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Topic: closed universe, flat space?
Replies: 48   Last Post: May 5, 2013 2:45 PM

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 dan.ms.chaos@gmail.com Posts: 409 Registered: 3/1/08
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 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
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.