```Date: Apr 29, 2013 3:05 PM
Author: dan.ms.chaos@gmail.com
Subject: Re: closed universe, flat space?

On Apr 29, 9:22 pm, RichD <r_delaney2...@yahoo.com> wrote:> On Apr 24, 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.>> um yeah> Finally, somebody gets it ->> > 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 ->> --> RichFirst of all, it's more like folding a napkin and gluing its edgesthan it is folding a 'cylinder' (you can try it if you want, great wayto learn topology) .http://en.wikipedia.org/wiki/File:TorusAsSquare.svgSecond , 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' .Whatmatters is the relationship of the "inside geometry" to itself .Let's say I have a flat , plastic blanket, and some people livingpurely within the world of the plastic blanket , with normal time(same as our time ) . Now , I proceed to 'fold the blanket' . Whatwould the observers living 'inside the blanket' notice? Has anythingchanged 'inside the blanket' ? Light along the blanket still travelsits shortest path , that is , along whatever fold I made in theblanket , as to be  a straight line in the 'unfolded blanket' . Theobservers 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 , sothat it 'expands' , and is no longer as flat as the rest of theblanket . Would the observers notice? Most definitely . How so?http://www.geometrygames.org/CurvedSpaces/index.htmlThis is a great program to learn how it feels to live in asignificantly curved universe .What properties of a space can you deduce purely from living 'insidethe space'? Well, clearly, you can't deduce it's 'outside shape' to anarbitrary degree , as our blanket example illustrates . But , you canfind out about it's 'intrinsic curvature' , something independent ofthe shape you fold it it . (a blanket is still a blanket, having thesame 'internal geometry' no matter how you fold it)Let's say our observers are living in a perfect sphere (or a surfacewith 'sphere-like' internal geometry ) . That means it has the samenon-zero 'intrinsic curvature' everywhere .  But,  can our observersnotice the 'intrinsic curvature' ?Yes . Inside a sphere , they can build a triangle with three angles of90 degrees . That clearly means something funky is going on with thespace .http://qph.is.quoracdn.net/main-qimg-c1baf06b22a9cc1325585d1099a9bf63Hoverer , inside my folded paper example, they can only build normaltriangles, who's angles sum up to 180 degrees . That's why the spherehas curvature while the folded paper has none . In fact, curvature canbe defined starting from the 'excess degrees' in some small trianglearound the region . If it has more than 180 degrees ,then you'redealing with spherical geometry (positive curvature ).If it has less than 180 degrees ,  then you're dealing with hyperbolicgeometry (negative curvature ) .http://en.wikipedia.org/wiki/Differential_geometry_of_surfaces#Surfaces_of_constant_Gaussian_curvaturehttp://upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/250px-Hyperbolic_triangle.svg.png
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