Strings, Branes, and Their Many Dimensions

[chipmunk stew]

I don't know that much about the math involved in string/brane science, and I have at best a slippery grasp of what little I know of the ideas the math is describing.

But the way this many dimensions idea is depicted, as extra spatial dimensions, puzzles me. I always assumed the extra dimensions are simply necessary to make the math work, but does that mean they must necessarily be interpreted as spatial dimensions?

We consider time to be one of four dimensions that we understand. Might one or more of these string/brane math dimensions be time dimensions? Might they simply be string/brane dimensions that can't be thought of in terms of space or time, rather only in terms of the way they affect the behavior of sub-sub-sub-atomic features of the universe?

 

[Schneibster]

Well, it's easiest to start right where you did: with the math, and with dimensions. A "dimension" is necessarily an attribute of space. There is no essential difference between one dimension and another. There may be differences in the way different dimensions are configured, but these are details of the construction of space (actually spacetime), not details of the dimensions themselves. (Since I mentioned it parenthetically, I'll point out for those who are pedantically inclined that when I speak of "space" here I actually mean "spacetime;" as those same pedantically inclined folks will know, there is nothing inherently different about the time dimension, it is in fact an excellent example of how dimensions are all the same, but the geometry in which they are arranged can make them seem different.)

Mathematically speaking, a dimension is a degree of freedom; IOW, if an object exists in a space, each dimension of that space is a direction (actually a pair of directions diametrically opposed to one another, and "diametrically opposed" can have some VERY unusual meanings depending on the exact geometry into which the dimensions of that space are configured) in which that object can move. This is the simplest and most basic definition of "dimension."

There is another way to define "dimension," which has to do either with the size of an object, or with the specification of a position in a space.

There are other definitions of "dimension;" in some advanced mathematical treatments of various areas of science, it is useful to deal with multiple variables that do not vary with a standard relation to one another in a "multi-dimensional manifold." The mathematics that describe the behavior of data points in such an array give powerful tools to the analysis of behaviors of various sorts with respect to the various "dimensions" of the manifold, and the mathematics used are equally applicable to more traditional conceptions of these manifolds as actual spaces, or to abstract representations of various sorts of parameters by these means.

So that gives you some idea of what is meant when you see discussion about "dimensions." You should take care to find out whether these are realistic or abstract dimensions before you get too far into the conversation.

Now let's get down to the specific point of your post. You are basically asking whether the "extra dimensions" of string theory are of this abstract type, or are concrete as the four dimensions of spacetime we see all around us are. The answer is, they are concrete; however, they are so small that they do not directly affect the observed degrees of freedom of objects that we observe.

"How," you ask, "can a dimension be small?" This is a good question, because all of the dimensions we are familiar with extend outward pretty much as far as we can see (there are exceptions and provisos, but for this discussion that is a sufficiently accurate description of the situation). The answer, it turns out, is that this geometry we observe may not be the only geometry that the universe uses in the construction of space from dimensions.

Now your first objection may be, "well, the universe uses one geometry for all the dimensions we see; why should there be another?" This is basically an Occam's Razor type of argument. But it fails, because as a matter of fact, there are already two geometries in use. The relationship between the three spatial dimensions is spherical; that is, one can use spherical geometry and spherical trigonometry to describe events that involve motion and rotation. However, the relation between each of the spatial dimensions and time is NOT spherical- it is hyperbolic, and we must use hyperbolic geometry, and hyperbolic trig, to describe it. If there's only ONE way, then proposing another way is an excursion; but if there's TWO ways, there's no reason to stop there.

The particular geometry of the relation of these extra dimensions to one another and to the four spacetime dimensions is highly complex. First of all, unlike the spacetime dimensions we normally deal with, these dimensions do not spread out (relatively) limitlessly in all directions; they are small and confined. They connect back on themselves, so that one can only move a certain distance before one finds oneself at the same location one was at before.

But much more importantly, unlike the relations among the three space dimensions, or between them and the time dimension, physicists do not have many clues to help them determine what these relations precisely are. In other words, they don't know what the correct mathematics is to describe something moving not only in our normal three large space and one large time dimensions, but also in these small dimensions.

It's important that you understand what happened the last time we came to a new understanding of the dimensionality of the universe, to understand how important this may be. The last great advance in our understanding of such matters was the final proof (or as close to proof as science ever comes) of the hyperbolic relation between the spatial dimensions and the time dimension, and the consequent identification of time as intrinsically no different from the spatial dimensions, merely related to them differently, which I discussed above. This understanding led directly and unambiguously to the Theory of Relativity. You can understand from this that if we are able to eventually prove the existence of these dimensions, and find the geometry that they are related to the spacetime dimensions and to one another by, it will likely lead to revelations as profound as that earlier theory did.

Now, from a practical standpoint, let me say something that may surprise you: these extra dimensions are in fact so small that your idea that we might think of them only in terms of their effects is in fact the only way we can think of them at this time, because we do not know the geometry. They are far smaller than an atom, far smaller than a nucleus. In fact, they are so small that if a nucleus were the size of the Earth, they would be smaller than grains of sand.

And last, you asked something interesting: might one or more of these dimensions be time dimensions? Actually, since the only difference between space and time is the geometry by which they are related, in fact, they ALL might be thought of that way! However, they all share one major difference from the four spacetime dimensions: they are small. They may share another: that they are recursive. (Some people say they are "circular," but that implies the spherical geometry that we are relatively certain is not true of them, so I prefer "recursive," to indicate that they are not limitless, but connect back on themselves.) It is not however clear beyond a shadow of doubt that the spacetime dimensions don't curve back on themselves; the possibility remains that they might (although most cosmologists at this time believe they have sufficient evidence to strongly conclude that they do not).

HTH.

 

[uruk]

Wow. I hope someday to be able to understand that.
Thanks!

 

[Beth]

Me Too.

 

[chipmunk stew]

...HTH.

Immensely. Thanks.

If I understand your explanation correctly, it seems like a misinterpretation or misrepresentation to talk about "where" particles "go" when they "leave" spacetime, then. It seems (if strings are real) as though we just have an incomplete picture of the universe, and we should just be referring to something like spacebranetime. And speculations about parallel universes and such are simply not implied by these ideas.

There's one thing I still can't quite grasp. Are these extra dimensions literally "small" and "bundled" (which would imply zillions of bundles each residing independently at zillions of 4-D places at the heart of matter, which to me implies n extra dimensions times zillions) or is this just an analogy created by minds nurtured on a 4-D macro reality, representative of the possibility that these extra-dimensional effects show up at zillions of places in our 4-D macro reality?

 

[SpaceFluffer]

The theory is that the dimensions are just as real as the rest. As you say, they are literally small and bundled up such that they are present at every point in our 4D spacetime.

I like the analogy (I think it's Brian Greene's) of a bug on a tightrope. To the bug there are two directions - along the rope and around the rope. But to the perspective of somewhere of a much larger size, like a tightrope walker, the rope only has one dimension (along the rope). The other dimension seems to be small and curled up to the tightrope walker.

You're quite right that it is misleading to talk about particles 'leaving' spacetime, since they no more leave than the bug does when taking a trip around the one extra dimension in our analogy. Of course, it's more complicated when we're talking about 6 more dimensions, but the flavor of the analogy holds.

 

[Schneibster]

Immensely. Thanks.

If I understand your explanation correctly, it seems like a misinterpretation or misrepresentation to talk about "where" particles "go" when they "leave" spacetime, then. It seems (if strings are real) as though we just have an incomplete picture of the universe, and we should just be referring to something like spacebranetime. And speculations about parallel universes and such are simply not implied by these ideas.

According to my best understanding, this is correct. However, you should be aware that string theory also does not falsify descriptions of parallel universes, specifically the Everett or Many Worlds interpretation of quantum mechanics; whether it will do so in the future is unknown.

I'll leave a discussion about ekpyrotic theory (which you also obliquely could have been referring to) alone, except to say that it is based on the idea that there is indeed another large dimension, and that there are at least two universes like ours and they come into contact and create the conditions we describe as the Big Bang on occasion. This idea is not in conflict with string theory, but it also is not an integral part of it despite being based on some of its ideas.

There's one thing I still can't quite grasp. Are these extra dimensions literally "small" and "bundled" (which would imply zillions of bundles each residing independently at zillions of 4-D places at the heart of matter, which to me implies n extra dimensions times zillions) or is this just an analogy created by minds nurtured on a 4-D macro reality, representative of the possibility that these extra-dimensional effects show up at zillions of places in our 4-D macro reality?

At every point in space these extra dimensions should represent extra degrees of freedom of movement- but only over very short distances, and remember that because these dimensions are so small we could not directly measure such movement or observe it. We might, however, observe its effects; and proponents of the theory maintain that that is in fact the case.

Also, remember that they're not "little bundles" that are separate at each point in spacetime- for example, if I decided to move an object in the x direction (however we agree to define that in our chosen frame of reference), and I also move another object in a different location in the x direction, you wouldn't argue that they were somehow "different x directions." Similarly, if I moved one object in the x direction now, and another two minutes later, that wouldn't make you argue it either. So your statement that imagining these dimensions as little bundles at every point in spacetime is just an analogy that we have to use because we think in 3D+time is correct.

One more thing, and this is a pedantic matter: no proof that the universe is actually configured this way is currently available. From one point of view, it is incorrect to call string theory a theory; it has not produced testable predictions. This is not to say that it will not or cannot; but it has not so far. Another interpretation, which defines a theory not in terms of what it does but of what it is, specifically an internally and externally consistent mathematical model of reality, says that it is a theory, but makes no more claims for its predictive abilities. It is possible that the Large Hadron Collider, currently coming on line at the European Center for Nuclear Research, which goes by the non-English based acronym CERN, may show evidence that will bolster string theory. But unfortunately, the predictions of string theory are not sufficiently well-developed that it would be possible to falsify it, so this is not a true test of its predictive abilities.

String theory's progenitor is ultimately the General Theory of Relativity. Not long after the introduction of this theory, a man named Theodor Kaluza brought a startling proposal to Albert Einstein. Kaluza had applied the techniques and mathematics Einstein developed to describe our four-dimensional spacetime to a spacetime that had an extra spatial dimension, and from this he had derived the equations that James Clerk Maxwell had developed in the mid-nineteenth century to describe electromagnetism. Maxwell's equations emerged smoothly and naturally from this formulation, just as the ten field equations that describe gravity emerge smoothly and naturally from GRT. Einstein ultimately encouraged Kaluza to publish, since this result was so startling, but was never entirely convinced, primarily because the addition of an extra large dimension like the three space dimensions we already have would have discernable other effects than electromagnetism, and these effects are not seen.

The theory languished for a few years until Oscar Klein revived it with another startling idea: what if this fifth dimension were not large, like the four spacetime dimensions, but small, like I have been describing above? Klein had done the necessary mathematics to show that this made no difference in the derivation of Maxwell's equations.

This theory was unprovable then, because of the extreme smallness of the proposed extra dimension, and remains so now for the same reason. However, this idea has had two interesting progeny: first, supergravity, and second, superstring theory. Recently, supergravity has been incorporated into superstring theory; you may be aware that there are five separate string theory "threads," now believed to be descriptive of various possible collections of laws of physics for different possible geometries of the small dimensions. Each of these threads is dual to another of the threads in certain mathematical ways, and it was discovered that one of these dualisms is a dualism to supergravity, which thus forms the sixth thread. These six collections of possible laws of physics are believed (but not yet proven) to be combined by a single underlying theory, called "M-theory," but the details of this "master superstring theory" are not mathematically defined, because (like its progenitor, GRT) this theory is incredibly mathematically complex. Work to try to define this theory is underway.

It is still not clear whether any of these six threads yields the physics that we see around us. This is again due to the extreme mathematical complexity of the theory. String physics has already yielded two new fields of mathematics, and it is probable that it will require more of them, which we obviously don't currently have, before we understand all its implications and can extract testable predictions from it.