questions:
----------
1) does the
cosmological principle introduce any restrictions on the topology of the
universe.
2) does the topology science relies on some multidimentional theory
- like string theory
now look here,
the problem:
------------
I finally figured it out. It is about that
transparency problem. The problem was how it is possible to roll up the
transparency into a torus not stretching it so it's curvature remained the
same. I was stubborn to say that this was not possible, so we came to the
conclusion that the 2-torus in three dimenitonal eucliean space is not an good
example of two dimentional closed space of zero curvature. But if we introduced
the fourth spital dimention than it becomes possible. Like for example we
imagine a string which has one pair of zero dimentional ends, we need at least
one more dimentional space (it's two) to make it closed (to joing the ends), so
in case of a flat plane (or paper) which has two pairs of one dimentional
ends which we have to join, then we need two more dimentions to do that. That
would be 4 dimentions. Thus in four dimenitons matematically we can do with the
topology of the space whatever we want and it's curvature will still be the
same.
But the problem now is how to put this idea
into the real world. All these considerations about the topology of space were
made from "the higer groud" - I mean from the point of view of a spce with at
lest two dimentions more that the space (earlier a transparency or just a piece
of paper) we were thinking of. The problem appeares if we talk about the
topology of the Universe where there are only three spital dimentions. To make
it close like 3-torus it is essential to introduce 3 more dimentions (it's
six) to make it possible. Then I say that, that all topoloy thing implies that
out phisical space is in fact more than three dimentional, otherwise we would
stand before question: what is that thing that out space (universe) lies in.
Since this question is considered to be sensless, or at least redundant, the
only solution is to assume the first option.
It leads to the link with the
science of the nature of the space itself - like in the string theory, which
assumes that our spacetime is 10-dimentional - time plus three known spital
dimentions and 6 other which were reduced to the lenght of Planck during the
very early stages of evolution of the universe. In that case that bending our
unverse into a 3-torus around those additional axies could be easily (al least
theoreticaly) done but the "radius" of that bend would be very small. But
whatever.
Is this way of thinking of topology ok
?
further
questions:
------------------
For the curvature matter, there is general theory of relativity which
connects the energy density with curvature.
What about the topology ? What is that force that shapes the topology of
space, and can the topology change in time ?
pozdrawiam
Bartek