From: Anders Sandberg (asa@nada.kth.se)
Date: Mon Aug 19 2002 - 06:00:40 MDT
On Sun, Aug 18, 2002 at 06:31:53PM -0700, Lee Corbin wrote:
> Rafal writes
>
> > Anders Sandberg wrote:
> >
> > Actually, it turns out that you can have infinite volume closed universes
> > and finite volume open universes if the topology is non-trivial.
> >
> > ### Tell us more! This is very cool.
> >
> > Rafal
>
> I second the motion! :-)
As the saying goes, "If you have said A, you have to say B".
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). The equations only tell us how spacetime
bends locally, but there are very natural ways of interpreting these
curvatures into the global shape of spacetime - a 3-sphere for closed
universes (i.e. like the surface of a globe but in 3D - finite but you
will get back to where you started if you travel straight) or an infinite
space in open universes.
But this is just the simplest case. One could imagine other
possibilities with the same curvatures but very different properties.
For example, one could have a multiply connected open universe: only a
small part of it exists, and when you get to the border you end up on
the other side. 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. 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.
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.
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
-- ----------------------------------------------------------------------- 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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