Extra Dimensions and Manifolds

6/15/15

Let me start with the conclusion from General Relativity. GR claims that the effects commonly attributed to gravity are the result of the curvature of space-time. The curvature is the result of the presence of mass in space-time. The important part of this for my present purposes is simply the fact that space-time is curved.

Now if you haven't thought about this, it might be a little hard to grasp to start with, but you probably already buy into it. In case this is your first introduction, let me explain a few important concepts, the first of which is the concept of dimension.

In mathematics, dimension is simply an independent variable. That is, dimension is a quantity that can change (i.e. it is variable) irrespective of what other variables might do (i.e. it is independent.) In Physics, a dimension is usually thought of as a degree of freedom. So, for example in a plane, there are two degrees of freedom for the position of a point, on a line there is only one, and in a room, there are three.

For lay people, we use terms usually occurring in pairs like up/down or East/West to designate what really are coordinate axes in space. And, as we well know, we experience our physical existence in a three D space. (By adding North/South to the previous two we have a complete coordinate system for our bodies' existence.)

Note that each dimension has two directions, each directly opposite the other. Of course there are many other directions which do not lead to additional dimensions. For example, we have the direction North-North-East among many others. In both Physics and Mathematics, the basic directions used in establishing a coordinate system are called "basis vectors" and the number of basis vectors is exactly the same as the number of directions. All other directions can be arrived at by combining the basis vectors. So for example, we might have the basis vectors Up, South, and East. That would establish a completely usable coordinate system.

That is probably enough as an introduction to dimension except to pose the question, are there more than three dimensions? In mathematics there certainly are. Mathematicians have no trouble positing many extra dimensions and deriving facts about them. Scientists aren't so quick to do so. Einstein added a fourth dimension to his system, but instead of an additional spatial dimension, he included time as the additional dimension. So he called the space we inhabit "space-time" instead of simply "space".

More recently, String Theorists have openly suggested the possibility of the existence of additional spatial dimensions, but they have succumbed to a psychological stumbling block, in my humble opinion. That psychological problem is elegantly treated in Edwin Abbott's Flatland which I have already mentioned. (And again, I recommend that everyone read it.) The problem is that people expect to be able to see the extra dimensions if they actually exist.

The people who are afflicted with this psychological problem include every science author I am aware of who has written anything on the subject for public consumption. It is for this reason that I feel compelled to spend so much effort in explaining why Abbott's remedy for this psychological problem needs to be widely applied. The remedy is to come to realize that you should not expect to be able to see extra spatial dimensions simply because all the objects available for you to see, your eyes nerves and brain, the aspects of the light used to transmit information from the objects to your brain, and any instruments, such as microscopes, telescopes, or eyeglasses, are all 3-D structures. More importantly, all of these structures exist in what mathematicians call a 3-D manifold. And just as Abbott's A. Square is physically, logically, and conceptually unable to access or "see" anything "above" his 2-D flatland manifold, we are likewise unable to access or "see" anything residing in higher dimensional space that is outside of our 3-D manifold.

For those who are struggling with this, let me explain what a manifold is. Simply put, it is a nice smooth subspace of lower dimension than the space that contains it. So, for example, a sheet of paper is a 2-D manifold in the 3-D space of my office. The paper is nice and smooth and it can contain 2-D structures, like circles and squares. But those structures cannot get up off the paper and move around in the office without the entire sheet of paper moving under them.

Similarly, a line, or a circle, drawn on a 2-D sheet of paper are examples of 1-D manifolds, as long as the line is reasonably smooth. (Mathematically, the "smoothness" has to do with the function describing the line must be differentiable in a certain way.)

For our purposes the importance of the "smoothness" is that if you magnify any portion of the manifold to higher and higher powers, the manifold begins to look flatter and flatter (or in the case of a 1-D manifold, straighter and straighter.) This is important for reasons that we should be aware of but needn't get too bogged down with.

The "flatness" has to do with the second of the important concepts concerning the curvature of space-time. The first concept was dimension and we now move on to curvature. What does it mean to curve space?

The idea is easy to see when you consider lower, e.g. 1-D and 2-D, spaces like lines or surfaces. Curved lines, like circles, are commonplace. Curved surfaces, like the surface of a beach ball, are also commonplace. But not all curvature is the same. Let's look at some differences.

Let's start with a sheet of paper lying on a table. The paper is 2-D and the tabletop is 2-D so we can say that the paper is not curved. Now, let's lift one corner of the paper up off the table a little. Now the paper is curved. If there were a triangle drawn near the corner we lifted up, the triangle would still be there and the sum of its angles would still be 180. What that means is that the theorems of Euclidean Geometry still hold true for the manifold (the sheet of paper) even though it is curved.

Now, instead of a sheet of paper let's consider a sheet of rubber lying on the table, also with a triangle drawn on it. Then we drape the sheet of rubber over our beach ball and stretch it enough so that it is "smooth". This gives us a second example of a curved 2-D manifold but this time the theorems of Euclidean Geometry no longer hold. If we measured, we would find that the sum of the angles of our triangle is greater than 180 if the triangle is in the region that got stretched over the sphere.

There are other ways of curving a manifold, but these two are enough for our purposes. What we see in both cases of curvature is that the curved manifold can no longer lie flat on the tabletop. That is, the manifold has to some extent moved into a part of the office that is "outside" of, or "off of" the 2-D tabletop. In short, at least one extra spatial dimension is required in order to curve the manifold in either of these ways.

The important conclusion to draw here is that if a manifold is curved in one of these ways (the Euclidean and the non-Euclidean) then it must of necessity be "embedded" in a space of at least one higher dimension. You can use your vernacular understanding of the mathematical term 'embedded' and be all right.

The next step is to move up one dimension from the previous examples. Can a 3-D space be curved in the same way? Well, according to what we just concluded, it can't unless the 3-D space is a manifold embedded in a higher dimensional space. In other words, there must be at least one extra dimension. And, keep in mind, that if there is a higher dimension, bringing with it an enormous increase in the total amount of space that exists, we cannot in principle "see" or access any part of it that is not part of our 3-D manifold even if it is curved. We shouldn't expect to be able to "see" the extra dimension(s).

So is our 3-D space (manifold) curved? Well, that's where General Relativity comes in. And, yes, in fact GR predicts, and many experiments have confirmed that our 4-D space-time continuum is curved.

It is a little annoying to have that fourth time dimension in the picture because it adds what I think is unnecessary complexity to the question we are examining. For example, could it be that the three spatial dimensions remain rigid and unbent while it is the time dimension that is really curved leading to the explanation of gravity effects? I don't think so. I don't think GR claims that mass bends time which then causes gravitational effects. If anything is bent, it is one or more of the spatial dimensions, i.e. the spatial components of our manifold.

For our purposes, we simply need the fact that our 4-D space-time continuum is curved and from this we can conclude that our manifold (the same 4-D space-time continuum) is embedded in at least a 5-D space-time continuum. And we know that we can't see "outside" of our manifold.

This conclusion, by itself, provides for and accounts for a huge 'place' where any or all of the additional worlds (i.e. the mental, Platonic, and Spiritual) can be and it explains why those extra worlds are simply inaccessible to us confined here in our manifold the way Abbot's A. Square is confined to his flatland.

I'm debating whether or not to go off on a tangent and address the psychological problem shared by most scientists which Abbott addressed and I mentioned earlier. Instead of going into a lot of detail, I'll just make two points: First, that the commonly used explanation for the invisibility of the extra dimensions, i.e. that they are "curled up" so tiny as to be invisible, is wholly unnecessary, and second that the all-to-frequently used "hosepipe analogy" is specious.

The invisibility of the extra dimensions is completely explained, and it is easy to see why we shouldn't expect anything else, by the discussion of dimensions and manifolds above.

The speciousness of the various "hosepipe analogies" is easily seen by taking any one of the explanations and carefully keeping track of the analogs used in the analogy. What you need to match up are the various observers so that a particular observer in our real case of living in our 3-D world and wondering about the possible existence of a 4th dimension is matched against an observer in the "hosepipe analogy". For example are we analogous to the ant crawling around the hosepipe, or are we analogous to the observer in the distance who sees the hosepipe as 1-D? You will find that in each case the analogs are not analogous so the entire argument falls apart. These arguments might sound good, but on close examination they do not make sense. They are all specious.

I am hungry and it is now lunch time so it is time to summarize. I began by asking you to consider the possibility of as many as three additional possible "worlds" that cannot be found in the familiar physical world. One of these additional worlds certainly exists because it is undeniable. That is the mental world we all experience. We know it exists but it can't be found in the physical world. The other two, the Platonic world and the Spiritual world, you may consider or not as you choose. At this point the simple establishment of Cartesian Dualism will be sufficient.

The next step is to explain where any of these additional "worlds" might exist. It is a common tactic of dismissal on the part of scientists to mockingly ask, "OK, so just exactly where is this extra world?" We answered that question by demonstrating that there is at least one extra dimension in which our physical world is an embedded manifold and which provides not only plenty of room but enormous possibilities for greater complexity than we have here in our physical world. It also explains why the additional worlds remain so exasperatingly out of reach of us and our instruments.

Once we finally accept these conclusions, it becomes obvious that we should re-visit the advice of Occam and his preference for simplicity and the advice of Popper whose limits based on falsifiability are probably too restrictive.



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