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More on infinity

  1. Aug 19, 2008 #1
    Why does the balloon analogy not accurately describe the universe?

    Common sense would say that an expanding isomorphic homogenous space would have to be curved in a higher, infinite, spatial dimension.

    Astronomers seem unsure if the universe is finite or infinite.
    Physicists seem to, according to wiki, wish to avoid the concept of infinity.
    Mathematicians are a little opaque, for me, on most issues; so, I don't know if they are saying infinity can expand.


    Can anyone tell me what the "official" teaching on the balloon analogy is?
     
  2. jcsd
  3. Aug 19, 2008 #2
    I would say that the notion of infinity expanding goes against the definition of infinity. I think it's quite obvious that the three spatial dimensions we are bound to are finite. Although, higher dimensions, like the one the balloon model suggests, may or may not be finite; who knows, maybe they even have the predisposition of being infinite.
     
  4. Aug 19, 2008 #3
    I suppose something would either be infinite or have a boundry beyond which there is nothing.
     
  5. Aug 19, 2008 #4

    Hurkyl

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    How could it, or anything else on this topic, possibly be common sense?
     
  6. Aug 19, 2008 #5

    Chronos

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    The universe is not constrained by the limits of human comprehension.
     
  7. Aug 20, 2008 #6

    HallsofIvy

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    No, that's not true. To use, again, the "balloon" analogy, the surface area of sphere is finite but has no boundary.

    If the universe has positive overall curvature, then it could be finite without having a boundary.
     
  8. Aug 20, 2008 #7

    marcus

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    You sound as if you've already made a pretty good start towards understanding this business.

    You probably understand the point that HallsofIvy just made----a balloon surface is finite area.

    You may be going to get tripped up by the idea of intrinsic curvature. Ever since around 1850 it has been possible to understand how a surface can be curved without living in higher dimensional surroundings. You can test curvature by measuring the angles of a triangle, or testing the pythagoras right triangle rule, or various other ways.

    We could be living in the 3D analog of a balloon surface and measure the curvature and yet there might not be a surrounding space. Our 3D space could be the only space there is and yet it could be curved. This (in my experience at PF cosmology forum) is probably what newcomers find hardest to grasp.

    The thing to remember is that we have no right to expect that the angles of a triangle add up to 180 degrees. Maybe with very large triangles they add up to more! It is the case with Euclidian geometry that they add up to 180 degrees. But we have no guarantee that the geometry of our world is Euclidian.
    ==================

    About what you said about physicists not liking infinity----you have to be careful what you mean here. There are two ideas of infinity: infinite spatial extent and infinite volume is a harmless commonplace type of infinity. It is one possibility in cosmology and in my experience physicists and everybody else are comfortable with it.
    The other kinds of infinities are where a mathematical model breaks down or blows up and stops computing reasonable numbers. then it is just broken and it is time to fix it. Say some model starts giving answers like in some bounded region there is infinite energy or infinite density or infinite radiant power or electric charge. That would be a singularity (a mathematical breakdown of the theory).

    If you are still bothered by the idea that physicists don't like infinities, that you found in Wikipedia, you could bring a link to the Wikipedia article and we could look at it. They probably are not talking about infinities like infinite spatial extent or volume---it's probably the other kind.
    ==================

    The most recent WMAP report (the 5 year CMB data, implications for cosmology) gave a lowerbound estimate for the radius of curvature of space.

    They said it was at least 100 billion lightyears, if I remember correctly---some estimate of that magnitude. At least means it could be infinite too, which would be the flat Euclidian case. Anyway a finite but very large RoC fits in with the balloon picture you mentioned. If you want a link to that WMAP report let us know. I think it's neat that they gave an estimate.
     
    Last edited: Aug 20, 2008
  9. Aug 20, 2008 #8
    Wouldn't there be a boundry between the surface and what is "above" the surface?
     
  10. Aug 20, 2008 #9
    And the limit of human comprehension is what?
     
  11. Aug 20, 2008 #10
    Because the balloon analogy is common sensical.
     
  12. Aug 20, 2008 #11
    We could be living in the 3D analog of a balloon surface and measure the curvature and yet there might not be a surrounding space. Our 3D space could be the only space there is and yet it could be curved. This (in my experience at PF cosmology forum) is probably what newcomers find hardest to grasp.

    This is what I'm not understanding. I can see how there may be nothing beyond our space but wouldn't that mean there is a boundry between our space and nothing (or what might not be nothing)?
     
  13. Aug 20, 2008 #12

    marcus

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    Indeed there COULD be some surrounding higher dim'l space---though we have no evidence of it so far.

    And there also might not be. To repeat: Our 3D space could be the only space there is.
    In that case there is obviously no boundary.
    I can't say it any clearer Pj, no boundary because no other space---you either get it or you don't. :smile:
     
  14. Aug 20, 2008 #13

    I wonder what book or article would explain this. It sounds like a contradiction in terms if the "offical" teaching is that something is finite but has no end point (infinite).
     
  15. Aug 20, 2008 #14
    In general relativity the question is a simple one, spacetime is either open or closed and that implies that the universe is finite if spacetime is closed and infinite if spacetime is open.
     
  16. Aug 20, 2008 #15
    Is the spacetime concept of a closed universe mean that there is a boundry in time (the big crunch)? But I'm still confused as to where the boundry in space would be.
     
  17. Aug 22, 2008 #16
    Boundaries have no meaning in a boundless system. Weather infinite or not, curved, closed, or flat, the picture seems to be one of no "boundaries in space".

    "The analogue of the two-dimensional sphere is called a three-sphere. In the rather unlikely event of four-dimensional creatures existing, they would be able to visualize the curvature of three-dimensional space in just the same way we can visualize that of two-dimensional space. However, like the two-dimensional sphere, the possible curvature of three-space is an intrinsic property and there is no actual need of a higher-dimensional space for it to live in. Obtaining a correct mental picture of this is one of the big challenges in understanding our Universe!
    A Universe with a spherical geometry, like the surface of the Earth, has a finite size but no boundary. All points are equivalent. If we live in a spherical geometry, and travel in a straight line, we would not go on for ever and ever. Rather, eventually we would come back to where we had started from, from the opposite direction, exactly in the manner that someone travelling outward from the North Pole on the Earth eventually returns there from the opposite direction." An introduction to Modern Cosmology 2nd ed, Liddle

    So even if the geometry of the universe turned out to be one of finite scope (the 'closed' positive curvature case), the concept of boundaries, on a cosmologic scale, seems to be omitted from reality.
     
  18. Aug 25, 2008 #17
    the possible curvature of three-space is an intrinsic property and there is no actual need of a higher-dimensional space for it to live in


    This is the part I don't understand. Is there an accessible, to a lay person, literature on the subject?

    I can see how space could be curved like the iso-bars on a weather map without recourse to a fourth dimension. But I don't see how a finte object can be without boundry without recourse to a higher dimesion.
     
  19. Aug 25, 2008 #18

    Chronos

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    Unbounded, but, finite - in the words of Stephen Hawking. It is well known the universe is observationally finite [surface of last scattering]. No experiment capable of discerning whether it is 'spatially' finite has been proposed and executed.
     
  20. Aug 26, 2008 #19
    It is an intrinsic mathematical consequence, that being the maths of the theory of General Relativity. Not too much tough stuff, but does rely on differential calculus.

    I suppose this would be your best bet for a good cursory introduction. He also gives some links to other materials for further study:

    http://www.astro.ucla.edu/~wright/relatvty.htm" [Broken]
     
    Last edited by a moderator: May 3, 2017
  21. Aug 26, 2008 #20
    However, like the two-dimensional sphere, the possible curvature of three-space is an intrinsic property and there is no actual need of a higher-dimensional space for it to live in. Obtaining a correct mental picture of this is one of the big challenges in understanding our Universe!

    This is the understanding I'm trying to grasp. I understand 'intrinsic' to mean that the (for example) sadle shape of universe can be discovered by people within the universe without referencing a higher dimension (because, for example, angles in a triangle don't add up to 180).

    What I don't understand is how there can be a curved space without being embedding in a higher dimension (like the balloon analogy).
     
  22. Aug 26, 2008 #21

    PhanthomJay

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    I am in full agreement with you. The 2D curved surface of a balloon, although finite and without edge, is bounded , both within and without, by the third (and higher order)dimension. That is not common sense; it is fact. It's curvature could not exist without a third dimension; it's curvature is into the third dimension. Now whether that analogy holds for our multidimensioned universe, is I guess open to question. However, the curvature of light in 3-space, in the presence of mass, has been measured at least once during the famous experiment carried out during a solar eclipse, where such curvature from light of a star was observed and measured. My question is: was that curvature into the 4th spatial dimension? Or just an ordinary ho-hum curve in the boring 3rd dimension? I think not.
     
  23. Aug 26, 2008 #22
    There is a way that the time dimension acts as the "4th spatial dimension". Differences in the way time advances in different regions of space brings about a curvature that appears like an extra spatial dimension. The role of time as a 4th spatial dimension can be seen in the mathematical equations of relativity. It is easier to visualise this in the analogy of 2D creatures living on a 2D surface. With a bit of imagination, you could place the 2D creatures on a circular piece of paper that is perfectly flat (in 3D), and by placing suitable time dilation on the concentric circles you could convince the 2D creatures by every measurement that they can make in 2 dimensions, that they are living on a 3D sphere. In the same way, it is a matter of personal taste whether you choose to see the universe as 3 spatial dimensions + 1 time dimension or as 3 visible spatial dimensions + 1 invisible spatial dimension. Mathematically they are the same thing. Everything that can be explained by an additional invisible spatial dimension can be explained by time dilation and length contraction.
     
  24. Aug 26, 2008 #23

    Chronos

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    A concise explanation of unbounded surfaces is not trivial. It requires very complex geometrical concepts to convey with mathematical precision. The common sense, logical explanation you are seeking, Pjpic, does not exist. Analogies are good, but, logically imprecise - as you have deduced. That defect does not exist in the more rigorous mathematics behind them.
     
  25. Aug 27, 2008 #24

    PhanthomJay

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    Kev, that is a very helpful explanation, thanks. But, what about M-Theories 5th and 6th etc. extra dimensions? Are they postulated to be time-like or space-like? Or both? Could not our 4D spacetime universe be curved into these dimensions?
     
  26. Aug 30, 2008 #25
    I don't see how the balloon analogy is common sensical. If one assumes some form of a Big Bang conjecture wrt the origin of our universe, then common sense would seem to dictate that our universe is some sort of expanding wave structure, and that the contents (the flotsam and jetsam created in the wake of the expanding universal wave front) of that expanding wave structure is bounded by an expanding universal wave front.

    We, and all other material phenomena that we collectively refer to as our universe, are the inside of the isotropically expanding universal wave structure. The surface (and beyond) of this universal wave structure would seem to be necessarily off limits to us or anything else that originated inside it because the rate of expansion would circumscribe an absolute limit wrt the rate of propagation of any disturbance. This is my grossly oversimplified, common sensically speculative view of what our universe is (it's not meant as an analogy).

    I realize that the balloon analogy is offered as one way to reify (or at least visualize) the idea of curved space or spacetime which arises via a geometric interpretation of general relativity.

    But one doesn't have to take this interpretation literally. More likely, in my common sensical view, GM is itself a gross simplification of the deep reality of gravitational behavior -- which might eventually be modeled in terms of complex wave interactions.
     
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