Main Question or Discussion Point
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?
Let's consider "curling up" one of our existing spatial dimensions. (just geometrically -- without worrying about how it might affect physics)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?
Yes, I remember asteroids. Used to play it on the original bar tables. Used to stand in line to feed quarters into Pong. I told you I was almost old.Let's consider "curling up" one of our existing spatial dimensions. (just geometrically -- without worrying about how it might affect physics)
Do you recall old games like asteroids or pac-man 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 two-dimensional object than a three-dimensional one!
Sounds to me like you've described a relativistic space. In that case, why are there any big problems? Not long ago, wasn't it a question in cosmology whether our universe was open, closed, or flat? Locally it looks flat, but on larger scales all the evidence was for open, until they introduced dark matter. Then there was the supernova data and dark energy, and now the universe is way open and accelerating. All this always sounded speculative and I take it with a big grain of salt. But in any case, how would a curled up dimension be any different from such a closed universe, just on a seemingly larger scale?First, it's obvious that a curled dimension is different from a non-curled 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 non-flat 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.
Don't you need some assumptions, like space being entirely homogenous in that direction, and in some sense static with respect to time?If you have a flat, closed dimension, you can tell whether or not you are moving.
I don't see how this follows. I thought it was correct to say that there is no preferred reference frame, at least in the dimensions I occupy (meaning, not in some higher dimension contained the curved one), and I didn't think the curvature of space qualified that statement.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).
Well I just made them up to make a point.What's a semi-curled 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.
My problem is that I don't know enough to be sure when you're making stuff up and when you aren't. Without such an indication, I'd have no choice but to simply not pay attention to you at all, since I'm really trying to learn something here.I wouldnt pay much attention to me when I am making stuff up OFF the top of my head
I wish I could help you here, but I haven't even found a satisfactory description of what a dimension is, or is like, yet. This might develop into an interesting thread if you could start at the beginning and describe an ordinary dimension, say one of the ones we call "space dimensions" (the time one is too difficult). Then one might be able to get a handle on how to describe a curled-up dimension.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?
You learn BEST by making stuff up off the top of your head.My problem is that I don't know enough to be sure when you're making stuff up and when you aren't. Without such an indication, I'd have no choice but to simply not pay attention to you at all, since I'm really trying to learn something here.
Not at all! But you clearly don't understand science at all. Science requires that you provide experimental evidence for your theories. Or at the very least you must show that experimental evidence could be provided, in priciple, given an appropriate state of the art.You learn BEST by making stuff up off the top of your head... ...Are you afraid to improvise in science? thats what science is supposed to be for it to progress.
What is a dimension? The flippant answer is "anything you want it to be." It's almost a religious question to some. Poetically it's a measure of a fundamental reality about the universe. In other words, it's one necessary measure of an observable. Mathematically (I think, help me out here) it is an orthogonal parameter in a mapping from the universal set onto the universal set. You just have to define your universal set.I wish I could help you here, but I haven't even found a satisfactory description of what a dimension is, or is like, yet.
Hurykl mentioned something about charge being a velocity in the direction of another U(1). If that's so, then it seems something like the above is going on.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.
Your first would be have been admired by Lewis Carroll. I like it. The second has an operational flavour and would have suited Percy Bridgman. The last, involving concepts like "orthogonal" "parameter","set" , "mapping", "vectors" etc are also somewhat operational, but disguised with that amazingly useful invented language, namely mathematics. It's the kind of answer that might later lead to further inventions, like differentiable manifolds and fiber bundles.What is a dimension? ......"anything you want it to be." .... it's a measure .....one necessary measure of an observable. Mathematically ..... it is an orthogonal parameter in a mapping from the universal set onto the universal set.
Your familiarity with undergraduate mathematics and your clear technical exposition of what theorists mean by the "dimension" of a vector space, and how they model space as a manifold that includes in one of its dimensions a circular character is helpful.As someone who knows the majority of the undergraduate mathematics curriculum and wants to learn some physics (as well as more mathematics), I'd just like to say:
Sorry, I haven't learned much about manifolds yet. I've just learned a definition, defined how to do integration on a manifold, a proved a few simple things about it. It was just as an end of the year thing so that we could see some more advanced topics. I probably won't actually do anything with manifolds for another 2 years at least. I'll try to explain things a bit, but I can't promise it'll be that great because what you are asking is a question about both differential geometry and topology, neither of which I've been more than just barely introduced to.Your familiarity with undergraduate mathematics and your clear technical exposition of what theorists mean by the "dimension" of a vector space, and how they model space as a manifold that includes in one of its dimensions a circular character is helpful.
But I don't think it goes very far in answering the second part of the original question, namely "what would a (curled-up) dimension be like?" This needs a simpler explanation, first, perhaps, of what an ordinary dimension is, or is "like", and how this would be changed were this dimension to be curled-up. I can't supply such an explanation. Perhaps you could take a less blinkered technical approach, and attempt a non-technical discussion, maybe by considering the possible topologies of manifolds?