Why do extra dimensions have to be "curled up"?

Paul Martin

This is my first post to this forum. I'll start with a question that has bothered me for a long time.

T. Kaluza suggested to Einstein that the equations describing fundamental forces and fields could be simplified if the possibility of real extra dimensions of space were considered. It seems to me that from the outset of that suggestion, people rejected the idea because they thought that if such extra dimensions exist, we should be able to see them or otherwise detect them.

Klein addressed this "problem" by suggesting that the extra dimensions were compactified or rolled-up. This was the Kaluza-Klein idea of a hyperspace in which all but three spatial dimensions were rolled-up so we can't detect them.

As far as I can tell, there have been only two objections to Kaluza's original idea which require contaminating (IMHO) it with Klein's addition:

1. If large extra dimensions exist, we should be able to detect them, and,

2. Large extra dimensions would require that inverse square laws would necessarily become inverse cube (or higher) laws.

I think these objections can easily be dealt with as follows:

1. If our 4-space (i.e. Einstein-DeSitter 4-D space-time continuum) is in fact a 4-manifold (equivalent, I believe, to what cosmologists currently call a 4-brane), then structures and features in the manifold could have the same properties as they would if the 4-space were not a manifold embedded in higher-D space.

An analogy would be that geometric structures drawn on a sheet of paper cannot, in principle, somehow "get up off the paper" and achieve access to any part of the 3-space, in which the sheet of paper is embedded, that is not on the paper. The structures are the same whether or not the "paper" is embedded in 3-D space.

Since everything (except possibly the observer's consciousness) that is involved in any observation of our world is a 3-D structure (objects, apparatus, eyes, etc.) it is reasonable to expect that we would not be able "get up out of our 3-D space" in order to access anything outside our manifold.

2. The topology of a 4-manifold could be identical to that of a 4-space which does not happen to be an embedded manifold. Thus there is no reason why inverse square laws shouldn't hold in the 4-manifold.

An analogy would be that the density per acre of a fixed number of sheep in a circular pasture varies inversely with the radius. The fact that the pasture is a 2-manifold embedded in 3-space does not require that the sheep density follows an inverse square law.

It seems possible to me, as Plato suggested, that extra-dimensional objects may produce effects in our manifold, just as 3-D objects can cast 2-D shadows. If this is the case, it would seem possible that some of our more elusive "objects" such as electrons and photons might be manifestations of such effects.

My questions to you are:

1. Are there reasons, other than the two I listed, for requiring Klein's approach of "curling up" extra dimensions?

2. Are there errors in my argument for dismissing those two reasons?

Hurkyl

AFAIK, the concept of a manifold was originally developed to help study surfaces embedded in an ambient Euclidean space, by eliminating any references to the ambient space.

It was eventually shown that any manifold has an embedding into some euclidean space. (not just one)


Kaluza's idea is not that the embedding of the universe viewed as a 4-manifold in a higher dimensional space explains electromagnetism... it was that viewing the universe as a particular type of 5-manifold explains electromagnetism.


I don't know if these address your post or not, but I hope they help.

Paul Martin

Thanks, Hurkyl. Your answer helps a little but it doesn't answer my questions.

I am not very familiar with the details of Kaluza's idea, but from what you say, I think there is even more reason to pursue it in his original form. That is, I think the notion of 5 or more large Euclidean dimensions should be seriously considered without encumbering it with the complex constraint of Klein's that the extra dimensions be curled up. This constraint, as far as I know, leads to such horrendous complexities as Calabi-Yau spaces.

It seems to me that if our 4-D universe is a subspace (manifold) in a higher dimensional space, we wouldn't be able to access or detect anything outside of our subspace just because of the mathematical properties of subspaces (manifolds). Furthermore, inverse square laws within our manifold would be natural consequences of the geometry of the manifold. I see no need to suppose that the extra dimensions should be curled up. Do you?

Paul

Mk

To throw something out... The analogy with the 3D power line, if you look at it from far away, it looks 2D. From an ant's point of view (who is on the power line) it is for sure 3D.

Hurkyl

Well, if the dimension isn't curled up, I think you have to explain why all observed matter is at roughly the same place in this dimension. (Because they bump into each other) In particular, this is a problem because, I think, that positively charged particles are travelling in one direction while negatively charged particles go the other way.

Paul Martin

That analogy is frequently used to try to explain the convoluted (literally) suggestion of Klein that extra dimensions are "curled up". A slight correction, though: from far away it looks 1D and from an ant's point of view the surface of the cable is 2D in terms of his freedom of movement. If the ant is perceptive enough, he may realize the cable is 3D the same as we know that our earth is 3D even though we are confined (pretty much) to its 2D surface.

But, IMHO, this analogy is not satisfying. It doesn't help make sense of Klein's suggestion. Here's why.

In this analogy there are three distinct observers. The ant, the person viewing the power line from far away, and you (or me) picturing this whole scenario from a yet-higher vantage point. (I'll refer to the third observer as 'we' so we can each separately think about what is going on here.)

We see that on the surface of the cable there is an ant who thinks it is 2D. He is able to crawl not only along the length of the cable, but also around it. He may be smart enough to discover that he returns to his starting point after going around the cable, but maybe not. If his path is a helix, for example, he may think he is on a vast 2D surface.

We also see that there is another observer far enough away from the power line that he can't see the ant or the fact that the cable has a non-zero thickness. To him it looks to be a 1D structure. If he knew the position of the ant, he would only know it as a single scalar number representing the ant's distance from some reference point on the cable.

We, as the third observer, see all this in 3D space. We see that the second observer is some distance from any point on the cable and that the surface of the cable is really a 2D manifold in 3D space with one of its dimensions big (the length of the cable) and the other dimension curled up and small as suggested by Klein.

Now, here are the problems I see with this analogy and with Klein's idea. We would like to use this analogy to explain how there could be extra spatial dimensions without our being able to see them.

First of all, to make the analogy, we need to increase the number of dimensions. That's easy. We simply add 1 to each dimension in the analogy. Next we need to identify ourselves with one of those observers. We start with the ant. The ant is crawling around on a 2D manifold and doesn't realize that in "reality" space is really 3D. To make the analogy, then, we would say that we are crawling around here in our 3D space not realizing that "reality" consists of 4 dimensions of space.

But this misses the whole point of the cable appearing to be a 1D space to that distant second observer. So let's identify ourselves with that second observer. From that observer's point of view, the universe (i.e. the power cable) is a 1D manifold in his 3D space (i.e. a line), but if he could only get a closer hi-resolution look at it, he would see that the cable is really a 3D structure after all (the surface of it is a 2D manifold embedded in his 3D space).

To complete the analogy from this second observer's point of view, we again add 1 to each dimension. This says that the second observer (I suppose Klein would say that is us) really exists in 4D space and he sees what looks to be a 2D structure in the distance. (Not far off so far. We really only see surfaces and from the point of view of quantum distances, all of our observations are very low-resolution from a great distance.) But if we could somehow get a high-enough-resolution view, we would see that things are really 4D after all or at most 3D manifolds in 4D space.

The problem with this is that that second observer needs a large 4th additional dimension to get him sufficiently far away from that cable that it gives the illusion of being of fewer dimensions. The ant cannot get this view because he is confined to the surface of the cable.

I think proponents of the Kaluza-Klein model would have us believe (back to the analogue) that the cable is all of existence - that there is no large 3D space in which the cable is an embedded 2D manifold. The cable is all there is.

But then there is no place for that second observer who is supposed to represent, in the analogy, our point of view with respect to the reality we observe. If you give that second observer the large extra dimension required for his point of view, then you don't need to confine the ant to that small rolled up surface. The whole "rolled up dimension" idea is unnecessary and only complicates things.

I apologize for the number of words, but I have found that my question is so easily and consistently misunderstood that I am desperately trying to articulate it. I hope this has helped clarify it.

Paul

Paul Martin

I think explaining why all observed matter is at roughly the same place in this dimension is easy. It is because that is where we, the observers, are. It is the same as explaining why in the vast reaches of space we get this close-up view of earth but not of other planets or galaxies. It's because we happen to be here. Most "observed matter" is nearby while most matter is far away and unobservable by us.

I may not be understanding you because I don't know what you are referring to when you say "Because they bump into each other." Looking at your previous sentence for possible antecedents of "they" it seems you must be referring either to the extra dimensions or to the observed matter.

If you mean that the extra dimensions bump into each other I don't see the problem. I guess you could consider two dimensions to "bump into each other" at the origin of any coordinate system that you establish. I suppose you could say that there is some mystery about why, for example, the x-axis and the y-asis should be in roughly the same place only near the origin. But there is really no mystery about it. They "bump into each other" there simply because that is where you chose to locate the origin of your coordinate system.

If you mean that particles of observed matter bump into each other then I don't see the problem either. The reason most particle interactions we see occur here on Earth is because that's where we observers are. If we were in some other galaxy, we would see most of the interactions occurring over there.

Whichever of these situations you intended, the addition of large extra dimensions doesn't introduce any new problem. Observations will naturally be localized around the observer in however many dimensions the observer exists.

Your suggestion that there "is a problem because, I think, that positively charged particles are travelling in one direction while negatively charged particles go the other way" is an interesting one. Since both positive and negative particles can travel in any combination of the three basic directions (North/South, East/West, and Up/Down) you must be talking about electric current flow in an approximately linear conductor like a wire. This is an approximation of a 1-D manifold in our 3-D space. I'm not sure what problem you see here. You might mean that in a truly 1-D manifold particles going opposite ways wouldn't have room to pass each other, whereas in a wire they seem to. But since wires are really 3-D and don't have zero diameter, this is not a real problem.

I probably misunderstood you completely, but from what I just wrote, you might be able to figure out where I missed the boat and correct me.

But in any case, you still haven't answered the questions I posed in my original post.

Paul.

Tom McCurdy

ok i am probably missing what your asking here because i am not reading the 4o page intro but let me take a whack at it. First of all Klein introduced the 4th spacial dimensiion in 1919 trying to unify GR and EM but it didn't work because of factors he missed. The added dimension needs to be curled becasue it works out mathmatically that way, plus there would need to be an reason we haven't detected another extended dimension. However the theory has kind rooted in the same idea of string theory (m-theory) which requires 11 dimensions curled up into a specific fashion. They must be curled into an exact calabi-yau manifold with three holes, if it isn't the strings don't get the right viberations and they wouldn't produce a universe like the one we live in. Basically the simple answer to why in almost any theory is so the math works out. I hoped I somewhat helped, tom.

Paul Martin

Sorry about the verbosity.

A reference or details please on why/how the mathematics works out better for curled dimensions.

Where is the error in my explanation of why extra dimensions are non-detectable?

My understanding is that M-theory requires 11 dimensions, but I don't believe they need to be curled up except for the mistaken "problem" of why they are undetectable.

If some specific topological structure, e.g. Calabi-Yau spaces, are required for an explanation it would, IMHO, be much simpler and more straightforward to posit those structures as manifolds. It seems excessive to assume that all of space must conform to that structure.

Paul

Hurkyl

Remember I mentioned that Kaluza's idea was that space-time is a 5-manifold. I was speaking of this 5-th dimension when I said that positively and negatively charged particles travel in opposite directions.

In terms of the 5-th dimension, all matter we observe remains "here" at roughly the same coordinate (because it is able to interact)... despite the fact that the positively charged particles travel away in one direction (in the 5-th dimension) while negatively charged particles go in the other direction. (I think - again, I'm not an expert on the subject, it would be nice if someone who is could comment)

John

A TV screen is a 5 dimensional surface with 2 dimensions that are obvious, and 3 more in a separate manifold below the two. The underlying three don't affect the two that are above them. How can all that be? A TV screen has to be made of points of colored light. Since each point of light is an actual thing, when you arrange the points of light, they can only be arranged in triangles, or they could be arranged as squares, which would make the underlying space consist of square structures that line up in two directions like on this computer sceen. If triangles, they line up in three directions. Can we detect these underlying “dimensions”? Yes.

The underlying structure of triangles can only produce lines that go in three directions. So if you have a circle on the computer screen, like the letter “O” it is made of straight lines and then squiggly lines and then straight lines as you go around the “o” because the underlying structure can only produce lines in three, or two directions.

String theory, in essence, says points cannot touch. If points cannot touch, they have to be arranged in structures. That means the underlying space, which is smaller than the space we are aware of, only lines up in a limited number of directions. Can we detect it? Yes. Look at a TV screen when there is a bright light. The light spills out along the underlying dimensional lines as a crisscross of three lines (draw a horizontal line, then draw and X). That is how a strong light spills out along the pixels of light that make up the screen.

Now look at a star, or any point of light. If you count carefully, you can see six lines crisscrossing, spilling out from the star or point of bright light. Those are the underlying dimensions in the world we live in. In 3D space, points that cannot touch line up in six directions.

Add 3 dimensions that are in a separate manifold above them, and one dimension of time and you have ten dimensions. Add the M dimension (which are the membranes that make up molecules, by the way) and you have 11.

Paul Martin

I understand. ... I agree. ... But you haven't addressed my questions.

Thanks anyway,

Paul

Paul Martin

Your description of a TV screen is close. There are 5 dimensions but they are not all spatial dimensions. Two of them are, obviously - as you say. The other three are color dimensions: Red, Green, and Blue. Each "point" in this space consists of a cluster of 3 dots on the screen, one of each color. The cluster itself has an (x,y) location on the screen and each of the three dots has a value within a color spectrum.

Although the entire structure exists within our 3D world, mathematically speaking it is not a 5D manifold. The 2D surface, however, is a manifold because any of its possible spatial coordinate systems can serve as a part of a basis for the 3D space. (e.g. you could orient the TV so that North/South and Up/Down plus some origin could be used as a coordinate system for the screen. Then by adding on the East/West axis you get a basis for the entire 3D space.)

Your description of an underlying triangular structure follows the traditional app\roach of assuming that everything that is responsible for phenomena is contained wholly within our 4D space-time continuum. Kaluza's contribution was the suggestion that something real might exist outside of that 4D continuum. Klein imposed the condition that if something existed "outside", then it couldn't be very far away.

It is Klein's restrictive condition that I think is unnecessary. I am looking for arguments for why Klein's condition should be retained. Do you know of any?

On a different subject, I think you are mistaken about the six lines you sometimes see radiating from a light source. In my case they are a result of my radial keratotomy scars. The answer I have always gotten when I asked about them in photographs is that they are an artefact of the optical system in the camera. I strongly doubt they have anything to do with extra dimensions. I have the same doubts about Kurlean Photography. (or is it Curlean?)

Thanks for your post, John.

Paul

Fredrik

Paul,

I don't think I can answer your questions, but I would like to recommend an article that I'm sure you'll find interesting: "Out of the darkness", by Georgi Dvali, Scientific American, february 2004.

Dvali suggests that the universe is a 3-dimensional membrane in a higher-dimensional space, where at least one of the extra dimensions is infinite in size. He suggests that particles are open strings with their ends tied to the 3-dimensional membrane, that gravitons are closed strings that can move around freely in all dimensions, and that this hypothesis might actually explain the accelerating expansion of our universe.

Paul Martin

Hi Fredrik,

Thanks for your post. It is encouraging to me that at least some scientists are beginning to take Kaluza's suggestion seriously. At least I have hope that they are. What has dismayed me for 40 years or so is that scientists in general have denied the possibility that anything exists outside of our 4D space-time continuum. When they reluctantly admit extra dimensions into the mathematics of their equations, they have been quick to explain that it is strictly a formality and doesn't have anything to do with reality.

I think there are two potential benefits to come from abandoning that traditional stance. One is that, from what I hear, the mathematics works out better. This would suggest that we might make better progress in coming up with workable theories if we consider large, real, extra spatial dimensions. (I happen to believe that we should also consider large, extra temporal dimensions but I'll wait with that. One small step at a time.)

The second potential benefit is that if we do acknowledge that something exists that is beyond our ability to access it, we can begin theorizing about what it must be like "out there". Some people seem to reject this possibility because they see no hope of learning anything about the contents of hyper-space if we can't interact with it. In my view, we should have learned from the examples of the declarations that we would never know the chemical constituents of stars, or that we would never be able to get to the moon.

I think Maxwell showed the way to learn something about inaccessible reaches of hyper-space, and that is via mathematics. Mathematicians routinely examine and discover properties of hyper-space, although they don't comment on any connection to reality. I think we can do exactly what Dvali seems to be doing and that is make some guess as to what some structures might be like in hyper-space, do the math to predict how they behave, and in particular how they might influence embedded 3D spatial manifolds, and finally see if we can't detect those influences by experiment. In my view, that would enable science to make a huge leap forward in our understanding of the cosmos.

As for the details of Dyali's approach, or of string theory, M-theory or any other, I am not competent to discuss them. I have a hunch, though, that nothing in reality is infinite so I wouldn't be so quick to assume that any dimension is infinite. Instead I suspect that all spatial directions close on themselves eventually, just as latitude and longitude do on the surface of the earth, which must have seemed infinite in extent to some ancients. I'll be satisfied to let people smarter than me work out the details. I'm just eager to see the results.

Thanks for your thoughts.

Paul

jtolliver

It also implies that electric charge is quantized(this is basically the particle in a ring problem in quantum mechanics).

Extra temporal dimensions(whether or not they are large) would imply the existance of tachyons; so far experiments have not detected tachyons.

Paul Martin

Thanks jt but your answer is a little too terse. What implies that electric charge is quantized? Are you saying that large extra dimensions don't require charge to be quantized but curled up dimensions do? (I am not familiar with the particle in a ring problem. Please explain it to me. My knowledge of QM is at the level of "The Quantum World" by Kenneth W. Ford. If you could explain it at that level I would be most grateful.)
That would be consistent with my argument. What I am saying is that the only things our experiments can detect in principle are things that are part of our 4D manifold. Anything that exists in hyperspace outside of that manifold we shouldn't expect to detect. If tachyons exist, that would seem to me to be where they are.