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When someone says that the universe is flat it should be infinite. But, this contradicts the Big bang because infinity can never be reached from below. If the universe is in fact infinite then shouldn't it be infinite always, I mean all the 'time'? Is Big bang a lie or universe does have a curvature?

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    $\begingroup$ It seems to me that you're missing a third option, which is that it's you who is wrong and not everybody else. See physics.stackexchange.com/q/136860. $\endgroup$
    – Javier
    Sep 23, 2017 at 12:46
  • $\begingroup$ Also related: physics.stackexchange.com/q/9419/2451 $\endgroup$
    – Qmechanic
    Sep 23, 2017 at 12:57
  • $\begingroup$ The flat FLRW universe is infinite in space, but not in time. Time still starts 13.8 billion years ago when the universe is infinitely large and has an infinite density. This way the universe does not grow from small to large, but only stretches into a lower density. This model IMHO has no physical meaning, because it is singular (infinite) in size and mass in addition to density and beyond the zero time. Ironically this model is viewed as "correct" by the consensus while in fact it only shows the failure of the FLRW model. $\endgroup$
    – safesphere
    Sep 23, 2017 at 15:38

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There are two misconceptions in your question:

First, flat does not necessarily imply infinite: Topologically, a torus is just a rectangle with oppposite edges identified, and thus can be as flat and finite as said rectangle. However, in the specific case of Friedmann universes, spatial flatness does indeed imply an infinite universe, so I'll let this slide.

Second (and more importantly), the Big Bang is a metric degeneracy (distances go to 0) that can happen whether or not your space is finite or infinite. It is not an explosion located at a certain point within space growing to finite size within finite times, but something that happens everywhere all at once. Therefore, the term 'everywhere stretch' has been proposed as a more descriptive name in certain videos aimed at the general public.

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  • $\begingroup$ Could you please elaborate on torus being a rectangle? Thanks! $\endgroup$
    – safesphere
    Sep 23, 2017 at 15:41
  • $\begingroup$ @safesphere: take a sheet of paper and roll it up into a tube, connecting the longer edges; if you could also connect the shorter edges by bending the tube into a ring, you'd get a torus which still had the same flat intrinsic geometry as the sheet; in the real world, you of course cannot do so because such a flat torus can't be smoothly embedded into $\mathbb R^3$ $\endgroup$
    – Christoph
    Sep 23, 2017 at 16:41
  • $\begingroup$ Can it be embedded in R4? And if yes, then would it be a true statement that the 3D equivalent of such a "curved in a flat way" space would be a hyper-torus that could be embedded only in R5? Well... thinking of it, you'd have to circle the hyper-torus 3 ways, so it could only be embedded in R6. Am I close or way off? $\endgroup$
    – safesphere
    Sep 23, 2017 at 16:48

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