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 non-infinitesimal 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 non-periodic 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 n-sphere, 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.