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Question about topology

  1. Nov 11, 2007 #1
    Hi, I've read this article (http://www.space.com/scienceastronomy/mystery_monday_040524.html), which says:

    "In the new study, researchers examined primordial radiation imprinted on the cosmos. Among their conclusions is that it is less likely that there is some crazy cosmic "hall of mirrors" that would cause one object to be visible in two locations. And they've ruled out the idea that we could peer deep into space and time and see our own planet in its youth."

    My understanding is that the Universe must have some kind of topology in Hyperspace. It was widely believed to be some kind of toroid, which would make the universe finite, but unbounded. Correct? If so, this new study says the topology is not toroidal, but then what is it? Does this mean now we can talk about the "edge" of the universe in whatever sense the new topology might imply?

    Thank you in advance for clearing up the confusion.

    Pavel.
     
  2. jcsd
  3. Nov 11, 2007 #2

    Wallace

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    The overall topology of the Universe is unknown. Current data is consistent with a Universe that is spatially flat everywhere on large scales, which implies that the Universe is infinite in extent. However, within the uncertainty regions permitted by current data, it is possible that the Universe has a small amount of positive or negative spatial curvature. If the curvature is positive, then the Universe would be finite and allow the 'Hall of mirrors' effect in principle. However given the constraints on any curvature are pretty tight, the Universe is very very big whatever the curvature actually is.

    Of course of all this depends on the model that is being fitted to the data being correct, which is also a subject of debate.
     
  4. Nov 11, 2007 #3
    I thought the curvature was really a "local" thing and had little bearing on the overall topology. IN other words, all possible three curvatures are still consistent with a torus. So, even if the CMBR shows flatness as far as we can see, the overall universe can still be toroidal. Can it not? And if so, how can they say "they ruled out the idea of hall of mirrors"? Perhaps they mean that as far as we can see and we will ever be able to see, there won't be any Halls of Mirrors. Is that it?


    Pavel.
     
  5. Nov 11, 2007 #4

    marcus

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    that's an old article
    it is space.com
    full of confusion because popular journalism

    you can easily find Neil Cornish journal article (peer-review professional grade)
    free for download on arxiv.org
    then you see what Cornish, Spergel et al REALLY said
    not what the space.com popular journalist THOUGHT he heard them say

    just go to arxiv.org and put in Cornish for the author and 2004 for the year or something like that

    In the space.com article, if you read it, it quotes Cornish saying
    they did NOT rule out
    http://www.space.com/scienceastronomy/mystery_monday_040524.html

    "Our results don't rule out a hall-of-mirrors effect, but they make the possibility far less likely," Cornish told SPACE.com, adding that the findings have shown "no sign that the universe is finite, but that doesn't prove that it is infinite."

    You ask the question "how can they say they rule out?"

    The answer is that they didn't say this.
    =================

    If I remember right we had a PF thread about this back in 2004, anyway I remember discussing it and reading the journal article.

    He has written more recently (2006) on the same thing, extending the distance bound outwards with new data, IIRC,
    or reducing the uncertainty.

    IMHO it is not very interesting.
     
    Last edited: Nov 11, 2007
  6. Nov 11, 2007 #5

    Wallace

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    Nope, if the Universe is flat then it is flat. It can't be flat and a torus at the same time! Curvature is neither a strictly 'local' or 'global' thing, the Sun causes curvature that causes the Earth to orbit it, yet the overall average (spatial!) curvature of the Universe may be zero.
     
  7. Nov 11, 2007 #6

    Hurkyl

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    A torus can certainly be flat...
     
  8. Nov 11, 2007 #7

    Wallace

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    More information and application to cosmology?
     
  9. Nov 12, 2007 #8

    marcus

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    that's for darn sure!
    the typical example cosmologists always give of a flat universe which is compact (finite volume) is a toroidal one.

    in 2D it is easy to see. you take a flat piece of paper and declare N and S edges to be identified-----so you have a cylinder topologically but it is still a piece of paper lying flat on the table
    and then you declere the E and W edges identified and you have a torus topologically, but it is flat.

    the only time you would need to consider a curved torus surface is if you EMBEDDED the 2D torus into a higher dimension surrounding space

    however our universe is not considered to need to be embedded in some higher D surrounding space. it is sufficient unto itself.

    so it could be a flat 3D cube like the flat piece of paper, and you identified the N and S faces, and the E and W faces, and the U and D faces----and it would be a 3D toroid thing, and it would be flat, metrically speaking.

    but all that fancy topology stuff is silly IMHO, probably the U is just infinite R-three or finite S-three
     
  10. Nov 12, 2007 #9

    Wallace

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    Sounds silly to me, if you can't measure it, it doesn't exist!
     
  11. Nov 12, 2007 #10

    pervect

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    Consider the topology of the arcade game "Asteroids". If you're not familiar with the game, you have a flat square screen, and the top of the screen wraps around to the bottom, the left side wraps around to the right hand side.

    The topology of this screen is that of a torus. However, it has no curvature. It's flat both in the intuitive sense, and in the mathematical sense that the curvature tensor vanishes everywhere.

    I don't know if anyone is seriously considering this as a possible topology of the universe, but you can't rule it out from Einstein's field equations.
     
  12. Nov 12, 2007 #11

    Wallace

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    Really? That surprises me, I would have thought that this would violate something from standard GR. Something doesn't feel right. I'll have a think...
     
  13. Nov 12, 2007 #12
    It doesn't feel right to me either - I can't help feeling that such a topology would make the twin paradox into a real paradox, and this could only be resolved by it having some local effect. But that doesn't mean that people aren't thinking about it.

    One easy to read book about this topic is How the universe got its spots by Janna Levin. I've written a review of this book at http://www.chronon.org/Science/How_the_universe_got_its_spots.php
     
  14. Nov 12, 2007 #13

    Hurkyl

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    You're referring to the the cosmological twin 'paradox', I presume? It's only a pseudoparadox -- like the usual twin paradox, it arises from making a conceptual mistake.

    A geodesic is a local maximum for the proper duration along a timelike path between two points -- i.e. when travelling between a given pair of points in space-time, an inertial traveller will always age more than a non-inertial traveller that follows a similar trajectory.

    In special relativity, space-time is affine -- there is a unique geodesic joining any pair of points. This implies that it must actually be a global maximum, and you can conclude that you will age more along an inertial trajectory than any other trajectory with the same endpoints.

    In general relativity, space-time is not affine, and you can have more than one geodesic joining a given pair of points. The above shortcut doesn't work.




    There is another common pseudoparadox involving a closed universe. Consider a flat, 1+1-dimensional space-time in the shape of a cylinder. It still makes sense to think of a "line of simultaneity" in this space-time, but it's very easy to make the mistake of assuming that lines of simultaneity are circles. In actuality, most are helixes.

    If you try to analyze this space-time in a way as similar to special relativity as possible, you find that in any particular frame, each time you go "around the universe", you have to introduce a correction factor to coordinate time. Furthermore, this correction factor is a frame-dependent quantity.

    In fact, while this space-time is locally isotropic, it is not globally isotropic -- space-time has an axis, and if you could perform an experiment involving the entire universe, you could determine whether or not a given observer is "stationary" with respect to the axis of space-time.
     
  15. Nov 12, 2007 #14

    Wallace

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    Can you give any more details of the flat but toroidal space-time? I've never heard of that and can't see how it would work, particularly in an expanding FRW metric? I'd be interested any more info?
     
  16. Nov 12, 2007 #15

    Garth

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    Does this not actually deepen the paradox?

    If the twins take separate and distinct geodesic paths (inertial trajectories) between two events, how do you decide which one is going to age more?

    My resolution is that it is necessary to involve the distribution of the matter causing the space-time curvature, ie. invoke Mach's Principle.

    Garth
     
  17. Nov 12, 2007 #16

    Hurkyl

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    You integrate [itex]d\tau[/itex] along the trajectories and compare the results.
     
    Last edited: Nov 12, 2007
  18. Nov 12, 2007 #17

    pervect

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    The serious proposal that comes to mind isn't quite a torus:

    Dodecahedral space topology as an explanation for weak wide-angle temperature correlations in the cosmic microwave background.

    This uses much the same trick of gluing together edges, but rather than gluing the edges of a square (as in the screen on the game Asteroids in my earlier example), or the faces of a cube a cube (the obvious 3-d generalization of the 2-d example above), they glue together opposite sides (faces) of a dodecahedron to form something called the "Poincare dodecahedral space".

    There's some info on this in the wikipedia.

     
    Last edited: Nov 12, 2007
  19. Nov 12, 2007 #18

    Garth

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    Yes, obviously, but the paradox only exists if we consider the twins not being able to locally distinguish between their separate inertial frames of reference.

    Integrating along the trajectories requires a global knowledge of the curvature of space-time that can only be done with knowledge of the source of that curvature - the matter in the rest of the universe.

    I liked your helical surfaces of simultaneity, can you explain more - for example, how are such surfaces determined?

    Garth
     
    Last edited: Nov 12, 2007
  20. Nov 12, 2007 #19

    Hurkyl

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    This only involves the part of space-time lying along the trajectory. The corresponding physical experiment would be "each observer uses their wristwatch to measure the time elapsed between the endpoints of their journeys".

    I don't need to see the Earth in order to feel its gravitational pull! :smile:


    I think the Euclidean geometry of the cylinder has all of the same relevant properties as the Minkowski geometry on the cylindrical 1+1 space-time... so I will propose some experiments you can do with the Euclidean geometry of real-life cylinders to get an idea for what's going on.



    The easiest way to write down coordinates for a cylinder is, of course, cylindrical coordinates. Each point of the cylinder can be given [itex](z, \theta)[/itex] coordinates. Infinitely, in fact, since [itex](z, \theta)[/itex] and [itex](z, \theta + 2\pi)[/itex] describe the same point.

    You could draw "coordinate axes" for this coordinate system -- the z axis is the line [itex]\theta = 0[/itex] parallel to the axis of the cylinder. The [itex]\theta[/itex] axis is the circle [itex]z = 0[/itex].

    There are three kinds of straight lines on the cylinder:
    (1) Ordinary lines parallel to the axis of the cylinder
    (2) Circles whose plane is perpendicular to the axis of the cylinder
    (3) Helixes

    Cylindrical coordinates are very convenient... and very special... because the axes are a line and a circle. What happens if you try to set up a different orthogonal coordinate system for a cylinder?

    This is something you can do yourself: pick any point on the cylinder and draw two perpendicular "lines" through it, neither one parallel to the axis. They will be helixes. You can use these as some sort of "coordinate axes". They are periodic like cylindrical coordinates, but the periodicity involves both coordinates: there is some nonzero a and b such that [itex](z', \theta')[/itex] and [itex](z' + a, \theta' + b)[/itex] define the same point.



    Special relativistic reference frames are orthogonal coordinate for Minkowski geometry -- for the 1+1-dimensional cylindrical space-time, they will have the same kind of qualitative properties (but with a sign flip) as the nonstandard Euclidean coordinates for a cylinder that I described above. This space-time is locally flat, so for any observer, you can draw a local line of simultaneity which is orthogonal to his trajectory -- such lines usually extend to helixes. Such observers cannot view the universe as being "circular space with linear time", because the periodicity of the universe involves both his spatial and his temporal coordinate.
     
    Last edited: Nov 12, 2007
  21. Nov 12, 2007 #20
    Thank you for your help. Would I be correct then to summarize what you say, in down to earth terms, as following:

    The recent observations, such as done by WMAP suggest that, as far as we can see, the Universe is flat, save a few very small irregularities here and there. However, such observations do not warrant any particular overall topology of the Universe, which might very well be a torus exhibiting the "Hall of Mirrors" effect, albeit on a scale beyond any empirical verification. There are reasons to believe though, according to WMAP, that the topology might be Poincare's dodecahedral sphere.

    Thanks,

    Pavel.
     
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