Is the universe actually expanding? Up until recently I was fairly sure that the universe is expanding, i.e. the (spatial) metric is changing proportionally to the scale factor, such that the distance measured between objects is increasing over time. However I read this article and it has really made me doubt my understanding. So, I'd really like to clear my doubt up. Is the universe actually expanding (in the sense that the observational data agrees with the hypothesis that the universe is expanding), or is something else going on (from what I've read, a large number of scientists do think that the universe is expanding, and at an accelerated rate)?
The aforementioned article essentially states that we should not conclude that the universe is expanding due to the wavelength of photons being stretched (N.B the author considers an FRW universe in their analysis), since the wavelength of photon is not a property of the photon, but of the "photon + observer" system, thus whilst observers in a comoving FRW reference frame will observe a photons wavelength being stretched and hence observe it redshifting, those in a locally flat inertial reference frame at each point along the trajectory of a comoving observer will just observe this as a series of Doppler shifts.
The author states that expansion of space only has meaning to observers in a comoving reference frame relative to the expansion of the FRW metric and hence one should not conclude that the space is expanding, since, with an appropriate change of coordinates, the expanding of space [as an explanation for the change of wavelength of a photon] can be extinguished and replaced with the simple Doppler shift.
At least this is the gist that I got from it (I may very well have misinterpreted it though)!
 A: The whole premise of the paper is wrong. Citing Synge in 1960 is irrelevant, the Big Bang was questioned to some extent, by a minority of physicists, until the cosmic microwave background was discovered in 1965. The first citation is a diatribe, in arxiv, and the reference to Weinberg saying there was no space expansion was not cited - Weinberg wrote his Gravitation book and completely describes the Big Bang cosmology and more. 
The author of that paper is confused
He says it can all be explained as Doppler shifts for observers moving in comoving coordinates, but explain it all as successively small Doppler shifts for observers not too greatly far apart. But we can do that ourselves, see progressively further galaxies and the larger red shifts, and so each is moving faster. It certainly looks like something expanding, sort of a strange thing that as they are further they travel faster if not. That author says nothing about that. The Hubble law measured has distances proportional to velocities. He does not explain that. 
Doppler is two observers moving wrt to each other. Sure as he explains below he can measure the Doppler of nearby observers, and then implies you add them all up. But redshift due to relative constant velocity motion is different than space expanding, because the light is then received when the relative velocities of the two has changed, and so the results are different. He is wrong. 
The case he makes is below. The fallacy in the referenced paper is clear in the following paragraph from p99 of the paper, second paragraph: "However, if we consider one of the intermediate observ- ers, we can ask what they see. To them, their adjacent observers are moving away in locally  at space-time, and that the redshifting they see is simply the Doppler shift due to motion. So the entire redshift between A and B can be considered just a long series of Doppler shifts. But, again, this is di cult to visualise without inserting our long chain of observers into the picture"
Well, good enough, but the two observers actually have a different velocity when transmitted than when received. He does not account on that. They were not inertial wrt the other during the light propagation. If he places them very close to ignore that he is ignoring the main effects - yes, you cannot measure space expanding if you are just 100 Kms form another observer, practically. His argument is thus totally bogus.
It is unfortunate it was published in that Australian journal, but it did not seem like a scientific paper anyway. He mixed up the idea of explaining something to the public in a simple way with one of debunking a completely accepted scientific explanation. Let's hope somebody writes to that journal. 
I suggest you read one of the good General Relativity books. Some of them explain the difference, but in all cases it is useful to see the derivation from the FRW metric (on which that author was also wrong, it is the only solution that is isotropic and homogeneous). 
A: The issue is that concepts like distance or relative velocity are problematic in curved spacetimes.
Personally, I'd stress that the meaning of metric expansion of space in FLRW cosmology is that matter is distributed in homogeneous spacelike layers of constant age, and the distance as evaluated within these layers between any two particles of same age increases over time.
Now, on to the redshift. The article is correct that there isn't really a fundamental difference between doppler shifts, gravitational and cosmological frequency shifts: All of them can be explained kinematically in terms of the relative orientation of the velocity vectors of light source, photon and observer at times of emission and absorption. If you evaluate relative velocities at time of absorption only (which can be achieved by parallel transporting the source velocity along the light path), you will in fact recover Doppler's formula, which remains valid for generic frequency shifts.
The thought experiment from the article using accelerating rockets is a nice illustration. Another one is exchanging baseballs instead of photons: If pitcher and catcher are both at rest relative to the Hubble flow, they will not agree on its kinetic energy even if there's no energy loss due to friction. However, I doubt many people would attribute this to the baseball's de Broglie wavelength being stretched by spatial expansion...
A: The paper is entirely correct.
The expanding-space picture of cosmology can be useful, but will lead you to wrong conclusions if you don't understand its nature and its limitations.
If I've got a hunk of matter, I can model it as a continuum solid, or as a bunch of atoms, but not both. If I take the predictions of the continuum model and the atomic model and add them together, I'll get wrong answers. At best, I'll get twice the right answer; but if I try to apply the continuum picture at a scale where it isn't even an accurate approximation, I'll get an accurate answer plus nonsense—i.e., nonsense. The only way to get an accurate answer at small scales is to use the atomic model only.
The relationship between the expanding-space picture of cosmology and the local picture is the same. The FLRW model with its scale factor is not a background on which the local physics happens. It's an approximation to the local physics. That's why the cosmological redshift (from the FLRW picture) can also be derived as a relative-motion Doppler shift between a chain of galaxies, as mentioned in the paper. They're two descriptions of the same phenomenon.
Many people think that there are extra effects of expansion of space that must be added to the ordinary local physics to get a complete description of nature. That idea has even made it into published papers in respectable journals, but it's just as wrong as the same belief about continuum and atomic solids. Here's an example of a published paper that makes the mistake, with some more discussion.
