Why must a flat universe be infinite if S^1×S^1×S^1 is flat and bounded?

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Hurkyl
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What exactly is the problem with a flat, finite universe?

I've often heard it quoted that a flat universe must be infinite... but it's easy enough to design space-times that are perfectly flat yet are bounded. For example, S^1*S^1*S^1. What additional information is used to suggest a flat, finite universe is not feasible?
 
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No your correct a flat finite universe is perfectly feasible in the shape of a torus. It's just observational evidence for a torus shape universe is non-existant (that's not to say tere's evidence against it).
 
But is a torus the only possible shape?
 
Your stretching my knowledge of cosmological models, but it's the only one I've heard tht meets observational evidence but is bounded and finite.
 
Originally posted by jcsd
No your correct a flat finite universe is perfectly feasible in the shape of a torus. It's just observational evidence for a torus shape universe is non-existant (that's not to say there's evidence against it).

this came up earlier at PF---astronomy forum I think.
someone posted a link to a paper by some folks who
were looking for evidence (I think it was repeated galaxies
but my memory is not clear about this) of some finite flat shape.

I wonder how careful one should be? Maybe one should only
say "spatially flat" and never mention infinite extent as probable consequence. I've noticed one cosmologist Michael Turner being careful about this---but others, like Ned Wright IIRC, being
unrigorous about it and talking as if flat implied infinite.
(my sympathies are with the unrigorous but I will pull the socks
up on this if you all think it best)
 
The reason I bring it up is that when I've read up on big bang theory, it is terribly "obvious" to me that it is describing a finite expanding universe... so obvious that I once read something on the horizon problem and thought that it actually said the above! (I just recently found it again and discovered it was claiming an infinite universe, which prompted this post)
 
Originally posted by Hurkyl
The reason I bring it up is that when I've read up on big bang theory, it is terribly "obvious" to me that it is describing a finite expanding universe... so obvious that I once read something on the horizon problem and thought that it actually said the above! (I just recently found it again and discovered it was claiming an infinite universe, which prompted this post)

delicate issue
the main thing is that the classical singularity
is thought of as having spatial extent
(it is not a point)

if space has a toroidal topology
(a cube where when the little guy goes out the top
he reappears at the bottom, and when he goes offstage
on the right he reappears coming in at stage left, and
front and back)
then I guess, tell me if I'm making an unjustified leap)
that the topology was probably like that at the time of
the classical singularity.

Do I really suffer any downside if I go on thinking of
the classical singularity as infinite in extent?

I'm ready to consider informed advice on this. Maybe
there is some way the singularity could be finite!

But so far it seems simplest if I assume it infinite in extent.

My experience with what I've read agrees with yours. The
authors tend to assume space is flat and infinite. I instinctively
go along with the crowd---but am not quite sure why.

So if the classical singularity is resolved it would (by this assumption) be across a spatially extended front.
the resolution would not be localized to a point.
divergence would be controlled throughout an extended 3D volume [?] which could however have a compact topology
along the lines you suggested
 
Hurkyl, as I recall, Eh posted this same question on the old PFs. It was side-tracked, but one of the points that was brought up was this:

If the Universe is flat and infinite, then there could be many (possibly infinite) "little" Universes popping up all over the place; while, if the Universe altogether is finite, and expanded from a smaller point, then it would take on a spherical shape.

I don't know how consistent this is with current cosmological models (which discard the idea of expanding from one point, for the idea of all parts of spacetime expanding), but it's all I can remember from the thread (I'll have to look it up again on my trusty Archive C.D. :smile: (that is, as soon as I get my home computer working again ).