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Properties of 6 Dimensional Spheres

  1. Mar 8, 2009 #1
    It occured to me that most unification theories propose at least 6 dimensions, and that the 6th dimension would look like a sphere or a doughnut. For the sake of greater simplicity, I will use a sphere for my question. My question is, if an object were travelling in the 6th dimension, would they be travelling 'in' or 'on' the the 6D sphere?

    Also, is the 6D sphere a 2D sphere embedded in 3D space plus time, or is it a 6D sphere, as in 5D + Time? For instance, if you wanted to find the surface area of the sphere, would you use the regular equations of a 2D sphere, because it is a sphere embedded in 4D space-time, or would you use the equation for surface area of a 5D sphere and add time?

    Thanks in advance.
     
  2. jcsd
  3. Mar 9, 2009 #2
    A single dimension cannot look like something 2-dimensional, such as a sphere or donut - so I suppose you are either referring to 5+1-dimensional spacetime with two spatial dimensions compactified on a sphere, or to the 6 'extra' (compact) dimensions in superstring theory.
    Anyway, in the first case, try to imagine (:)) 3+1d spacetime with a sphere at each point in spacetime. If an object were travelling on the sphere it is indeed restricted to those two dimensions.
    In the http://en.wikipedia.org/wiki/Superstrings#Extra_dimensions" the situation is actually the same; supposing we compactified the extra dimensions on a (6-)sphere, you would be travelling on that sphere in any of those compact (periodic) directions.

    Referring to the superstring case, your 6d sphere is actually embedded in 9+1-dimensional spacetime.
     
    Last edited by a moderator: Apr 24, 2017
  4. Mar 9, 2009 #3
    Also, don't confuse spheres with balls (giggle).

    A ball is a solid, spherical object, while a sphere is the surface (only!) of the ball.

    So, for example, a 2-ball is a solid disk. The boundary of the disk is a one dimensional circle, or a 1-sphere. A 3-ball is a solid object, like a bowling ball. The boundary of this object is an infinitesimally thin shell, called a 2-sphere. And so on.
     
  5. Mar 9, 2009 #4

    So, going into an unrelated topic, if you were to establish that there is the sphere of 2 dimensions embedded in 4D Space-Time, what would the inside of the sphere be like, if there was one? From what I have concurred, the sphere is defined by having surface area, but if you were to travel through 'surface,' would you find that the area inside the boundaries of the sphere to be hyperspace?
     
  6. Mar 9, 2009 #5
    Now that you have taught me the right diction on this subject, can you answer my questions from my first post in a concrete way? Would one use the regular equation for a regular sphere to find the surface area of the higher dimensional sphere?
     
    Last edited by a moderator: Apr 24, 2017
  7. Mar 9, 2009 #6
    I am interested in the language used here, as in phrases like 6-sphere and 5+1 D spacetime. Is there a guide to phrases of this kind? What branch of the math or physics tree do they come from?

    Thank you for this and many past kindnesses.
     
  8. Mar 10, 2009 #7
    They're just fancy (short) names for '6-dimensional (hyper-)sphere' and '6-dimensional http://en.wikipedia.org/wiki/Minkowski_space" [Broken]', which treats the time-direction as something special (as opposed to 6-dimensional Euclidean spacetime, where all directions - including time - are on an equal footing).
    I don't know of any guide to that kind of language :smile: but I'm sure the mathematics->geometry/topology section of Wikipedia can help you out.

    At least in higher-dimensional Euclidean spacetime one would use the http://en.wikipedia.org/wiki/N-sphere#Volume_of_the_n-ball". To be honest though, I wouldn't know how this works out in a general spacetime - I guess it's just the same equation, but you'd have to be careful in stating what the radius of the sphere is, because the notion of distance can be different for each direction.

    [edit]http://www.absoluteastronomy.com/topics/Sphere" [Broken] might be useful.
     
    Last edited by a moderator: May 4, 2017
  9. Mar 10, 2009 #8
    Now, I'm pretty sure that if you had an object on a 5+1 D sphere, the object would travel on the sphere, and not inside of it, because the sphere by definition has no inside. My next question, which I've posted above already, is this:

    If you were to establish that there is the sphere with 5 + 1D Space-Time, what would the inside of the sphere be like, if there was one? From what I have concurred, the sphere is defined by having surface area, but if you were to travel through 'surface,' would you find the area inside the boundaries of the sphere to be hyperspace?
     
  10. Mar 10, 2009 #9
    I think it is probably incorrect to say that a sphere by definition has no inside. The definition of a sphere, including an n-sphere, starts with a defined origin point, from which every point on the sphere is equidistant. That point, as far as I can see, would be 'inside', for a 2-sphere anyway. None of the points of the sphere itself are 'inside', but there is still a definable ball, definably inside the sphere.

    But perhaps you are saying the 5+1 sphere has no inside. I am still ignorant enough to imagine that as a possibility.

    Hyperspace, if I understand correctly, is any space of higher dimension than 3+1 or 4. Or maybe it includes 4, but not ordinary Euclidian 3+1.
     
    Last edited: Mar 10, 2009
  11. Mar 10, 2009 #10

    When I posted the last comment, I was looking off of what the others had responded. They have concluded that the sphere would have only surface area, and no volume. I actually can honestly say that I disagree with that. I think that the sphere has an inside, because it has an origin.

    If you go further ahead, I have actually thought about the properties of these higher dimensional spheres, and if an object were to travel, is it traveling 'on' or in the 'inside' of these spheres embedded in 3+ 1 dimensional space. Is it possible to assume that there is an inside? From what I know, people say that objects would travel on the sphere, but what I have asked throughout this post is; can anything travel inside the sphere?
     
  12. Mar 10, 2009 #11
    I need time to think about this.

    I would normally respond now, but I got married this afternoon and my bride is waiting.
     
  13. Mar 10, 2009 #12

    Congragulations!!!!!

    Best wishes for you and your wife!
     
  14. Mar 10, 2009 #13

    apeiron

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    The correct answer I believe would be that a sphere in the pure sense being discussed here has no physical origin point or interior. This would require the sphere to be embedded as an object in a higher-D space rather than simply being that space.

    So think of it this way. The sphere surface has a curvature which at every location "points off" towards a virtual origin. There is no actual interior or origin, but there is an actual curvature which looked at in a certain light could be said to be angled off towards some imaginary reference point. That warpage might also be experienced by inhabitants of the surface as some other kind of physical constant. An "energy" perhaps.

    To get a deeper appreciation of the range of possibility here, think about what is the "opposite" of closed curvature. It is hyperbolic or open curvature.

    So you can have 1) flat space (not "spherical" but infinite and straight in its directions).

    You have 2) a self-closed space like a sphere where every point shares the same curvature that points towards a common connection. A single global value (looking liking the co-ordinates for a phantom origin) acts as a constant that describes an extra quality to be found at all the locations.

    Then you could have 3) a space of hyperbolic curvature at every point, so with every point then pointing away from any common space. So not a very connected world but more a quantum foam, a dimensional roil.

    So out of this, we have the alternatives of cohesive spaces and uncohesive spaces. With flat space being the exact balance point between the two.

    None of these need to be embedded in a larger dimensional realm to have curvature, a positive or negative "tension" of some kind. Or flatness even.
     
    Last edited: Mar 10, 2009
  15. Mar 10, 2009 #14

    I do see what you're saying. I have to wander though why, if one thought of the space being curved, why it would be curved in such a manner as to give the same angle of curvature to everyone at every reference point. I find this appearance of 'energy' a little hard to believe.

    If what you're saying is correct, you are basically eliminating the sphere entirely, because one would have no need for the model of a sphere unless they were to take all measurements in very precise ways as to have a uniform curvature arise as the solution, which, from all vantage points, would reveal a sphere.

    Even then, after all of the virtual origins, wouldn't the energy have to radiate from the 'center' of the sphere in order to make it warped in the precise and uniform manner you propose?
     
  16. Mar 10, 2009 #15

    apeiron

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    When I say energy, I am trying in an abstract way to indicate how a tension at a location could be interpreted as a curvature towards something from a perspective outside - and then all that may actually exist is that tension!

    And this is not far from general relativity is it? We can model gravity as a force (complete with particles) or a spacetime warp. And the warp can then be reduced to a tension, a potential acceleration, existing at some point.

    I'm not actually thinking of an energy that can be radiated away from a surface. Though brane theories do sometimes venture into this territory with stories of unconfined gravitinos leaking away and making gravity so "unnaturally" weak.

    Is the sphere eliminated? Yes, conventional ball sitting-within-a-space mental images have to be got past to talk about issues of naked dimensionality, naked topology. But some essential ball-like properties remain.

    It would after all be meaningful to contrast a hyperspherical universe with a hypertorus or some other non-sphere geometry. The tensions across all the points in this case would not be so constant, so symmetrical, but would vary because not all would be pointing in the same common direction. Well, maybe the curvature of a torus is constant - a question others might be able to quickly answer.
     
  17. Mar 10, 2009 #16
    The idea of a tension causing the curvature is nice, but in simplistic general relativity, a mass causes curvature of space-time. From what I know, nothing can be simplified to 'just' tension. The curvature is caused by more than a potential acceleration existing at one point. It is caused by the mass, which creates the tension in space. So, if the tension is the curvature, then what causes the tension in space-time. I don't understand how a tension could be 'all there is.'

    By the way, I'm just curious, is this idea of tensions a theory of publication or your own thoughts?
     
  18. Mar 11, 2009 #17

    apeiron

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    What I was saying was not that one thing causes the other but what you might read off in one reference frame (as an an external observer) as a being a curvature might then be read off as something apparently quite different from another frame (as now an observer internal to the sphere surface).

    Tension would be an example of an alternative reading, not necessarily the "right" one under GR, or indeed any other physical modelling. Although I feel it is certainly in the spirit of GR (tensor equations, etc).

    So let's not get too caught up in the physical reality of "tension" (which may be a valuable idea or not) and just accept it as a way that you can understand why not to worry about the physical reality of origins to hyperspheres.

    It is the reason I don't worry - and so it would be useful if someone else knew a flaw in this particular line of argument.
     
  19. Mar 11, 2009 #18
    I'm sorry Benk, it must be me, but I still don't quite get your question. Let's for simplicity drop the whole hyperspace-thing for the moment, and just consider an 'ordinary' 2-sphere embedded in three spatial dimensions (no time).
    Then, if one lived on the 'flatland' (or more perhaps more appropriately, 'curveland' - because of the curvature of the sphere) of the 2-sphere, one would indeed be confined to that surface of the sphere. However, due to the embedding in (i.e. the being-part-of) the higher dimensional space, it actually makes sense (at least for 'higher-dimensional creatures' living in the 3D (spatial) world) to talk about the inside of the sphere: it's a 3-ball.
    You could of course wonder whether or not it makes sense for the 'curve-landers' to talk about the radius of the sphere, supposing they dont know about the existence of the additional (third) dimension that they cannot see. But at least they can calculate the curvature radius, and if they found it to be the same everywhere on the sphere, they could think of that as some kind of virtual origin, as was pointed out by Apeiron.

    Now I think that by analogy this can simply be extended to the higher-dimensional case (still ignoring the time-direction for the moment, though). So creatures living on a 5-sphere could calculate the (for them invisible, virtual) origin in the 6-dimensional space.
    I hope that this at least partially answers your question...
     
  20. Mar 11, 2009 #19
    benk99nenm312, thanks for your good wishes.

    Apeiron, are you talking about tensors? Vector spaces? Covector spaces? Covariance and contravariance? I am studying these things but still not entirely comfortable with them. But maybe this conversation helps my understanding.

    I think I understand that a torus has two directions of curvature. But I am running into language again. A point included in the set of points defined as a torus could also be said to be an element of a two dimensional surface, just as a point included in the definition of a 2 sphere is part of a two dimensional surface. The language difficulty seems to be centered on double meanings for the word "in": are we using 'in' to refer to a point being included in the set of surface points, or included in the ball, enclosed by the surface? It seems well-defined in a sphere, and we see the origin as not in or part of the surface, but as in or part of the ball. But look at the torus. Where is the origin? Is it even a point at all?

    Or should we say the origin of the torus is a circle, with the torus defined as a set of points equidistant from the circle? This last works for me except I need to add the constraint that the distance from the circle which generates the torus must be less than the radius of the circle. Otherwise I get confused, and lose the idea of the torus shape.

    I suspect the language used here, such as 2-sphere and 5+1 space, must have been defined so as to eliminate the double meanings. That's great, but I need to understand the grammar to understand the conversation. A phrase book might help.

    If I am working in the right direction, I think I understand that the tangent space of a point on a 2-sphere is a disk or plane. The point is the origin of the disk, or the (0,0) of the plane. Each point on the surface of the 2-sphere has its own unique tangent plane. The line normal to the plane or disk through the origin of the disk also passes through the origin of the 2-sphere.

    Not so for the torus. The tangent space is not always a plane or disk. It ( the tangent space) may sometimes be constrained by the curvature at some points to be a single line rather than a plane. The lines normal to the tangent spaces of a torus do not converge to a single point as they do in the 2-sphere.

    Sorry to go on and on but I feel my understanding has moved forward during this conversation and I hope for confirmation or correction. I also hope my thoughts are useful to the original poster.

    Thanks!
     
  21. Mar 11, 2009 #20
    It actully answers all of my question. When I posted the quote that you quoted on your post, I was bluntly stating what the others before me had answered. I actually agree with your statement, that it would be more of a 3-ball.

    Thanks for the answer. This really clears things up.
     
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