0
$\begingroup$

Trying to make sense of small and large extra dimension(s) of phyiscal space in a simple intuitive example.

Consider a two dimensional manifold like $\mathbb{R}^2$ and we are trying to add a small and a large extra dimension.

Do we mean by small extra dimension in this case something like $(0,1) \times \mathbb{R}$ (the flat case) or $S^1 \times \mathbb{R}$ (the curved case)?

Do we mean by large extra dimension something like $\mathbb{R^2} \times \mathbb{R}=\mathbb{R}^3$?

Do we mean in the case of our three dimensional space that basically we have a base space of our phyiscal three dimensional space with a total space built by adding a fiber and thus creating a fiber bundle or a even more general an arbitrary total space?

Does the extra dimension need to be real or can we even consider the complex manifolds, in the case of adding extra dimension to the phyiscal space, for example $\mathbb{C} \times \mathbb{R^3}$ or (Riemann surface) $\times \mathbb{R^3}$

$\endgroup$
1
  • 2
    $\begingroup$ 1. Yes, basically, keeping in mind that we need a metric to make sense of "small." 2. Yes, basically. 3. Yes, something like that. 4. Complex manifolds are in particular real manifolds, and "extra dimensions" arising from complex manifolds are considered in string theory, so yes. $\endgroup$
    – Qiaochu Yuan
    Commented Jan 26, 2023 at 21:48

1 Answer 1

Reset to default
2
$\begingroup$

"Small" and "large" dimension are rather vague terms. I assume this comes from a string theory or something similar. The point is that "small" refers to things at quantum sizes, waaay below things we experience every day.

So for example when I look at a sheet of paper, it is two dimensional for me. But if we magnify it, then it clearly has three dimensions. The third dimension of a sheet of paper is small compared to the other two dimensions though. And in many (practical) scenarios it is enough to think about it as a two-dimensional object (although you quickly find out that paper is three dimensional when you have to move 100 books :)).

This analogy goes to the string theory. If we magnify things very very very much, it turns out that there might be more than just three spatial dimensions. But the additional dimensions are very small, as far as I remember assumed to be on the order of the Planck length. This not only means that we don't experience them every day, but in fact (again: as far as I know) no tech exists that is capable of detecting them.

Either way: in order to talk about "small" and "large" you need some kind of measure and a point of reference.

The shape of those additonal dimensions (whether it is $S^1$ or $\mathbb{R}$ or something else) is interesting, but not really related to the small/large distinction.

There is also another way to look at things, which I personally prefer. Instead of thinking about small and large dimensions, as some properties of reality that you can touch, I often think like that: there is reality and there are mathematical models that describe the reality. If those models require say 12 parameters, then we have 12 dimensional space. And that's it. It is a very pragmatic way of thinking. For example we used to think (thanks to Newton) that gravity is some force. And then Einstein came and enlightened us that gravity is not really a force, but rather a spacetime curvature. This is quite counterintuitive: gravity doesn't exist, it is just spacetime bending around us.

But, hey, Einstein's models actually explain a lot more than Newton's. And they work. And we have for example GPS thanks to this.

The moral is: do not stick to some interpretation and intuition. These can and do change. Intuition is good, empiric evidences better and actual proofs the best.

Do we mean in the case of our three dimensional space that basically we have a base space of our phyiscal three dimensional space with a total space built by adding a fiber and thus creating a fiber bundle or a even more general an arbitrary total space?

I can't answer that. This is a very specific question for a specialist in the field of string theory. I'm not one of them, unfortunately.

Does the extra dimension need to be real or can we even consider the complex manifolds, in the case of adding extra dimension to the phyiscal space, for example $\mathbb{C} \times \mathbb{R^3}$ or (Riemann surface) $\times \mathbb{R^3}$

We, mathematicians, can consider whatever we want. And we often do, regardless of whether it has applications or not. I mean, what can stop imagination, right? But if you ask whether physicians actually consider complex manifolds, then the answer is: yes, they do.

$\endgroup$
3
  • $\begingroup$ So, small S1 has nothing to with being compact? Can you further explain how S1 can be either large or small? Is it related to the metric? $\endgroup$
    – VVM
    Commented Jan 26, 2023 at 22:25
  • $\begingroup$ So are you saying that the S1 being small or large depends on how large it is measured relative to the other component in the product topology? For example, we could have or imagine a manifold S3=S1 (large) X S1 (large) X S1 (small)? $\endgroup$
    – VVM
    Commented Jan 26, 2023 at 22:41
  • $\begingroup$ @VVM yes, I'm saying small or large depends on how large it is measured relative to other components. $\endgroup$
    – freakish
    Commented Jan 27, 2023 at 0:33

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .