```Date: May 3, 2013 7:48 PM
Author: Rich Delaney
Subject: Re: closed universe, flat space?

On Apr 29, Dan <dan.ms.ch...@gmail.com> wrote:> > > >Supposedly, our universe is closed and finite,> > > >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.> > > >So, space is closed, but also flat...  back in my> > > >day, they had something called a logical> > > >contradiction ->> > > 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 .>> > And to do that, you have to twist the paper into  a cylinder... twist,> > flat... see the problem here?>> > > 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.>> > wooosh!  Over my head ->> First of all, it's more like folding a napkin and gluing its edges> than it is folding a 'cylinder' (you can try it if you want, great way> to learn topology) .> http://en.wikipedia.org/wiki/File:TorusAsSquare.svg> Second , it doesn't matter what it's "outside geometry" looks> like . What matters is what the observers living "inside" the> space notice.  The "outside geometry" is inaccessible to> the 'inside observers' .What> matters is the relationship of the "inside geometry" to itself .>> Let's say I have a flat , plastic blanket, and some people living> purely within the world of the plastic blanket , with normal time.>  Now , I proceed to 'fold the blanket' .Into what?  A toroid?> What would the observers living 'inside the blanket' notice?> Has anything changed 'inside the blanket' ? Light along the> blanket still travels> its shortest path , that is , along whatever fold I made in the> blanket , as to be  a straight line in the 'unfolded blanket' . The> observers wouldn't notice anything has changed . In fact, for them ,> nothing has changed .>> Let's say now , that I heat up a small portion of the blanket , so> that it 'expands' , and is no longer as flat as the rest of the> blanket . Would the observers notice? Most definitely .> http://www.geometrygames.org/CurvedSpaces/index.html> This is a great program to learn how it feels to live in a> significantly curved universe .I tried, got nothing but the usual computer aggravations.> What properties of a space can you deduce purely from living 'inside> the space'? Well, clearly, you can't deduce it's 'outside shape' to an> arbitrary degree , as our blanket example illustrates . But , you can> find out about it's 'intrinsic curvature' , something independent of> the shape you fold it it . (a blanket is still a blanket, having the> same 'internal geometry' no matter how you fold it)>> Let's say our observers are living in a perfect sphere (or a surface> with 'sphere-like' internal geometry ) . That means it has the same> non-zero 'intrinsic curvature' everywhere .  But,  can our observers> notice the 'intrinsic curvature' ?> Yes . Inside a sphere , they can build a triangle with three angles of> 90 degrees . That clearly means something funky is going on> with the space.> http://qph.is.quoracdn.net/main-qimg-c1baf06b22a9cc1325585d1099a9bf63> Hoverer, inside my folded paper example, they can> only build normal triangles, who's angles sum up to> 180 degrees . That's why the sphere has curvature while> the folded paper has none.So you're saying, that an inhabitant of a toroid orcylinder - on its 2-D surface - cannot draw any shapewhich will tell them it's warped?  I find that hard to swallow.> In fact, curvature can> be defined starting from the 'excess degrees' in some small triangle> around the region . If it has more than 180 degrees ,then you're> dealing with spherical geometry (positive curvature ).> If it has less than 180 degrees ,  then you're dealing with hyperbolic
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