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Suppose the universe is a torus

#1 User is offline   helene_t 

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Posted 2006-March-30, 03:47

In another thread, Arend and David told me that according to my estetic requirements for a model for the universe, a torus would be ok since it's symetric, and euclidian geometry applies. This is pretty cool since a torus is bounded and therefore the distribution of matter in the universe could be homogenous without the (somewhat dizzying) assumption of an infinite amount of matter in the universe. Also, it occurs to me that if the universe was very small at the time of the big bang, it could be somewhat bigger today but it's hard to imagine it could be unbounded.

Since I don't understand modern physics, this all raises some stupid questions for those forum members who do:

1) Consider the space [0:2pi]*[0:2pi]*[0:2pi] with the identification 0=2pi. Now the cyclus length of a light beam fired parallel to either axis is 2pi. The cyclus length would be longer, but finite, if the ratios between the three coordinates of the direction vector of the beam are rational. Otherwise, it would be infinite. Is this correct? If so, does it show that the universe is not isotropic? If so, is this a problem (for example, would it mean that non-isotropy could somehow be seen in the cosmic background radiation?)

2) Euclid had an axiom saying that two non-parallel lines in two-dimensional space intersect each other in exactly one point. But if you consider two of my non-axis-parallel light beams, they may intersect in many, even an infinity, of points. Is this true? If so, does the word "euclidian" have different meanings whether one is discussing classical geometry or modern, dizzying manyfold-theory? If so, which concept should be in my list of estetic requirements for a model for the universe?

3) When cosmologist talk about a bounded universe I allways think of a sphere rather than a torus. Is that correct? If so, why?

4) The attractive thing about the torus is that it's finite. Suppose time is finite, i.e. cyclical, too (I prefer a model in which time has the same geometry aa the other dimensions). Now I wonder if it matters whether time is finite or not. Consider this:
boundedtime = ]0:2pi[
unboundedtime = cotan(boundedtime/2)
Now you could formulate the laws of physics with the substitution
A: time=boundedtime
or with the substitution
B: time=unboundedtime
Locally, it doesn't matter of course, but if the constants of physics are constant in (say) model A then they will not be constant in model B. Which would lead us to prefer model A for estetic reasons. But I here cosmologists talk about changing constants of physics anyway. Which sounds to me as if the tow models are not only logically equivalent but also that even from an estetic point of view it doesn't matter whether time is finite or not. Is that correct?
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#2 User is offline   Gerben42 

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Posted 2006-March-30, 04:20

Quote

1) Consider the space [0:2pi]*[0:2pi]*[0:2pi] with the identification 0=2pi. Now the cyclus length of a light beam fired parallel to either axis is 2pi. The cyclus length would be longer, but finite, if the ratios between the three coordinates of the direction vector of the beam are rational. Otherwise, it would be infinite. Is this correct? If so, does it show that the universe is not isotropic? If so, is this a problem (for example, would it mean that non-isotropy could somehow be seen in the cosmic background radiation?)


This universe would not be isotropic. There would be one or more directions in which the distance from a point to itself would be longer and shorter. I'm not sure if this could be observed in any way if the universe is homogeneous.

Quote

2) Euclid had an axiom saying that two non-parallel lines in two-dimensional space intersect each other in exactly one point. But if you consider two of my non-axis-parallel light beams, they may intersect in many, even an infinity, of points. Is this true? If so, does the word "euclidian" have different meanings whether one is discussing classical geometry or modern, dizzying manyfold-theory? If so, which concept should be in my list of estetic requirements for a model for the universe?


This is only true in R^2. Now if you make an identification, topologically what you have done is created the space R^2 / Z^2, i.e. points are the same if they differ by an element in Z x Z. Now two lines in this space may suddenly cross in points where they didn't cross in Euclidian space.

The manifold meaning of euclidian geometry is: Curvature = 0. Hyperbolic geometry is when the curvature is negative, elliptic if the curvature is positive.

Examples of hyperbolic geometry were drawn by M.C. Escher. This is for example Circle Limit III:

Posted Image

Quote

3) When cosmologist talk about a bounded universe I allways think of a sphere rather than a torus. Is that correct? If so, why?


The de Sitter solution for a homogeneous and isotropic universe is given geometrically by R x S^3, i.e. the real time axis and a 3-sphere. On the time axis "anything" is allowed, there is a time-variable which denotes the curvature at any given time and thus defines the "size" of the 3-sphere.

As mentioned above a torus would not be isotropic.
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#3 User is offline   helene_t 

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Posted 2006-March-30, 05:58

Gerben42, on Mar 30 2006, 12:20 PM, said:

Quote

2) Euclid had an axiom saying that two non-parallel lines in two-dimensional space intersect each other in exactly one point. But if you consider two of my non-axis-parallel light beams, they may intersect in many, even an infinity, of points. Is this true? If so, does the word "euclidian" have different meanings whether one is discussing classical geometry or modern, dizzying manyfold-theory? If so, which concept should be in my list of estetic requirements for a model for the universe?


This is only true in R^2. Now if you make an identification, topologically what you have done is created the space R^2 / Z^2, i.e. points are the same if they differ by an element in Z x Z. Now two lines in this space may suddenly cross in points where they didn't cross in Euclidian space.

Now I'm confused. If the parallel axiom does not hold on a 2-dimensional torus, I suppose it will not hold on a 3-dimensional one either, since a 2-dimensional subspace of the 3-dimensional torus is a 2-dimensional torus. Am I wrong?
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#4 User is offline   Gerben42 

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Posted 2006-March-30, 06:23

No it won't hold for a 3-dimensional torus either. But still the geometry can be locally Euclidian.

A better definition of "flat space" is that every triangle has an angle sum of 180 degrees.

BTW the 3-dimensional counterpart is that the intersection of two planes is one line if they are not parallel and empty if they are parallel.
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#5 User is offline   AceOfHeart 

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Posted 2006-March-30, 12:27

some how i dont understand a thing. Bridge players are sometimes more complex creatures than I give them credit for. B)

BTW , The universe is square.

I stand by my hypothesis until I am proved wrong.
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#6 User is offline   Al_U_Card 

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Posted 2006-March-30, 12:32

Complicated and complex are not the same thing, especially in a holographic universe....where we are not what we appear to seem to be... B)
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#7 User is offline   joshs 

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Posted 2006-March-30, 12:40

There are lots of ways of thinking about geometry and defining geometries.

In Synthetic Geometry (e.g. euclid) you start with some almost set theoretic definitions, make some axioms and reach conclusions.
So for instance:
A line is a set of pairs of points in your space such that:
if (a ,b ) is on the line and ( b , c ) is on the line so is ( a , c )

Thus a line can be thought of as an equivalance relationship between pairs of points. (It determines which pairs of points are "the same-e.g. determines the same line")

The euclidian axiom is that 2 points uniquely determine a line.

Note there is no reference to any notion of "distance" or "straigtness" here.

A metric space is a set with a distance measure. The distance measure must satisfy the triangle inequality - D( A , B )+ D( B , C ) >= D( A , C )

Given a distance measure you get a definition for a line. Its the path that minimizes distance.

You can also define a manifold. A manifold is definined as a space that is"locally" euclidean (more formally you have a set of local coordinates {a map] that look euclidean and smooth transition functions that tell you how to convert one map to an nearbye overlapping map) - That is if you were a tiny ant on the manifold you could not tell that you were not on euclidian space.

A manifold can be given a metric or other structures.

When we define a torus, we can:
a. use a metric inherited from embedding the torus in n-space (that is literally the euclidean distance. In this metric I don't think the torus is flat.
b. use a metric inherited from the R^n/Z^n mapping. Take two points on the torus. Find the coordinates on R^n that they came from. Then measure the distance. Here the torus is flat. [Note: points come from many places on Z^n, so find the points that are closest to each other]

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#8 User is offline   mike777 

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Posted 2006-March-30, 18:18

I thought it is something between Hilbert space and classical phase space. Some mathematical space hitherto undiscovered which is intermediate of the two?

On a seperate subject how is Inflation (see Guth) rectified with nothing crossing the light speed barrier. If I understand, Inflation is still going on in the distant parts of the universe? If parts of the universe are expanding faster than the speed of light and the vast remaining parts expanding at a rate slower than the speed of light how we get any of these shapes.

http://web.mit.edu/physics/facultyandstaff.../alan_guth.html
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