Matt Strassler [10 Jan 2012]
It’s time now to learn something new from a two-dimensional space that we couldn’t learn from a one-dimensional space. In particular, I want to focus your attention on the strip – a space that looks like a ribbon, with one very long (perhaps infinite) dimension and one very short one. (Most of what I’ll say for the strip is also true for the tube, but I’ll stick with the strip because it is easier to draw.)
First, let me remind you of a concept I introduced at the end of Worlds of 1 Spatial Dimension. That was the idea that even when a physical world is three-dimensional, such as the one we are used to, it is possible for certain aspects of the world to behave as though they are one-dimensional. For example, a person walking on a high-wire truly exists in three dimensions, but her motion is going to be essentially one-dimensional. In this case, it is the constraint of her own safety that makes her world one-dimensional. Well, on the strip we have another reason why the world could be effectively one-dimensional. It’s because there is a large dimension and a small one, and whether you can move around in the small dimension depends upon what your own shape is… and how big you are relative to the distance across the small dimension.
Here’s an example of a strip: a ship canal. A long strip of water allows a small boat to move around in a two-dimensional way: it can move up and down the canal, or back and forth across it. But a giant freighter that barely fits inside the canal can only move forward or backward. For the freighter, the canal is effectively one-dimensional. If you watched it move, you would see no sign that the canal is truly two-dimensional. The motion of a large object — large compared to the width of the strip — reveals the long dimension but does not show signs of the second, short dimension; the second dimension is only revealed by the motion of a small object, like the small boat. This is an important lesson!
What’s another example of the same thing? For you and me, the surface of a human hair is essentially a one-dimensional object, as far as our eyes can see. We could cut a hair into two strands of half the length, but we would have a tough time slicing a strand of hair the long way: indeed, in English we refer colloquially to “splitting hairs” as something not worth the time and trouble. A tiny dust mite, however, can crawl around on the surface of a strand of hair as though it were a giant tube. For the mite, the surface of the hair seems two-dimensional, just as the surface of a flag pole seems two-dimensional to us.
Let’s get back to the freighter. Imagine for a moment that the freighter were a conscious creature, with a brain that obtained information from sense organs, just as do all animals, but whose only sense organ was one that allowed it to detect its own motion. [Why am I introducing consciousness here? It has nothing to do with the dimensions of space that we’re talking about, which are present whether the freighter is conscious or not. What I am doing here is trying to explain to you how a conscious creature — such as yourself — could be misled by its senses and brain as to the true features of nature, and could consequently be completely unaware of a dimension of space.] The only information coming in from that sense to the freighter’s brain would be something to the effect of “forward at five kilometers per hour” or “backward at two kilometers per hour” or “stationary”. What picture of the world would the brain of the freighter build? With no information coming in from outside about the presence of the second dimension across the strip, it would naturally build a one-dimensional image of the world, as though the canal had but one dimension. The freighter would have no knowledge of the second dimension, even though a much smaller boat could easily detect it.
This is one example of how “extra” dimensions can work (though not the only one, as we’ll see later.) What we mean by “extra” is this: it is a dimension that the freighter didn’t (and couldn’t, given its senses) know was there.
Imagine now that something similar applies to us. Imagine we live in a sort of “strip” that, in addition to three large spatial dimensions that we normally move around in, detect with our senses, and conceive of in our minds, there is also a fourth spatial dimension, so incredibly small that we fill it up just like the freighter, so that we can’t move across it. Moreover, neither our sight, hearing, touch, smell or taste faculties can detect the presence of this extra dimension. As far as our brains know, it isn’t there. But maybe it really is there, and sufficiently tiny creatures might be able to move across it and sense it.
To suggest that one or more extra dimensions exist is just speculation — pure imagination . We certainly do not know that this is true. But what I am trying to convey to you is that we do not know it is not true.
So then the next question is this: how could human scientists, whose senses and brains cannot detect this fourth dimension of space, discover it is there? That is where a mixture of human intelligence and advanced scientific instruments comes into play.
[New additions as of 10 January begin here]
And as we’ll see later, scientific study reveals that although one or more extra dimensions of this type may exist in nature, if they do they must be extraordinarily small. The distance across any such dimensions can be no more than 1/10,000 the distance across a proton, which is about 1/1,000,000,000 times smaller than the distance across an atom, or if you prefer, about 0.000,000,000,000,000,000,1 meters (and a meter is just over 3 feet.) Research at the Large Hadron Collider [LHC] is gradually pushing out this frontier.
The type of extra dimension that I’ve just described here is just one of several general possibilities. Here’s another example. Instead of a ship of huge width that spans the entire canal, let’s consider a much narrower ship that is tethered tightly to the side of the canal, so that although in principle it could move across the canal it is unable to do so in practice. The relation of the boat to the extra dimension is different, but the effect — that the short dimension of the two-dimensional ship canal is undetectable in the boat’s motion — is the same.
Specifically, as shown in Figure 3, imagine a boat that moves forward or backward along the canal, using its own engine, but also with the assistance of vehicles that run on rails alongside the ship canal — a modern form of a tow-path. In contrast to the freighter, this boat does not does not fill the canal from side to side. In principle it could move across the canal from one side to the other. But in practice, there are forces — exerted by the ropes that attach it to the vehicles on rails — that hold it tight against the left wall of the canal. So strong are these forces that the boat never strays from the left wall. And so, as it moves forward and backward along the canal, its motion is, like that of the freighter, entirely one-dimensional. And as we did for the freighter, we may imagine that if this boat were conscious, and its senses provided information only about its motion, its brain would likely construct a one-dimensional image of the canal, despite the fact that, if it were possible for the ropes to be cut, the boat would then be able to turn and move into the center or across to the right-hand wall of the canal.
The same could be true for us. We — more precisely, every particle from which our bodies are made, and from which everything that we can see and feel or even detect by modern scientific instruments is made — may be attached, by forces of extraordinary strength that we cannot counter, and of which we are not aware, to the three-dimensional wall of a space with four spatial dimensions. And because our motions are constrained to the wall, and our senses detect only objects that are similarly stuck to the wall, our brains build for us a three-dimensional image of the world, despite the fact that in truth the world has an extra dimension.
Again, there’s nothing that assures us that this is true. It’s pure speculation. But neither are we sure that it is not true. The only way to find out is to design and carry out scientific experiments, whose nature I will explain soon. It turns out (as was pointed out, somewhat to the shock and incredulity of the community [including yours truly], in a famous 1999 paper by Nima Arkani-Hamed, Savas Dimopoulos, and Gia Dvali) that this second type of extra dimension can be much larger than the type I described earlier — maybe almost as large across as the width of a human hair! [In fact when the paper was first written, experiments still allowed for extra dimensions to be as large as a millimeter; but new experiments have been done, as I’ll describe later, that have put better constraints on this possibility.] This is rather amazing. Despite all the experiments that we perform using microscopes, and the tiny electronic equipment that we build into our computers, we would still not know about an extra dimension of small but still macroscopic size.
Once we start going down these roads we find many more possibilities than just these two.
For instance, there’s no reason there couldn’t be more than one extra dimension. We can get an example using our ship canal — which, if it is deep enough, should really be thought of as three-dimensional, very long in one dimension but with a finite width and a finite depth. Then we can have all sorts of objects for which the canal has zero, one or two extra short dimensions, as shown in Figure 4. A submarine can move around three-dimensionally within the canal. A small boat is stuck to the two-dimensional surface of the canal, and moves unaware that the canal has depth. Its two-dimensional view of the world would miss one extra dimension. And three different types of boats would be unaware of two of the dimensions. The towed boat, stuck at the wall and at the surface, is one example. A wide barge that fills the canal from side to side but is stuck at the surface is a second. And finally, the giant freighter, whose keel reaches to the bottom of the canal, essentially filling it completely, gives a third example. For three very different reasons, these three boats would have motions that show no sign of the depth and width of the canal, and were they conscious creatures they might well be surprised to learn that there are two extra dimensions to their world.
More examples will follow over time. But at this point it is important to learn more about how the scientific experiments to search for these and other types of extra dimensions are actually carried out.