Dimensions: Perpendicular vs. Coiled Up?

[wittgenstein]

TL;DR

Dimensions are defined as perpendicular, and I have read that some are coiled up. How can something be perpendicular and coiled up?

I am confused! Dimensions are defined as perpendicular, and I have read that some are coiled up. How can something be perpendicular and coiled up?

 

[wittgenstein]

Yes, I understand that gravity is curved spacetime. But is that the dimension that is curved?

 

[Ibix]

Dimensions don't have a notion of perpendicularity. They're just a count of how many linearly independent directions there are.

An example of a rolled up surface is a drinking straw. Anywhere on the surface you can always draw two arrows in different directions. But there's one particular direction where you get back to where you started in minimum time. In a quantum universe, the existence or not of a tiny rolled-up direction is not noticeable until you get things with wavelengths on a similar scale to the distance around the straw or smaller. So (if rolled up dimensions exist) they would not have any effect on anything that isn't at really high energies. We've never seen any such effect.

Note that this "rolling up" is extrinsic curvature. Gravity is all due to intrinsic curvature, so this is a distinct phenomenon.

 

[PeterDonis]

Moderator's note: Thread moved to the Beyond the Standard Models forum.

 

[PeterDonis]

I have read

Where? Please give a reference.

 

[arivero]

The paradox here is that a straw is not curved

The intuition is, a surface is curved if it can not be developed in a plane. The straw, as any cylinder, you can cut longitudinally and develop into a perfectly flat surface. As a bonus, in this case the notion of perperdicularity works perfectly.

 

[wittgenstein]

Where? Please give a reference.

"The three spatial dimensions we are familiar with from daily life are essentially infinite, while compactified dimensions are curled up, and have a finite size that ranges from a few microns (10-6 m) down to the Planck …" FROM https://www.learner.org/series/physics-for-the-21st-century/string-theory-and-extra-dimensions/

 

[wittgenstein]

Where? Please give a reference.

Still, a couple of decades later another physicist, Oskar Klein, tried to give Kaluza's idea an interpretation in terms of quantum mechanics. He found that if this fifth dimension existed and was responsible in some way for electromagnetism, that dimension had to be scrunched down, wrapping back around itself FROM https://www.space.com/more-universe-dimensions-for-string-theory.html

 

[wittgenstein]

Yes, I understand that gravity is curved spacetime. But is that the dimension that is curved?

My question is IF dimensions are defined as perpendicular (4- d would have to be perpendicular to a cube just as width is perpendicular to length) then how can something such as a dimension that is defined as perpendicular be coiled up? How can that be perpendicular?

 

[Ibix]

IF dimensions are defined as perpendicular

Dimensions don't have a notion of perpendicularity. They're just a count of how many linearly independent directions there are.

 

[wittgenstein]

With TOTAL RESPECT! I am still confused. So, width is not defined as perpendicular to length? Suppose there are only 2 dimensions (width and height) height would not necessarily be perpendicular to width?

 

[PeterDonis]

" FROM https://www.learner.org/series/physics-for-the-21st-century/string-theory-and-extra-dimensions/

This is talking about string theory, i.e., it is speculation. We have no actual evidence that there are compactified extra dimensions.

a couple of decades later another physicist, Oskar Klein, tried to give Kaluza's idea an interpretation in terms of quantum mechanics. He found that if this fifth dimension existed

No, he constructed a model that had a fifth dimension, and found that it could reproduce some of the predictions of electromagnetism. But that model, like string theory, was speculation. There was never any actual evidence for it.

 

[PeterDonis]

width is not defined as perpendicular to length?

"Width" and "length" are not dimensions. They are defined measurements of an object. That's not the same thing.

 

[arivero]

Now moderator should mode this to geometry/math forum Indeed dimensions are a very polysemic word. Perhaps the confusing point the plural, as it has been colloquial to say that the dimension of x is the number of dimensions of x . That is abbreviation for the number of vector is a basis of x. And not even need to be an orthogonal basis.

Here we are speaking about dimensions of a connected manifold, that are the dimensions of the euclidean vector space used to map each chart. Of course we can not use a single chart. Not even for the surface of earth, we need at least two charts.

 

[Ibix]

So, width is not defined as perpendicular to length? Suppose there are only 2 dimensions (width and height) height would not necessarily be perpendicular to width?

Those aren't dimensions in the sense that the word is being used here.

In a one-dimensional space you can only go forwards and backwards. In a two dimensional space you can go forwards and backwards, but there are place you can't reach by doing that. You need to define another direction - but it need not be perpendicular to the first one. (Indeed, it's possible to define spaces without notions of direction at all but still consider them to have multiple dimensions, although these are typically more interesting to mathematicians than physicists.) That's all that the number of dimensions means.

Mathematically, I can just say "consider all the real numbers" (the reals are all decimals, including whole numbers and infinitely long ones like $\pi$), and I've defined a one-dimensional space (usually called $\mathbb{R}^1$). Now I can say consider all pairs of real numbers, like (1.32, 6.98427) or (1.90, $e$), and I've defined a 2d space ($\mathbb{R}^2$). But the numbers aren't perpendicular to each other - how could 7 be perpendicular to 3? You need to add more structure to this space to get notions of distance and angle - that's actually what the metric provides.

If I take "all pairs of reals where the second one is between 0 and 1" then I have a 2d space with a finite extent in one dimension. If I add "...and the label (x, 1) refers to the same point as (x, 0)" then I have a space that's infinite in one direction and rolled up in another (called $\mathbb{R}^1\times\mathbb{S}^1$). But until I define a metric, I don't have any meaning to the angle between "moving so that each point I meet has the same first number" and "moving so that each point I meet has the same second number". If I add a standard Euclidean metric, then I've defined those two motions as perpendicular - and only then do I have a mathematical model of a drinking straw (more or less - there's some other stuff needed). Carroll covers this a bit more carefully in chapter 2 of his GR lecture notes.

Getting back to the question you originally asked, you are free to add fifth, sixth, seventh, (etc.) dimensions to your mathematical models if you wish, and make them rolled up or not. You then need to add a $5\times 5$, $6\times 6$, $7\times 7$ (etc.) tensor to describe the metric, and then you will always be able to find 5, 6, 7, etc perpendicular directions at each point - but perpendicularity isn't a thing until you add the metric, and which directions are perpendicular will depend on what metric you pick.

Adding those dimensions may or may not help anything. Kaluza-Klein's model naturally describes an electromagnetic field, but IIRC also describes an un-named scalar field that would be easily detectable if it existed. And the jury is still very much out on the utility of string theory.

 

[Ibix]

polysemic

Good word!

<Adds to dictionary.>

 

[PeterDonis]

moderator should mode this to geometry/math forum

The OP appears to be interested in physical models such as string theory and Kaluza-Klein theory. That makes this the appropriate forum.