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This question (as the previous one) is mostly arose from such pictures:

enter image description here

As explained by Brian Greene, this is something what our Universe should look like at a Planck scales in superstring theories.

Now, from an article on Wikipedia:

The central idea is that the visible 3D Universe is restricted to a brane inside a higher-dimensional space, called the "bulk". If the additional dimensions are compact, then no reference to the bulk is appropriate.

So here goes my questions (all for the compact extra dimensions):

  1. If we imagine our whole Universe as a giant 3D Volume, am I correct that this compactified "extra space" is a "complete subset" of that 3D Volume? I.e. (since I am terrible with math and it's terms), there're no "points" inside them (like coordinates) that do not belong to (shared with) our 3D Volume, right?..
  2. Are these shapes (called Calabi-Yau manifolds) meant to be connected with each other? Like, if you were small enough to place yourself inside 1 Planck volume of our Universe (or even smaller if it needed), could you use 1 (or any) of these compactified dimensions to travel from 1 Planck volume to another Planck volume?.. Or prof. Greene did picture them separately on purpose, to mean specifically that they're unconnected? And if he did, how a String can get from 1 Planck (or any) volume to another if there's no "bulk" - ?..
  3. If there's no higher-dimensional space (as suggested by any recent experiments, including this one), how compactified dimensions could be even considered "dimensions", if 3D (4D) is all we need to specify any point inside them? May be "solutions" (of some equation, like $\text{Lagrangian}$, or other) is the better term then?..
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    $\begingroup$ If string theory is correct, our whole universe is 10 dimensional. 6 of dimensions are small Calabi-Yau's and they are indeed "connected". Greene pictured them separately for convenience, but there is in fact one such CY at every point in spacetime. The remaining 4 dimensions are large (and that's what we live in). $\endgroup$
    – Prahar
    Jul 8, 2021 at 13:30
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    $\begingroup$ I'm not a string theorist but I believe a helpful analogy might be to picture a (thin) cylinder. This has one 'non-compact' dimension along its length and one (small) compact dimension around the circumference. The Calabi-Yau's are analogous to circles then, and Greene would draw a cylinder as a straight line with a circle drawn on top of every point, $\endgroup$
    – jacob1729
    Jul 8, 2021 at 13:35
  • $\begingroup$ @jacob1729 if these circles are not placed in the "bulk" (a "true" higher dimension), then to me they look like subsets of a cylinder. Can't see how any lesser dimension (like, say, a 2D slice of 3D volume) can be considered "true" on its own: that's why I suggested term "solution" instead. $\endgroup$ Jul 8, 2021 at 14:22
  • $\begingroup$ @VictorNovak You are misunderstanding Jacob's point. Think about an ant living on a 2 dimensional cylindrical universe. There is no 3rd dimension or anything, just the two dimensions of the surface of cylinder. If the ant is very small compared to the radius of the cylinder then the ant will move around as if it lives in 2D. If the ant is very very very very large compared to that radius (e.g. if the ant were a human), then the human will not be able to access the circular direction and can only walk along the length of the cylinder $\endgroup$
    – Prahar
    Jul 8, 2021 at 14:35
  • $\begingroup$ @VictorNovak - Perhaps to give a more physical picture. Think of a tightrope walker. They can walk along the wire, but obviously are too big to walk in a circle on that wire. This is because the radius of the wire is very small compared to the size of a human. But an ant on the same tightrope can walk along the wire, but also in a circle around it. $\endgroup$
    – Prahar
    Jul 8, 2021 at 14:36

1 Answer 1

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I cannot really answer this from a string theory point of view, just from the mathematical perspective of extra dimensions:

  1. I am not sure your intuition is correct, or if I just misunderstood you. The points in the compact dimensions are not a subset of our large 3D space, but what we call 'points' in the large dimensions are actually a subset of all of space, where the latter includes the compactified dimensions. Mathematically spoken, there are NO points that are either in compactified dimensions OR in the large ones. Every point is just a point in 9D space (ignoring the time-dimension for the whole answer, but taking the 6 compact dimensions from superstring theory) and if you want to give a coordinate of such a point in local coordinates, every point always has 9 coordinates, which you can write as $x=(x^1,x^2,x^3,...,x^9)^T$, where, if we have 6 small dimensions, w.l.o.g. we can say the first three coordinates can have arbitrary real values, while the other 6 can only have 'small' values, or just loop very fast (meaning larger values corresponds to the same point as smaller values), because the dimensions are so small. Since we are so large compared to the compactified dimensions, it seems we can never distinguish for any point with the same first 3 coordinates any of the coordinates from $x^4$ to $x^9$, so we only use the first three to specify points in 3D space. So the points in the compactified dimensions are not a subset of the 3D volume, and also not the other way around, but the 3D world we experience is actually a subset of the whole 9D universe, while it not relevant in our three dimensional macroscopic world at which coordinates $x^4$ to $x^9$ any object is, if that is even defined, since even what we call elementary particles seems to be larger than the compactified dimensions, at least that is how I understand it.

  2. All the small dimensions are connected, and the picture you posted is misleading in that sense. If you would be small enough to move around in compactified dimensions, you would need to move along a large dimension to get to 'another' Calabi-Yau manifold (at another 3D point) (which is quite obvious if you think about it). Think about the example with the straw, which is a small cylinder: The length of the straw is the large dimension (let's assume the straw is infinitely long), while the circle around the 'hole' of the straw is the small dimension. You can specify a point on a cylinder by two coordinates, one that can have arbitrary real values (since the straw/cylinder is infinite), and one that loops as soon as you circled around the straw. Mathematically this is equivalent to gluing a circle to every point of the number line (remember we said the straw is infinite), which could be written as $\mathbb{R}\times S^1$. All these circles, $S^1$, are definitely connected. How? No matter at which point on the circle you are, you can always get to 'another circle' just by moving along the 'large' dimension, which does not change your coordinate on the circle, i.e. in the small dimension, but nevertheless could be pictured as moving you to 'a ('different') circle glued to another point on the straw'. It is analogous with 6D-CY-manifolds and 3 large dimensions, just harder, or impossible, to picture.

  3. I do not understand this question. If there are no additional dimensions additional to our large space-time dimensions and time, then there cannot be any compactified dimensions. But I do not think that was your question. :-D

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  • $\begingroup$ Thanks a lot! And yeah, I've mistaken "higher-dim" with "large-dim" (which is the "bulk") in my 3rd question. Thank you very much again! $\endgroup$ Jul 8, 2021 at 17:29
  • $\begingroup$ Btw, I've learned recently from folks here and on Quora, like Ron Maimon, that all these questions I asked have absolutely nothing to do with reality that String theory describes)) So, I gonna read Zwiebach once I'll get more urgent things done (I just decided to leave that note here). $\endgroup$ Jul 12, 2021 at 20:41
  • $\begingroup$ You may have learned nothing about string theory from this, but since you asked quite general questions about space-time with additional compact dimensions, I would now know, why you impression should be true, since 10D superstring theory operates exactly in such space-time. $\endgroup$
    – Koschi
    Jul 13, 2021 at 11:34
  • $\begingroup$ Your answer was really helpful! I perhaps should've better just edited original post to leave my previous note; but I've left it here for some reason. $\endgroup$ Jul 13, 2021 at 11:43

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