Why do extra dimensions have to be curled up ?

[Paul Martin]

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

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]

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

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 traveling in one direction while negatively charged particles go the other way.

 

[Paul Martin]

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.

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]

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 traveling in one direction while negatively charged particles go the other way.

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 traveling 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 spatial 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]

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 spatial 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.

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]

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

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]

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)

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

Thanks anyway,

Paul

 

[Paul Martin]

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.

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]

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.

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]

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?

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

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.)

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

 

[Paul Martin]

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

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.)

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

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.

 

[4newton]

Postulate:

There are an unlimited number of possible dimensions and all independent dimensions have the same properties. All actions in a dimension can only affect a limited number of associated dimensions.

If you take a point that is going around in a circle, like the moon going around the earth, and view it from the axis of rotation you will see the path of the point make a circle.

If you now move one dimension away and view the action from the side you will see the point going back and forth side to side from your point of view.

Again if you take side to side action and move one more dimension away you will see no motion but you will still be able to detect the motion toward and away from you using the proper detectors.

If you start at this point and move another dimension away you will see no motion and not be able to detect any action.

You are not able to make this move in our three-dimensional space. We are restricted to just repeating back to the last dimension. The same as the ant on the wire but our restrictions is not the restriction of nature.

As you move more and more dimensions away from the action you will never be able to know that any action is taking place. This is a simple example of the nature of dimensions.

From this it is easy to see that there is no need to resort to curled up dimensions to explain our inability to detect anything past three independent dimensions. Any action beyond three-dimensions has no effect. This is true for 3 * 3 or a max of nine-dimensions.

 

[Chronos]

Why assume that 3 spatial dimensions plus a time dimension are not enough to describe the universe we observe? The concept works well. It agrees with observation and is the simplest model. I need a lot more theory and supporting observational evidence before I will be willing question that assertion.

 

[4newton]

Hi Chronos:

Why assume that 3 spatial dimensions plus a time dimension are not enough to describe the universe we observe? The concept works well. It agrees with observation and is the simplest model. I need a lot more theory and supporting observational evidence before I will be willing question that assertion.

When you examine the nature of dimensions there is no indication of a limited number. Mathematics places no limit on the number of possible dimensions and as shown the number of dimensions have no effect past nine. A fifth and sixth dimension must be used to explain gravity and electric charge. I have not yet posted the theory of forces that use more dimensions.

The observation of the nature of dimensions indicates that there are independent and limited independent dimensions.

In independent dimensions you may move up to the speed of light in two directions at the same time. The vector sums do not add.

The spatial dimension is an independent dimension and has three limited independent dimensions.

In the spatial limited dimensions you may only move in one direction at a time but you do have independence in actions in the limited spatial dimensions. All actions in the limited spatial dimensions vector sum and the vector sum is limited by the speed of light.

Magnetic fields that are perpendicular to each other are independent and have no effect on each other. In the spatial dimension you can only have three limited independent dimensions that are perpendicular to each other. You may however move in any direction or rotate the limited spatial dimensions through out all angles with any degree of resolution desired. This same observation is seen in polarization of light and electromagnetic waves.

At this time we only know of two independent dimensions, the spatial dimension and the time dimension and the three, limited independent spatial dimensions.

All these theories are very simple, the math is easy and the ideas are easy to understand. They are however hard to accept because they go against our conditioning and physical experience.

We can not perceive of anything more than three-dimensions and many deny a fourth-dimension, time. How many times have you heard “time is not real”. It just takes time to accept these concepts, about 10 years.

 

[Paul Martin]

Hi 4Newton,

You Postulate that "There are an unlimited number of possible dimensions and all independent dimensions have the same properties. All actions in a dimension can only affect a limited number of associated dimensions."

That sounds like a perfectly legitimate mathematical starting point, but from it, you can infer only mathematical consequences and nothing about reality. Of course a mathematical result may suggest an hypothesis about reality, but nothing is proved about that hypothesis unless and until experiment supports or denies it. So let's be careful not to mix up the two.

Mathematically, your description is not quite right. If you view a point moving in a circle from its axis, you see a circle. And if you move to a line containing a diameter of the circle, you see the line segment you described. But if you move to a third axis which is mutually perpendicular to the other two axes, you again see that same line segment with the point oscillating between the two end points.

The way you get what you are talking about is to take a projection of each successive view and view that projection from the side. In other words, you have to reduce the dimensions by one each time before you change your viewpoint. If you do that, then the number of dimensions goes to zero in the manner you suggest. It isn't the same as the ant on the wire. (In choosing your starting point, you have already reduced the general 3D case to a 2D circle. To be general, you should have had your point following a precessing orbit so that it covers the surface of a sphere. That way, your first view would reduce to a point moving in a disc, and the subsequent views would be as you described them.)

As you move more and more dimensions away from the action you will never be able to know that any action is taking place. This is a simple example of the nature of dimensions.

I don't think this conclusion is quite accurate. You were not moving more and more dimensions away from the action. You were just moving from one orthogonal viewpoint to another. You have three such viewpoints in three dimensions, and if you are viewing a 3D structure, you will simply see three views of the same object. If you are viewing a manifold, however, either 2D, 1D, or 0D, then you might see the degenerate views you describe from some viewpoints.

At first I thought you might be correct that the more dimensions there are, the more degenerate views there would be, but I don't think that is right. Consider a line segment for example. There is only one degenerate view in 1D (actually two if you consider looking at the segment from the other end in the opposite direction). The same segment in a plane offers the same one (or two) degenerate views. And again, in 3D space, the same segment offers the same one (or two) degenerate views. I think it is reasonable to expect that this would hold true for any number of extra dimensions. The extra dimensions just give you additional views of the object, which it seems to me give you more information about the object rather than less.

From this it is easy to see that there is no need to resort to curled up dimensions to explain our inability to detect anything past three independent dimensions. Any action beyond three-dimensions has no effect. This is true for 3 * 3 or a max of nine-dimensions..

I agree with your conclusion, but not at all with your method of arriving at it.

We need to talk more.

Paul

 

[Paul Martin]

Why assume that 3 spatial dimensions plus a time dimension are not enough to describe the universe we observe? The concept works well. It agrees with observation and is the simplest model. I need a lot more theory and supporting observational evidence before I will be willing question that assertion.

Hi Chronos,

Here's why I think 3 spatial dimensions and 1 temporal dimension are not enough:

1. The concept does not work well - not well enough at least IMHO. For one thing, it doesn't seem to permit a sensible interpretation of QM. But to me, the most obvious failure is that it doesn't, and it seems to me that it can't, explain consciousness. I think extra dimensions will be required in order to come up with a sensible, complete, and useful theory of consciousness. Such a theory will explain the role of the observer in physical interactions and provide a sensible solution to the mind/body problem.

2. It does not agree with observation - at least in one respect. We seem to have evidence that our 4D continuum is "curved" or "bent". I don't think you can bend a space unless it is an embedded manifold in a space of at least one dimension greater than that of the manifold. Take a 2D space like a sheet of paper for example. If it is flat, it can exist in 2D and nothing more. But if you bend the paper, you cause it necessarily to occupy part of a 3D space. If you lift a corner of the paper up off the table, you have your original two dimensions of the paper and you have moved part of it into the 3D space above the table.

3. 4D being simpler than 5D or 11D or whatever is a virtue only if it answers the same questions. As I said, I have a hunch we will be able to answer more questions if we are willing to question the assertion that 4D is all there is. I am hoping a lot of smart people will begin doing a lot of theorizing about large, extra dimensions, and figuring out ways to detect their presence.

Thanks for your comments, Chronos.

Paul

 

[Paul Martin]

Hi 4Newton,

I have not yet posted the theory of forces that use more dimensions.

Please do. I am eager to see it.

Paul

 

[4newton]

Hi Paul Martin:
It is great to have your comments.

That sounds like a perfectly legitimate mathematical starting point, but from it, you can infer only mathematical consequences and nothing about reality. Of course a mathematical result may suggest an hypothesis about reality, but nothing is proved about that hypothesis unless and until experiment supports or denies it. So let's be careful not to mix up the two.

You are absolutely correct. These are concepts based on very little firm observation. I am trying to understand the pattern of nature. With the assumption that there is a basic pattern. Then with a starting point of direct observation or experiment in the area of interest extrapolate a theory. It is then necessary to fill in all parts of the theory. First to make sure that any part does not conflict with any observation, or at least be able to explain the difference. Then the theory should offer the most rational and simple description. Not conflicting with the mathematics, I think, is a very important start.

Mathematically, your description is not quite right. If you view a point moving in a circle from its axis, you see a circle. And if you move to a line containing a diameter of the circle, you see the line segment you described. But if you move to a third axis which is mutually perpendicular to the other two axes, you again see that same line segment with the point oscillating between the two end points.

You are correct my description did not say very well what I was trying to get across. You did however get the concept. Let me see if I can state it better.

The total concept must be viewed from the math of unlimited dimensions. You must take yourself outside of any dimensional restrictions.

When you are in a three-dimensional frame and you take degenerate views (that is a great way of stating the action, thanks) you are only able to cycle through the dimensions of your restriction.

This relationship only applies and is only necessary to apply to motion or transition not static objects. Static objects are unable to transfer information between dimensions. We only see static objects because of dynamic objects, like photons, are providing information about the static objects.

The way you get what you are talking about is to take a projection of each successive view and view that projection from the side. In other words, you have to reduce the dimensions by one each time before you change your viewpoint

Yes, that is what I thought I was saying but going back and reading it after your comment I see that was not made clear. Also if you are restricted to the three-dimension you are not able to take the next degenerate view but if you are still one dimension removed and there is action to transfer energy you may then have an affect on the next degenerate dimension.

The use of mechanical action in three-dimensional construct does not have any results to prove the function. I only use this to illustrate the concept of degenerate views.

There is only one function I can think of that shows a result. If an electron is moving it produces a magnetic field in a plain perpendicular to the direction of motion. The first field will affect any field that is in the same plain. Any field that is perpendicular to the first field will show no effect. You can have only two plains, or two dimensions that are unaffected by the first field. These are end degenerate views.

As you can see this is the starting point to explain all forces in the universe.

 

[Mk]

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.

Err... that's not how I remember it, one in a cafe can see it as having wideness, and length. The ant, I think a gecko would be more appropriate, but the ant, can crawl a full 360˚ around the cable.

 

[Paul Martin]

Hi 4Newton,

In addition to being about the same age, you and I seem to think a lot alike. We also seem to be in the same predicament. That is that we are both "trying to understand the pattern of nature" and neither of us is completely satisfied with the current scientific explanations. Furthermore neither of us is well enough educated in math or physics to come up with a new theory on our own.

That leaves us in this frustrating pickle of having some vague ideas about how nature might work, but not being able to articulate what we have in mind, not even to mention being able to demonstrate or prove anything about those ideas.

You gave me credit for getting the concept you were trying to explain, and if you are right about that, then I think my vague ideas about the fundamental structure and operation of nature are very similar to yours. We may be able to get somewhere if we keep refining our explanations to each other, and hopefully, some younger readers with better math and science skills might step into the conversation. If we are on to something, they might pick up on it. If we are totally out to lunch, they might be able to straighten us out.

So let me take a crack at explaining my ideas and you can tell me what you think of them. I'll list some of my hunches or beliefs, and hopefully together they will convey my vague ideas about how nature works. These are all simply hunches or speculations on my part. I am not asserting any of them as facts. If you like, you can take them to be postulates, as you identified yours in your earlier post. Here goes:

1. Reality consists of more than 3 space and 1 time dimension. (I explained one of my reasons for thinking that as point #2 in my response to Chronos above. There are a few more reasons I won't go into here.)

2. It is reasonable to expect that we have no access to anything outside of our 4D space-time manifold. Thus, traditional experimentation is powerless to directly confirm or deny the existence of additional dimensions.

3.We can mathematically deduce many of the properties of hyper-space-time. By figuring out how hypothetical hyperspatial events might interact with our 4D manifold, we might be able to indirectly confirm or deny the real existence of those hypothetical events.

4. Considering an electrically charged point moving rectilinearly in our 4D manifold, we in a sense "use up" one spatial dimension and the temporal dimension. That is, we can establish the direction of motion of the point as one spatial axis, and, since the point is moving, we need the temporal dimension to contain or describe this motion. In addition, the right-hand rule of EE tells us that associated with that moving charge, there is an electric field perpendicular to the direction of travel of the charge, and a magnetic field perpendicular to both the electric field and the direction of travel. These two mutually perpendicular directions in a sense "use up" the remaining two spatial dimensions. It is remarkable that two seemingly different forces (electric and magnetic) can be unified as a single force by combining the effects in mutually perpendicular directions, i.e. in two different dimensions.

5. Unifying the other three known forces (strong, weak, and gravity) has not been as easy. It seems to me that it might be fruitful to approach the problem, as Kaluza suggested long ago, by considering that the other forces manifest themselves in hyper-space and that those manifestations interact with our 4D manifold to yield what we observe.

6. The admission of extra dimensions opens up the possibility for vastly greater complexity and variety in structures and configurations of "things" than is possible in our 4D manifold. This runs afoul of Occam because even though extra dimensions might be "simple" in concept, it would be like opening Pandora's box unleashing a wide range of new complexities. But, science has routinely done that in expanding our notion of the cosmos from the relatively small celestial spheres of the ancients to the mind-boggling expanse of space as science now describes it. The same is true for the new complexities of the quantum world as we describe ever smaller systems and their behavior.

To be continued, ...

Paul

 

[Paul Martin]

As I was saying,

7. Consciousness is the most baffling remaining mystery of our experience. Science - in particular Physics - has virtually nothing to say on the subject. The best we have done is to classify and name some phenomena and properties associated with consciousness. We have no idea about what it is or how or why it seems to appear in the presence of live brains. Current brain research is analogous to a study of a working cell phone in isolation. If you didn't have knowledge of, or access to, any of the other parts of the cell phone system, including EM radiation and including the person talking on the other end - i.e. you had access only to the one instrument you are studying - you would be completely baffled at the apparent conscious entity inside that phone that you could talk to. (Imagine Isaac Newton being presented with this problem knowing only what he knew when he was alive.) You could attach all the instruments you wanted to the cell phone, or use any amount of MRI scanning while you were talking on that phone, or you could understand in detail the flow of each and every electron in the circuits of the cell phone, and you would still have no clue to the existence of the things that make a cell phone work. I happen to think that the brain is a cell-phone-type-device which communicates with a vastly more complex system that is resident in hyperspace-time outside of our 4D manifold.

8. Time is mysterious. A significant aspect of time that, IMHO, doesn't get enough press is the different roles it plays in Physics and in consciousness. In Physics, time appears as a variable in virtually all equations describing dynamic phenomena. Most of these equations work just as well with time running forward or backward. In none of these equations is there any notion of "now" or the present moment. In fact relativity says that you can't even define the notion of "now" in any consistent way. Time also appears in consciousness as inexorably flowing. But by contrast to Physics, time always flows forward in consciousness. And in complete contradistinction to Physics, there is a very certain unavoidable "now" in consciousness. In fact, it is only in the present moment that consciousness exists. It does not exist in the past or the future. From the conscious viewpoint, time is clearly divided into the past, present, and future - concepts which don't appear anywhere in Physics.

9. The metric in our 4D space-time is not symmetric wrt time and space dimensions. The spatial metric, i.e. the formula giving distance in terms of spatial coordinates, is the Euclidean metric. That is, distance is the square root of the sum of the squares of the component distances. But when you introduce the time dimension, the square of the time component is multiplied by the imaginary number -i. (Or something like that. Someone please correct me if I am wrong.)

10. The metric in our 4D space-time gives distances (more correctly the separation of events) as we measure them here in our 4D manifold. There might be a different metric measuring the separation of events in hyperspace. (E.g. consider two points on a sheet of paper. The Euclidean metric is the familiar Pythagorean Theorem giving the length of the diagonal. Now, roll the paper in the shape of a tube and imagine a string inside the tube stretched tightly between those two points. This could be considered a "more true" measure of the separation of those points. Physicists who consider wormholes in space are tacitly admitting the reality of the picture I am trying to present.)

11. Considering the possibility of a different metric over our manifold from the perspective of hyperspace, one possibility is that our time dimension becomes another spatial dimension. In order to maintain the possibility of dynamism, an additional time dimension would be necessary. This would mean that our world lines would become geometric lines in 4D space (different from 4D space-time). It would also mean that if there were a consciousness resident in hyperspace (the guy who was on the other cell phone), that consciousness could, in the new dimension of time, attend to one of our world lines by following it wherever it goes over its 4D spatial path. If you think about this, it would be tantamount to that consciousness living and experiencing exactly what we seem to experience as we live our lives.

12. This picture presents what seems to be an even more complex problem in explaining consciousness. It would require vastly more complex "brains" and "organisms" (i.e. 4D instead of 3D as ours are), at least if we try to stay within the paradigm we are familiar with for the manifestation of consciousness in our world.

13. Rather than jump to the conclusion that this sort of explanation leads to "infinite" regress, consider the possibility that consciousness itself is the fundamental constituent of reality, as Berkeley suggested, and that what we perceive as physical reality is nothing more than thoughts of that consciousness (Wheeler's "It from bit"). Consciousness alone is capable of conceiving of mathematics. Mathematics is nothing more than consistent patterns of thought. Physicists are beginning to conclude that physical reality is nothing more than sets of properties (quantum numbers) associated with coordinates in some geometrical system (points in space-time) evolving over time according to some (sort of) mathematical rules. In this view, the number of dimensions could be limited to (or at least currently fixed at) some small number, like 11.

14. I said the rules are only sort of because they have an inherent level of uncertainty and can at best describe the probability of a particular outcome of an interaction. This, IMHO, allows consciousness to deliberately influence the outcome of certain interactions as long as it fell within the bounds of the uncertainty. This opens up myriad possibilities for the explanation of mysterious behavior of the type Rupert Sheldrake has been investigating.

That's probably enough for now. I would appreciate anyone's comments on this.

Paul

 

[Paul Martin]

Err... that's not how I remember it, one in a cafe can see it as having wideness, and length. The ant, I think a gecko would be more appropriate, but the ant, can crawl a full 360˚ around the cable.

Hi Mk,

Thanks for your thoughts. Should I interpret "Err..." as simply your indication of hesitation as you composed your thoughts? Or was it a one-word sentence telling me that I made an error?

Any time we use an analogy we have the problem of being clear on what parts correspond and what parts don't apply. We usually have the additional problem of allowing some leeway for parts that don't correspond exactly. I think we have all those problems here. Let me try to clear it up.

The point of the analogy is to demonstrate that from some point of view, dimensions may seem to be missing even though they are really there. The mechanism for this "disappearing act", according to Klein is that the hidden dimension is rolled up too small to see.

In the power cable analogy, all but one dimension is too small to see from a distance, but from close up (i.e. from the point of view of the gecko) the rolled up dimension can be seen. So you are right: the gecko would notice the second dimension which wraps around the cable giving a 2D surface. In this half of the analogy we have two real dimensions and one of them is hidden from one observer.

The other half of the analogy is us here in 3D space wondering if there might be a 4th dimension hidden from us. So are we analogous to the distant observer or to the gecko? As I pointed out, neither one works. If we are analogous to the distant observer, then we are positing a real large extra dimension outside of our 3D space from which point we do our observing. (The cable is analogous to our world and that is where we really are when we do our observing.)

On the other hand, if we are analogous to the gecko, then all the existing dimensions are visible to us, just as are the walls in the cafe, and no dimension is hidden.

The analogy fails either way.

Paul