Defining Superstring's Additional Dimensions

[IqbalGomar]

Hello everyone. This is my first post here, though I've stumbled upon the site on several occasions while investigating various physics concepts over the years.

I've decided to post now because of a maddening dearth of information regarding the definition of each of the additional dimensions predicted in Superstring Theory. I realize this is a topic which has been opened here many times before, but it seems all those threads have long since died... mostly because no conclusions could be reached. So here we go again.

Usually when this question comes up, the extent of the answer is that the additional dimensions are curled up at every point in our familiar spacetime into a compactified Calabi-Yau manifold. This is all well and good, and to this extent I understand it. My question is in regards to the specific identities of the extra degrees of freedom represented in this manifold. I know that this is sort of the central problem in string theory, but we should be able to talk about it in layman's language and understand something about it intuitively, don't you think?

My understanding so far:

Dimensions 1 - 4 represent our classical spacetime. They are defined in our experience by our relationship to Earth's surface.

1: Length (forward - backward, or longitude/latitude)
2: Breadth (left - right, or latitude/longitude)
3: Height (up - down, or altitude)
4: Time (past - future, or duration)

So far so good. All this basically means that at our ordinary mundane scale, there are four meaningful values by which any point in our experience is defined: where it happens, in three dimensions, and when.

It then follows that the additional dimensions should correspond to meaningful values. The four classical dimensions are large; probably infinite in extension. A moving object has momentum in spacetime. At relativistic speeds, a large portion of its ordinary fixed movement through time is converted through a Lorentz transformation into movement in a spatial dimension.

But the additional dimensions are small, somewhere between the TeV scale and the Planck scale. As far as I understand it, objects (particles) can have momentum in these dimensions just as they have momentum in space. But since the dimensions are small, a particle moving in one of them will circumscribe the universe in that dimension. The higher the frequency of its vibration as it travels in that direction, the higher its energy.

For instance, an electron's movement in one of these additional dimensions corresponds to its charge. Its movement in another corresponds to its magnetic polarization. Therefore, two of the additional dimensions correspond to meaningful values of the electromagnetic force.

5: electric charge
6: magnetic polarization

Momentum can be translated readily between these two dimensions, and also out of them into our familiar four as photons moving in spacetime. The photon has no momentum in those two dimensions, and therefore no mass or charge. All its momentum is in spacetime.

From here on my intuitive understanding gets a little fuzzy. Does the value of quantum spin correspond to momentum in a higher dimension?

7: spin (?)

Dimensions higher than this will have much smaller diameters, between the 10^16 TeV scale and the Planck length. Momentum in these dimensions can not be readily transferred into others, therefore the forces residing there are confined to their own scales, which is what we see in the real world.

8: weak gauge interactions
9: strong force
10: higgs field interaction (?)

At the Planck scale, rotation of 10D unit spacetime in any dimension should look identical, representing grand unification and perfect symmetry.

Does this seem like a reasonable description to those of you who know more about these topics on the level of their actual experimental values? Is this a consistent layman's description of higher dimensional spacetime? Please respond with any corrections or criticisms of this model. Thank you.

 

[Unit]

i have never thought about this but i love the idea of 5: electric charge and 6: magnetic polarization existing as "dimensions". it makes me think like however "far" the string is in this dimension, or however "much" the string vibrates in the direction of this dimension determines the particle's charge or whatever other property. it's an appealing idea, but I've never heard of it before.
now i might be wrong, but, with 8 and 9, it sort of seems like you are starting to say that it would be possible for a string to represent a particle that can exert the strong force, and the weak force, and exert the electromagnetic force. is that possible? i am not sure.

and i also am thinking, the "depth" of the higgs dimension would equal the amount by which the particle can be influenced by the higgs field, yes? i.e., the depth of the higgs "dimension" = mass. however, are higgs bosons themselves affected by higgs bosons? do they have mass? or would massless particles, like photons, have zero length within this "higgs" dimension?

this is so cool!

 

[IqbalGomar]

Yes, that's pretty much how I've been thinking about it. It is indeed an appealing idea, which is why I am loath to presume its accuracy. It seems unlikely that interactions as complex and strange as the nuclear forces could be fully represented as momenta in higher dimensions, but maybe it's just not a way people have thought to consider it. I don't know. I'm hoping to get the input of people who have some specific understanding of QED and QCD so as to know if the way I've been thinking about it is all wrong.

But yeah, it's intuitively a pretty solid model. If mass = energy, and that energy takes the form of momentum around the diameter of one of the small dimensions, then it would make sense that that momentum would subtract from classical momentum in 4-spacetime. A particle that must move in additional dimensions could never have all of its total energy converted to momentum through space, and therefore could not move at C. The more momentum a particle has confined to small dimensions, the greater the mass and the more resistant to movement through space. By that line of reasoning, the Higgs interaction actually seems irrelevant, so I'm not sure it's a valid way of thinking.

One key concept here is the idea of string harmonics. I'm presuming that all the small dimensions have a constant positive curvature and loop back around on themselves at a small distance, rather than being bounded by branes. Therefore all strings are closed and encircle the universe around small dimensions. Only an integer number of wavelengths could exist on a single diameter. The greater the number, the higher the energy and the higher the mass. A quantum of energy transferred would be equivalent to the loss of one wavelength, but only certain exchanges would be allowed. If a string circles the universe in one small dimension, say the strong force dimension, it would only be able to trade energy with strings that circle it in the same way. For this reason, a gluon can not be readily exchanged with an electron for instance, as they are qualitatively different kinds of energy, interacting on different scales in different dimensions. However, at a small enough scale and a high enough energy, any type of interaction would be allowed, corresponding to unification of all forces; free exchange of energy between strings circling the universe in any small dimension.

As for the Higgs if it exists, I assume that like other bosons, they are able to exist simultaneously in the same quantum state and do not interact with each other as fermionic matter does. I may be wrong about this. Does anyone happen to know the answer?

 

[MTd2]

Out of 10 dimensions, 4 means the usual space time and gravity are manifest, that is, space is not compactified, the other 6 are the curled ones, where the other forces are realized in weird geometric shapes of the compactified dimentions.

 

[IqbalGomar]

Out of 10 dimensions, 4 means the usual space time and gravity are manifest, that is, space is not compactified, the other 6 are the curled ones, where the other forces are realized in weird geometric shapes of the compactified dimentions.

Right, that's what we've been talking about. That's the typical response I mentioned in the original post. I'm hoping to find out something more specific about all this, such as the specific parameters of the Calabi-Yau space which comprises the compactified dimensions. I know that this is a key problem string theorists are dealing with, and there's no simple answer (though there may be 10^500 complex ones). I'm just trying to figure out if it makes sense to think about electromagnetism and the nuclear forces as momenta confined to one or more of the compactified dimensions. Anyone?

 

[OB 50]

Why is it necessary for these additional dimensions to be compactified?

Wouldn't it be make more sense to view the additional dimensions as analogous to how 3D space relates to 2D space? If we lived in a 2D universe, the concept of 3D space would appear to be compactified, when in actuality it encompasses the 2D space. Why should we assume that this relationship somehow reverses once we go "higher" than 3D space?

Compactification seems completely backwards. Is this literally the way these dimensions are posited to exist, or is it more of an accounting trick to make calculations possible?

 

[IqbalGomar]

That's a good question, OB 50. To be fair, there are models which have emerged more recently as a consequence of M theory and various braneworld conjectures that present the extra dimensions as large in extension. Though regardless of how large these bulk dimensions could be, it still stands that our universe's meaningful presence in them is confined to a very small region. The idea that they are small and curved is the most appealing model to me, but it could also be that particles operating in those dimensions have wavefunctions which localize them on the 4-brane of our universe. I'm not familiar enough with the maths involved to form an educated opinion of which model is more likely, but it seems most reasonable to me from an intuitive standpoint that they are small and curved. It seems to be a more elegant solution. Whatever the case, our univese's extension in the extra dimensions must be small and localized, because we do not see the spontaneous loss of energy that would be associated with particles escaping into higher dimensions. The only force which, as I understand it, must extend into all dimensions equally is gravity, hence its weakness relative to other forces.

There is little doubt that these extra dimensions exist, whatever form they take. I remain curious about what their specific identities and properties are. I am curious to see if anyone can respond with any insight into the information I am seeking. Will anyone bite?

 

[Hurkyl]

Why is it necessary for these additional dimensions to be compactified?

Wouldn't it be make more sense to view the additional dimensions as analogous to how 3D space relates to 2D space? If we lived in a 2D universe, the concept of 3D space would appear to be compactified, when in actuality it encompasses the 2D space. Why should we assume that this relationship somehow reverses once we go "higher" than 3D space?

Among other things, you have the analogy wrong. The analogy would be "We live in a 3D space, but it appears to be a 2D space when we look around. What can we infer about the geometry of 3D space?"

Or, put differently, "How can a three-dimensional space look like it's two-dimensional?"

 

[ExactlySolved]

These parameters are not thought of as extents in higher spatial dimensions:

5: electric charge
7: spin (?)

The reason is that spacetime is Lorentz-invariant, so in particular the space dimensions are rotationally invariant. This means that I should be able to rotate, for example, dim 1 into dim 5. It is not possible rotate distance into charge, or into spin, since there are interactions that conserve/charge and (intrinsic) spin: these are Lorentz invariants, and so they do not transform in the way that the components of a position vector do.

8: weak gauge interactions
9: strong force

Gauge fields are just geometry, so your on some kind of correct track, I'll come back to this.

10: higgs field interaction (?)

The Higgs field is not a gauge field, and in particular it is a Lorentz scalar field (this is why rest mass is a lorentz scalar invariant).

As for gauge fields, each one takes more than one dimension. The gauge group of the standard model is SU(3) x SU(2) x U(1). The color SU(3) is an 8 dimensional manifold. SU(2) is a 2d manifold, and U(1) is 1D. Notice 8 + 2 + 1 = 11, which is only heuristically meaningful (this is not a real mathematical connection, just suggestive of the number of dimensions it takes to unify the forces).

Therefore the standard model, interpreted as geometry, attaches an 11 dimensional manifold to each point of spacetime, and all of these 11 dimensional manifolds have to be connected together in a smooth way; the curvature of this spacetime + gauge groups manifold, a construction which is in general called a principle gauge bundle, is the field strength tensor of the (unified) forces (just as curvature replaces the gravitational force in Einstein's GR).

In the standard model, the dimensions corresponding to the gauge groups are not thought of as spatial, for the same reason I gave above that it makes no sense to do rotations between these dimensions and normal spacetime.

String theory (in the narrow sense e.g. no M-theory, F-theory) is about quantized relativistic strings in 10 dimensional spacetime. A string has charge, spin, etc but these do not have to do with how much of the string is extended in a particular direction e.g. the charge direction, there is nothing like that in string theory. For some more suggestive numerology we have 4d spacetime and SU(3) x SU(2) x U(1) gauge symmetry yielding 4 + 3 + 2 + 1 = 10.

 

[Phrak]

"Therefore the standard model, interpreted as geometry, attaches an 11 dimensional manifold to each point of spacetime, ..."

I think you mean, attach 7 additional dimensions to each event on the spacetime manifold.

 

[IqbalGomar]

Many thanks for your response, ExactlySolved! I feel like I'm getting somewhere here. Of course I'm only left with further questions, but that's only natural. Perhaps you or someone else would be willing to indulge them.

I assume that in the case of the electromagnetic force, your explanation must mean that without parameters or extension in additional dimensions, the force is confined entirely to spacetime. Is this correct? Would this indicate that the field lines of an EM field correspond in some way to the 4D curvature of a gravitational field? If my interpretation is not correct, would you be able to explain further in regards to where or in what dimensions the properties of charge and spin reside?

I am interested by your explanation of gauge fields. If the dimensions required to describe them are not considered to be classically spatial, then what exactly are they? Surely they are not timelike dimensions. Also, what are the properties or parameters of gauge fields that rely on extra dimensions for their description? I am used to thinking of dimensions as degrees of freedom, or as the number of coordinates necessary to describe something's location. If the extra dimensions are not spatial or spacelike, what do these coordinates represent?

By your explanation, I am getting the sense that all the quantities defined by identities in the Calabi-Yau manifold relate to the nuclear forces. Is this correct? Are gravity and electromagnetism fully described by a theory in four dimensions, and only the gauge fields reliant on the extra ones?

It is possible that over the years I've tried to put together a complete intuitive picture of string theory, I have relied excessively on heuristics and taken the map for the territory. In your understanding, are the extra dimensions posited by string theory actual physical spaces, or simply mathematical abstractions required to make the equations work?

Just one more question which I'm hoping to have answered, though I know it may be impossible to describe in intuitive terms: why is Lorentz invariance a necessary ingredient for these models? Its necessity in relativity is intuitively obvious, as relativistic travel transposes time and space dimensions; but I fail to understand why it would be necessary to preserve rotation invariance between spacetime and its compactified dimensions, as that type of rotation would have no real-world analog. Is there any way to make this requirement comprehensible to a layman?

 

[ExactlySolved]

The gauge group for electromagnetism is U(1), which is literally just a circle i.e. rotations specified by one angle. This goes all the way back to Kaluza and Klein in the 1920s and 1930s, who discovered that if you consider a 5d spacetime made of ordinary 4d minkowski space plus one extra dimension that is equivalent to a circle attached at each spacetime point, then Einstein's classical field equations of general relativity imply the classical maxwell equations of E&M.

Kaluza-Klein: (General Relativity) + (extra circular dimension at each point) = Gravity and E&M.

Incidentally, Kaluza-Klein theory has various unphysical ghosts which is why the idea was put aside for so long.

Note that in the Kaluza-Klein theory the extra dimension is described by an extra parameter, as you suspected, something like (t,x,y,z,theta).

Gauge fields are due to curvature in an abstract mathematical space i.e. a SU(3) x SU(2) x U(1) x (spacetime). This space is rather like (temperature) x (wind velocity) x (position): only part of the space is spatial, the other 'dimensions' are just parameters. Yes, the 'dimensions' are coordinates that specify a point in an abstract space, but No, they are not additional dimensions in the spatial sense of adding another axis like up-down etc. Gauge fields describe the strong force, the weak force, and the electromagnetic force.

but I fail to understand why it would be necessary to preserve rotation invariance between spacetime and its compactified dimensions, as that type of rotation would have no real-world analog.

Quantized relativistic strings cannot exist in dimension D = 4 spacetime, and tacking on extra non-Lorentzian dimensions will not change that. Therefore if quantized relativistic strings exist, they must do so in a higher dimensional space.

I think you mean, attach 7 additional dimensions to each event on the spacetime manifold.

Nope, SU(3) is 8d, SU(2) is 2d, and U(1) is 1d, so SU(3) x SU(2) x U(1) is 11d. Therefore SU(3) x SU(2) x U(1) x (spacetime) is a 15 dimensional manifold. Heuristically, there is no difference between spacetime dimensions and gauge dimensions in string theory and so only 10 or 11 dimensions are needed, with four of them serving 'double duty' as 'gauge dimensions' and as observable dimensions that we move around in.

 

[arivero]

Nope, SU(3) is 8d, SU(2) is 2d, and U(1) is 1d, so SU(3) x SU(2) x U(1) is 11d. Therefore SU(3) x SU(2) x U(1) x (spacetime) is a 15 dimensional manifold. Heuristically, there is no difference between spacetime dimensions and gauge dimensions in string theory and so only 10 or 11 dimensions are needed, with four of them serving 'double duty' as 'gauge dimensions' and as observable dimensions that we move around in.

But you must consider the smallest non trivial homogeneus space where the group acts. For U(1) it is pretty trivial, U(1)/1. For SU(2) it is SU(2)/U(1). For SU(3) it is SU(3)/U(2) (or SU(3)/SU(2)xU(1). Now you count dimensions: 1+2+4 = 7, right in the target of supergravity/M-theory.

This fact was pointed out by Witten in "Realistic Kaluza Klein theories". He did a classification of all the spaces of the kind SU(3)xSU(2)xU(1) / SU(2)xU(1)xU(1)

 

[OB 50]

Among other things, you have the analogy wrong. The analogy would be "We live in a 3D space, but it appears to be a 2D space when we look around. What can we infer about the geometry of 3D space?"

Or, put differently, "How can a three-dimensional space look like it's two-dimensional?"

I think you misunderstood what I was trying to say.

Let's say we live in Flatland (2D space). If a 3D sphere intersects and travels through this space, it will appear to originate as a point, which will then grow to the maximum radius and then recede back to a point.

To someone living in flatland, it would appear that the higher dimension must be compactified, as it originated from a single point, or "inside". The actual direction of the sphere's origination does not exist for the residents of Flatland.

Is it such a stretch to think that we would perceive higher dimensions in the same way?

 

[Hurkyl]

I think you misunderstood what I was trying to say.

No, you're talking about what I thought you were talking about. You're talking about how things would look to a person constrained to live and observe within a 2D slice of an (uncompactified) 3D space.

But that's not the situation here. The observer really and truly gets to move and look in all three dimensions without restriction; the challenge is to figure out what circumstances would lead such an observer to think that space was only two-dimensional.

To someone living in flatland, it would appear that the higher dimension must be compactified, as it originated from a single point, or "inside".

I can't make sense of this.

 

[OB 50]

No, you're talking about what I thought you were talking about. You're talking about how things would look to a person constrained to live and observe within a 2D slice of an (uncompactified) 3D space.

I'm sorry, but that's not exactly what I'm trying to say.

A 2D space would not simply consist of a "slice" of 3D space. It has to contain information and be bounded in some fashion. Take the sphere that defines the surface of a water droplet suspended in space as a simple example. I'm not talking about the water atoms, and I'm not talking about the air around it, I'm talking about the boundary that defines the shape. The 2D space consists entirely of this boundary and the information contained within.

But that's not the situation here. The observer really and truly gets to move and look in all three dimensions without restriction; the challenge is to figure out what circumstances would lead such an observer to think that space was only two-dimensional.

Why would someone in 3D space think they were in 2D space? Where are you getting this?

A "resident" of 2D space would encounter a completely different geometry from someone in 3D space. First of all, a "point" would have no meaning to them, as the simplest geometric element would be a line. The very idea of a point implies 3D space, so I regret wording it as such in my previous post. I was trying to reference "Flatland" for simplicity's sake, but I guess that backfired.

A resident of 2D space would not perceive their degrees of freedom to be constrained in any way, because the directions in which they are constrained do not exist for that resident. Likewise, our degrees of freedom may be similarly constrained, of which we are similarly unaware.

If a 4D hypersphere intersected our 3D space, it would appear to originate from a point. We can naively assume that the dimension of origination must be very small, but that assumption isn't necessary at all. It is simply impossible for us to point in the right direction.

 

[ExactlySolved]

This fact was pointed out by Witten in "Realistic Kaluza Klein theories". He did a classification of all the spaces of the kind SU(3)xSU(2)xU(1) / SU(2)xU(1)xU(1)

Thanks for the reference, I look forward to reading the paper!

 

[Hurkyl]

The 2D space consists entirely of this boundary and the information contained within.

No, that's a 3D space... and it would look three dimensional to anybody living in it.

The 2D space consists only of the boundary.

Why would someone in 3D space think they were in 2D space? Where are you getting this?

That's the whole point of "compactified dimensions".

Once upon a time, Kaluza was playing around with the mathematics of General Relativity, and discovered that he could make Maxwell's equations appear when considering four spatial dimensions.

While this provided some hope that electromagnetism could be incorporated directly into GR, his discovery had one major flaw -- it described a universe with four spatial dimensions, rather than a universe with three spatial dimensions and one electromagnetic field.

So, to proceed down this line of thought, the challenge is how to make 4 spatial dimensions look like 3 spatial dimensions plus a field. Klein's idea was to compactify one of the dimensions -- curl up space-time so that in three of the directions it extended 'normally', and in the remaining direction it was a tiny loop.

A "resident" of 2D space would encounter a completely different geometry from someone in 3D space.

No... a resident of 2D space would encounter plane geometry, something which would be very familiar to a resident of 3D space.

First of all, a "point" would have no meaning to them, as the simplest geometric element would be a line. The very idea of a point implies 3D space,

This makes no sense. Spaces are made out of points, no matter what kind of spaces you're talking about.


(Assuming you don't do something weird, like study pointless topology. But even those topoi have points...)

 

[marcus]

That reference to pointless topology leads to a pretty darned distinguished 1983 Bulletin American Math Society paper
http://projecteuclid.org/DPubS/Repo...w=body&id=pdf_1&handle=euclid.bams/1183550014
by Peter Johnstone.
About doing topology using the lattice of the open sets.

 

[OB 50]

No, that's a 3D space... and it would look three dimensional to anybody living in it.

The 2D space consists only of the boundary.

You seem really convinced that I'm saying the exact opposite of what I'm actually trying to get across. I'm talking about the information contained within the boundary itself. Any perturbations of the surface boundary contain information. This is what I'm referring to when I say "within". I am not referring to anything contained within the volume of the described sphere.

That's the whole point of "compactified dimensions".

Once upon a time, Kaluza was playing around with the mathematics of General Relativity, and discovered that he could make Maxwell's equations appear when considering four spatial dimensions.

While this provided some hope that electromagnetism could be incorporated directly into GR, his discovery had one major flaw -- it described a universe with four spatial dimensions, rather than a universe with three spatial dimensions and one electromagnetic field.

So, to proceed down this line of thought, the challenge is how to make 4 spatial dimensions look like 3 spatial dimensions plus a field. Klein's idea was to compactify one of the dimensions -- curl up space-time so that in three of the directions it extended 'normally', and in the remaining direction it was a tiny loop.

I get that. My point is that the 4th spatial dimension doesn't literally have to be compactified. Higher dimensions will always appear compactified, since we are limited to observing the universe exclusively from a 3D vantage point.

No... a resident of 2D space would encounter plane geometry, something which would be very familiar to a resident of 3D space.

I disagree. There is a huge difference between looking at plane geometry, and living in plane geometry.

This makes no sense. Spaces are made out of points, no matter what kind of spaces you're talking about.

Consider how we experience a point in our 3D space. It can be circumnavigated by passing it along any axis. In order for us to experience the same point as a resident of 2D space, we would have to remove one degree of freedom, such as "up". the only way to do this would be to extend the point infinitely along one of its axes, thus effectively creating a line. Of course, this would apply to every single point within the 2D space as well. To live within 2D space would be to live in a universe where the most basic geometric element is effectively a line.

The important point to remember is this: To anyone within this 2D space, the direction parallel to the "line" does not exist. Nothing within the 2D space is free to move in that direction.

Now, is it possible that residents of 2D space would perceive these lines as "points"? Of course. I don't see how they could perceive them otherwise. It seems a lot like how we perceive photons as point particles, even though they behave more like lines extending through spacetime.

The key thing is that although within each space, whether 3D or 2D, it is possible to experience an internally consistent 3D world, the elements don't carry over from one to the other at a one-to-one correspondence. You can't take a point from 3D space, drop it into 2D space, and expect it to be the same thing.

This is why the idea that the extra dimensions are just really really tiny seems kind if absurd. If they are in fact extra dimensions, then there is no way that we could ever look in the correct direction to "see" any of them.

Simply put, compactification seems like an accounting trick designed to constrain the universe to 3D space and allow calculations to make sense. Maybe this is the only way we can do it and make sense of anything, but I don't think it's necessarily the literal truth.

 

[arivero]

Thanks for the reference, I look forward to reading the paper!

Indeed the paper is a "must read". Late in the history it was presented as a "non go theorem", because it addresses also the question of having chiral theories (and fails). But it is really a very positive paper, the only one really justifying the use of 11 dimensions.

Now, string theory has 10 dimensions, you can tell. But on other side, non chiral standard model, if you think about, is only SU(3)xU(1), so the same counting gives 4+1 extra dimensions. So string theory lives between the non chiral model, in 9 dimensions, and the full unbroken standard model, in 11 dimensions.

Going back to the original topic, if Kaluza Klein were the way to go, we should account:

3 dimensions for space
1 dimensions for time
4 dimensions for strong force
1 dimension for electromagnetism
1 dimension for chiral SU(2) and/or broken symmetry (it should be 2 dimensions for unbroken non chiral theory)

 

[Hurkyl]

None of that really makes sense.

Higher dimensions will always appear compactified, since we are limited to observing the universe exclusively from a 3D vantage point.
...
If they are in fact extra dimensions, then there is no way that we could ever look in the correct direction to "see" any of them.

I admit that I don't have great depth of knowledge of general relativity, but I'm pretty sure you are in direct contradiction -- an observer living in 4+1-dimensional spacetime (4 spatial 1 temporal) gets to look in all of the spatial directions.

Same goes for quantum mechanics.

To live within 2D space would be to live in a universe where the most basic geometric element is effectively a line.

You learned plane geometry in school, right? Points, lines, triangles, et cetera. Reflect upon the fact that your description of plane geometry doesn't resemble what you learned in school.

The key thing is that although within each space, whether 3D or 2D, it is possible to experience an internally consistent 3D world,

How exactly were you planning on embedding 3D space into a 2D space?


Of course, the other direction is easy -- geometrically we all know how to embed 2D space into a 3D space: it's called a 'surface'.

 

[OB 50]

None of that really makes sense.


I admit that I don't have great depth of knowledge of general relativity, but I'm pretty sure you are in direct contradiction -- an observer living in 4+1-dimensional spacetime (4 spatial 1 temporal) gets to look in all of the spatial directions.

Same goes for quantum mechanics.

Where did I ever say that someone living in 4+1 spacetime is restricted from looking in all spatial dimensions? If one lives in 3D space, then by definition, they are free to look in all directions afforded by six degrees of freedom. If that 3D space is embedded within 4D or higher space, the residents of the 3D space are not capable of looking in the "direction" of 4D space or higher dimensions.

This is a simple concept.

Take a clock face. The hands are constrained to a 2D system of movement. They can freely rotate to point in any direction within their 2D constrains, but no matter which way they rotate, they will never point "out", or towards the outside (3D) observer reading the time. That direction is always 90 degrees away from any possible orientation of the hands.

Consequently, we would not be able to look in the direction of anything the exists in 4D or higher space because we would encounter the same situation.

You learned plane geometry in school, right? Points, lines, triangles, et cetera. Reflect upon the fact that your description of plane geometry doesn't resemble what you learned in school.

You are completely missing an important distinction I've been trying to make regarding this, and I don't appreciate the condescension.

There is a huge difference from an observer in 3D space looking at a 2D plane and the geometry contained within, and the experience of an observer viewing the same thing from within the 2D space itself. The conceptual perspective shift required is substantial.

By using the point-to-line analogy, I was attempting to illustrate how one might take the standard view of 3D space and artificially remove one degree of freedom, thus simulating the experience of 2D space. In order to artificially remove one degree of freedom from the elements within 3D space, one axis would need to extended to infinity. Thus, a point becomes a line.

A 3D space which consists of parallel lines as its most basic geometric element is essentially the equivalent of 2D space. This is simply a thought exercise meant to illustrate how one would experience the constraints of 2D space.

How exactly were you planning on embedding 3D space into a 2D space?

Of course, the other direction is easy -- geometrically we all know how to embed 2D space into a 3D space: it's called a 'surface'.

I'm pretty sure I bolded the word experience.

If it is possible for an actual observer to exist within 2D space, that observer would have no concept of his own "flatness". Very possibly, that observer would experience that 2D universe much the same as we experience our 3D universe; completely unaware of the additional degrees of freedom afforded by higher dimensions.

All I'm trying to say is that regardless of the actual fact, any additional dimensions higher than our 4+1 spacetime will always appear to be compactified. Consequently, we should allow for the possibility that they are not.

Also, the very definition of an additional dimension precludes us from ever being able to look at it.

 

[Hurkyl]

Where did I ever say that someone living in 4+1 spacetime is restricted from looking in all spatial dimensions?

I assumed your comments were referring to the situation at hand.

There are (at least) two sorts of ways to have extra dimensions in general relativity.

(1) Physics takes place on a 3+1 dimensional spacetime manifold, governed by the Einstein field equations (EFE). However, the spacetime manifold is located in a higher dimensional 'ambient' space. The ambient space can tell us about geometry, but is ultimately irrelevant because we don't need it to do geometry, and it has no bearing on the physics of our manifold.

(2) Physics takes place on a 4+1 dimensional spacetime manifold, governed by the EFE.

Except for Occham's razor, there is no problem with suggesting case (1) describes the real world.

But Kaluza's discovery is a case (2) thing. And there is a problem with suggesting case (2) describes the real world, because physics is actually happing in all 4+1 dimensions. Something extraordinary has to happen to make a resident think the manifold is 3+1 dimensional when, in reality, he has access to all 4+1 dimensions.

You are completely missing an important distinction I've been trying to make regarding this, and I don't appreciate the condescension.

I understand that 3D affine space modulo translation along a vector is a model 2D affine space. But it really did look like you were trying to say that's the only possible model. (e.g. that there is no such thing as "Newtonian physics on 2 dimensions" -- only "Newtonian physics on 3 dimensions where all matter is made from 'line particles' with the same orientation")

Also, the very definition of an additional dimension precludes us from ever being able to look at it.

"Dimensions beyond 3 cannot be looked at" is not part of any definition I've seen. Even if we restrict to models that could plausibly resemble the real world, Kaluza-Klein geometry is a counterexample to your proposition.

 

[OB 50]

"Dimensions beyond 3 cannot be looked at" is not part of any definition I've seen. Even if we restrict to models that could plausibly resemble the real world, Kaluza-Klein geometry is a counterexample to your proposition.

Maybe I need to clarify a bit here. I don't mean to say there is an arbitrary limit where anything higher than a third dimension can't be looked at. It would be more accurate to say that for any observer occupying a given space (whether 2D, 3D, etc.), that observer is incapable of orienting himself in a direction necessary to observe any subsequently higher dimension.

A resident of 2D space (a plane dweller) can never orient himself to observe anything outside the plane (x,y). He may infer that a higher dimension(z) exists, but it can never be directly observed. He may even correctly detect the curvature of his own plane through 3D space, but his observations are still exclusively limited to the plane itself.

This isn't to say that a 3D object intersecting 2D space would not be observable. It would be, but only as a 2D "slice". I hesitate to use the word "slice" since that is how it would appear from the perspective of the 3D object, and not that of the observer within the 2D space.

It's as simple as stating that a clock hand can never rotate to a position where it points parallel to it's own axis of rotation. The degrees of freedom which define a given space will prevent all observers within that space from observing subsequently higher dimensional spaces.

 

[apeiron]

It's as simple as stating that a clock hand can never rotate to a position where it points parallel to it's own axis of rotation. The degrees of freedom which define a given space will prevent all observers within that space from observing subsequently higher dimensional spaces.

Chipping in, I too find it illogical to presume that there can be both extra dimensions and that these dimensions be observationally unavailable. The logic would have to work the other way round. If you have four degrees of freedom, then by definition, an observer would have to be constrained in some fashion to only exist in three of them.

Compactification would be one such example of a constraint. The space becomes too small for our motions in it to be observable (to us...as motions...).

A clock hand is an example of constraint as well. We are presuming a rigid body that will rotate in a plane and won't flop out into the third dimension, even though 3D forces like gravity may be acting upon it at the time.

 

[Hurkyl]

A resident of 2D space (a plane dweller) can never orient himself to observe anything outside the plane (x,y). He may infer that a higher dimension(z) exists, but it can never be directly observed. He may even correctly detect the curvature of his own plane through 3D space, but his observations are still exclusively limited to the plane itself.

But this describes a case (1) situation. (using the terminology from my previous post)



Going back to compactified dimensions -- for the sake of argument let's fix a coordinate chart that on space-time that is organized into 1 temporal dimension, 3 'normal' spatial dimensions, and 1 compactified spatial dimension.

At each point in the remaining 1+3 dimensions, the fiber described by the compactified dimension is a loop. Mathematically, a loop is pretty much the same thing as a line upon which we consider only periodic things.

So what would our 1+3+1 space-time look like if we unrolled the compactified dimension into a periodic line?

If we assume a continuous matter distribution, if the compactified dimension has a sufficiently small circumference, one would expect that matter would tend to be roughly uniformly distributed around the loop. In the unrolled picture, matter would be roughly uniformly distributed along the entire line. (With periodic perturbations)

Mathematically this is pretty much the same thing as the picture you described in your example of 3-dimensional space in which all matter was composed of infinite lines (rather than points), all oriented in the same direction.

The big overwhelming difference is that the roughly uniform distribution appears naturally in the compactified dimension picture, whereas the unrolled picture is thermodynamically impossible, and essentially requires a massive cosmic coincidence to happen.

 

[OB 50]

Chipping in, I too find it illogical to presume that there can be both extra dimensions and that these dimensions be observationally unavailable. The logic would have to work the other way round. If you have four degrees of freedom, then by definition, an observer would have to be constrained in some fashion to only exist in three of them.

As human observers, we exist in 3D space. We possesses only 3 degrees of freedom (x,y,z). There are elements in our universe which demonstrate additional degrees of freedom, like photons and electrons, but we as observers are not photons or electrons. Although they may not be bounded by 3 degrees, we as observers are. As such, our direct observations are limited exclusively to 3D space.

This is not to say we cannot detect the presence of additional or compact dimensions, but we cannot directly observe them. The compact dimension which appears to be a line (but is actually a "tube") will always appear to be a line, no matter how close or magnified we get.

Compactification would be one such example of a constraint. The space becomes too small for our motions in it to be observable (to us...as motions...).

A clock hand is an example of constraint as well. We are presuming a rigid body that will rotate in a plane and won't flop out into the third dimension, even though 3D forces like gravity may be acting upon it at the time.

Maybe the clock analogy is a bit sloppy, since it is a 3D object. The basic principle I'm trying to illustrate is that no object can rotate to point in the direction of its own axis of rotation.

 

[Hurkyl]

The compact dimension which appears to be a line (but is actually a "tube") will always appear to be a line, no matter how close or magnified we get.

(Just checking:) A (tiny) tube is two dimensions, one compactified, one not.

 

[Haelfix]

"This is not to say we cannot detect the presence of additional or compact dimensions, but we cannot directly observe them."

Yikes, ok, obviously there's a confusion going on somewhere.. Might I suggest picking up 'Warped Passages' by Lisa Randall. She does a good job explaining the details in simple conceptual terms.

 

[OB 50]

(Just checking:) A (tiny) tube is two dimensions, one compactified, one not.

Simply explain how the following scenario is possible, and I'll shut up:

There is a 2D observer who exists in 2D space (a plane, surface of a balloon, what have you). There is also a 3D observer looking on from a surrounding 3D space.

How does the observer within the 2D space directly observe anything in the 3D space that is not intersecting the 2D space?

 

[Hurkyl]

Simply explain how the following scenario is possible,

There is a 2D observer who exists in 2D space (a plane, surface of a balloon, what have you). There is also a 3D observer looking on from a surrounding 3D space.

I don't see how it's relevant to anything I've been talking about... in particular, it has nothing to do with my correction that the tube is two-dimensional, not one-dimensional.

I was assuming you meant the surface of the tube -- i.e. a cylinder. If you were referring to the interior of the tube, then that's three-dimensional, not one-dimensional, and none of them are compactified (just incomplete).


Anyways, the first thing to pay attention to is that we're making up laws of physics as we go along -- I'm pretty sure this scenario cannot be the product of, e.g., general relativity.


If we assume that everybody's vision works via photons, then the 2D observer really gets to see all of 3D space -- a photon emitted from any point in 3D space can reach the eye of the 2D observer. The only difference is that field of vision is a curve, rather than the surface we get to view. If there was a rectangle with red and white stripes, then our 2D observer would just see a pink interval -- unless the rectangle had certain orientations which would let our 2D observer see alternating red and white intervals, possibly with pink blotches at the boundaries.


However, the fact the 2D observer interacts with photons means that he absorbs their momentum. But since we've assumed that he is constrained to live in some 2D surface, this means that interaction is limited to deflecting photons, not absorbing them -- any photon that passes through the body of our 2D observer retains all of its momentum in the normal direction.


If we assume 3D conservation of momentum, that means either the 2D surface is perfectly flat, or it means that whenever our 2D observer passes through a curve, he emits some sort of energy into 3D space to conserve momentum.

If his 2D surface is curved, then he could take advantage of them to change the angle at which he is oriented in 3D space, and use them (at least to some extent) to resolve the position of things perpendicular to his surface.


Okay, I've had enough stream of consciousness speculation for the moment, so I'll stop here.

 

[OB 50]

The process you just described has the 2D observer inferring the properties of 3D space through interactions and measurements exclusively within his own 2D space. Never does he directly observe any element of 3D space.

It is still impossible for the observer to orient himself within the 2D space such that he can look in the direction of travel of the photons as they are traveling through the 3D space. He can never look directly "outward" into the surrounding 3D space.

This seems self-evident and basic, not stream of consciousness speculation.

I'll leave it at that.

 

[apeiron]

This is not to say we cannot detect the presence of additional or compact dimensions, but we cannot directly observe them. The compact dimension which appears to be a line (but is actually a "tube") will always appear to be a line, no matter how close or magnified we get.

.

OK, so the question comes down to whether we can or cannot in practice observe the compact dimensions. Planck-scale limits would seem to constrain us - we cannot expend the energy that would be required to "magnify" our observations to the point the dimensions are directly visible.

Probably correct. Though there may be indirect effects that are observable at more achievable temperatures. So the particle zoo and its symmetries may indeed be the outward expression of these hidden from us dimensions.

 

[Haelfix]

"Never does he directly observe any element of 3D space."

And we've been trying to tell you that's incorrect for 2 pages worth of completely elementary physics. If I'm confined to live on the 2d plane of my computer screen, I still absorb photons from the full 3d geometry around me. I not only indirectly deduce that the world is 3dimensional, I also observe it (whatever that means) and I further can infer that I am simply living on a constrained subspace. In fact, your very eyes (the surface of which is more or less 2 dimensional) does the same exact thing, and its your brain that models and extrapolates the information it receives as 3dimensional.

In the compactified dimension case (say a 2d cylinder or tube), one possible sublety is that sometimes physics as we know it isn't necessarily allowed to propogate along the compactified dimension. So for instance EM/Strong/Weak force might only be allowed to live on the uncompactified dimension, whereas say gravity might be allowed to propagate along both dimensions. The observation of the extra dimension is to simply conduct an inverse square law experiment (or whatever the r dependancy is), to probe the radius of the hidden dimension. So, if we had eyes that could interpret gravitational information (rather than electromagnetic information), we would see 2 dimensions if we looked closely enough, but if we had normal eyes that are sensitive to photons, we would only see 1.