Cześć Bartek, Michał, everyone, I'm happy to see this discussion starting up! :)
On 12 Nov 2002, Michal Frackowiak wrote:
On Sun, 2002-11-10 at 17:12, szajtan odwieczny wrote:
questions:
- does the cosmological principle introduce any restrictions on the topology of the universe.
This is a *big* question ;).
Firstly, it depends what you mean by "cosmological principle"
If you mean local homogeneity and local isotropy, which is (more or less) consistent with observations, then it allows all constant curvature 3-manifolds.
If you mean global homogeneity and global isotropy, which is not (yet or maybe never?) measured, then many multiply connected 3-manifolds are inconsistent with this strict definition, since some points in space or directions can be "favoured".
See http://de.arxiv.org/abs/astro-ph/0101191
- does the topology science relies on some multidimentional theory - like string theory
The physics explaining topology *might* be related to higher dimensional theories, but the mathematics does not require it, and the physics certainly does not *require* a higher dimensional 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
Agreed. Unless we redefine the metric. For example: Let's put the 2-torus in the X-Y plane, centred on (0,0,0), with radius R.
Then the metric of E^3 (euclidean 3-space, i.e. R^3 with flat metric) is
ds^2 = dx^2 + dy^2 + dz^2 = (dr^2 + r^2 d\theta^2) + dz^2 where r=\sqrt{x^2+y+2}
Under this metric, the 2-torus does not, in general, have zero curvature, and it is inhomogeneous (curvature different on different points).
So let's just make a new metric on the 2-torus:
ds^2 = (dr^2 + R^2 d\theta^2) + dz^2 where r=\sqrt{x^2+y+2}
only difference: "r^2" -> "R^2".
(Exercise: Is this a metric on R^3?)
the fourth spital dimention than it becomes possible. Like for
possible: yes; necessary: no.
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
Only if you *assume* that the only spaces which are physically possible are n-dimensional Euclidean spaces.
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 ?
I think there's still some work to do on your intuition, sorry...
the main problem is that real-world intuition is EXTREMALLY ILLUSIONARY AND DECEIVING when dealing with spacetime and its curvature. If you would think in terms of manifolds with given metric (riemann manifolds) there is ABSOLUTELY NO PROBLEM with 2-torus, 3-torus and so on. they do not even have to be embedded in other spaces.
I would say that ordinary intuition can be used as *one* possible framework for building intuition of other possibilities.
But you have to accept that part of your intuition consists of arbitrary assumptions, and consider using other ways of thinking about the space or spacetime which interests you which avoid making these assumptions.
so i strongly recommend http://arxiv.org/abs/gr-qc/9712019 with an excelent introduction to modern general realtivity
Nice :) I added it here: http://www.wikipedia.org/wiki/General_relativity
ps. you will see that you do not have to stretch a transparency to create a 2-torus, which is a perfect example of a zero-corvature manifold (PROVIDED that you imply a proper METRIC). thats the METRIC that is responsible for the curvature, not the SHAPE!!!
Bartek, please read:
http://www.wikipedia.org/wiki/Shape_of_the_universe
and maybe if you write:
http://pl.wikipedia.com/wiki.cgi?action=edit&id=Forma_Wszech%B6wiata
in po polsku, then this would help you clarify the question. You could try translating the English version, but rewriting it in the way you understand it and want to explain it.
If Michał or I disagree with you, don't worry, we'll correct it sooner or later ;).
na razie boud
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Boud Roukema