what's with that transparency

[Boud Roukema]

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:
> > ----------
> > 1) 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


> > 2) 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