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