From: Lee Corbin (lcorbin@tsoft.com)
Date: Mon Aug 19 2002 - 22:01:50 MDT
Anders writes
> > I second the motion! :-)
>
> As the saying goes, "If you have said A, you have to say B".
Perhaps in Europe. What does that mean, exactly? It's not
a saying in English. Perhaps it means, sort of, that you
have to do what you said you would? But you never said you
would.
> The normal derivation of the shape of spacetime in elementary cosmology
> works by solving the Einstein equation in a homogeneous and isotropic
> spacetime (every point is alike every other). This produces the usual
> closed universes (recollapses after a time) and open (doesn't
> recollapse), as well as flat ones (in between - never collapses but
> expands more and more slowly).
Okay, but in the excellent paper to which you directed us, (see
below), there is so much talk about topology. Now, I had always
supposed that "open" and "closed" vis-a-vis universes meant only,
as you just said, whether they're going to collapse or not.
So I supposed that this had nothing to do with the topological
meanings of open and closed. (For the non-mathematician, "closed"
in topology means that a set contains all its limit points, that
is, if it has any edge, then it's a hard edge and the edge is in
the set being called "closed". An open set is a set whose
complement is closed. An entire space X is both open and closed.
It's open because of course it contains all its limit points (it
contains everything), and it's closed because its complement---
the null set---also contains its limit points (it doesn't have
any)).
What is the relationship, if any, between topological open and
closed, and open and closed universes?
> ...A simple case would be if the "real" universe was a cube
> where light and particles leaving one face would reappear on
> the other. It would still seem infinite, but one would only
> have a finite volume.
Yes. The acid test of having a finite volume, to me, has always
been filling up the space with balloons measuring a meter on a
side. If after finitely many balloons the space is entirely
full, then the volume is finite. But if it takes more than
that many balloons, the space is infinite.
> In the cubic example the curvature of space is zero, but if you use a
> dodecahedron or icosahedron then you instead get a negative curvature
> (see the great animation "Not Knot" from the Geometry Center for
> flythroughs; a simple picture can be seen at
> http://www.american.edu/academic.depts/cas/mathstat/MAA/fall98/notknot.gif).
> So you could have a finite but open universe.
So here you definitely are using "open" in some topological sense.
How so (if you didn't already explain above)? For surely here you
aren't talking about collapsing universes.
> On the other hand, you could have a closed universe where the volume is
> infinite. Think about the globe analogy and imagine a earth map wrapped
> twice around the globe: when you move across the zero meridian from the
> east you don't cross over to the normal Atlantic but a new fantasy
> Atlantic. Continuing westwards you pass fantasy America, the fantasy
> Pacific, the fantasy Asia and Europe until you again pass the zero
> meridian and end up in the real Atlantic. This world would be a
> 2D-sphere with constant positive curvature but twice the area of the
> normal sphere. One could imagine this wrapping continuing even further
> and in all directions - when you circle the Earth you end up in more and
> more remote versions of the geography. In the same way our universe
> could be a 3D version of this, with constant positive curvature but
> nowhere repeating.
Yes, but from topology I would expect that of course such a universe
is both open and closed. Is there a cosmological space that is open
but not closed? (Maybe a "relative topology" or something? But that
seems cheating.)
> There are some issues of what kinds of topologies are self-consistent,
> but in general there seems to be no obvious reason except simplicity to
> assume the "classic" topologies. But the world could be strange...
>
> http://www.maths.lse.ac.uk/Personal/mark/topos.pdf
I'm only half-way through this eminently readable article (and for me
unfortunately, diagram 5 did not appear). The most remarkable part
so far: "It has only been in the previous five years [from 1999] that
serious investigations into the topology of the universe have been
undertaken [!]."
Lee
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