From: Anders Sandberg (asa@nada.kth.se)
Date: Tue Aug 20 2002 - 03:36:40 MDT
On Mon, Aug 19, 2002 at 09:01:50PM -0700, Lee Corbin wrote:
> 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.
It is a Swedish saying that means roughly that if you have said the
first element of a series or given a hint you make an implicit promise
to say the rest too.
> What is the relationship, if any, between topological open and
> closed, and open and closed universes?
As far as I understand, there is no strong relationship. GR seems to
imply that spacetime is an open set (since the singularities can't be a
part of it), but that is just local. But I'm not knowledgeable in global
relativity, which is the discipline to look to here.
In general people seem to prefer (ignoring black holes and stuff) entire
spaces. But M-theory seems to imply that our spacetime is actually an
open set embedded in the entire bulk of the universe.
> > 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.
I meant "having negative curvature"; this is probably an improper usage
of the word and only bound to confuse, but the identification is made so
strongly in all textbooks that I just forgot to distinction.
> > 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 [!]."
Yes, it is a bit surprising. But maybe people were too fond of the smooth
differential geometry of spacetime to even consider group properties and
other discrete features.
-- ----------------------------------------------------------------------- Anders Sandberg Towards Ascension! asa@nada.kth.se http://www.nada.kth.se/~asa/ GCS/M/S/O d++ -p+ c++++ !l u+ e++ m++ s+/+ n--- h+/* f+ g+ w++ t+ r+ !y
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