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 or
cylinder - on its 2-D surface - cannot draw any shape
which 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