How to prove the stretching of space

[timmdeeg]

Hi,

according to several scientists, among them

J.A.Peacock: A diatribe on expanding space
M.J.Chodorowski: The kinematic component of the cosmological redshift
E.F.Bunn&D.W.Hogg: The kinematic origin of the cosmological redshift

space doesn't expand, instead the cosmological redshift is due to a combined kinematic/gravitational effect. Peacock is author of the book "Cosmological Physics".
An interesting consequence is e.g. "Summing up, the expansion of the universe is never superluminal", Chodorowski.

The Maxwell-Equations don't have a "stretching" term, but it seems a plausible assumption that the streching of space goes along with the stretching of a photon's wavelength, though - at least to my knowledge - there is no fundamental physical explanation for that. Or am I wrong?

This leads to my question, if and how the stretching of space could in principle be proved experimentally.

One can imaging long-living physicists in the center of a large void (they don't see galaxies), which supposedly expands because of its subcritical energy density, equipped with all necessary tools like watches, metersticks, ropes, light-pulse-generators, redshift analysis, whatsoever.

With which kind of experiment could the physicists prove the stretching of space?

 

[Chalnoth]

There are multiple ways of describing the universe that are mutually-compatible. The Bunn & Hogg paper that you cite makes this explicit:

"We show that an observed frequency shift in any spacetime can be interpreted either as a kinematic (Doppler) shift or a gravitational shift by imagining a suitable family of observers along the photon’s path."

In other words, it is perfectly-valid to talk about the redshift either as coming from the motions of galaxies, or as coming from the stretching of space. The stretching of space interpretation is, however, mathematically simpler in many respects, and doesn't have weird effects at the edge of the visible universe. But it is always useful to realize that there are other ways of describing the universe.

 

[bapowell]

The Maxwell-Equations don't have a "stretching" term, but it seems a plausible assumption that the streching of space goes along with the stretching of a photon's wavelength, though - at least to my knowledge - there is no fundamental physical explanation for that. Or am I wrong?

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.

 

[Chronos]

The integrated Sachs-Wolf effect is the clearest, independent [of supernova], evidence of dark energy [the strange anti-gravity effect that powers expansion]. The sloan digital sky survey [SDSS] confirmed this effect to better than 4 sigma - which is almost good enough to be considered conclusive even by particle physics standards. Photons entering a large gravitational well [like a galactic supercluster] get a gravitational 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]. See http://arxiv.org/abs/0805.3695 for discussion.

 

[timmdeeg]

There are multiple ways of describing the universe that are mutually-compatible. The Bunn & Hogg paper that you cite makes this explicit:

"We show that an observed frequency shift in any spacetime can be interpreted either as a kinematic (Doppler) shift or a gravitational shift by imagining a suitable family of observers along the photon’s path."

In other words, it is perfectly-valid to talk about the redshift either as coming from the motions of galaxies, or as coming from the stretching of space. The stretching of space interpretation is, however, mathematically simpler in many respects, and doesn't have weird effects at the edge of the visible universe. But it is always useful to realize that there are other ways of describing the universe.

Bunn & Hogg talk about the kinematic shift (motion of galaxies) and the gravitational shift as an alternative view and then identify the former as more natural. However none of these views coincides with the "stretching of space" notion.

In contrast, Bunn & Hogg criticize this notion (page 8):

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

Supposing that stretching belongs to the "properties of space itself", shouldn't this be measurable?

 

[timmdeeg]

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.

This is very interesting approach. Usually the geodesic equation deals with the momentum of particles having a rest mass. A photon has momentum, but no rest mass. Is it compulsory that this equation is valid for photons too, via de Broglie? Could you show an article? Thanks.

 

[bapowell]

Usually the geodesic equation deals with the momentum of particles having a rest mass.

Why do you say this? The geodesic equation describes massless particles as well.

 

[timmdeeg]

The integrated Sachs-Wolf effect is the clearest, independent [of supernova], evidence of dark energy [the strange anti-gravity effect that powers expansion]. ...

Yes, the discovery of accelerated expansion seems to support the stretching of space view.
But I suspect the authors I named would argue that during the parallel transport of the velocity four-vectors along the path of the photon any changes of the rate of the expansion of the universe are included. Thus, it seems still possible to argue that the cosmological redshift is due to kinematic/gravitational effects. However I am not sure myself at all.

 

[timmdeeg]

Why do you say this? The geodesic equation describes massless particles as well.

Yes, sorry I was wrong (forgot the null geodesic). So, your explanation why λ goes with a(t) is very convincing and new to me, thanks.

 

[Chalnoth]

Bunn & Hogg talk about the kinematic shift (motion of galaxies) and the gravitational shift as an alternative view and then identify the former as more natural. However none of these views coincides with the "stretching of space" notion.

In contrast, Bunn & Hogg criticize this notion (page 8):

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

The "stretching of space" picture is precisely the picture under which the redshift is a gravitational phenomenon. What they are saying is that this is simply an interpretation, and there are other ways of understanding the redshift. They claim that their way is somehow more natural (this is open to debate: which is more natural is more about personal preference than it is about the universe).

My main point is that the "stretching of space" picture is mathematically simple for many calculations, and so most people working in the field will naturally default to this view out of simple pragmatism.

The Bunn & Hogg paper is, to me, mostly useful in terms of showing that we should be careful as to whether some apparent fact of the universe is a real fact, or simply a result of a particular interpretation that disappears if we look at the universe a bit differently. The recession velocity of galaxies is one such fact: what the recession velocity of a given galaxy is depends entirely upon your coordinate choice. You'll get very different answers for the "stretching of space" and "things moving apart" views.

 

[timmdeeg]

Thank you for answering, Chalnoth,

I agree with most of that, but have still a problen with your statement:

The "stretching of space" picture is precisely the picture under which the redshift is a gravitational phenomenon.

Why do I have this choice? With regard to the Pound-Rebka experiment we talk about a fixed distance.

The result - the photon's stretched wavelength - is the same. But the physical reasons, climbing in a gravitational field versus traveling through stretching space is much different and hence not subject of personal preference.

I must have missed something, where am I wrong?

 

[Chalnoth]

The result - the photon's stretched wavelength - is the same. But the physical reasons, climbing in a gravitational field versus traveling through stretching space is much different and hence not subject of personal preference.

Except it is the subject of personal preference, because the ultimate cause of the redshift is entirely a result of the coordinate system you choose to use. That's the entire point: whether the redshift is a result of gravitation or velocity is not a physical question at all, as the answer depends upon your coordinates. The redshift itself is physical, but the distinction between the doppler effect and gravitational redshift is not.

 

[timmdeeg]

That's the entire point: whether the redshift is a result of gravitation or velocity is not a physical question at all, as the answer depends upon your coordinates. The redshift itself is physical, but the distinction between the doppler effect and gravitational redshift is not.

And presumably the distinction between the doppler effect and the stretching of space is not a physical question as well. Here I would like to mention, that both, Peacock and Chodorowski don't interpret the cosmological redshift as purely kinematic (in contrast to B&H). Chodorowski, page 9: "in a non-empty Universe there is a gravitational field, inducing a gravitational shift. Therefore, with an exception of the empty model, the origin of the cosmological redshift must be partly gravitational." To my understanding Chodorowski thus distinguishes very well different physical reasons of the redshift. Also Peacock's Equation (16) shows the combined effect of velocity and gravity.

All this in mind I still miss the meaning of your remark "whether the redshift is a result of gravitation or velocity is not a physical question at all, as the answer depends upon your coordinates." You seem to say that the physical result, the stretched wavelength, originates from mathematics, not from physics.

 

[Chalnoth]

And presumably the distinction between the doppler effect and the stretching of space is not a physical question as well. Here I would like to mention, that both, Peacock and Chodorowski don't interpret the cosmological redshift as purely kinematic (in contrast to B&H). Chodorowski, page 9: "in a non-empty Universe there is a gravitational field, inducing a gravitational shift. Therefore, with an exception of the empty model, the origin of the cosmological redshift must be partly gravitational." To my understanding Chodorowski thus distinguishes very well different physical reasons of the redshift. Also Peacock's Equation (16) shows the combined effect of velocity and gravity.

All this in mind I still miss the meaning of your remark "whether the redshift is a result of gravitation or velocity is not a physical question at all, as the answer depends upon your coordinates." You seem to say that the physical result, the stretched wavelength, originates from mathematics, not from physics.

What I am saying is that it is only coordinate-invariant quantities that are truly physical. Anything else depends upon the numbers we use to describe the space-time, and thus must be at least partly due to our choice of those numbers. As it turns out, neither the strength of the gravitational field nor the velocity of far-away objects is a coordinate-invariant quantities.

 

[timmdeeg]

What I am saying is that it is only coordinate-invariant quantities that are truly physical.

Like mass. And the redshift itself, as you mentioned. If I understand you correctly energy then is not truly physical. But I am free to measure energy in a given coordinate-system.

At least in that sense, the stretching of space should be measurable. When a gravitational wave passes by, the change of spacing of two testmasses relative to coordinates provided by meter-sticks is measurable using interferometry techniqes. An analogue measurement should be possible in a void. What happens to two testmasses, which initially are at rest relative to each other and then are allowed to move freely. They don't get a kick. Will their distance measured with meter-sticks increase over time?
In the case of a receding galaxie we have nothing but a redshift which we can interpret. In the void we have distances additionally. Is the moving apart from each other (provided that happens) still a matter of interpretation?

Thanks for your valuable comments.

 

[Chalnoth]

Like mass. And the redshift itself, as you mentioned. If I understand you correctly energy then is not truly physical. But I am free to measure energy in a given coordinate-system.

Well, energy is at least a coordinate covariant quantity, meaning that if you measure energy in your coordinate system, I can figure out what it is in my coordinate system just fine.

At least in that sense, the stretching of space should be measurable. When a gravitational wave passes by, the change of spacing of two testmasses relative to coordinates provided by meter-sticks is measurable using interferometry techniqes. An analogue measurement should be possible in a void. What happens to two testmasses, which initially are at rest relative to each other and then are allowed to move freely. They don't get a kick. Will their distance measured with meter-sticks increase over time?

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.

In the case of a receding galaxie we have nothing but a redshift which we can interpret. In the void we have distances additionally. Is the moving apart from each other (provided that happens) still a matter of interpretation?

The increase in measured distances is real. But whether you interpret this increase in distances as a velocity is, well, up to your interpretation.

 

[timmdeeg]

The increase in measured distances is real. But whether you interpret this increase in distances as a velocity is, well, up to your interpretation.

Ok, thanks for clarifying my questions.

 

[Old Smuggler]

Except it is the subject of personal preference, because the ultimate cause of the redshift is entirely a result of the coordinate system you choose to use. That's the entire point: whether the redshift is a result of gravitation or velocity is not a physical question at all, as the answer depends upon your coordinates. The redshift itself is physical, but the distinction between the doppler effect and gravitational redshift is not.

The question of interpretations of the cosmic redshift in Robertson-Walker (RW) models is not a
question about a choice of coordinates. 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. The FOs are those observers always moving orthogonally to the "preferred" hypersurfaces. The cosmic redshift is then defined as that obtained by exchanging pulses of electromagnetic radiation between the FOs.
This means that the cosmic redshift is in principle an observational result defined via specific observers, and that cannot be dependent on a choice of coordinates.

Moreover, it is possible (at least for sufficiently small regions) to change the geometry of the RW-models from curved to flat but holding the world lines of the FOs and the coordinate system fixed. One may then compare the cosmic redshift calculated in the two cases, and in general the two results will differ (these calculated results are of course independent of the choice of coordinate system). In particular, it is possible that the redshift may vanish in the flat space-time case (this happens for all RW-models with flat or spherical space sections). In these cases it is obvious that the cosmic redshift is entirely due to space-time curvature so that any interpretation in terms of a Doppler shift in flat space-time is mathematically inconsistent with the RW-model.

In sum, the question of interpretations of cosmic redshifts as described by the RW-models is not a subject of personal preference, but rather depends on the geometrical properties of the particular RW-manifold under consideration. This is a mathematical fact, and no arguments based on personal gut-feelings can change that.

 

[Chalnoth]

The question of interpretations of the cosmic redshift in Robertson-Walker (RW) models is not a
question about a choice of coordinates. 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;

Just because one choice of observers makes the universe more symmetric doesn't mean you can't choose some other set of observers instead. The math may not be quite as nice if you do that, but it is an equally-valid thing to do.

 

[Old Smuggler]

Just because one choice of observers makes the universe more symmetric doesn't mean you can't choose some other set of observers instead. The math may not be quite as nice if you do that, but it is an equally-valid thing to do.

But the fact is that the cosmological redshift in the RW-models is DEFINED in terms of a set of particular
observers (the FOs). This means that choosing some other set of observers is simply irrelevant and confuses the issue. That is, in principle the redshifts defined by these alternative observers have nothing to do with cosmological redshifts.

 

[Chalnoth]

But the fact is that the cosmological redshift in the RW-models is DEFINED in terms of a set of particular
observers (the FOs). This means that choosing some other set of observers is simply irrelevant and confuses the issue. That is, in principle the redshifts defined by these alternative observers have nothing to do with cosmological redshifts.

Only the observer who is actually measuring the redshift and the rest frame of the emitting matter. But the coordinates do not matter, and the coordinates determine whether we think of that redshift as being gravitational or doppler (or some mixture of the two).

 

[timmdeeg]

But the fact is that the cosmological redshift in the RW-models is DEFINED in terms of a set of particular
observers (the FOs). This means that choosing some other set of observers is simply irrelevant and confuses the issue. That is, in principle the redshifts defined by these alternative observers have nothing to do with cosmological redshifts.

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.

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.

 

[Chalnoth]

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.

Milne is actually a change in the geometry, and does lead to real changes in redshifts. Those changes are small out to some pretty impressive differences, but they are there.

 

[Old Smuggler]

Only the observer who is actually measuring the redshift and the rest frame of the emitting matter. But the coordinates do not matter, and the coordinates determine whether we think of that redshift as being gravitational or doppler (or some mixture of the two).

Coordinates are irrelevant for interpretations of redshifts in the RW-models. What matters is the choice of observers emitting and receiving electromagnetic radiation being redshifted (but these observers are not chosen arbitrary since they are specific observers determined from the symmetry of the RW-manifolds), plus the space-time geometry of the RW-manifold under consideration. Nothing else matters and only confuses the issue.

In particular, it does not make sense to choose some other observers and define the redshifts measured by those as "cosmological redshifts". For example, given a RW-model with flat space sections, one may choose some arbitrary FO and approximate the scale factor with a Taylor series truncated after the linear term in a small region around the chosen FO. This yields a velocity field mimicking the Hubble law in flat space-time in the small region. But the observers defining this velocity field cannot be identified with the FOs since their world lines are different from those of the FOs. (The FOs yield no cosmological redshift in the flat space-time approximation for RW-models with flat space sections.) This means that one gets something else than the Hubble law if one uses these alternative observers in the curved RW-manifold one started out with. So said procedure is indeed irrelevant for interpretations of the cosmological redshift found from the given RW-model, and choosing other observers than the FOs to define "cosmological redshifts" does not yield consistent results.

 

[Chalnoth]

Coordinates are irrelevant for interpretations of redshifts in the RW-models. What matters is the
choice of observers emitting and receiving electromagnetic radiation being redshifted (but these
observers are not chosen arbitrary since they are specific observers determined from the symmetry of the RW-manifolds), plus the space-time geometry of the RW-manifold under consideration. Nothing else matters and only confuses the issue.

There is only one emitter and one receiver for a given redshift observation. And you don't actually have to make a coordinate choice that is stationary with regard to either one, let alone stationary with regard to hypothetical observers along the path of the light ray.

 

[Old Smuggler]

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".)

 

[timmdeeg]

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.

 

[Old Smuggler]

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.

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.

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.

 

[timmdeeg]

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.

 

[timmdeeg]

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 truly 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 truly 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).

 

[Naty1]

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.)

 

[Naty1]

The stretching of space being not something truly 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 gravitational 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.

 

[timmdeeg]

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.

 

[timmdeeg]

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.

 

[Naty1]

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.

 

[timmdeeg]

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.

 

[Old Smuggler]

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).

 

[Naty1]

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.

 

[timmdeeg]

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.

 

[timmdeeg]

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)?

 

[Old Smuggler]

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.

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.

 

[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.

 

[Naty1]

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

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).

 

[Chalnoth]

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

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).

 

[Old Smuggler]

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.

 

[Old Smuggler]

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.

 

[Chalnoth]

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.

Ahh, okay, I missed that bit. This does seem like a somewhat-reasonable way of distinguishing between redshift and Doppler shift, as it is sort of a means of estimating the space-time curvature along the path of the photon. But as you mention, it's not the only way, so I still think it's worth keeping in mind that the distinction between gravitational and kinematic redshift isn't completely cut-and-dried. Some prescriptions are easier to interpret than others, of course.