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Re: closed universe, flat space?
Posted:
Apr 29, 2013 3:05 PM


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  > >  > Rich
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 (same as our time ) . Now , I proceed to 'fold the blanket' . 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 . How so? http://www.geometrygames.org/CurvedSpaces/index.html This is a great program to learn how it feels to live in a significantly curved universe .
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 'spherelike' internal geometry ) . That means it has the same nonzero '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/mainqimgc1baf06b22a9cc1325585d1099a9bf63 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 . 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 geometry (negative curvature ) .
http://en.wikipedia.org/wiki/Differential_geometry_of_surfaces#Surfaces_of_constant_Gaussian_curvature
http://upload.wikimedia.org/wikipedia/commons/thumb/8/89/Hyperbolic_triangle.svg/250pxHyperbolic_triangle.svg.png



