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Re: closed universe, flat space?
Posted:
Apr 24, 2013 5:11 PM


On Apr 23, 10:23 pm, RichD <r_delaney2...@yahoo.com> wrote: > Supposedly, our universe is closed and finite, although the > geometry and topology isn't precisely known. This means > a straight line (geodesic) traveler must return to his starting > poiint, yes/no? Hence, curved space. > > At the same time, astronomers claim, that space is flat, to > the precision of their measurements. Usually, this is > accompanied by the comment that the universe is balanced > between infinite expansion, or eventual collapse, we can't > tell which. > > So, space is closed, but also flat... back in my day, they > had something called a logical contradiction, which was > considered a bad thing, but I guess times change... > viva progress  > >  > Rich
Space can be 'closed' , and also , 'locally flat' , in the sense that the Riemann tensor vanishes ,or there exists ,for any point of the space, a noninfinitesimal spherical section around that point that's indistinguishable from flat space .
Consider a piece of paper: flat? Yes . Closed? No. You can go off the edge . Now make it so that when you go trough the 'up' edge you end up coming from the 'down' edge , and when you go go trough the 'left' edge you end up coming from the 'right' edge . More specifically, this space is the factor group (R^2) / (Z^2) . The space is still flat, as far as definitions tell . However, it's closed.
What's interesting is that there are directions where you can travel in a straight line and newer return to your starting point , even though you might end up arbitrarily close to it . This is similar but not the same as what happens to a ball in an idealized game of billiards : It can follow a periodic trajectory , and eventually return to its starting point, but it usually follows a nonperiodic one, and goes all over the place : http://mathworld.wolfram.com/Billiards.html
The only metric I can think of where the traveler in a straight line *must always* return to his starting point for every possible starting point is some sort of nsphere, though it would have to have exceedingly small curvature to match experiments. It would be interesting to find another metric where this is valid, though I doubt another one exists .
I for one think of the universe as infinite in extent, and mostly flat, though my reasons tend to be philosophical : Our world should be the most perfect, that is "the one which is at the same time the simplest in hypothesis and the richest in phenomena" (Leibniz) . And , like it or not, an infinite (and 'roughly isotropic') universe matches these criteria.



