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:
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> > > 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 -

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> > 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 .

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> > Consider a piece of paper: flat? Yes. Closed? No.
> > You can go off the edge.

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> um yeah
> Finally, somebody gets it -
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> > 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 .

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> 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.

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> 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 '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 . 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/250px-Hyperbolic_triangle.svg.png