PeterDonis said:
After reading through the paper, I don't think the claimed demonstration is correct.
I believe you are missing several things. Before substantive comments, I will make an argument by authority - by definition, not substantive. The referenced paper has been cited two dozen times by a wide range of authors, with no suggestion there is a problem with the paper. Both the paper itself, and many of the papers that cite it are published in peer reviewed journals.
PeterDonis said:
First, a key preliminary observation: the surfaces of constant FRW coordinate time are everywhere orthogonal to the worldlines of comoving observers. That means that, if we take the set of spacelike geodesics orthogonal to a comoving worldline at some event, and extend them indefinitely, they will span a surface of constant FRW coordinate time.
The first statement is true, the second is false. The constant cosmological time surfaces
are orthogonal to each comoving world line but geodesics in these surfaces
are not spacetime geodesics. Pretty much the rest of your arguments fall down because of this mistake. The simplest demonstration is using the Milne case of linear expansion (but the same is true of
all FLRW solutions, and this fact is rather well known). The hyperbolic surfaces are orthogonal to every comoving word line, but they obviously do not consist of spacetime geodesics. Instead, the spacetime geodesics orthogonal to a given comoving world line are constant time lines in the Minkowski coordinates based on that world line, that intersect the 'big bang' (light cone bounding the Milne solution). This captures the whole essence of the general demonstration in the paper - the cosmological constant time slices are all unbounded, while the Fermi slices are all bounded, yet the latter are still a global foliation.
Note, it is in some sense obvious that Fermi constant time surfaces cannot be orthogonal to any comoving observer except the defining one. This is because, in Fermi-normal coordinates, every comoving observer other than the origin is moving, therefore not orthogonal to spacelike geodesics orthogonal to the defining world line, Instead, each comoving observer defines a
different set of constant Fermi-time surfaces.
PeterDonis said:
It also means that the "comoving proper distance", i.e., the geodesic distance within a surface of constant FRW coordinate time between two comoving worldlines, is also the spacetime geodesic distance--i.e., it is what we would get if we did the "naive" construction of picking the spacelike geodesic orthogonal to the first worldline and pointing in the direction of the second worldline, and extending it to the second worldline, and measuring arc length along it.
Simply false, as noted above.
PeterDonis said:
What does this mean in terms of the construction given in the paper? Unfortunately, it appears to me to mean that the construction goes wrong right at the very start. Equation (10) in the paper says that ##\dot{t} \neq 0## for a spacelike geodesic orthogonal to a comoving worldline. But that is saying that such a geodesic does not lie in a spacelike surface of constant FRW coordinate time. That seems obviously wrong. The 4-velocity of a comoving worldline is ##(1, 0)## everywhere. Since the metric is diagonal, any vector orthogonal to ##(1, 0)## cannot have any ##t## component, i.e., it must have ##\dot{t} = 0##.
Again, wrong as noted above.