Why "expanding space"?

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.
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) + [tex] \dot a [/tex](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 [tex] \dot a [/tex](t_0) (as is the case, in general). This simple misunderstanding may be appropriately called "the Bunn/Hogg fallacy", and you have endorsed it.
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 [tex]\dot a [/tex], 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 [tex]\dot a [/tex] 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 [tex] \dot a [/tex]. 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 [tex] \dot a [/tex](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.
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.
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

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

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 [itex](1+z)^2[/itex]. 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:
[tex]D_A = \frac{c}{H_0}(1-(1+z)^{-2})/2[/tex]

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

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:
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,

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

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

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.
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.
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.
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

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.
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.
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 + [tex]\Delta[/tex] 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[tex]\Delta[/tex] 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

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.
Well, I think yes. It's simply a different description of the same thing, mot concurring theories.
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.
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".
Hey, it is proven (I think). It's just a matter of calculus, it must be true.
Ok, I'll come back to that later.
I don't know of such proofs, but a proof limited to z~0 is sufficient.
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.
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.
Ok, but it's trivial that relativistic doppler shift agrees with the classical one in the low speed limit. No big deal.
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.
That's interesting. I've read this assertion once, in a paper called "Expanding Space: the Root of all Evil?". 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

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

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

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

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).
Use a gravitational potential of [tex]1/2 (\ddot a / a) x^2)[/tex] in otherwise flat space. That works at the post-Newtonian level.
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.
I don't see this "obvious" contradiction. Please show a proof.
No. I'm saying that they are the same to leading order, and that is all that counts in the limit.
Sorry, I didn't read exactly what you wrote. I think we can go on using the setup of Francis and Barnes.
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.
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.
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

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

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:

.

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!