Rudi Van Nieuwenhove said:
Hi, I'm grateful for all the information I got and I tried to read most of the referenced articles. I think I understood the bound systems (also due to the information in the References).
In post #8, I wrote "If the universe is finite, then I can quote J. Peacock from ref 5 (you indicated); "This is most clear-cut in the case of closed universes, where the total volume is a well-defined quantity that increases with time; so undoubtedly space is expanding in this case"."
I still do not see what is wrong with this reasoning. In my mind it is contradicting the statement that "expanding space notion is an inessential feature of understanding GR cosmology, and has no associated physics". Could you please explain what is wrong with Peacocks simple argument?
I agree now that the notion of vacuum is irrelevant within the framework of GR based cosmology. However, in the inflation theory (Alan Guth), it is an essential ingredient.
I will attempt to address this. However, the capsule summary is that 'expansion of space' means different things to different people and has no precise definition.
With regard to Peacock, note that his discussion of the closed universe is to justify 'expansion of the universe', which he then treats as equivalent to 'global expansion of space' as distinct from 'local expansion of space'; the latter he argues is fundamentally misleading in all cases. He further argues that galaxies recede in an expanding universe simply because they always were and are able to continue the motion from their initial conditions - NOT because of space expanding between them. Rather than join Peacock in using two flavors of expanding space (one of which he, of course, argues against), I prefer expanding universe, as he titles his first section, and not bother with expanding space at all.
I will make an analogy for 1 x 1 spacetime, that (like any analogy) is limited, but still gets across many points. Consider a spacetime shaped like a cone with the apex down, and cosmological time being distance up from the apex. We will compare this to the a similar model where the sides of the cone bend to asymptote an imagined enclosing cylinder, and also to the case where they bend outward faster than linear.
Considering first the simple cone, first consider the family of geodesics moving up from the apex. They recede from each other because they started receding from each other and in this geometry, such geodesics continue to have growing distance between them. Consider also other geodesics, e.g. one parallel to one of these. Such geodesic will remain parallel to one of the apex family, without growing distance between between them (this feature actually carries over to cosmologies with non-accelerating expansion). Also note that there is no difference in nature of the growing distance between the apex geodesics and the growing distance between any two arbitrary, non parallel geodesics you can imagine on the cone. There is no basis to isolate the apex geodesics as not really moving, with space growing between them, while other non-parallel geodesics are really moving relative to each other. For me, the best description of all of this is that 'expansion of the universe' is a statement about global geometry of spacetime allowing for the existence of an isotropically expanding congruence of world lines. The red shift and growth of distance between them is no different from the so called peculiar motion against the same geometric background. To me, adding the notion of expanding space is as if one insisted on imagining a rubber band placed on the cone that stretches as you move it up the cone. It doesn't add anything to understanding the geometry of the cone, or to why the apex geodesics have growing distance between them. The latter is simply due to the geometry of the cone. One final note is that one can slice the cone many different ways, getting complex patterns of 'growth' and 'shrinkage' depending on how you do it (you can even find a slicing where the elliptical cross sections oscillate between growth and shrinkage). Thus, even the 'expanding universe' notion is dependent on how you slice the manifold. However, invariants like the existence and behavior of an isotropically expanding congruence remain the same.
Now consider the case of the cone with sides bending to asymptote a cylinder. In this geometry, the apex geodesics have ever slower growth of distance between them, and a geodesic initially parallel to an apex geodesic starting near the apex will ultimately intersect the apex geodesic. To stay the same distance from the apex geodesic, such an initally parallel world line will have to deviate from being a geodesic (physically, it would have to maintain proper acceleration away from the apex geodesic to maintain its distance). Rather than talking about declining expansion of space, I simply see this case as a different overall geometry, with different geodesic behavior.
Finally, consider the case cone modified to bend outward ever faster than linear. The only point I want to make in this case is that a geodesic initially parallel to an apex geodesic will have its distance from that apex geodesic increase. A non-geodesic world line would be required to maintain fixed distance, in this case requiring proper acceleration toward the apex geodesic. This is analogous to the similar fact about cosmologies with accelerated expansion.
One final technical caveat - it is well known that distance in general relativity has no unique, preferred definition. In the above, I am using a specific geometric construction known as fermi-normal distance, which is different from what is commonly used in cosmology.
I have no idea how much all of this will help or confuse.