Recently a student asked about the concept of extra dimensions of space. This is certainly one of the hardest ideas in physics to visualize, although there have been attempts, in a purely mathematical way, such as Calabi-Yau shapes which attempt to represent higher dimensional surfaces to our three dimensionally (3D) constrained eyes and brains. What follows is my response to the student, edited and augmented a bit.
First, I think you can say that “dimension” is an attribute of space in the sense that “volume” is the attribute of a sphere. Dimension is a way to partially define space. Space can have other attributes as well, to account for whatever is in it (like particles, planets, etc). At any rate, dimension helps to define how we experience space, whether we are limited in traveling through it and so on.
A dimensional space, such as our 3D universe, really has shape only in the sense that it is defined by how far you could move in all the allowable directions. If you are inside that space, you could not directly perceive the shape, just as someone floating inside a hollow sphere out in space could not directly perceive the shape but could only guess at it. In order to fully view the shape, you would have to be outside of it, which for us means that you would have to exist in an additional dimension, like someone looking at the sphere from the outside. (Note that if the sphere represents our entire 3D world, anyone outside it would necessarily exist with at least one additional dimension.)
In general, the concept of dimensions refers to our ability to define our position, whether that be in space or time or some other, perhaps as yet undefined, parameter.
Imagine a straight line. If you were confined to a straight line, and could only move forward or back, you could define your position by one number representing the position along the line. That would be one-dimensional. This is like being stuck on a ruler, where your positions could be defined as 5 cm or 134 mm. While it seems strange to us, there is no reason in physics to forbid a universe with only one dimension. There are equally strange aspects of quantum physics and relativity that we know are true.
Now let’s say that you are not confined to a straight line, but can also move from side to side. Assuming that you could move only forward, back or to the left or right, or any combination thereof, you would be in a two-dimensional universe. It would be as if you were living on a flat piece of paper (a plane). Again, although we do not experience real situations like this, there is nothing in physics to forbid it. To define your precise location, you would need two numbers…. one to define how far you are to the front or behind your original starting point, and another to define how far to the left of right you are. Our familiar coordinate systems can define our location, say, by the intersections of two streets. Our position along one street is one coordinate, the position along the other is the other coordinate.
However, if you also could move up and down, you would need three numbers, the third to define how far you are above or below the (geometric) plane. An airplane flying across country needs to be defined by its geometric coordinates (two) with the third coordinate being its height.
We also would define a fourth coordinate, time, so we know where the airplane was located, and when. Time is not a space dimension, but it is considered the “4th dimension.”
Thus our familiar world has four dimensions we can easily define and understand, three of space and one of time. Furthermore, we do not experience any fewer or any more dimensions, but as I have said, physics does not forbid the existence of universes of one, two, five or even more dimensions.
In fact, certain questions in physics could be solved if additional dimensions do, in fact, exist. For example, if we assume the existence of extra dimensions, mathematics implies that we can find a way to unite all four known forces of our Universe. Today the four forces act separately (two nuclear forces, electromagnetism, and gravity), but as temperatures and pressures rise (approaching those of the Big Bang) electromagnetism and the weak nuclear force are known to combine into one force called the “electroweak” force. “Grand Unified Theories” (GUTs) predict that at even higher temperatures (even closer to those of the Big Bang), the strong nuclear force would combine with the electroweak force.
In that case, there would be only two forces, the Grand Unified Force and Gravity.
It would seem natural, then, that at some point gravity also would join the other three so that all the forces would be united into one. This is the goal of “Theories of Everything” (TOEs). Physicists have a preference for theories that are simple or elegant, and the most elegant theory would combine those four forces into one at the time of the Big Bang. There is good theoretical thinking to support this. The problem is, however, that in all these theories, there must be additional dimensions in order for these four forces to unite.
Also, the existence of additional dimensions could help resolve some physical situations whose mathematics return infinite, or negative, or otherwise puzzling results.
The bottom line is that existence of additional dimensions of space:
- is not forbidden by physics
- would solve some perplexing problems
- would open up to us many knew ways to study, and perhaps travel through, the Universe (or Megaverse)
Although we are limited to 3D vision, mathematicians and scientists have devised many ways to visualize the additional dimensions, but they are all limited simply by virtue of the fact that we are limited in our experience to 3 dimensions. For me, the easiest way is to remember the “rabbit hole” from “Alice in Wonderland,” or Dorothy’s emergence into Oz, or even something akin to the stargate in “Stargate.” In all of these cases, the characters pass from one specific location in our 3D world into a completely different world. In the first two cases it is into another world with certain similarities to our own, but with major differences. In the case of “Stargate,” the new world is really just a different location within our own 3D Universe.
In some modern ideas, the familiar space we know is in fact filled with tiny little such dimensional portals. They are too small for us to see or directly sense in any way. We move through them all the time with no interaction. I am hard pressed to come up with a good analogy for this, but consider that all the time we pass through electromagnetic fields (or they pass through us, if you will) without any direct interaction at all. A familiar example are radio waves. There is clearly a difference however, since we do have technology that can detect such fields. [Again, this is just a crude example to give you an idea of what I mean. I do not mean to imply that the array of possible extra-dimensional “portals” is similar to a electromagnetic field, although the concept of them being arrayed as a field may have some merit.]
But imagine that such sub-microscopic extra dimensional portals really do exist. If they do, we have no way currently to open them up so that we could not probe them even if we found them. But they could lead to a world that has our familiar front-back, left-right, up-down, and time dimensions, along with other characteristics completely foreign to us. Sadly, I cannot really describe these additional dimensions because we haven’t been there to know. Perhaps it would be possible to walk through solid walls, or see in all directions at once, or travel from galaxy to galaxy as easily as we move from one room to another. I simply do not know.
There are some suggestions that the additional dimensions have something to do with alternate versions of the future, or how certain things in our world (e.g., quantum effects) may be tied to events in the other dimensions. Just as a thought, it may be that some of the “weirdness” of quantum effects is simply because we are not seeing the whole picture or event, because some of it actually takes place in another dimension. (We have no proof of this, but it is one possible answer to the questions about why certain quantum effects happen that cannot otherwise be explained in our 3D world. It could even be related to the “hidden variables” proposition, held by only a relatively few physicists, posited to explain away aspects of quantum “weirdness.”)
Still the problem boils down to visualizing the additional dimensions, and especially to finding ways of discovering these additional dimensions (if they really exist).
It is not hard to understand why most people dismiss the idea out of hand. But before you say it is just fantasy, remember the many things in today’s world that were considered impossible or not even imagined a few hundred years ago — human flight, television, computers, near instantaneous intercontinental communication, and interplanetary missions as just a few examples.
And keep in mind that the very fact that you are reading this on a computer, tablet, cell phone — or indeed that you even have any of these things or any of today’s electronic devices — depends on proven facts of quantum physics (e.g., quantum tunneling) that are as fantastic and unbelievable as the idea that ghosts can walk through walls. To be clear, we have solid and unmistakable evidence that quantum tunneling does in fact happen all the time at the subatomic level, but no good evidence at all of “ghosts.”
Because many of these ideas are based on scenarios that are not proven, that they lead to conclusions that well outside our normal experience, including Einstein’s “spooky action at a distance,” some folks may think that they may confirm metaphysical or paranormal beliefs (ghosts, ESP, precognition and so on). I want to make it very clear that I am not supporting this interpretation. While I cannot absolutely deny the possibility of much of anything, I do not “believe” in ghosts or other supernatural or metaphysical entities or events (i.e., I have no evidence of them nor any reason to suspect that such evidence exists) and so I do not believe that such fantasies have anything to do with extra dimensions.