May 26, 2013

essay 0169

by: Paul R. Martin

The idea of extra dimensions is not new. It has been written about for a long time, but until recently it had never really been seriously considered by science. Before most of the idle speculation on extra dimensions was written by science fiction writers, Edwin Abbott published his famous book "Flatland". This book not only explores how extra dimensions would relate to us in our 3D world, but it explains the psychological reasons why people, scientists in particular, are so prone to reject the possibility of the real existence of hyper-dimensional space. Those reasons have a strong kinship to religious faith. This essay will be an attempt to remedy that situation.

We'll begin with an argument which demonstrates evidence for the real existence of large extra dimensions of space in which our 3D world is an embedded manifold. We'll proceed from there to explore implications of this idea, then the counter-arguments, and the refutation of these counter-arguments. We'll conclude with a challenge to all who refuse to consider the reality of large extra dimensions.

THE EVIDENCE

With the many confirmations of Einstein's General Theory of Relativity, it is believed by most scientists that the space-time in which our universe exists is curved, or bent. Incidentally, the curvature is caused by the presence of mass, although that won't concern us in this essay. The important thing is that our four-dimensional space-time is bent.

There is a theorem in mathematics which I remember seeing proved in a Differential Geometry class in graduate school, that says a space cannot be bent unless it is a manifold embedded in a space of at least one higher dimension. Unfortunately I can no longer find a reference to that theorem, but the assertion of the theorem is intuitively obvious. For example, you can't bend a sheet of paper (2D) lying on a flat tabletop unless you lift part of it up into the 3D space above the tabletop.

Putting the observation of space-time being bent together with the theorem, it seems clear that our universe must be a 4D manifold embedded in a space-time of at least 5 dimensions.

IMPLICATIONS

According to the mathematics of manifolds, the geometry of a manifold is, or can be, identically the same as that of a space of the same dimensionality which is not a manifold. For example, if you have a 2D space, say the surface of a sheet of paper, and that is all there is, structures found in Euclidean Geometry, e.g. circles and triangles, can exist on that sheet of paper and they must conform to the theorems of Euclidean Geometry.

But now suppose that instead of limiting ourselves to that 2D universe (that of the surface of the sheet of paper), we considered a "larger" universe, say a classroom with that same sheet of paper lying on a desk in the classroom. In this case, the surface of the sheet of paper is still a 2D space, but it is also a 2D manifold embedded in the 3D space of the classroom. The geometry of whatever is constructed on that sheet of paper remains the same as it was in the previous case where the surface of the paper was all that existed. In other words, an embedded manifold can be considered to be a space by itself.

Now, if we lift one corner of the sheet of paper up off the desk, we will bend or curve the 2D manifold, but we will leave all the geometrical structures that we have drawn on the paper unchanged. All the theorems of Euclid still apply to those figures even though the "straight lines" are no longer "really" straight, that is if they happen to be on a part of the paper near the corner that we lifted up.

Moreover, using this example of a bent sheet of paper, it is not possible for any theoretical inhabitants of the paper, such as the characters in Abbott's book, to detect the fact that we have curved their space by lifting up a corner of their manifold.

This last fact does not hold in general, however. Instead of a sheet of paper forming the 2D space let's say we used a sheet of rubber or plastic wrap, and instead of bending it by lifting up a corner, we stretched the rubber or plastic sheet over the top of a basketball so as to form a spherical bulge on the sheet. Now the 2D inhabitants of the manifold would be able to detect the curvature by making measurements strictly from within their manifold.

The trick they would use would be to check out the Pythagorean Theorem on any right triangle. They would find that it fails. Or, equivalently, they could sum up the angles of any triangle constructed in that bulge area and they would find that the sum is greater than 180º. Since these results contradict the Theorems of Euclidean Geometry, they would be able to conclude that their space is bent.

This is similar to the way in which scientists have deduced that our 4D space-time continuum is bent under the influence of massive bodies.

So it would seem pretty clear, in the face of the evidence, that our 4D space-time continuum must be a bent manifold embedded in at least a 5D-space-time continuum.

Next we want to draw one more conclusion from the results we have developed so far. Let's make another analogy between what 2D inhabitants of a 2D space might experience and what we as inhabitants of a 3D space do experience. (For this consideration we can ignore time whether we consider it a dimension or not. So instead of talking about our 4D space-time continuum, we will simply talk about our 3D spatial environment.)

Let's suppose that the 2D inhabitants have a way of "seeing" each other. Again, for an excellent description of what these inhabitants might be like see "Flatland" by Edwin Abbott. And for simplicity and familiarity let's suppose that they have something like our light which is actually electromagnetic radiation. So let's say that their light is some kind of wave vibrating in the 2D space of the paper surface, and that these waves travel outward from a source in straight lines, or rays, just as our light does. Of course, their light must stay on the surface of the paper.

If their manifold is still flat, i.e. Euclidean Geometry still holds, these "light rays" will be straight. If their manifold is curved, as over the basketball, then the "light rays" will curve to follow what is called a geodesic. This is no problem—it is analogous to what our light does in our 3D spatial universe. The important thing is that in either case, the path of the "light rays" must stay on the surface of the paper or rubber sheet. Light is confined to the manifold.

Finally, the important conclusion we can draw is that using the "light", and the "eyes", and whatever else the 2D inhabitants use in order to see one another, it is not possible for them to see outside of their manifold and "see" books on the shelves that might be in the classroom alongside that table or basketball with the manifold lying on it. It is not possible in principle for structures within an embedded manifold to "see" or detect structures which exist in the embedding space that is not part of the manifold.

So if we apply that conclusion to our 3D situation, it is not possible for us to "see" or detect anything that exists in higher dimensions outside of our 3D manifold.

Another way to look at this, so to speak, is to consider that dimensions give us "directions". For example if we have only one dimension, we have only two "directions", say left and right. This can be reduced to only a single direction, say "right", if we allow positive and negative movement in that direction.

Moving up one dimension to the 2D case, there are a large number of directions we could go from any starting point. For example we could go in the direction of any of the 360 degrees of the compass. But here again, we can reduce this to only two directions, say East and North, and make up composite directions by combining these two and, as before, considering positive and negative travel in each "basis" directions (that's what East and North are.). For example, going South is simply going North negatively. Now it turns out, that in 2D, we need two of these "basis" directions to be able to "point" at every possible direction in our space. And, in the 1D case, we need only one such "basis" direction. (Mathematically these "basis" directions are called "basis vectors", but we needn't use such formal language for this essay.)

Now, moving up one more dimension to our 3D case, we not only can "point" to any of the directions of the compass, we can also point up and down. And, we can point in any combination of these directions. We can point to any point on the surface of a sphere that we can imagine surrounding us with us at the center. And, it turns out that three basis directions, say North, East, and Up, are sufficient to point to any point on the sphere. Or, equivalently it takes three numbers to aim an artillery piece at a target if we expect to hit it, viz., bearing, range, and azimuth.

This means that in our 3D manifold, we can point to anything in the manifold, but we cannot point to anything in the higher-dimensional space that is outside of our manifold.

These implications seem to offer a new way of thinking about the fundamental nature of reality. String theory is a recent attempt to find this fundamental theory, and their early work led them to conclude that multiple extra real dimensions do in fact exist. Unfortunately, however, they succumbed to the siren's song of the expectation of being able to see the extra dimensions if they exist, and they fell into the trap of adopting Oskar Klein's ostensible remedy.

As a result, the string theorists have been hard at work trying to identify the complex geometrical structures that would not only satisfy solutions to the Einstein and Maxwell field equations, but that would also be curled-up microscopically tiny.

Shing-Tung Yau succeeded in categorizing all such structures, which are known as Calabi-Yau spaces (Calabi conjectured their existence prior to Yau's proving them). Not only are these C-Y spaces fantastically complex, but there is a huge number of them. String theorists have been combing through these myriad C-Y spaces for decades now, trying to find the magic one which will make string theory work.

But they are overlooking the C-Y spaces which are not curled up, and which have large dimensions like our familiar 4. The simplest C-Y spaces, which have large dimensions, are tori of various dimensions. A 2-torus, for example, the familiar donut shape, is the simplest (and I think the only) 2D C-Y space. Even though large 5 or 10-D tori are daunting, if not impossible, to visualize, they are still far simpler than the curled up C-Y spaces. It would seem to me that considering large real dimensions would greatly simplify the string theorists' searches, and may even lead to a successful Theory of Everything.

COUNTER-ARGUMENTS

Now comes the great mystery that this essay is intended to address. This mystery has baffled me for many years and I am at a loss to explain it. The mystery is, Why, in the face of these rather obvious and straightforward conclusions should anyone expect to be able to "see" extra dimensions, if in fact they do exist?

That unwarranted expectation is equivalent to asking Why can't I aim my artillery piece to some point outside of our 3D space? Well, it's because you only have 3D artillery pieces at your disposal.

The failure of our expectations to be able to "see" extra dimensions is by far the most expressed objection to the idea of large, real extra dimensions. All writers that I am aware of have said something like, "If extra dimensions really exist, why can't we see them?" That is the most common objection but it is not the only one. Let me mention two others before we proceed.

I have read the argument that if we lived in higher-dimensional space, our inverse square laws would instead be inverse cube, or higher, laws. This argument is defeated by the fact that structures and actions in a manifold behave the same whether or not the manifold is embedded in higher dimensional space. So the inverse square laws could remain unchanged with the addition of, or in the presence of, higher dimensions.

The second objection was made by Roger Penrose on page 923 of his monumental The Road to Reality where he is overwhelmed by "enormous increase in functional freedom" that extra dimensions would bring, and as a result he refuses to consider them seriously. To which I say to Sir Roger, "Acknowledge and relish that extra freedom. It implies enormously complex structures which might exist and act outside of our 4D manifold. The asymmetry among dimensions allows only some manifestations of those actions to be detectable in our manifold. This would be exactly like 2D shadows existing in our 3D environment. The shadows exist only on 2D manifolds, but they are manifestations of 3D structures and actions."

But let's return to the main objection: Why we can't see the extra dimensions?

We can't see outside of our 3D manifold because everything involved in our "seeing" is a 3D structure, or less. I say less because some things, like our retinas are 2D. And, some things, like a serial port used to transmit images to devices that help us "see" parts of our world, are only 1D. In order to "see" anything outside of our 3D manifold, we would need at least some 4D piece of apparatus, and all such instruments are unavailable to us.

The mystery, of why people seem to expect to be able to see 4D or 5D objects, if they exist, would not be so baffling to me if those people were all mathematical and scientific lay-people. They could be excused. The real mystery is why respectable scientists and mathematicians seem to have the same expectations. It is to these people I am really directing this essay, although my expectations are low that any of them will ever see it. I will be happy if anyone at all reads this essay, and I sincerely thank you, dear reader, for reading at least this far. I hope you read to the end, and maybe even give me some help in getting some scientist to read it.

And now, let me name names. In my reading, I have noticed that all of the following illustrious scientists and mathematicians have expressed an expectation to be able to "see" higher dimensions if they exist: Stephen Hawking¹, Martin Rees², Lisa Randall³, John D. Barrow⁴, Michio Kaku⁵, Brian Greene⁶, Shing-Tung Yau⁷, and even Roger Penrose⁸.

There is a glimmer of hope, however. Among these scientists, John D. Barrow has acknowledged that if our phenomenal world is indeed an embedded manifold, then the extra dimensions need not be "compactified" but instead could be large.⁹

When these eminent scientists attempt to explain the existence of higher dimensions in the face of our inability to see them, they typically resort to an idea introduced by Oskar Klein in which the extra dimensions are compactified, or "rolled up" so tightly and small that we are thus unable to see them. The point of this essay is to suggest that such a ploy is unnecessary.

To the extent that the eminent scientists explain this ploy further, they usually use what is called the "Hosepipe Analogy". I have selected two of these and I include them verbatim here:

From Knocking on Heaven's Door, by Lisa Randall, page 312:

"Clearly, since we don't see them, these new dimensions of space must be hidden. That could be either because they are too small to directly influence anything we could possibly see, as physicist Oskar Klein suggested back in 1926. The idea is that as much that is hidden from view at our limited resolution, the dimensions might be to [sic] small to discern. ... [W]e might not notice a curled-up dimension that we cannot travel through—much as a tightrope walker would view his path as a one-dimensional, whereas a tiny ant on the wire might experience two, as illustrated in Figure 61.

"Another possibility is that dimensions can be hidden because space-time is curved or warped, as Einstein taught us will happen in the presence of energy. If the curving is sufficiently dramatic, the effects of the additional dimensions are obscured..."

Her Figure 61 shows a picture of a man on a tightrope and another of an ant on a thick section of the rope. It is followed by the caption,

"[FIG 61] A person and a tiny ant experience a tightrope very differently. For the person, it appears to have one dimension, while for the ant, it seems to have two.

The second example is from The Road to Reality, by Roger Penrose, beginning on page 326:

"The analogy is often presented of a hosepipe (see Fig. 15.1), which is to represent a Kaluza-Klein-type modification of a 1-dimensional universe. When looked at on a large scale, the hosepipe indeed looks 1-dimensional: the dimension of its length. But when examined more closely, we find that the hosepipe surface is actually 2-dimensional, with the extra dimension looping tightly around on a much smaller scale than the length of the hosepipe. This is to be taken as the direct analogy of how we would perceive only a 4-dimensional physical spacetime in a 5-dimensional Kaluza-Klein total 'spacetime'. The Kaluza-Klein 5-space is to be the direct analogue of the hosepipe 2-surface, where the 4-spacetime that we actually perceive is the direct analogue of the basically 1-dimensional appearance of the hosepipe.

His Fig. 15.1 Shows a picture of a garden hose with a magnifying glass enlarging a part of it. It is followed by the caption,

"Fig 15.1 The analogy of a hosepipe. Viewed on a large scale, it appears 1-dimensional, but when examined more minutely it is seen to be a 2-dimensional surface. Likewise, according to the Kaluza-Klein idea, there could be 'small' extra spatial dimensions unobserved on an ordinary scale."

On page 800, Penrose continues: "How is it that physicists could take seriously the possibility that the dimensionality of spacetime might be other than the four that we directly experience (one time and three space)? ... Before we dismiss this idea as a total fantasy we must recall, from §15.1, the ingenious scheme, put forward in 1919 by...Kaluza, and...Klein... Provided that the extra dimensions (in excess of 4, that is) are taken as small dimensions, in some appropriate sense, then we might not be directly aware of them. What does 'small' mean in this context? Recall the 'hosepipe' analogy of Fig. 15.1. When looked at from a great distance, the hosepipe appears to be 1-dimensional, but if we examine it more closely, we find a 2-dimensional surface. The idea is that some being, inhabiting the hosepipe universe, would not 'know' that the extra dimension wrapping around the pipe is actually 'there', provided that the physical dimensions of that being are much larger than the circumference of the hosepipe. Similar remarks would apply to a higher-dimensional 'hosepipe universe' of 4 + d dimensions, where d of the dimensions are 'small' and not directly perceived by a much larger being inhabiting this universe, who perceives only the 4 'large' dimensions; see Fig. 31.3."

His Fig. 31.3 shows a section of a garden hose with the length labeled as "Normal spacetime", a painted band around a section of the hose labeled "being", and the circumference of one end of the hose labeled "Extra dimensions". It is followed by the caption,

"Fig. 31.3 Hosepipe model of a Kaluza-Klein-type higher-dimensional spacetime (see Fig. 15.1), where the dimension along the length of the hosepipe represents normal 4-spacetime and the dimension around the pipe represents the 'small' (perhaps Planck-scale) extra dimensions. We imagine a 'being' who inhabits this world, as straddling these 'small' extra dimensions, and so is not actually aware of them."

REFUTATION

My refutation comes in two parts. First, I claim that we should not expect to be able to see additional large dimensions which are outside of our manifold, so the gyrations suggested by Oskar Klein are completely unnecessary. Secondly, I claim that the hosepipe analogy is fatally flawed as an argument.

The first part has already been covered: In the IMPLICATIONS discussed above, we concluded that it is not possible for us to "see" or detect anything that exists in higher dimensions outside of our 3D manifold.

This conclusion should obviate the need to even consider the "Hosepipe Analogy", but since it is so frequently cited, I will show how the analogy is fallacious.

I apologize in advance for the tedium required to go through this, but since my target audience is the very scientists I am quoting, I will go through this tedious refutation for their benefit. The rest of you may follow along if you like.

Let's start with Lisa Randall's version. The tightrope walker is not an analogue for the ant.

The tightrope walker exists in a 3D body (as we all do) which exists in a 3D spatial world. The tightrope is a 3D structure in his 3D world right along with his body. The tightrope walker, however, is subject to the illusion that the tightrope is a 1D manifold as long as his perspective is coarse enough to obliterate perception of the 2D cross-section of the tightrope. So rather than the illusion hiding an additional dimension, the illusion hides two dimensions of a 3D manifold (the tightrope) making it appear as a 1D manifold, which is still part of his 3D world.

The ant also exists in a 3D world, but the intent is to consider the ant to be confined to the surface of the tightrope so that it, the ant, is considered to be a 2D structure residing on the 2D structure of the surface of the tightrope. The illusion for the ant is that it seems to live in a 2D world, but in reality it lives in a 3D world, one dimension of which is hidden due to the cylindrical nature of the surface of the tightrope.

In the ant's case, the size of the curvature, and indeed the curvature itself, has nothing to do with the illusion the ant experiences. If instead of a tightrope, the ant were confined to a flat sheet of paper, the illusion would be the same: the ant would perceive the 2D paper surface and be oblivious of the fact that the surface is a manifold embedded in the 3D space of a room.

The analogues between the ant and the tightrope walker do not correspond so the "analogy" gives us no coherent explanation for the inability to "see" higher dimensions outside of our manifold.

Next we look at Penrose's version of the analogy. His analogy is more nearly correct in terms of spaces, dimensions, and beings, but not with respect to the observer who is deceived by the illusion of an invisible dimension. Penrose says,

"Viewed on a large scale, it[, the hosepipe,] appears 1-dimensional, but when examined more minutely it is seen to be a 2-dimensional surface."

The observer, who views the hosepipe on a large scale, is necessarily outside and separate from the hosepipe, presumably in a 3D world like our own. The hosepipe, whether conceived as 1D, 2D, or even 3D is not an analogue for the world lived in by the observer.

In order to make the observer analogs make sense, we would have to consider an observer who is confined to the hosepipe, as is the painted band in his Fig. 31.3, which he labeled "being". This being would experience one degree of freedom as it could slide either direction along the hosepipe. But it would not see itself as a 2D structure unless it were aware of the cylindrical nature of the hosepipe.

If the being were aware of the cylindrical nature of his 2D world, then he would know, by the reasoning earlier in this essay, that his 2D world is bent, and that it therefore must be a 2D manifold embedded in a 3D space. Which is exactly what the hosepipe is with a being confined to a portion of its surface.

And, in this case, there is no need for the diameter of the hosepipe to be small relative to the size of the being.

CONCLUSION

There are many good reasons for considering the real possibility of our 4D spacetime world being a manifold embedded in a higher (5D or greater) dimensional spacetime. First, is the suggestion made by Kaluza that the mathematics of describing the phenomena of our 4D world is simplified. John D. Barrow has written, "[P]article physicists have discovered that the most elegant and complete theories of elementary-particle processes,...predict that there are many more than three dimensions of space (perhaps a further six, or even another twenty-two in some cases)."¹⁰.

Aside from those tantalizing suggestions, we have the, to me, obvious evidence I presented above that our world is indeed an embedded manifold in a larger spacetime. And, as I hope I have shown, the expectation that we should be able to "see" the extra dimensions is a false expectation, so there is no need to consider them to be compactified or curled-up.

If the need to compactify the extra dimensions is abandoned, the problem of string theorists of trying to sort through the enormous number of fantastically complex Calabi-Yau spaces should reduce to the simplest case. String theorists should consider only the simplest Calabi-Yau spaces, those of n-dimensional tori.

In summary, since the universe seems to be more mysterious than we can imagine, we should at least stretch our imaginations as far as we can in order to try to understand it. The possibility of large, extra, dimensions of space-time is within our power to imagine, so we should seriously consider the idea and exploit it for whatever usefulness it might provide.

Footnotes

1 Stephen Hawking, A Brief History of Time, p. 163, "Why don't we notice all these extra dimensions, if they are really there? Why do we see only three space and one time dimension? The suggestion is that the other dimensions are curved up into a space of very small size, something like a million million million million millionth of an inch. This is so small that we just don't notice it; we see only one time and three space dimensions, in which space-time is fairly flat."

2 Martin Rees. Just Six Numbers, p. 153

3 Lisa Randall. Knocking on Heaven's Door, p. 8, "Extra dimensions of space might exist, but they would have to be tiny or warped or otherwise currently hidden from view in order for us to explain why they have not yet yielded any noticeable evidence of their existence."

4 John D. Barrow, New Theories of Everything: The Quest for Ultimate Explanation p. 33 "Of course we do not live in a ten- or eleven-dimensional space so in order to reconcile such a world with what we see it must be assumed that only three of the dimensions of space in these theories became large and the others remain 'trapped' with (so far) unobservably small sizes."

5 Michio Kaku, Parallel Worlds, p. 200 "...there was the much more disturbing physical question: why don't we see the fifth dimensions?" and p. 386 "Perhaps the reason why we don't see these other dimensions is either that they are curled up or that our vibrations are confined to the surface of a membrane."

6 Brian Greene, The Fabric of the Cosmos, p, 372, "Since we don't see the extra dimensions, they must be small".

7 Shing-Tung Yau and Steve Nadis, The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions, p. 2

8 Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe, p. 326, "The extra dimension, enables Maxwell's superb theory of electromagnetism ... to be incorporated, in a certain sense, into a 'spacetime geometrical description'. However, this '5th dimension' has to be thought of as being 'curled up into a tiny loop' so that we are not directly aware of it as an ordinary spatial dimension."

9 John D. Barrow, New Theories of Everything: The Quest for Ultimate Explanation, p, 126 "If we exist in a manifold, the extra dimensions could also be large."

10 John D. Barrow, New Theories of Everything: The Quest for Ultimate Explanation, p. 125, "We observe there to exist three dimensions of space, but particle physicists have discovered that the most elegant and complete theories of elementary-particle processes,...predict that there are many more than three dimensions of space (perhaps a further six, or even another twenty-two in some cases)."