Exploring Compactified Dimensions: What Happens When 1D Curls Up?

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When you curl 1 dimensional thing like a line.. won't it become 2D?

I'm trying to imagine how a compactified dimension in superstring theory actually look like in our world.

Let's take our 3D world and say the depth got compactifed or curl up to Planck length or a millimeter. What kind of bosons can you produce that can navigate the 1 millimeter curl up dimension? Would it also travel at speed of light?

 

[mfb]

When you curl 1 dimensional thing like a line.. won't it become 2D?

For large length scales it will become 0 D (it looks like a single point), for small length scales it stays 1D (you can only move forward and backward).
If you curl up a line in our three-dimensional world you will use the second dimension, but this is an embedding of the line in some other space - it is not a property of the line (curled or not) itself.

What kind of bosons can you produce that can navigate the 1 millimeter curl up dimension? Would it also travel at speed of light?

In general, everything with wavelength much smaller than 1 mm will behave as in a regular 3-dimensional space, everything with a wavelength much larger will behave as in a 2D space, and everything in between is more complicated.

 

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For large length scales it will become 0 D (it looks like a single point), for small length scales it stays 1D (you can only move forward and backward).

But if you curl the line, won't it become circular in small length scale so you don't just move forward and backward but in circle too?

If you curl up a line in our three-dimensional world you will use the second dimension, but this is an embedding of the line in some other space - it is not a property of the line (curled or not) itself.

In general, everything with wavelength much smaller than 1 mm will behave as in a regular 3-dimensional space, everything with a wavelength much larger will behave as in a 2D space, and everything in between is more complicated.

can the moduli field inside calabi-yau manifold able to affect us? Is there a mediator field where you can couple the moduli field with our large scale field like EM?

 

[mfb]

But if you curl the line, won't it become circular in small length scale so you don't just move forward and backward but in circle too?

Can you move forward and backward (and sideward) on the surface of Earth?
Same concept. Sure, if you go 40,000 km in one direction you'll end up where you started, but that is very long compared to a typical walk.

can the moduli field inside calabi-yau manifold able to affect us? Is there a mediator field where you can couple the moduli field with our large scale field like EM?

I don't understand that question.

 

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Can you move forward and backward (and sideward) on the surface of Earth?
Same concept. Sure, if you go 40,000 km in one direction you'll end up where you started, but that is very long compared to a typical walk.

But the point is.. if you bend any 1D line.. it becomes 2D.. you may say at large length scale it becomes 0D.. while at small length scale.. it is same 1D.. but it's not the same.. because the line becomes curve (when you curl (or compactify it) so it becomes 2D. Gets?

Anyway. Any graphics to illustrate what it's like to compactify the dimension of depth in the 3D.

I don't understand that question.

 

[mfb]

But the point is.. if you bend any 1D line.. it becomes 2D.

No it does not, that is the point. This is a statement about the geometry of the line, not of an embedding of the line in some higher-dimensional space. These are different things.

 

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No it does not, that is the point. This is a statement about the geometry of the line, not of an embedding of the line in some higher-dimensional space. These are different things.

Ok. Got it. Anyway when a macroscopic dimension has size of say 1 light year.. how can you curl it into Planck size? how many turns of curling would there be?

Remember in 10D string theory 6 of the macroscopic dimensions curled up into Planck size.. so how many turns of packing it in or near Planck scale can you fit an infinite size dimension?

 

[mfb]

Anyway when a macroscopic dimension has size of say 1 light year.. how can you curl it into Planck size? how many turns of curling would there be?

This is not a process. A dimension is either curled up or not.

Multiple turns are not a meaningful concept. See the surface of Earth as analog again: You can only go around the equator once before you reach your origin again. If you continue walking in the same direction, you just go along the same path again.

 

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This is not a process. A dimension is either curled up or not.

Multiple turns are not a meaningful concept. See the surface of Earth as analog again: You can only go around the equator once before you reach your origin again. If you continue walking in the same direction, you just go along the same path again.

Let's say you curled up one of the 3D dimensions in your living room to 1mm.. so there is no more vertical above 1mm and everything is 2D on the floor (seen from large length scale).. would your table or bed still occur inside the 1mm?

 

[mfb]

Let's say you curled up one of the 3D dimensions in your living room to 1mm.

See above: It is not a process. You either have a living room with a curled up dimension or you do not.

so there is no more vertical above 1mm

There is no "above 1mm" if the height dimension is curled up.

would your table or bed still occur inside the 1mm?

This question is meaningless.

 

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See above: It is not a process. You either have a living room with a curled up dimension or you do not.There is no "above 1mm" if the height dimension is curled up.This question is meaningless.

I mean if you initiate the curling up in an already existing large dimension. Maybe your table or bed would just implode or would it somehow end up in the 1mm curled up dimension (from large infinite extend) intact? Just want idea and versatibility of the concept.

 

[mfb]

It is not a process.

I have no idea how to make it clearer than that, sorry. If you don't understand "it is not a process" I cannot help here.

 

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It is not a process.

I have no idea how to make it clearer than that, sorry. If you don't understand "it is not a process" I cannot help here.

I see. So you mean when the Big Bang happened.. the 6 (or whatever) dimensions were already started as compactified?

I'll read the book now called "The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions"
https://www.amazon.com/dp/0465028373/?tag=pfamazon01-20

Thanks for your assistance. It can make me visualize (or conceptualize) compactified dimensions better.

 

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I see. So you mean when the Big Bang happened.. the 6 (or whatever) dimensions were already started as compactified?

I'll read the book now called "The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions"
https://www.amazon.com/dp/0465028373/?tag=pfamazon01-20

Thanks for your assistance. It can make me visualize (or conceptualize) compactified dimensions better.

mft, I got your point.. that the curl up dimension was created right at the Big Bang (supposed these exist). No problem about that. Now supposed there were Calabi-yau manifolds with 6 curled up dimensions. Is the 6 curled dimension in my room directly connected to your room? If so, how do you send a boson that can only travel in the curled up dimensions and doesn't use the existing larger dimension? This is just to visualize the whole idea. Thank you.

 

[mfb]

Is the 6 curled dimension in my room directly connected to your room?

"connected" doesn't make sense. If there are extra dimensions, your room has them, and every "place" in the room is actually a small 6D space in the 9D space. The spatial dimensions are so small that most particles are indifferent to them - their wave function is the same at all places along these extra dimensions.

If so, how do you send a boson that can only travel in the curled up dimensions and doesn't use the existing larger dimension?

You don't. With sufficient energy a boson can differ along these small dimensions, and then motion along these dimensions becomes possible.

 

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"connected" doesn't make sense. If there are extra dimensions, your room has them, and every "place" in the room is actually a small 6D space in the 9D space. The spatial dimensions are so small that most particles are indifferent to them - their wave function is the same at all places along these extra dimensions.

If there are extra dimensions, the Earth has them, and every "place" on Earth is actually a small 6D space in the 9D space. Therefore my room is connected to yours via the 6D space.. so why do you say "connected" doesn't make sense?

You don't. With sufficient energy a boson can differ along these small dimensions, and then motion along these dimensions becomes possible.

Ah. You mean Planck energy bosons. It is said you need cosmic size particle accelerator to inject the photon into the compactified dimension.. no problem about that. But what do you mean a boson can differ along these small dimensions, and then motion along these dimensions becomes possible, any example or reference?

If there are compactified dimensions. It is implausible that they should rely on Planck size bosons to communicate. It's like Earth relying on the energy of pulsars to communicate in the surface. Maybe there are native scalar particles inside the compactified dimensions that are non-herztian, non-vectorian and has other mode of propagation?

 

[mfb]

Therefore my room is connected to yours via the 6D space

Are our rooms "connected via height" as well? What does that even mean?

Ah. You mean Planck energy bosons.

No I do not.
The extra dimensions, if they exist at all, could be large enough to be found by the LHC with more data. Likely? Depends on who you ask. Possible? For sure.

You seem to have some strange imagination of how compact extra dimensions would look like.
Let's try an analogy: 2 large space dimensions (x,y) and one small one (z). Let's make it 5 meters wide. You can freely move in all directions, space is three-dimensional for you. If you move in x and y direction you can explore other regions of space. If you move in z direction, after just 5 meters you return to where you started. If you move in a 45 degree angle to x and z, after 5*sqrt(2) meters you end up at the same z coordinate but now 5 meters shifted in x. You can explore space that way as well, you are just a bit slower. This is how the world looks like for particles at high energy - with a wavelength shorter than the length of the compact coordinate.

Now let's make the extra dimension smaller - 0.4 m. If you try to orient yourself along the z direction you'll hit yourself, you have to "lie" in the x/y plane. How does a house get designed in this world? The room layout will simply be an x/y-map. The z direction is too small to make walls or anything similar there. For someone designing this the world looks 2-dimensional. This is how the world looks for particles at low energy - with a wavelength longer than the length of the compact coordinate. For an ant, it still looks three-dimensional, because it is so much smaller.

 

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Are our rooms "connected via height" as well? What does that even mean?No I do not.
The extra dimensions, if they exist at all, could be large enough to be found by the LHC with more data. Likely? Depends on who you ask. Possible? For sure.

You seem to have some strange imagination of how compact extra dimensions would look like.
Let's try an analogy: 2 large space dimensions (x,y) and one small one (z). Let's make it 5 meters wide. You can freely move in all directions, space is three-dimensional for you. If you move in x and y direction you can explore other regions of space. If you move in z direction, after just 5 meters you return to where you started. If you move in a 45 degree angle to x and z, after 5*sqrt(2) meters you end up at the same z coordinate but now 5 meters shifted in x. You can explore space that way as well, you are just a bit slower. This is how the world looks like for particles at high energy - with a wavelength shorter than the length of the compact coordinate.

Now let's make the extra dimension smaller - 0.4 m. If you try to orient yourself along the z direction you'll hit yourself, you have to "lie" in the x/y plane. How does a house get designed in this world? The room layout will simply be an x/y-map. The z direction is too small to make walls or anything similar there. For someone designing this the world looks 2-dimensional. This is how the world looks for particles at low energy - with a wavelength longer than the length of the compact coordinate. For an ant, it still looks three-dimensional, because it is so much smaller.

This is great description! Brian Green depicted space as having many Cabali-Yau manifolds in his book Elegant Universe like the following I saw at google

It confused me for over 15 years...

So the universe has only one say 6-dimensional Calabi-Yau manifold?

But then since the dimensions are very small.. is it possible that in our solar system and andromeda, there could be different Cabali-Yau shapes? Or if you can alter the flux and moduli, can you change the shape of the Cabali-Yau at will at a small test locality?

 

[mfb]

The image can be misleading.

So the universe has only one say 6-dimensional Calabi-Yau manifold?

If it has that at all.

But then since the dimensions are very small.. is it possible that in our solar system and andromeda, there could be different Cabali-Yau shapes?

I'm not aware of any model that would predict this.