explore Ich's assertion that cosmological redshift is nothing but an accumulation of classical Doppler shifts.

Time dilation of the interval between two events (such as the beginning and end of an emitted light wave packet) is an inherent and commonly accepted outcome of applying the RW line equation. As Longair says, distant galaxies are observed at an earlier cosmic time when a(t) < 1 and so phenomena are observed to take longer in our frame of reference than they do in that of the source.

I don't understand what physical action would cause an accumulation of incremental classical Doppler shifts to occur locally all along the light path, while also causing an accumulation of incremental elongations of the entire wave packet (photon stream) as it will eventually be observed in our observer frame of reference. The only purely kinematic cause I can see for such an elongation would be an ongoing acceleration of the wave packet (relative to our frame of reference). In that case, the leading edge of the wave packet would progressively "pull further ahead" of the trailing edge, because the leading edge experiences each successive temporal increment of acceleration before the trailing edge does.

If such an ongoing acceleration is a real physical phenomenon, mustn't it be caused by the same cosmic gravitational spacetime curvature that causes gravitational blueshift (when the observer is considered to be at the center of the coordinate system)? I can't see any other kinematic explanation for ongoing incremental acceleration. However, an accumulation of gravitational blueshifts along the entire light path ought to reduce the total amount of cosmological redshift, as compared to a global classical Doppler shift calculation. But this is not what we observe. At high z's, the cosmological redshift is dramatically larger than the classical Doppler shift when calculated on a global basis. Thus gravitational blueshift seems to cut in the opposite direction it needs to.

 

[Ich]

I think we all agree that cosmological redshift includes no accumulation of SR time dilation, when considered in cosmological time coordinates.

It's a bit more complicated. In general spacetimes, there is no exact definition of the relative velocity of two observers at different positions.
For a measurement of redshift, both observers are connected by a unique path, the path that the light ray actually took.
You can transport the wave vector along this path, and see that it got redshifted on arrival.
Or, alternatively, you can compare the four velocities of the two observers by transporting the velocity vector of the emitter to the absorber. If you apply the SR doppler effect (including time dilatation) to this velocity, you get the same result. Both approaches always work.
In the special case of a FRW spacetime, you can skip the procedures and get the result by simply comparing the scale factors at both events. The underlying symmetries make sure that it works. Cosmological coordinates reflect these symmetries, that's why they are so useful for this kind of calculation.
But that does not mean that the other approaches, one of which including SR doppler and time dilatation, are no longer valid. You are still free to interpret the result as you like, and there is an exact mathematical framework for these different interpretations.

And I see no explanatory benefit in translating to global SR time coordinates in a hypothetical "empty" universe, as an alternative coordinate system, because isotropy and homogeneity require a distinctly hyperbolic (negative) spatial curvature in SR coordinates, which is inconsistent with actual observations.

I know that the universe is not empty. And I do not propose the Milne model as a model to describe our universe.
But it has great explanatory power as a toy model. Not for predicting observations, but to make clear that cosmological coordinates are quite different from minkowski coordinates, even if one uses x=a*r as a spatial coordinate.
No big deal, one should think, but I've seen that it's a common misconception among experts to neglect the difference and invent new physics to describe coordinate effects. I bet there are quite a few professionals who think that "cosmological proper distance" reduces to "(SR) proper distance" in an empty universe.

So I next want to explore Ich's assertion that cosmological redshift is nothing but an accumulation of classical Doppler shifts.

Just to get it straight: It's the assertion of a peer reviewed paper, not mine. I somehow came to play the role of the lone defender of this - rather natural - claim.

The only purely kinematic cause I can see for such an elongation would be an ongoing acceleration of the wave packet (relative to our frame of reference).

No, not an acceleration of the wave packet. It's rather an acceleration of "the observer".
We observe the wave packet in a succession of different reference frames. To get from one frame to the next includes a translation of the origin as well as a boost to the next velocity. That's effectively the acceleration you mention.
I hope that clarifies your further points.

 

[nutgeb]

If you apply the SR doppler effect (including time dilatation) to this velocity, you get the same result. Both approaches always work. ...
But that does not mean that the other approaches, one of which including SR doppler and time dilatation, are no longer valid. You are still free to interpret the result as you like, and there is an exact mathematical framework for these different interpretations.

In any single coordinate system, such as the FRW system based on cosmological time, by definition it is impossible for accumulated classical Doppler shift to yield the same result regardless of whether SR time dilation is included or excluded, unless the accumulated SR time dilation over the light path equals zero.

I think you are saying that SR time dilation can be part of the correct answer only if we transform from FRW coordinates to Milne or other non-FRW coordinates. I don't disagree with that limited conclusion, but I think in the particular context of the point I'm trying to make, it is unhelpful in nailing down the physical kinematic basis for cosmological redshift. First because as I said, an empty Milne SR universe depends upon distinctly hyperbolic spatial curvature which is inconsistent with actual observations. And second because no viable alternative global SR coordinate system exists (nor could it exist) which accurately accounts for the effects of cosmic gravitation on worldlines while preserving spatially flat global geometry, homogeneity and isotropy all at the same time. Therefore your statement - that inserting accumulated SR time dilation into the calculation does not change the cosmological redshift mathematical calculation one way or the other (presumably even if the accumulated SR time dilation is non-zero in any single selected coordinate system) - cannot be proven in a realistic model. Vague statements such as that "the underlying symmetries of FRW mathematics" ensure equivalence do not add clarity.

Just to get it straight: It's the assertion of a peer reviewed paper, not mine. I somehow came to play the role of the lone defender of this - rather natural - claim.

Ich, I agree that it is frequently stated in scholarly works that cosmological redshift "seems to be" an accumulation of SR doppler shifts, although often it is suggested to be a combined effect with gravitational blueshift. But I have not seen published (a) any definitive and complete mathematical proof of that equivalence (often the proofs are limited to distances z << 1), (b) an explanation how accumulated SR time dilation (or cosmic gravitational time dilation, for that matter) does not logically conflict with the universal clock synchronicity of FRW fundamental observers, or (c) an explanation in explicit kinematic terminology of the physical action which causes both the wavelength and the wave packet length to stretch longitudinally in exact proportion to the scale factor.

And I think it's fair to say that you are the only author I've seen state that accumulated classical Doppler shift can be the sole basis for cosmological redshift.

No, not an acceleration of the wave packet. It's rather an acceleration of "the observer". We observe the wave packet in a succession of different reference frames. To get from one frame to the next includes a translation of the origin as well as a boost to the next velocity. That's effectively the acceleration you mention.

It is traditional in scholarly works on this subject that the observer's location is considered to be "stationary" as the origin of an FRW coordinate system. Then gravitational acceleration is deemed to be applied to an incoming wave packet by the total mass-energy contained within the sphere centered on the origin and with the wave packet located at the radius of the sphere. Gauss' Law is then applied to yield a Newtonian approximation (mathematically accurate only up to some distance) of the gravitational acceleration experienced by the wave packet, resulting in gravitational blueshifting.

Obviously if the emitting location were set as the origin of the FRW coordinate system, and the gravitational sphere were drawn with it as the center, the wave packet would experience gravitational redshifting instead. But this arrangement seems to reflect what would be observed in the reference frame of the emitter rather than the receiver, which presumably is why it is not generally used.

Moving ahead with the story, I want to further explore the kinematic action underlying cosmological redshift. Consider a scenario where a gun located at the emitting Galaxy "Ge" sequentially fires two massless test projectiles toward observing Galaxy "Go". Both projectiles have the same nonrelativistic muzzle velocity, which is far greater than Ge's escape velocity. Projectile 1 (P1) is launched at cosmological time t, and Projectile 2 (P2) at t + \Delta t. Time t happens to be at z=3 in Go's reference frame. The scale factor increases by 4 during projectiles' journey, so the RW line equation says that P2 arrives at Go at an interval of 4\Delta t after P1's arrival, in Go's reference frame. (Or at least the RW line equation would say that if the projectiles' velocities were relativistic.)

Did cosmic gravitational acceleration cause the 4x increase in the arrival interval compared to the launch interval? It doesn't seem so. During the interval between the launch of P1 and P2, it is true that the sphere of cosmic mass-energy centered on Go applies an acceleration to P1, increasing P1's velocity by the time P2 is launched. However, during the same interval the same cosmic gravitation applies an acceleration to Go, causing Go's recession velocity to decrease in approximately the same proportion as P1's velocity has increased. So when P2 is launched, its initial velocity toward Go should be approximately the same as P1's contemporaneous velocity. So this difference in launch times does not cause a significant increase in the distance between P1 and P2 at P2's launch time.

Once both projectiles are launched, they both are subject to ongoing cosmic gravitational acceleration toward Go. However, since at each discrete moment during flight P2 is always further away from Go than P1 is, P2's position at that moment defines a gravitational sphere of slightly larger radius than the sphere affecting P1. (Both spheres have the same density). So if there is any gravitational effect on the in-flight spacing between P1 and P2, it should be to decrease the distance between them because P2 experiences greater gravitational acceleration than P1.

I can't see any kinematic mechanism for gravitational blueshift to be the cause of the time dilation of the arrival interval which is inherent in FRW cosmological redshift. P1 and P2 are not locally accelerated relatively away from each other. Of course I analogize P1 and P2 to the leading and trailing edge respectively of a wave packet.

 

[Ich]

In any single coordinate system, such as the FRW system based on cosmological time, by definition it is impossible for accumulated classical Doppler shift to yield the same result regardless of whether SR time dilation is included or excluded, unless the accumulated SR time dilation over the light path equals zero.

Sorry, you lost me. As I tried to explain, in that specific coordinate system, redshift can be calculated without resorting to descriptions like doppler effect or gravitational effects. That doesn't mean in any way that these descriptions cannot yield the same result, e.g. if calculated in a different coordinate system or especially if calculated in a coordinate independent way like the two transport scenarios I described. Coordinates are one thing, physics is another thing. And it's physics that counts, no matter what desription you prefer.

I think you are saying that SR time dilation can be part of the correct answer only if we transform from FRW coordinates to Milne or other non-FRW coordinates.

Well, I think yes. It's simply a different description of the same thing, mot concurring theories.

I don't disagree with that limited conclusion, but I think in the particular context of the point I'm trying to make, it is unhelpful in nailing down the physical kinematic basis for cosmological redshift.

Ah ok, I think I didn't make my personal point of view clear enough: while Bunn and Hogg assert something like they proved that redshift is of kinematical origin, I'd say that they merely showed that it can be viewed as to be of kinematic origin. Personally, I'd prefer to include second order effects and explain it as a combination of kinematic and gravitational effects, as I said in a previous post. I explicitly refrain from "nailing down" the cause of redshift, I emphasize that different viewpoints are equally valid. And that one should know about as many viewpoints as possible, be it to pick the most appropriate one for a specific problem or simply to extend one's horizon.

And second because no viable alternative global SR coordinate system exists (nor could it exist) which accurately accounts for the effects of cosmic gravitation on worldlines while preserving spatially flat global geometry, homogeneity and isotropy all at the same time.

Well, that's a tautology. Of course SR does not include gravitation. But there are alternative coordinate representations of some FRW spacetimes that do include doppler and gravitational shifts as a "cause" of redshift, without "stretching of space".

Therefore your statement ... cannot be proven in a realistic model.

Hey, it is proven (I think). It's just a matter of calculus, it must be true.

Vague statements such as that "the underlying symmetries of FRW mathematics" ensure equivalence do not add clarity.

Ok, I'll come back to that later.

(a) any definitive and complete mathematical proof of that equivalence (often the proofs are limited to distances z << 1)

I don't know of such proofs, but a proof limited to z~0 is sufficient.

(b) an explanation how accumulated SR time dilation (or cosmic gravitational time dilation, for that matter) does not logically conflict with the universal clock synchronicity of FRW fundamental observers

Now, you have to prove that it is in conflict. Synchronicity is coordinate dependent, it's hard to imagine how this could disprove consequences of different coordinate representations.

(c) an explanation in explicit kinematic terminology of the physical action which causes both the wavelength and the wave packet length to stretch longitudinally in exact proportion to the scale factor.

By changing to a different coordinate system, you exactly give up the symmetries that lead to this result. You can't see it easily anymore. But as the physics is the same, the results must agree.

And I think it's fair to say that you are the only author I've seen state that accumulated classical Doppler shift can be the sole basis for cosmological redshift.

Ok, but it's trivial that relativistic doppler shift agrees with the classical one in the low speed limit. No big deal.

It is traditional in scholarly works on this subject that the observer's location is considered to be "stationary" as the origin of an FRW coordinate system...

Yes, but Bunn and Hogg explicitly do not use one single coordinate system, but are constantly switching. That's why gravitation is somebody else's problem.

Projectile 1 (P1) is launched at cosmological time t, and Projectile 2 (P2) at t + Delta t. The scale factor increases by 4 during projectiles' journey, so the RW line equation says that P2 arrives at Go at an interval of 4 Delta t after P1's arrival, in Go's reference frame.

That's interesting. I've read this assertion once, in a paper called "http://arxiv.org/abs/0707.0380" ". Now I'm again in the position to contradict a paper: this assertion is wrong.
Let's go back to the symmetry argument I mentioned earlier:
In the standard FRW metric ds²=dt²-a²dr², r does not appear explicitly. That means that at cosmological time t1 you can choose an arbitrary origin r1, start there a particle (say, a bullet), and it will be at r1+Dr at time t2. Consequently a particle started at the same time at arbitrary r2 under the same conditions will be at r2+Dr. Their comoving distance r2-r1 will not change over time, therefore their "proper distance" a*r will increase with the scale factor. The underlying symmetry is the one concerning transformations r -> r+dr.
If you talk about particles started at the same pale but different times, this symmetry does not apply, except for light, where the speed is constant. Nonrelativistic particles startes under such conditions will simply stay at a constant proper distance. Relativistic particles will increase their distance only as length contraction (wrt the respective observers) gets smaller and smaller, and will eventually maintain constant distance also.
Generally, the main contribution to the increasing distance in the symmetric ~a case is the relative velocity of the two starting points. If there is no such velocity difference, as in your scenario, the distance will not increase proportional to a.

 

[nutgeb]

That doesn't mean in any way that these descriptions cannot yield the same result, e.g. if calculated in a different coordinate system or especially if calculated in a coordinate independent way like the two transport scenarios I described.

Parallel transport is helpful as a conceptual description, but I am not aware of any published equation that uses parallel transport to provide a complete end-to-end calculation of how accumulated Doppler shift and gravitational shift equals cosmological redshift.

Personally, I'd prefer to include second order effects and explain it as a combination of kinematic and gravitational effects, as I said in a previous post.

Be my guest, I'd like to see a complete equation.

Hey, it is proven (I think). It's just a matter of calculus, it must be true.

I don't know of such proofs, but a proof limited to z~0 is sufficient.

Ich I don't want to take your statements out of context, but these two seem to me to be in conflict. I'll be satisfied to see a complete equation based on calculus. If integration of the accumulated Doppler/gravitation effects is too difficult to be directly calculated in a concise equation, then I'd even be satisfied if someone ran a manual integration in a spreadsheet to demonstrate a numerical result which roughly approximates the effects of cosmological redshift. If it's easy and obvious, why hasn't it been published?

I don't think a proof limited to z~0 is sufficient; even the authors who provide it don't claim that alone it is a complete proof.

Now, you have to prove that it is in conflict. Synchronicity is coordinate dependent, it's hard to imagine how this could disprove consequences of different coordinate representations.

Since a non-zero accumulated SR time dilation creates an obvious contradiction within the FRW metric, I don't see why it's necessary to show that the same contradiction occurs in other coordinate systems (especially when the other coordinate systems don't accurately and completely reproduce actual observations). Unless we want to concede that the FRW metric itself has a previously undisclosed limitation.

OK, but it's trivial that relativistic Doppler shift agrees with the classical one in the low speed limit. No big deal.

OK, then you are saying that SR and classical Doppler shift are interchangeable merely because over tiny spatial increments the SR time dilation approaches the limit of zero. If so, we don't disagree on this point. In that case, it's reasonable to conclude that SR time dilation in fact makes no contribution to the calculation of cosmological redshift.

Generally, the main contribution to the increasing distance in the symmetric ~a case is the relative velocity of the two starting points. If there is no such velocity difference, as in your scenario, the distance will not increase proportional to a.

I did allude to the change in Ge's recession velocity before P2 launches, but as I said this change is matched by the concurrent gravitational acceleration of P1.

If you talk about particles started at the same pale but different times, this symmetry does not apply, except for light, where the speed is constant. Non relativistic particles startes under such conditions will simply stay at a constant proper distance.

Can you point me to a specific mathematical analysis of that conclusion? I would appreciate it. As you point out, you are contradicting the peer-reviewed Francis, Barnes paper you cited.

Relativistic particles will increase their distance only as length contraction (wrt the respective observers) gets smaller and smaller, and will eventually maintain constant distance also.

Well of course I'm most interested in relativistic particles, specifically photons. Are you saying that the kinematic explanation for cosmological redshift is that: (a) the initial distance between fundamental observers Ge and Go is initially radially length contracted in Go's reference frame, and (b) the leading and trailing edges of the wave packet emitted by Ge move apart (as viewed in Go's reference frame) as the packet approaches Go because the intervening length contraction (as between the packet and Go) diminishes progressively, eventually to zero? Interesting explanation, can you point me to a published source for it?

Edit: What specific underlying "symmetry" would account for an exact correspondence between the change in length contraction and the change in the scale factor? That correspondence implies to me that the universe isn't expanding at all, that the true scale factor (after correction for SR-like length distortion) is fixed for all time. This in turn seems to pose a fundamental circularity: if the scale factor does not expand with time (except to the extent that deceleration of recession velocities over time causes global length de-contraction), then there wasn't a Hubble flow in the first place, and galaxies possessed no recession velocity with respect to each other; in which case the original justification for the occurrence of SR-like length contraction disappears!

 

[oldman]

Edit: What specific underlying "symmetry" would account for an exact correspondence between the change in length contraction and the change in the scale factor?...

Interesting question in an interesting thread.

If by the "change in length contraction" you mean the change in Lorentz contraction (due to acceleration) and by "change in scale factor" the change (with time) in the separation of two objects moving with the Hubble flow (due to gravitation), then I think that you are asking about a gauge symmetry (in the original Weyl sense of a change of length scale).

Here this gauge symmetry arises from a global uniformity of scale. In the case of SR this symmetry is uniaxial (along the axis of relative motion), in the case of a homogeneous FRW universe it is isotropic. The equivalence of these two symmetries is, I think, rooted in the Equivalence Principle of GR.

 

[nutgeb]

Interesting question in an interesting thread.

If by the "change in length contraction" you mean the change in Lorentz contraction (due to acceleration) and by "change in scale factor" the change (with time) in the separation of two objects moving with the Hubble flow (due to gravitation), then I think that you are asking about a gauge symmetry (in the original Weyl sense of a change of length scale).

Here this gauge symmetry arises from a global uniformity of scale. In the case of SR this symmetry is uniaxial (along the axis of relative motion), in the case of a homogeneous FRW universe it is isotropic. The equivalence of these two symmetries is, I think, rooted in the Equivalence Principle of GR.

Thanks for the clear description of the concept. However, before Ich's post I don't recall reading any source stating that, in a realistic gravitating FRW model, at the time of emission a distant galaxy's recession velocity causes that galaxy to be radially Lorentz contracted in the observer's rest frame at all, let alone by precisely the same amount as the FRW scale factor will expand during light's journey from the distant galaxy to the observer. That would be a very powerful symmetry if it existed. Can you point me to a published source describing it?

I see a reason why such a "symmetrical" cosmic Lorentz contraction seems to be completely ruled out. If the Lorentz contraction occurred, it would require that the duration of the aging of a supernova in the supernova rest frame at the time of emission would be at a factor of 1 (compared to the duration of aging finally observed in a distant observer's rest frame), rather than the factor of 1 / (1 + z) which has been widely confirmed by observations of low and high z supernovae and is currently accepted as standard.

Consider a supernova at z=3: In the supernova's rest frame at time of emission let's say the time between the first 2 spectra is 17 days, which is within the normal expected range. In the distant observer's frame that duration would initially be Lorentz contracted by 4x to 4.25 days, and then over the course of the wave packet's journey it would eventually "de-contract" back to the original 17 day duration which the observer would finally measure. But in this example, actual observations have led us to expect a 4x dilation from the original dilation in the supernova frame, resulting in a 68 day duration measured by the observer.

I think this exercise demonstrates that there is no place for ANY non-zero Lorentz contraction in lightpaths in the gravitational FRW model. So that idea for explaining a kinematic cause for FRW elapsed time dilation seems to be a dead end.

 

[Ich]

Parallel transport is helpful as a conceptual description, but I am not aware of any published equation that uses parallel transport to provide a complete end-to-end calculation of how accumulated Doppler shift and gravitational shift equals cosmological redshift.

Blame Old Smuggler, not me. He set me on the track and gave me the following reference (I confess, I didn't read it): J.V. Narlikar, American Journal of Physics, 62, 903 (1994).

Be my guest, I'd like to see a complete equation.

Use a gravitational potential of 1/2 (\ddot a / a) x^2) in otherwise flat space. That works at the post-Newtonian level.

Ich I don't want to take your statements out of context, but these two seem to me to be in conflict.

Ok, I know of Narlikar's proof concerning transport. The redshift thing is IMHO the same, but I don't know of a proof of this variant.

Since a non-zero accumulated SR time dilation creates an obvious contradiction within the FRW metri

I don't see this "obvious" contradiction. Please show a proof.

OK, then you are saying that SR and classical Doppler shift are interchangeable merely because over tiny spatial increments the SR time dilation approaches the limit of zero.

No. I'm saying that they are the same to leading order, and that is all that counts in the limit.

I did allude to the change in Ge's recession velocity before P2 launches, but as I said this change is matched by the concurrent gravitational acceleration of P1.

Sorry, I didn't read exactly what you wrote. I think we can go on using the setup of Francis and Barnes.

Can you point me to a specific mathematical analysis of that conclusion? I would appreciate it. As you point out, you are contradicting the peer-reviewed Francis, Barnes paper you cited.

It's fairly easy to show that F&B's setup does not lead to an increase in distance proportional to a. But I have to correct myself: my comments regarding Lorentz contraction and that the bullets stay at the same distance aplly exactal only to an empty spacetime. When I read the paper, I used the Milne model to calculate a specific example, and found that F&B's analysis does not work. My comments are based on that example, and I forgot to say that. Generally, gravitation of course plays a role and changes the results - but doesn't make F&R valid.
Draw a spacetime diagram of the gedankenexperiment (empty model) in minkowski coordinates, and you have two paralle worldlines of the bullets. Their distance is measured by comoving observers at any point in the trajectory. You'll see that (for tardyons) it's the same as a ruler measured by observers with different relative velocities to it, and that therefore its length is maximal in the frame (for the observer) where it comes to rest. It does not expand indefinitely.

Are you saying that the kinematic explanation for cosmological redshift is that: ...

Not at all. I merely wanted to point out why F&R'S setup does not follow the expansion, but I missed to point out that my counter-example is based on an empty spacetime.

Edit: What specific underlying "symmetry" would account for an exact correspondence between the change in length contraction and the change in the scale factor?

Again, sorry for the inconvenience, but the "underlying symmetry" was meant to be an easy deerivation of redshift, no matter what "causes" are invoked. It's clear that any valid description, even if it does not exploit that symmetry, must yield the same result.
In the empty model, the "change in length contraction" is not enough to give the result. It is important that there is an difference in velocity at the start, and that's exactly what F&R fail to account for.

 

[nutgeb]

Blame Old Smuggler, not me. He set me on the track and gave me the following reference (I confess, I didn't read it): J.V. Narlikar, American Journal of Physics, 62, 903 (1994).

Can someone please point me to a freely accessible version of this paper?

Use a gravitational potential of 1/2 (\ddot a / a) x^2) in otherwise flat space. That works at the post-Newtonian level.

I don't see how to use this equation to prove that gravitational blueshift and classical Doppler shift combine to calculate FRW cosmological redshift. Of course I'm familiar with the formula for FRW cosmological redshift, which alone does nothing to prove the point I'm interested in.

I don't see this "obvious" contradiction. Please show a proof.

This part of the dialogue is just going round in circles. The contradiction is "obvious" because all fundamental comoving FRW observers have synchronized clocks; inserting non-zero SR time dilation into light's worldline by definition requires the emitter's and observer's clocks to be running at different rates. Therefore non-zero SR time dilation is flatly contradictory to the FRW model.

By the way, non-zero SR time dilation would be inconsistent with the Milne model too, except that the homogeneous, isotropic Milne model admits that it applies physically unrealistic hyperbolic global spatial curvature distortion for the express purpose of exactly negating the mathematical/geometric effect of non-zero SR time dilation between fundamental comoving observers. Of course I'm aware that unrealistic hyperbolic global spatial curvature is a standard theoretical analysis tool of GR and cosmology, which unfortunately can introduce confusion between what is physically real and what is mathematically possible.

It's fairly easy to show that F&B's setup does not lead to an increase in distance proportional to a... When I read the paper, I used the Milne model to calculate a specific example, and found that F&B's analysis does not work.

I'll be especially interested in Wallace's response to your demonstration. Again, can you point to a published source which explains why the B&F approach is wrong?

Again, sorry for the inconvenience, but the "underlying symmetry" was meant to be an easy deerivation of redshift, no matter what "causes" are invoked. It's clear that any valid description, even if it does not exploit that symmetry, must yield the same result.
In the empty model, the "change in length contraction" is not enough to give the result. It is important that there is an difference in velocity at the start, and that's exactly what F&R fail to account for.

I'm pretty sure that any non-zero amount of Lorentz contraction would result in calculations of elapsed time dilation in a realistic FRW universe that are inconsistent with actual supernova observations, as explained in my post #82.

 

[oldman]

... I don't recall reading any source stating that, in a realistic gravitating FRW model, at the time of emission a distant galaxy's recession velocity causes that galaxy to be radially Lorentz contracted in the observer's rest frame at all, let alone by precisely the same amount as the FRW scale factor will expand during light's journey from the distant galaxy to the observer. That would be a very powerful symmetry if it existed. Can you point me to a published source describing it?.

No, I can't. It's just my own suggestion. I hasten to add that, in my view, one should never try and extend calculations of SR effects (such as the Lorentz contraction) to situations where gravity rules (as in FRW models), and where the the situation has a quite different geometrical symmetry. There the much more sophisticated mathematical machinery of GR is needed for obtaining numerical results. I therefore fully agree with you that:

...there is no place for ANY non-zero Lorentz contraction in lightpaths in the gravitational FRW model. So that idea for explaining a kinematic cause for FRW elapsed time dilation seems to be a dead end.

.

But remember that the eqivalence of acceleration and gravity is something raised to the status of a principle (the EP) because we don't understand why there is this equivalence; we like to conceal our ignorance in pompous ways. I'm suggesting that equivalence is due to an underlying gauge symmetry, namely the global uniformity of scale that seems to prevail in the universe we find ourselves in. But sadly I've not the least idea how or why this came about -- so this is just regressing further into the unknown!

 

[atyy]

Can someone please point me to a freely accessible version of this paper?

It's also discussed by Gron and Elgaroy, http://arxiv.org/abs/astro-ph/0603162.

But remember that the eqivalence of acceleration and gravity is something raised to the status of a principle (the EP) because we don't understand why there is this equivalence; we like to conceal our ignorance in pompous ways.

The EP is not a principle principle, it is a heuristic principle. Try Carroll's discussion around Eq 4.32 of http://nedwww.ipac.caltech.edu/level5/March01/Carroll3/Carroll_contents.html, or section 24.7 of Blandford and Thorne's http://www.pma.caltech.edu/Courses/ph136/yr2006/text.html

 

[Ich]

I don't see how to use this equation to prove that gravitational blueshift and classical Doppler shift combine to calculate FRW cosmological redshift

In the neighbourhood of any comoving observer, you can approximate any FRW spacetime by a post-Newtonian model with some gravitational potential. Gravitational redshift corresponds to potential difference over c². Doppler shift comes from the recession velocities that other comoving observers have in this frame. Within the accuracy of the approximation, the result is the same as the one derived in different (FRW-) coordinates.

The contradiction is "obvious" because all fundamental comoving FRW observers have synchronized clocks

No, they don't. Synchronization is coordinate dependent, that's really basic stuff. There are local frames, those in which all observers have different coordinate velocity, where the clocks are not synchronized anymore. In such frames, there is time dilatation.
If we can't get over this point, I fear that we'd better agree to disagree.

physically unrealistic hyperbolic global spatial curvature distortion

Now what's a "physically unrealistic hyperbolic global spatial curvature distortion"? That's simply different spacelike slices though spacetime. Day to day business in GR, you shouldn't have a problem with that.

...confusion between what is physically real and what is mathematically possible.

Do you think that "reality" cares about coordinates? That'd be a problem for our discussion.

I'll be especially interested in Wallace's response to your demonstration. Again, can you point to a published source which explains why the B&F approach is wrong?

I don't know about such a source. That's my claim, and I showed you you to follow its derivation.
I'd be happy to discuss this point with Wallace, if he likes to jump in.

 

[oldman]

The EP is not a principle principle, it is a heuristic principle.

I'd like to discusss this briefly, but not here, as it'll take us off the topic of this long thread, atyy; so I'll start another thread. Meanwhile, thanks for the references to Carroll and Thorne. They make me wish I'd attended Grad school in either Chicago or Caltech.

 

[nutgeb]

In the neighbourhood of any comoving observer, you can approximate any FRW spacetime by a post-Newtonian model with some gravitational potential. Gravitational redshift corresponds to potential difference over c². Doppler shift comes from the recession velocities that other comoving observers have in this frame. Within the accuracy of the approximation, the result is the same as the one derived in different (FRW-) coordinates.

Great, but you cannot claim that an equation which by definition is valid only at z << 1 is also valid at greater distances. What I requested was an equation that starts with Doppler shift (together with gravitational shift, if you like) and calculates cosmological redshift globally, at any distance and over any time duration. No equation which purports to do that has been published, despite the fact that a lot of really smart people have puzzled over it for many years.

No, they don't. Synchronization is coordinate dependent, that's really basic stuff. There are local frames, those in which all observers have different coordinate velocity, where the clocks are not synchronized anymore. In such frames, there is time dilatation. If we can't get over this point, I fear that we'd better agree to disagree.

I'm about at the point where I'll agree to disagee. I believe you are misapplying the concept of covariant diffeomorphism here. Clock synchronization is coordinate dependent, but so is the condition of fundamental observers having unsynchronized clocks. Since dis-synchronicity (is that a word?) vanishes in some coordinate systems, one could just as well argue that it isn't a "real" aspect of physics either. But I believe the covariance principle just doesn't apply in that way. I need some help in articulating this point.

In any event, I'm talking about internal "rules" consistency within an individual coordinate system, as distinguished from the translation of coordinates between different systems. The homogeneous, isotropic FRW model by definition prohibits unsynchronized clocks as between fundamental comoving observers, so you must corrupt the metric if you try to insert it. Similarly, the homogeneous, isotropic Milne model (with hyperbolic spatial curvature) also prohibits unsynchronized clocks as between fundamental comoving observers. More trivially, even the spatially flat Minkowski metric does not support a homogeneous, isotropic matter distribution if it is expanding: instead, the matter field must be entirely at rest w/r/t itself, meaning zero recession velocity as between particles, which in turn means that zero SR time dilation is required as between fundamental "costatic" (opposite of "comoving") particles. (Hmm, I wonder if this pattern can be generalized, and homogeneity+isotropy is impossible in ALL coordinate systems that permit non-zero time dilation as between fundamental observers?)

On the other hand, I think it's possible that SR time dilation and gravitational time dilation together could fit into the calculation of cosmological redshift. Since cosmic gravitational shift normally is interpreted by the observer as blueshift, it cuts in the opposite direction as SR time dilation. Yet for the same reason as for SR time dilation, the rules of the FRW metric rule out the possibility that non-zero gravitational time dilation could result (alone) as between fundamental comoving observers. So one is led to the thought that perhaps SR and gravitational time dilation exactly offset and negate each other mathematically in the FRW model. Each contributes an equal and opposite element of time dilation, such that when the two elements are combined, the net effect is zero. I'm skeptical that the math would work out so neatly, but I don't recall having seen any mathematical attempt to test this straightforward question.

 

[Ich]

Great, but you cannot claim that an equation which by definition is valid only at z << 1 is also valid at greater distances.

I didn't claim that.
I'll give an example of what the paper claims:
Consider an arbytrary function y=f(x). The claim is that, at each point, the arc length of the funtion between two nearby points can be approximated by rotating to a system where the two points lie parallel to x', and measure the difference dx'. To get the exact arc legth between two points at some distance, you repeat the procedure by applying it to infinitely many, infinitely small patches of the function.
The parallels are:
-Via a (not really specified) coordinate transformation, you get a simple formula valid in the vicinity of an arbitrary point
-The formula is valid to first order only, second order contributions (such as curvature or relativistic corrections to the doppler effect) are neglected
-it gives nevertheless definitely the correct answer
-it is completely useless for all practical purposes, such as actually doing the calculation.

The interesting point is the transformation. The authors specify it exactly, like I did here, by what it has to do. But they don't give its global mathematical form.
In this example, you can get the difference dx' by applying dx'²=dx²+dy² in the global coordinate system. Nothing has changed in principle, the procedure is correct whether you define the transformation globally or not. But now it's useful, too. This last step should be done in the paper Old Smuggler referenced to.

Since dis-synchronicity (is that a word?) vanishes in some coordinate systems, one could just as well argue that it isn't a "real" aspect of physics either.

The point is not about physical or unphysical. Synchronization simply depends on the procedure you use to establish it. Without specifying the procedure, "synchronization" is not defined and thus not a "real" aspect of physics. When you claim that fundamental observers are synchronized if you use a coordinate time that equals the proper time since the big bang, that's ok. And when I say that they are not synchronized if I use the standard procedure to establish synchronizity, that's also ok. The covariance principle surely applies here.
But it's not ok to pick one definition to establish synchronizity, and claim that procedures that give a different result are wrong. They aren't, they're simply different.

More trivially, even the spatially flat Minkowski metric does not support a homogeneous, isotropic matter distribution if it is expanding

Please be exact.
"Homogeneous" means that after a certain proper time since the big bang, each comoving observer measures the same matter density in his/her vicinity. None is privileged.
"Isotropic" means that thy universe looks the same to them in each direction. No direction is privileged.
Both principles are, of course, also true in the minkowski coordinate representation, because they are defined independent of coordinates.
It's just that FRW coordinates fully reflect that symmetry, while minkowski coordinates don't. But they have the advantage that space and time coordinates are defined the usual way, with velocities being velocities and such.
So, by exploiting the symmetry, there is a simple redshift formula in FRW coordinates, namely anow/athen.
But there is also a simple formula in minkowski coordinates, namely the SR doppler formula.