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Thread: Curled up dimensions?

  1. #1 Curled up dimensions? 
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    I'm just a lay person but i read a lot of science material and books by physicists. I have some questions I hope some of you could answer. From what I understand in string theory, there are tiny curled up dimensions that take on the shape of Calabi-Yau spaces located at every point in space. What does it mean every point in space? What defines the size of the point in space, the size of the Calabi-Yau shape? Also, how does a string move from one shape to the next? Are the Calabi-Yau spaces connected in some way, or are they separated? How does a string that is wrapped a particular way around the curves and holes of the curled up dimension retain it's configuration as it moves to a different point in space? I'm just wondering if it just "jumps" to the other shape and retains is frequency and winding properties somehow or if there is a continuous bridge of some kind that allows a smooth transition from one space to the next. Any help would be appreciated.


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  3. #2 Re: Curled up dimensions? 
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    Quote Originally Posted by Wildstar
    I'm just a lay person but i read a lot of science material and books by physicists. I have some questions I hope some of you could answer. From what I understand in string theory, there are tiny curled up dimensions that take on the shape of Calabi-Yau spaces located at every point in space. What does it mean every point in space? What defines the size of the point in space, the size of the Calabi-Yau shape? Also, how does a string move from one shape to the next? Are the Calabi-Yau spaces connected in some way, or are they separated? How does a string that is wrapped a particular way around the curves and holes of the curled up dimension retain it's configuration as it moves to a different point in space? I'm just wondering if it just "jumps" to the other shape and retains is frequency and winding properties somehow or if there is a continuous bridge of some kind that allows a smooth transition from one space to the next. Any help would be appreciated.
    First, you need to understand what they mean by a dimension.

    The ordinary definition of a dimension is a degree of freedom. So, in our normal expeerience we have length, width, and height, and you can throw in time, but there are three dimensions that we call "space". You can easily extend this idea to simply requiring four or five or six or whatever, numbers to designate a point. That is what is done with ordered pairs and defines ordinary Euclidean space of whatever dimension you might wish.

    When they talk about curled up dimensions they are talking about what is called space-time, the backdrop for modern physics. This is not a Euclidean space, but rather is a manifold. A manifold is a mathematical construct that locally "looks like" Euclidean space, but on a larger scale might be quite different. For instance the surface of the earth locally "looks like" a flat plane but in reality is a sphere. A sphere (the surface of a ball) is a 2-dimensional manifold or a 2-manifold. Another example of a 2-manifold is the surface of a garden hose. In some string theories the formulation requires an 11-dimensinal manifold.

    Now for curled up dimensions. Consider the garden hose. It is a 2-manifold. In fact it is the cartesion product of a line with a circle, which is a tube. From a long distance that tube appears to be just a line or a curve. A line or a curve is a 1-manifold. The other dimension, what makes it a tube close up, the circle, is compactified. You don't notice it except close up or under some magnification. But that dimension exists everywhere on the surface of the tube,

    It is the same idea in the compactified dimensions of string theory. The difference is that rather than a circle, the compactified dimensions are associated with a complicated manifold, a Calabi-Yau manifold. That added structure exists everywhere just as does the circle with respect to the tube. A point on the tube is still just a point, and the same thing applies in the more complicated case.


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  4. #3  
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    "The ordinary definition of a dimension is a degree of freedom"

    Okay I understand that the dimensions are degrees of freedom, are you saying that no matter where a point is in space there is a manifold? There is no minimum distance between the points? . No matter where the particle is there is a 11-manifold?
    Also I'm confused by the concept of space-time being a fabric. If it's just a degree of freedom and has no physicality how can it bend and stretch and so forth. The various shapes of the curled up manifold determines the masses and properties of the strings, but how can space-time influence and shape the vibrational patterns if space isn't a "something"
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  5. #4  
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    Quote Originally Posted by Wildstar
    "The ordinary definition of a dimension is a degree of freedom"

    Okay I understand that the dimensions are degrees of freedom, are you saying that no matter where a point is in space there is a manifold? There is no minimum distance between the points? . No matter where the particle is there is a 11-manifold?
    Also I'm confused by the concept of space-time being a fabric. If it's just a degree of freedom and has no physicality how can it bend and stretch and so forth. The various shapes of the curled up manifold determines the masses and properties of the strings, but how can space-time influence and shape the vibrational patterns if space isn't a "something"
    A manifold, like any other topological space is a set, the elements of which are called points. There is no minimum distance between ponts, just like there is no minimum distance between points on a ruler.

    The dimension of a manifold is the same everywhere.

    Space-time is not a fabric. That is simply a figure of speech. Space-time is a manifold. It is more specifically a 4-diimensional Lorentzian manifold. It does not bend and stretch.

    I am not an expert in string theory. But the structure of space-time should influence the vibrational modes os anything that is in space-time. An ordinary string of everyday expenience can vibrate in three dimensions. Presumably a string that is sufficiently small that the compactified dimensions are important could also vibrate in all of those dimensions as well. IF these additional dimensions really exist, than in fact an ordinary string would also vibrate in those dimensions, but the amplitude would be so small that we just would not notice.

    If you are confused by string theory, then you have a lot of company. String theory is really not a theory, but rather a collection of a lot of theories, none of which are fully formed. It is not so much a theory as a research direction, a direction that may or may not eventually bear fruit. But the mathematics involved is a combination of the very sophisticated and the poorly defined. Even professional physicists don't really understand what is going on. There is a lot of conjecture that is presented to the public as fact, but simply is not fact.

    If you really want to study it there are some books out there, but they require a lot of background. One that has received some good reviews is
    String Theory and M-Theory: a Modern Introduction by Becker, Becker and Schwartz
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  6. #5  
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    okay, thank you. I'm reading the elegant universe by Brian Greene right now and it doesn't really explain my thoughts about it. Good book though. I'll check out the one you mentioned.
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