How to prove the stretching of space

Old Smuggler

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

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

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

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

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

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

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

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

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.
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

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/~tamara...s_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

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.
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.

timmdeeg

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)In the sense, that there is no other choice, so my understanding.

timmdeeg

Thanks to everybody for your explanations.

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,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 issome 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

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

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

Chalnoth posed a wonderfully insightful physical explanation in another discussion:

timmdeeg

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.