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Curled up dimensions 
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#1
Apr3008, 10:22 PM

P: 226

What do they mean by a dimension that's curled up? I'm having a hard time with this notion. What would such a dimension be like?



#2
Apr3008, 10:29 PM

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Do you recall old games like asteroids or pacman where, if you go off of one edge of the screen, you reappear on the other side? Imagine that if you walked a mile East, you would up right back where you started. (Of course, this would mean the Earth is some weird sort of shape, but don't think about that) Now, imagine how it would look if you only had to go a few hundred feet East before you wound up where you started. What about just 10 feet? 1 inch? 1 millimeter? 1 angstrom? With these small sizes, it would be more like you were a twodimensional object than a threedimensional one! 


#3
May108, 12:03 AM

P: 35

String Theory is not scritctly a science yet  Its more of a mathematical philosophy that hasn't got any supporting evidence or observational data to conform any of the conclusions or predictions.
The mathematics in ST is logically consistent but that does not mean that it reflects reality in any way. When String Theorists make statements such as "curled up dimensions" or even an infinite number of dimensions they need to support these theoretical constructs with reproducible evidence of some kind  and they cant  so far! 


#4
May108, 11:05 PM

P: 226

Curled up dimensions
Tell me if I understand you right in these terms. I live inside this curled up dimension. Locally everything looks normal. I can survey a local map, and as I move around, overlapping at the edges of the pages I draw, I collect these maps into an atlas. After collecting, say nine maps, I notice the landmarks are familiar. I arbitrarily pick one map to be the center, and connect four maps, one to a side. That was pretty easy. It wasn't hard placing the last four corner sheets into a 3x3 grid either. As I'm making a list of distances to cities, I notice that cities on the opposite corners of this big map are actually close together. Everything on the extreme left is close to everything on the extreme right, but the top left is close to the top right, and bottom left is close to the bottom right. I devise a way to index to every point on the grid, and figure out a modulo function on the edges that let me calculate the distance between any two cities. I get up to write down this function and wonder what it means, when a breeze blows my maps apart. I reassemble them using a different map for the center, but now the mapping function I get is a little different for the corners. Any point on any sheet could be used as the origin, and I have a family of mapping functions at every point to every other point. the domain of the maps and the class of distance functions defines the topological space, right? I can generalize the family of functions into a single function in one added dimension, in the above case an oblate spheroid. It might have been a torus, but I suspect it'd take more than nine maps to figure that out. As long as that function is differentiable and conformal in my space, locally everything is guaranteed to seem normal to me. This also works even if my space is infinite, I just have to map it onto the extended Rieman sphere. Am I on track so far? In the example you used, when you shrunk the curled up dimension from 100 feet to 10, that 10 feet was measured in your dimension. In mine, the maps I drew might be smaller, but on my maps the distance from AnkMorpork to Quirm is still awfully far. I don't even think the feet that I use are the same kinds of feet as yours. So you might see me as becoming more two dimensional as you curl me up, but I'd still feel three dee. The functions that define how things work in my space are still locally valid and presumably unchanged and possibly as I can measure it infinite. It seems to me then that there isn't much difference between a dimension that's curled up and one that isn't. So unless I'm wrong in how I understood the above, I still don't understand what a curled up dimension means. If we were living inside a curled up dimension, could we tell? How would we detect it? What properties of it could we measure? 


#5
May108, 11:44 PM

P: 355

First, it's obvious that a curled dimension is different from a noncurled one. For one, you can send a message in one direction and it could come back and hit you. Second, using just that method, you can show that it's possible to determine whether or not you are moving in a curled dimension (i.e., there is a difference between moving and not moving in a curled dimension) Consider a person sending 2 light beams in opposite directions and asking where they collide. A person standing still will give you one answer. However, if there were really no difference between standing still and moving, then a person who is moving would see them collide at a different point. However, this is a contradiction (since they could communicate to each other where they saw the light beams collide and they'd disagree. If you allowed this, there could be a lot of weird causality breaking things). This is of course assuming that the curled dimension is also flat, but I'm pretty sure similar things happen for nonflat curled dimensions.
Other odd things happen when you have multiple dimensions, some of which are curled and some of which are not curled, but I don't know any of the details. 


#6
May208, 09:36 AM

P: 35

mathematical theories produce all sorts of weird and wonderful concepts  doesnt make them REAL though  even if they are logically consistent
What about a semicurled dimension? Or a dimension that does not interact with any other dimensions? Or a oscillating or wave like dimension? All mathematically consistent  BUT do they exist? 


#7
May308, 03:19 PM

P: 226

What's a semicurled dimension? Would that be one where the maps on the sides overlap but the ones on top and bottom go off to infinity?
What's an oscillating or wave like dimension? I've never heard of that. As for a dimension that doesn't interact with any other dimension, I guess that depends on what your definition of "is" is. If it doesn't interact with any other dimension, then I don't care if it exists or not. Makes no difference to me or to any theory. 


#8
May308, 03:49 PM

P: 226

You also said the curled dimension would have causality breaking if it was also flat. I don't know what that means. What is a flat curled dimension? What is a nonflat curled dimension? BTW, I've always found the notion of looking at the back of my head rather disturbing. I avoid that unpleasantness by presuming the curvature of space is an irrational number. That way my line of sight will never return to it's starting point. I take comfort that I'm probably 100% right about that. 


#9
May308, 08:41 PM

P: 355

No, no, there's no causality breaking in assuming that the universe is flat and curled. There's causality breaking in assuming that there is no preferred reference frame (i.e., that you can't tell whether you or not you are moving). What I meant by flat was that this is a simple special relativity calculation (i.e., that that you are ignoring any of the effects of mass and energy on spacetime and that the closedness of the dimension is just an intrinsic property of the dimension). If you have a flat, closed dimension, you can tell whether or not you are moving. I believe that something similar is true in general relativity as well unless certain constraints are put on the way dimensions curl and on the way they interact with mas, but I do not believe that this is done in string theory.



#10
May308, 08:57 PM

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Anyways, I'm of the (possibly wrong!) understanding that gauge theories can be thought of exactly as that  e.g. you can attach a U(1) dimension to classical spacetime, and electric charge manifests itself as something like velocity in that direction. (This is something I don't understand well) 


#11
May308, 09:49 PM

P: 226

Besides that, the argument about seeing the back of your head only applies with positive curvature and doesn't happen with negative curvature. I still don't understand why there should be causality problems in either case. 


#12
May408, 12:22 AM

P: 35

Not sure if they exist in mathematical theory either I wouldnt pay much attention to me when I am making stuff up OFF the top of my head (altough thinking about it now, I dont see why an OSCIALLTING dimension could not be a feasible mathematical construct. I would imagine it would be a dimesion that begns to curl one way and then curl the other way producing a wave like pattern  nonlinear but period if you like???) 


#13
May408, 10:40 AM

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Not sure if we should clarify the use of the word "flat" in this thread. It is usually interperted as zero "gaussian curvature", a property which is told of pairs of directions, but not of a single direction. Think about the plane, the cone, the cylinder, the torus and the sphere.
also, check http://mathworld.wolfram.com/FlatManifold.html and http://en.wikipedia.org/wiki/Curvature 


#14
May408, 10:43 AM

P: 226




#15
May508, 04:38 AM

P: 622

Come to think of it, I don't even know what "space" is. I'm almost too old, it seems. 


#16
May508, 08:59 AM

P: 35

If you wish to regurgitate accepted standard models of nature then just read dozens of text books  that a different sort of learning  although its important, its not critical to me. Free thinking is far more challenging than PLAIN LEARNING and it opens up many horizons and to enter this realm ONE must ponder the wrong or neglected questions Are you afraid to improvise in science? thats what science is supposed to be for it to progress. 


#17
May508, 01:27 PM

P: 168

Making things up off the top of your head is, at best, philosophy and, more likely, nothing more than nonsense. 


#18
May508, 10:10 PM

P: 226

The trick is in knowing everything about your universal set. If you know everything, you can map one point in reality (an element of the universal set, call it e) to any other point of reality within a neighborhood of x. If you know everything, the radius of that neighborhood might be unbounded. Does that seem right? Anyway, if you don't know everything and don't have enough parameters, you can only map to subspaces of the universal set, and it won't be sensibly ordered, at least in some ways (in exactly the number of dimensions you're short ways. I think). If you have too many dimensions, then n of them measure some combination of other dimensions, and not what you think it does. In this case some combinations of parameters will be able to predict some things but won't apply to others. This makes it very hard to decide which parameters to get rid of. You don't even have to pick just the exact right dimensions, either. As long as the n dimensions you pick contain all of the n independent orthogonal vectors, then everything will work out the same. I doubt the equations which defined the mappings would even be more cumbersome. So while it's not exactly "anything you want it to be" it's still sort of close to that. 


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