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