# How to prove the stretching of space

To my understanding the cosmological redshift in the Milne Cosmos is a Dopplershift, but is due to expansion in the empty FRW Universe. Comoving observers can be defined for both cases. If that is correct, the choice of the interpretation of the redshift depends on a transformation of coordinates, not on physics.
The Milne model is equivalent to an empty RW-manifold and is mathematically a subset of Minkowski space-time. The FOs are those observers moving orthogonally to the "preferred" hypersurfaces (with hyperbolic geometry) foliating this RW-manifold. There is no alternative set of observers involved here. Since this RW-manifold is flat, the corresponding cosmological redshifts must of course be interpreted as Doppler shifts in flat space-time. This has nothing to do with transformations of coordinates.
In this view (I might be wrong) I don't understand your remark, that the other set of observers (Milne, e.g.) is irrelevant. Arn't there just interchangeable discriptions for the same universe? Why then have a preference for one of these? If it contains mass, I guess these discriptions are more complicated which however shouldn't influence in principle the reasoning.
What alternative set of observers do you have in mind? Perhaps you are thinking of the set of (non-expanding) observers moving orthogonally to some foliation of Minkowski space-time into flat hypersurfaces? If so, yes, these observers are irrelevant for interpretations of cosmological redshifts in the empty RW-model.
(Note that the empty RW-manifold is only a SUBSET of Minkowski space-time, so the FOs and these alternative observers do not really describe "the same universe".)

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The Milne model is equivalent to an empty RW-manifold and is mathematically a subset of Minkowski space-time. The FOs are those observers moving orthogonally to the "preferred" hypersurfaces (with hyperbolic geometry) foliating this RW-manifold. There is no alternative set of observers involved here. Since this RW-manifold is flat, the corresponding cosmological redshifts must of course be interpreted as Doppler shifts in flat space-time. This has nothing to do with transformations of coordinates.
On the other side the recession velocities depend on the choosen coordinates. In RW coordinates they are a function of the Hubble Constant, whereas in Minkowskian coordinates they are distance/time (Special Relativity). So, according to that the cosmological redshift depends on H, or can be described by the special relativistic Doppler formula, respectively. Hereby I take reference to the thesis of Tamara Davis, Chapter 4 - The empty universe, http://www.dark-cosmology.dk/~tamarad/papers/thesis_complete.pdf.

But as you say, both models (Milne and empty RW) are equivalent, so, the redshifts should not depend on coordinates and are Doppler shifts for both models therefore. Somewhere my reasoning must be wrong. I appreciate any help.

Another point. The empty RW model is negatively curved (and therefore expands?). This curvature means the geometry of space, right? The spacetime is flat, which is common to both models.

On the other side the recession velocities depend on the choosen coordinates. In RW coordinates they are a function of the Hubble Constant, whereas in Minkowskian coordinates they are distance/time (Special Relativity). So, according to that the cosmological redshift depends on H, or can be described by the special relativistic Doppler formula, respectively. Hereby I take reference to the thesis of Tamara Davis, Chapter 4 - The empty universe, http://www.dark-cosmology.dk/~tamarad/papers/thesis_complete.pdf.
It is not a good idea to introduce coordinate-dependent quantities if they are not directly related to coordinate-free objects. That is, the proper objects to consider are the 4-velocities of the FOs and not some independently defined "recession velocities". The question of what speed to use in the SR Doppler formula is then answered by performing the following procedure (see J.V. Narlikar, American Journal of Physics 62, 903-907 (1994) for the mathematical details). Parallell-transporting the 4-velocity of the emitting FO along a null geodesic to the receiving FO and projecting the resulted parallell-transported 4-velocity into the local rest frame of the receiving FO yields a 3-velocity that can be put into the SR Doppler formula to give the desired redshift. This procedure is independent of any choice of coordinates; it depends only on the 4-velocities of the emitting and receiving FOs and on the space-time geometry.
timmdeeg:4205749 said:
But as you say, both models (Milne and empty RW) are equivalent, so, the redshifts should not depend on coordinates and are Doppler shifts for both models therefore. Somewhere my reasoning must be wrong. I appreciate any help.
The reason for the confusion is that the defined "recession velocities" are not components of any coordinate-free space-time objects. Therfore, any reference to such "recession velocities" is fraught with danger and should be avoided.
timmdeeg:4205749 said:
Another point. The empty RW model is negatively curved (and therefore expands?). This curvature means the geometry of space, right? The spacetime is flat, which is common to both models.
Right.

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The reason for the confusion is that the defined "recession velocities" are not components of any coordinate-free space-time objects. Therfore, any reference to such "recession velocities" is fraught with danger and should be avoided.

Right.
Thanks.

If I understood you correctly, the redshift observed between FOs depends in the non-empty RW model on whether these are closed, flat, or open and on the space-time curvature and thus not on the choice of coordinates (i). Would you please specify in which cases the redshift is purely gravitational and gravitational/kinematic respectively, including the Lambda-CDM model, the universe in which we live.

(i)
The reason for this is simple: in the RW-models there is a set of "preferred" observers (the so-called "fundamental observers" (FOs)) defining the cosmic redshift; i.e., the high symmetry of the RW-manifolds implies that they can be foliated in a "preferred" way such that the spatial hypersurfaces are homogeneous and isotropic.
In the sense, that there is no other choice, so my understanding.

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Thanks to everybody for your explanations.

The problem with doing the measurement in a void is you'd need to do it in an expanding void, which means having the test masses extremely far apart and far away from any other matter in the universe, which makes it an undoable experiment (at least for the forseeable future). This is because the local space-time around massive objects is not expanding.

The increase in measured distances is real. But whether you interpret this increase in distances as a velocity is, well, up to your interpretation.
Perhaps it is legitimate to overcome the problems in the void by imagining a gedanken experiment. But I guess, even then your final statement would be the same. I am free to interpret any increase in distance in this or that way.

So,
The "stretching of space" picture is precisely the picture under which the redshift is a gravitational phenomenon.
we talke about a picture. And therefore the answer is: The stretching of space being not something truely physical can not be measured.

Nevertheless there is
From general relativity (specifically, the geodesic equation), it is seen that the momentum of a particle is inversely proportional to the expansion (the scale factor, a(t)). From de Broglie, this becomes a statement about the wavelength of photons -- as space expands, the wavelength of light must increase.
some physical support for this "picture". It's still a bit confusing, "as space expands, the wavelength of light must increase." You didn't say, "as space is stretched ...", but I wonder, if this was meant. Perhaps one should careful distinguish between expansion and stretching. The universe expands according to a(t), but the expansion isn't necessarily a true stretching of space effect.

Please don't hesitate to correct if I said something wrong.

Chalnoth
So,we talke about a picture. And therefore the answer is: The stretching of space being not something truely physical can not be measured.
That's not an accurate take-away.

The correct statement is that there is a real physical phenomenon here, and one correct description of that phenomenon is that it is a stretching of space. There are other seemingly-different but nevertheless also completely correct descriptions of the exact same physical phenomenon.

This is one of the weird things about physics: it is sometimes possible to describe the exact same thing in seemingly completely different ways, while actually describing the same system. And sometimes the difference is so different that it is hard to believe that it's actually the same system being described (e.g. sometimes you can describe a system using different numbers of spatial dimensions and still be describing the same system).

If I understood you correctly, the redshift observed between FOs depends in the non-empty RW model on whether these are closed, flat, or open and on the space-time curvature and thus not on the choice of coordinates (i).
Choice of coordinates also DOES affect the outcome. The standard cosmological measure involved comoving coordinates, an observer at rest with respect to the CMBR.

The description of curved 4D spacetime as 'expanding' or 'increasing distances' over time depends on a choice of 3+1D split. We use one that is convenient but not unique.)

The stretching of space being not something truely physical can not be measured.
Chalnoth posed a wonderfully insightful physical explanation in another discussion:

The integrated Sachs-Wolf effect is the clearest, independent [of supernova], evidence of dark energy [the strange anti-gravity effect that powers expansion]… Photons entering a large gravitational well [like a galactic supercluster] get a gravitional energy boost upon entering the region causing a small gravitational blue shift. Upon exiting, they lose this free energy and redshift back to their original energy state upon exiting - almost. If the universe were flat and static, the net effect would be zero. In an expanding universe, the photon takes so long to pass through the gravity well that it gets to keep a small amount of the blue shift it acquired on the way in due to expansion and the resulting dilution of gravity. This extra energy shows up as a slight anisotropy in the CMB photons passing through a supercluster or supervoid [the effect is just the opposite for CMB photons passing through a supervoid]. Seehttp://arxiv.org/abs/0805.3695 for discussion.

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This is one of the weird things about physics: it is sometimes possible to describe the exact same thing in seemingly completely different ways, while actually describing the same system.
Yes, I will have to accept this truth, though being weird. But your remark reminds me strongly of an article of R.L.Jaffe, wherein he shows that the measurable Casimir force can not only be described by vacuum fluctuations (as usual), but without taking reference to the vacuum as a van der Waals force as well.

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If I understood you correctly, the redshift observed between FOs depends in the non-empty RW model on whether these are closed, flat, or open and on the space-time curvature and thus not on the choice of coordinates (i).
Choice of coordinates also DOES affect the outcome. The standard cosmological measure involved comoving coordinates, an observer at rest with respect to the CMBR.
My remark refers to explanations of Old Smuggler. From his perspective "Coordinates are irrelevant for interpretations of redshifts in the RW-models".

Chalnoth posed a wonderfully insightful physical explanation in another discussion:
Thanks, I wasn't aware of that but read the paper meanwhile. The independent confirmation of the dark energy is very surprising and confirms the Lambda CMB model. Also, the ISW-effect is in accordance with the stretching of space description.

Here is an interesting 8 page paper I stumbled across in my notes:

Expanding Space: the Root of all Evil?

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0380v1.pdf

Several of these 'issues' have been discussed in other threads......I found these are insightful:

...the expansion of space is neither more nor less than the increase over time of the distance between observers at rest with respect to the cosmic fluid in terms of the
FRW metric. With this metric..... the density and pressures of cosmological fluids must change over cosmic time, and it is this change that represents the basic property of an expanding (or contracting) universe.

The proper time for …..privileged observers at rest with regards to the cosmic fluid ticks at the same rate as cosmic time and hence the watches of all privileged observers are synchronised.

In an expanding universe, the change of the metric implies that the physical distance between any two privileged [comoving] observers increases with time...

The Hubble flow is then viewed as a purely kinematical phenomenon —
objects recede because they have been given an initial velocity proportional to distance.

the velocity of [a] particle due its motion relative to the Hubble flow (or equivalently the homogeneous fluid defining the FRW metric) must be less than the speed of light; its velocity due to the increase of the scale factor is not restricted in this way…..

cosmological redshift is not, as is often implied, a gradual process caused by the stretching of the space a photon is traveling through. Rather cosmological redshift is caused by the photon being observed in a different frame to that which it is emitted.

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Here is an interesting 8 page paper I stumbled across in my notes:

Expanding Space: the Root of all Evil?

http://arxiv.org/PS_cache/arxiv/pdf/0707/0707.0380v1.pdf
Thank you, this paper is very interesting and worthwile to be read. It moreover shows the controversy between cosmologists regarding thought experiments (one of my questions),

Expanding Space: the Root of all Evil? page 2: To illustrate how short this pragamatic formalism falls being platitude, one need no further than Abramowicz et al. (2006), in which a thought experiment of laser ranging in an FRW Universe is proposed to 'prove' that space must expand. This is sensibly refuted by Chodorowski (2006b), but followed by a spurious counter-claim that such a refutation likewise proves space does not expand.
which makes it not easier to improve one's own understanding.

To me this statement
Expanding Space: the Root of all Evil? page 2: The expansion of space is no more extant than magnetic fields are and exists only as a tool for understanding the unambiguous predictions of GR, not a force-like term in a dynamical equation.
sounds very agreeable.

Expanding Space: the Root of all Evil? page 7: The key is to make it clear that the cosmological redshift is not, as is often implied, a gradual process caused by the stretching of the space a photon is traveling through. Rather cosmological redshift is caused by the photon being observed in a different frame to that which it is emitted. In this way it is not as dissimilar to a Doppler shift as is often implied. The difference between frames relates to a changing background metric rather than a different velocity.
Is this proposal in accordance with the parallel transport of the 4-velocity vector?

And how about this thought experiment: Supposed the universe doesn't expand at the time of emission and absorption but expands during the photon's travelling. What kind of shift if any will be measured?

Chalnoth
And how about this thought experiment: Supposed the universe doesn't expand at the time of emission and absorption but expands during the photon's travelling. What kind of shift if any will be measured?
The observed redshift will be equal to the total amount of expansion between the emission and absorption of the photon, regardless of what the rate of that expansion was at different times.

Sorry for the late reply (no internet connection for the last week).
If I understood you correctly, the redshift observed between FOs depends in the non-empty RW model on whether these are closed, flat, or open and on the space-time curvature and thus not on the choice of coordinates (i).
The redshift depends only on the 4-velocities of the FOs and on the space-time geometry. (This is most easily seen by using said procedure of parallel-transport.) On the other hand, the INTERPRETATION of the redshift depends on the spatial geometry, since the spatial geometry is crucial for determining how well a flat space-time connection approximates the curved space-time connection.

Would you please specify in which cases the redshift is purely gravitational and gravitational/kinematic respectively, including the Lambda-CDM model, the universe in which we live.
As far as the Lambda-CDM model is based on RW-models, the properties of the RW-models apply (see below). If inhomogenities are taken into account, the effects of these come in addition.

In RW-models with flat or spherical space sections, the redshift is entirely due to the non-flat connection and thus indirectly to space-time curvature (i.e., "gravitational"). (See, e.g., arXiv:0911.1205.) For RW-models with hyperbolic space sections things are more complicated, and some part of the redshift is "kinematic" (meaning that some part of the redshift survives even if one replaces the curved space-time metric with a flat one). To decide how much of the redshift is "kinematic", a recipe for spectral shift split-up into "kinematic" and "gravitational" parts is necessary (this can be done unambiguously, at least for small distances).

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Would you please specify in which cases the redshift is purely gravitational and gravitational/kinematic respectively, including the Lambda-CDM model, the universe in which we live.
As I understand the consensus from earlier discussions on this subject, such a split in our universe, represented by the Lambda-CDM model, over cosmological distances is arbitrary.

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The observed redshift will be equal to the total amount of expansion between the emission and absorption of the photon, regardless of what the rate of that expansion was at different times.
Thanks for this clear and unambiguous answer.

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The redshift depends only on the 4-velocities of the FOs and on the space-time geometry. (This is most easily seen by using said procedure of parallel-transport.) On the other hand, the INTERPRETATION of the redshift depends on the spatial geometry, since the spatial geometry is crucial for determining how well a flat space-time connection approximates the curved space-time connection.
.
In RW-models with flat or spherical space sections, the redshift is entirely due to the non-flat connection and thus indirectly to space-time curvature (i.e., "gravitational"). (See, e.g., arXiv:0911.1205.)
These authors argue that in order to interpret the cosmological redshift in terms of a Doppler effect in non-expanding Minkowskian space-time the observer would have to move away from himself and thus claim (spatial curvature >= 0) "The Doppler interpretation is clearly self-contradictory (page 5). But this is relativised later (page 6):
"Hence, ironically in the context of the recent debate, parallel-transport of four-velocities along photons path can allow cosmological redshifts to be interpretet as a relativistic Doppler effect without the contradiction presented here, provided that the concept of expanding space is added to the Minkowski space-time ... and provided that the velocity is thought of as being tied to a path and not as a global concept."

But irrespective of such an ambiguous debate I have a problem to understand the cosmological redshift in the sense of a purely gravitational shift. It is quite clear that a photon looses energy und thus becomes redshifted as it climbs out of a gravitational field or in other words as it moves away from a mass (i). In contrast the photon traveling through homogeneous space doesn't move away from a gravitational center, but undergoes a redshift (= looses energy) as well. How shall I understand this (obvious?) discrepancy? You mentioned already the dependence on spatial geometrie ... . Is there any explanation besides the stretched wavelenght picture as simpel as (i)?

These authors argue that in order to interpret the cosmological redshift in terms of a Doppler effect in non-expanding Minkowskian space-time the observer would have to move away from himself and thus claim (spatial curvature >= 0) "The Doppler interpretation is clearly self-contradictory (page 5). But this is relativised later (page 6):
"Hence, ironically in the context of the recent debate, parallel-transport of four-velocities along photons path can allow cosmological redshifts to be interpretet as a relativistic Doppler effect without the contradiction presented here, provided that the concept of expanding space is added to the Minkowski space-time ... and provided that the velocity is thought of as being tied to a path and not as a global concept."
Yes, cosmological redshifts can always be interpreted as Doppler shifts in CURVED space-time. However, they cannot in general be interpreted as Doppler shifts in FLAT space-time, and it is the latter meaning that is usually understood with "kinematic" redshift.

timmdeeg:4216930 said:
But irrespective of such an ambiguous debate I have a problem to understand the cosmological redshift in the sense of a purely gravitational shift. It is quite clear that a photon looses energy und thus becomes redshifted as it climbs out of a gravitational field or in other words as it moves away from a mass (i). In contrast the photon traveling through homogeneous space doesn't move away from a gravitational center, but undergoes a redshift (= looses energy) as well. How shall I understand this (obvious?) discrepancy? You mentioned already the dependence on spatial geometrie ... . Is there any explanation besides the stretched wavelenght picture as simpel as (i)?
There is no obvious intuitive picture to decide the question of "kinematic" versus "gravitational" interpretations, I'm afraid. (If there were, this question would not have been debated so vigourously in the literature.) However, as I have mentioned earlier, there exists a general procedure to decide the matter for small distances, and for arbitrary space-times. That is, choose a pair of fixed ("close") observers with given world lines. Calculate spectral shifts obtained by exchanging photons between these observers. Then replace the space-time geometry in the relevant region with flat space-time (holding the chosen world lines and the coordinate system fixed). Calculate spectral shifts again, but now with the flat space-time geometry. If the latter calculation yields no spectral shifts at all, the spectral shifts obtained in the first calculation must be entirely due to space-time curvature, i.e., "gravitational".

For example, in the Schwarzschild metric, the chosen observers defining gravitational spectral shifts are observers with fixed spatial Schwarzschild coordinates. The flat space-time limit of this metric is obtained by setting the mass M=0. Now it is rather obvious that there is no spectral shift between the chosen observers in the Schwarzscild metric with M=0, so the spectral shift obtained when M is nonzero must be purely gravitational. A similar situation to that of the Schwarzschild metric occurs for RW-models with flat or spherical space sections, so the spectral shifts obtained between the FOs in these models must also be purely gravitational.

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Chalnoth
Yes, cosmological redshifts can always be interpreted as Doppler shifts in CURVED space-time. However, they cannot in general be interpreted as Doppler shifts in FLAT space-time, and it is the latter meaning that is usually understood with "kinematic" redshift.
I don't understand what you mean. In flat space-time, there is no curvature, and thus in general you don't expect there to be any gravitational redshift at all, meaning that any observed redshift would be purely kinematic (of course, you might still be able to impose what looks like gravitational redshift with an appropriate coordinate choice, such as Milne coordinates).

Either way, though, our space-time does have a definite degree of overall curvature, as it must due to the fact that our universe is not empty (more pedantically-stated, the average energy density of our universe is non-zero).

Regardless of the overall curvature, however, the amount of the redshift that is attributed to gravitation and the amount attribute to motion of the emitter or observer is still arbitrary. Some choices may seem more or less natural to some people, but many choices are possible in any event.

Chalnoth:
Regardless of the overall curvature, however, the amount of the redshift that is attributed to gravitation and the amount attribute to motion of the emitter or observer is still arbitrary. Some choices may seem more or less natural to some people, but many choices are possible in any event.
That seemed to be the conclusion from another discussion on this topic, with some insights that may be of interest:

[Note, especially the change in scale factor and,in Schwarzschild coordinates, the change in velocity, comments.]

https://www.physicsforums.com/showthr...nt+flow&page=4 [Broken]

edit: oops, that link no longer works???

[In the great 2007 thread Wallace, Chronos and Oldman take a different view than expressed here [and there] by Marcus...you can read the posts from the 40's thru 50's and see the pros and cons.]

I do think it is better to think of (photons) as being redshifted by being observed in a different frame ......Now as t ticks along, the scale factor a(t) increases. Therefore two observers who are both at rest wrt to the CMB, but who have different times t will therefore be in different frames (have different metrics). This is what leads to photons being redshifted when observed and emitted at different times.

I tend to agree, photons are not redshifted by traveling through the universe, they are redshifted only because they are observed in a different frame from which they were emitted.

Marcus: # 48] I am not comfortable with that because among other things I see cosmologists doing inventories of the energy density which are implicitly estimated IN A CMB FRAME....

These 'conflicting' viewpoints stem from this as explained by Chalnoth elsewhere:

" … You get some total redshift for faraway objects due to cosmological expansion. How much of that redshift is due to the Doppler shift# and how much is due to the expansion between us and the far away object is completely arbitrary."

# Doppler shift is based on [relative velocity] frame based differences, not expansion, Hence photon frequency and wavelength can be viewed as fixed just like in a static Spacetime.. Doppler shift is a particular explanation of redshift, with a particular formula.

Marcus:
Don’t think of the redshift as a Doppler [relative velocity] effect. It is not the result of some particular speed. The formula involves the entire [varying] factor by which distances have been expanded during the whole time the light has been traveling.

PeterDonis: The law governing the relationship of emitted to observed photon energies (or frequencies) is general and applies in any spacetime. The 4-momentum of the photon gets determined at the emitter; then it gets parallel transported along the photon's worldline from emitter to observer; then you contract that 4-momentum with the observer's 4-velocity to get the observed energy (or frequency if you throw in a factor of Planck's constant). That "parallel transport" process is actually where the "redshift" occurs in an expanding universe; the expansion alters the 4-momentum of the photon as it travels (or at least that's one way of looking at it), whereas in a static universe the photon's 4-momentum would "stay the same" as it traveled.

There's another complication here, btw; what about the gravitational redshift of photons in Schwarzschild spacetime? Here the "change" with changing radius is actually in the 4-velocity of the observer; the photon's 4-momentum stays the same, but the 4-velocities of "hovering" observers are different at different radii, so they contract differently with the constant photon 4-momentum.

PAllen:

Redshift is a measured shift in received frequency versus emitted frequency. Doppler [shift] refers to one of two formulas (pre-relativistic; relativistic) for relating redshift to velocity. Doppler shift is a particular explanation of redshift, with a particular formula. It is not a measure of redshift.

Where the speeds of source and the receiver relative to the medium are lower than the velocity of waves in the medium, the classical Doppler shift formula; in cosmology, where we deal with lightspeed 'c' and recessional 'velocities' greater than 'c' we need the relativistic version of the formula. [Doppler is like a radar speed trap: The radar signal goes out and returns and keeps the same 'color', but we record the difference in wavelength as a speed measure.]

Cosmological redshift is typically considered distinct from Doppler redshift because it is a relation between distance and redshift rather than speed and redshift, under the assumption that both source and target are motionless relative to center of mass of the local matter (here, local is quite large - galaxy or galaxy cluster).

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Chalnoth
https://www.physicsforums.com/showthr...nt+flow&page=4 [Broken]

edit: oops, that link no longer works???
You copied and pasted the shortened display text. Try right clicking and copying the link itself. Easier still if you copy the link of the post itself (which can be found by clicking the post number next to the post).

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I don't understand what you mean. In flat space-time, there is no curvature, and thus in general you don't expect there to be any gravitational redshift at all, meaning that any observed redshift would be purely kinematic (of course, you might still be able to impose what looks like gravitational redshift with an appropriate coordinate choice, such as Milne coordinates).
You misunderstand. If the redshift in a general RW-model were purely "kinematic", the procedure described in #43 would yield the same redshift for small enough distances both for the curved space-time geometry and for flat space-time. Since this does not happen in general, the nature of the redshift in a general RW-model cannot be interpreted as purely kinematic.

The empty RW-model is an exceptional case since the space-time geometry is flat, so in this case, the observed redshift would be purely kinematic. But this does not apply to a general RW-model where the space-time geometry is not flat. The interpretation of the redshift in a general RW-model depends on the spatial geometry. (Only if the spatial geometry is hyperbolic there will be a non-zero "kinematic" contribution to the redshift.)
Regardless of the overall curvature, however, the amount of the redshift that is attributed to gravitation and the amount attribute to motion of the emitter or observer is still arbitrary. Some choices may seem more or less natural to some people, but many choices are possible in any event.
This may seem reasonable, but a proper mathematical analysis shows that it is simply not true. For example, no "kinematic" interpretation is consistent with the fact that the procedure described in #43 yields no cosmic redhifts between FOs for e.g., an arbitrary RW-model with flat space sections if the space-time geometry is replaced with flat space-time. Please do this (simple) calculation to convince yourself.

Chalnoth
This may seem reasonable, but a proper mathematical analysis shows that it is simply not true. For example, no "kinematic" interpretation is consistent with the fact that the procedure described in #43 yields no cosmic redhifts between FOs for e.g., an arbitrary RW-model with flat space sections if the space-time geometry is replaced with flat space-time. Please do this (simple) calculation to convince yourself.
I think the problem is that the procedure in #43 is still an arbitrary way of distinguishing between gravitational redshift and kinematic redshift. And I'm not sure it works in any event, because the relative velocity of two objects separated by some distance is arbitrary. If I select some coordinates with an interpretation of velocity which precisely gives the relative velocity between two objects in FRW space-time which would correspond to a Doppler shift, and then replace the space-time with flat space-time in those same coordinates, I'll have nothing but a Doppler shift.

I think the problem is that the procedure in #43 is still an arbitrary way of distinguishing between gravitational redshift and kinematic redshift. And I'm not sure it works in any event, because the relative velocity of two objects separated by some distance is arbitrary.
Nothing is arbitrary with the procedure described in #43. That is, the world lines of the FOs and their 4-velocities are not arbitrary and neither are the null curves.
(For sufficiently small distances the effects of geodesic deviation can be neglected, so the world lines of the FOs are still geodesics and the null curves are still null when replacing the curved space-time metric with a flat space-time metric.) Since the redshift is obtained by parallel-transporting the 4-velocity of the emitter along a null curve to the observer, this shows that the redshift obtained using the procedure described in #43 is unambiguous, only depending on the space-time geometry. Thus, changing the space-time geometry from curved to flat will in general change the redshift, so it cannot be interpreted as purely kinematic. Any concept of "relative velocity of two objects separated by some distance" is not part of the procedure; this is irrelevant since the coordinate-free concept of parallel-transport makes it unnecessary.
If I select some coordinates with an interpretation of velocity which precisely gives the relative velocity between two objects in FRW space-time which would correspond to a Doppler shift, and then replace the space-time with flat space-time in those same coordinates, I'll have nothing but a Doppler shift.
Whatever it is you are thinking of here, it would not correspond to selecting fixed observers (the FOs) and then changing the space-time geometry so the argument is quite irrelevant.

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Chalnoth