As I've said elsewhere, I've not had the opportunity to take a class in general relativity. Nonetheless, I understand that two major aspects of the standard cosmological model are the cosmological principle and the observation of a flat space. To get where I'm coming from, I'll try to give a brief description of my understanding of these concepts:

  • cosmological principle - This principle states that there is no privileged position within the universe. In other words, wherever any observer is located, s/he will observe approximately the same thing. Obviously the specific celestial bodies observed will change, but the expansion of the universe will be judged the same and the universe will essentially appear isotropic.

  • flat space - This observation, tested and largely verified by the WMAP satellite, shows that the large scale universe is not curved.

My natural inclination is that these two things cannot be simultaneously true. The reason it seems this way to me is that if any observer can see roughly the same amount of the universe in any direction, and the universe is of finite size, the observable portions must overlap somewhere. If the observable portions overlap, it must be possible to continue traveling in one direction and eventually end up where you started. To me, this seems to be what curvature is.

How do we reconcile these two concepts?

$\dagger$ I have read some of the articles on the subject such as those on Minkowski space and multidimensional toruses. I believe I can reconcile the two concepts and imagine a higher dimensional flat torus, but the concept is still a difficult one for me and I would love some clarification.

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    $\begingroup$ Spacetime isn't flat--- just space. There is plenty of curvature. $\endgroup$ – Ron Maimon Oct 24 '11 at 6:20
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    $\begingroup$ Yes, there is a great deal of local curvature obviously, but I was speaking of the large-scale topology only... and right... just space. $\endgroup$ – AdamRedwine Oct 24 '11 at 11:11
  • $\begingroup$ "If the observable portions overlap, it must be possible to continue traveling in one direction and eventually end up where you started." why?? Observers situated at the integer locations in the real line (...-2 -1 0 1...) with a horizon of 1 unit in both directions have overlapping "observable portions" and yet if you travel in any one direction you will never end up where you started. $\endgroup$ – user5800 Oct 24 '11 at 12:05
  • $\begingroup$ @whistles: Mind the premise stated in the previous sentence: "...and the universe is of finite size..." The real line is infinite in extent. Imagine an observer on a one dimensional path S. The assumptions that at any point s on S there is roughly the same amount of path length observable in either direction necessitates that either the path is closed (e.g. without ends) or the path is infinite. If it were finite and open, there would be end points at which the original assumption would not hold. Also, see my comment to Frank below. $\endgroup$ – AdamRedwine Oct 24 '11 at 13:11
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    $\begingroup$ A sphere was an even more obvious example. However, I think that what Jerry is trying to tell you is that, quoting your original question, there's nothing to resolve between flat spacetime and cosmological principle. The 3D spaces are tipically assumed infinite. Voilà. Edit ok, settled then, you beat me by some seconds. $\endgroup$ – user5800 Oct 24 '11 at 14:57

I'm not sure I understand what you don't understand. I am adding a answer since this would be too long for a comment...

Given the cosmological principle and the flat space time observation, the idea is that the flat spacetime is infinite, or at least very much larger than our horizon. So, yes, an observer that is, say 7 billion light years away from us would see parts of the universe that are beyond our horizon, and we would see parts of the universe beyond his horizon. There is no contradiction in this. This other observer at 7 billion years would, for example, also see a CMB with properties similar to ours but all the ripples would be different.

It is true that this could also be a flat spacetime that has the topological property of a torus but that is not at all required. This would give a flat spacetime that is finite. Do you think the universe cannot be infinite?

By the way, the cosmological principle is more than a principle, there is observational support for it. For example the CMB looks the same in all directions and the large scale structure of galaxy cluster seems uniform on the largest scales.

EDIT: Thinking about it more, you probably are thinking of the big bang as being one point (or very small region) in space at $t=0$ that then "explodes" into our universe. I had that same problem / misconception when I started learning about this. The problem is, we cannot say anything about t=0 of the big bang, because that would effectively be like a singularity filling all of space. Instead think of $t=\epsilon$, just very slightly after the big bang. At that time, you would have an infinite space filled with extremely high energy density at a very high temperature and the space would be expanding extremely rapidly. As time goes on from there the expansion rate slows, the density and temperature go down and you get the infinite universe we live in. Does that help?

  • $\begingroup$ You are right that my conception is easily compared to the locality of the Big Bang, but it is slightly different. Going to your example, the observer at 7 billion light years out would see things beyond our horizon, but would they possibly see things as far as 7 billion light years further? And what about the observers 7 billion light years from them? How far do they see? If you continue this process of leaping 7 billion light years (or any other distance), either the universe is infinitely large or you must eventually get back to where you started. $\endgroup$ – AdamRedwine Oct 24 '11 at 11:10
  • $\begingroup$ So, @Adam, is everything resolved now? You say "I cannot imagine one "confined" to a singularity such as suggested by the Big Bang Theory" but as I said in the EDIT in the answer above, you just need to think about the singularity filling all of an infinite space at $t=0$ which really means we cannot model or have a theory about $t=0$ we can only do theory at t = ϵ on... $\endgroup$ – FrankH Oct 24 '11 at 19:41
  • $\begingroup$ Yes, I think it is thanks. I never conceived as the Big Bang happening at one point and, as I said, I have no difficulty imagining an expansion of an infinite manifold, my problem was in not understanding that the manifold is (believed to be) infinite. In this case, I can imagine how the universe might be up to $t=\epsilon$ and I see how it is impossible to get to $t=0$... it's starting to make sense now. $\endgroup$ – AdamRedwine Oct 24 '11 at 19:45
  • $\begingroup$ This has been extremely helpful to my understanding. So if space is infinite and it is also said that the universe has/is expanding, does that mean we're talking about different size infinities? $\endgroup$ – dllahr Nov 1 '19 at 7:22
  • $\begingroup$ @dllahr, not really. In this case, there are not different sizes of infinities. The number of points between 0.0 and 1.0 on a line is a strictly larger infinity than the number of integers. But the expanding universe is like comparing the number of integers to the number of even integers, which are exactly equal infinities since there is a 1-to-1 mapping between them. Similarly the number of points between 0 and 1 is exactly the same as the number of points between 0 and 2 since there is a 1-to-1 mapping between them. Is that clear? $\endgroup$ – FrankH Nov 1 '19 at 18:04

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