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In his Special and General theories of Relativity Einstein resolved two of the three main conflicts in physics. And his theories radically changed our understanding of the Universe. String Theory, in turn, allowed physicists to find the resolution to the third contradiction which had remained, arguably, the most important and most challenging of them all. And String Theory has demanded even more dramatic changes to our view of physics. The tremor of the basic concepts has been so strong that it put into question even such seemingly unshakable notions as the number of spatial dimensions in the Universe. This question is what we are going to be concerned with in this chapter. For those who would like a more detailed explanation of these concepts I suggest reading Brian Greene’s book “The Elegant Universe”, from which the main ideas of this article are taken.
A Typical, Seemingly Undoubtful View
Our intuition is being continuously fed by our daily experience. But the role of experience goes beyond that: it forms the basis on which we interpret and analyse events around us. The experience of two different people raised in two different cultures might be anything but similar. There are such phenomena, though, that are experienced by anyone regardless of where they grew up. Typically it is those beliefs which are underpinned by such universal experience that are extraordinarily hard to reassess. Let us consider a simple example. If you now stand up from your chair and decide to go somewhere, there are essentially 3 independent directions which you could move in, i.e. 3 spatial dimensions. Here you might object that actually you could move in much more directions, but all of them are going to be the combination of “left or right”, “forward or backward”, and “up or down”. This is what we mean by three independent directions, or three spatial dimensions. Each time you make a step, you are making three independent decisions regarding the direction of your next step in each dimension.
When we were considering Special Relativity we saw that each point in the Universe can be defined by using three parameters indicating its position in three dimensions. In New York, for instance, you can arrange a meeting with your boss by indicating a street (left-right direction), an avenue (forward-backward direction), and the floor number (up-down direction). Einstein’s theory also showed us that time can be treated as the fourth dimension, which can be thought of as the “future-past” direction. So when arranging a meeting you should also define time, apart from street, avenue, and floor. This extends the number of dimensions to four (three spatial and one temporal), thus events in the Universe are defined by the indication of where and when they occurred (or will occur).
This property seems so fundamental and obvious that rarely is it even mentioned. Nonetheless, in 1919 a Polish mathematician Theodor Kaluza had enough bravery to propose that there might be an additional spatial dimension that we just had never been aware of. Initially this proposal did not quite pan out because, as Carl Sagan’s famous quote goes, “Extraordinary claims require extraordinary evidence”, but later this idea hit stride. The extension of it is essential for String Theory to be mathematically consistent.
The idea of there being a greater number of spatial dimensions than three might sound bizarre, mystical and even pointless, but as we shall see, it is actually based on thorough reasoning. We can start thinking of it with a simple example. Suppose you are looking at a wire on the street from a distance of 500 metres. As you can imagine, from such a distance the wire looks like a line, i.e. it extends only in one dimension from our perspective (we won’t use our binoculars). If you imagine an ant living on that wire, you might think that it would have only one independent direction to travel in (up or down in the figure 1 below). If you were to define its position, you would only need to specify its distance away from the top or bottom point of the wire, and that would be enough. What I want to show in this example is that from the distance of 500 meters the wire seems to be one-dimensional object.
Of course we know that this picture is delusive; if we are to approach the wire, we would see that it has circumference, another independent direction for our ant to travel across. Although it’s hard to recognise the circumference from 500m distance, if we do use our binoculars we would certainly see that the ant is travelling in two independent directions (up-down and across the circumference). You can see this in the magnified part of the figure 1. Now we understand that we have to specify two independent numbers in order to define the ant’s position: one representing its position on the vertical axis (up-down), and another one being its position on the circumference (clockwise – counter-clockwise direction). This reflects the fact that the surface of our wire is two-dimensional. Of course if we are to consider the object (wire) itself, we know that it’s actually three-dimensional, but since in this example we were concerned only with its surface, the two numbers mentioned above were enough to explicitly define ant’s location.
These two dimensions have an apparent difference. One is well extended and easily noticeable. The other one (clockwise – counter-clockwise) is short, “curled up” and hardly recognisable. In order to notice this second dimension we had to use some tools with greater resolution.
What we’ve just seen in this example is that dimensions could be “lengthy”, extended, and be easily noticeable by an unaided eye, but they also might be “tightly packed” and hardly noticeable. Of course in our example we could easily find the hidden dimension solely by using binoculars, but if the width of the wire were much smaller, like the width of a human hair, then this task would be quite complicated.
In 1919 Kaluza wrote an article and sent it to Einstein. In that article he proposed an outstanding hypothesis: namely, that the Universe might have more than three spatial dimensions which we are familiar with from our everyday experience. The motive for such a radical proposal was the following: this hypothesis provided the way to construct an elegant and powerful mathematical apparatus combining Einstein’s General theory of Relativity and Maxwell’s theory of electromagnetic field into one conceptual system which we will examine a little bit later. But how this hypothesis could be conformed to the obvious fact that we see and perceive only three dimensions?
Kaluza’s work did not contain an explicit answer to that question, but a Swedish mathematician Oskar Klein improved initial Kaluza’s proposal in 1926 and expressed the answer to the question above explicitly. His work showed that the structure of our Universe may contain both extended and curled up dimensions. This implies that the three dimensions we know of are like the up-down direction in the figure 1, but apart from them the Universe might contain additional dimensions that are curled up so tightly that they remain inaccessible even to the most powerful instruments to date. Specifically, Kaluza and Klein suggested that the Universe might have 3 extended spatial dimensions familiar to us, but also one dimension whose presence we’ve had no chance to identify. Thus the overall number of spatial dimensions, according to Kaluza and Klein, is equal to four, which gives us a total number of 5 dimensions: 4 for space and 1 for time.
If we draw the picture of the Universe proposed by Kaluza and Klein, we are to find the following. Imagine that we have an extremely powerful apparatus that could magnify the resolution billions of billions times more than the most precise microscopes to date. If we take a part of completely empty space, and start increasing the resolution using our apparatus, we would see increasingly small packages of nothingness. After a few such magnifications, however, something interesting happens. An additional fourth dimension having the form of a flat circle unfolds, as shown in the figure 2 below.
Here we see the structure of space as was seen by Kaluza and Klein. There is one important simplification on this figure though. The circles are shown only at the particular points on the grid. What Kaluza and Klein actually proposed was that these loops are actually present at each point of space given by three extended dimensions, just as the additional circumferential dimension exists at each point of extended dimension of the wire shown in the figure 1.
In spite of the similarity with the previous example with the wire, this picture has a couple of things that clearly distinguish between them. The Universe contains three extended dimensions (we showed two of them in the picture) as opposed to one dimension which we were considering speaking about the wire. And what’s even more important, in the former example we showed one extended dimension of an object (the wire) existing inside the Universe, whereas in the latter we’ve been considering the structure of the Universe itself, which is quite a big deal. The main idea, however, remains the same: the curled up dimensions might be so exceedingly small that it would be extraordinarily hard to find a way to detect their presence.
As we’ve already mentioned, the circles representing an additional dimension in the figure 2 are placed at particular locations just for the sake of demonstrativeness, but the theory suggests that those circles exist at each point all around us. If our ant from the wire was small enough, it would be able to walk across this fourth dimension, so that we would have to provide four numbers in order to specify ant’s exact location. On top of which, we, of course, would have to provide the fifth number specifying the ‘location’ of the ant in the temporal dimension.
So we’ve got to quite an interesting conclusion. Following the reasoning of Kaluza and Klein we see that even though we can perceive only three spatial dimensions, this does not eliminate the possibility of additional curled up (or “compactified”, as string theorists call them) dimensions being hidden in the very structure of space. The Universe might have more dimensions than the eye is capable of perceiving. But how small should these dimensions be? In 1926 Klein combined the initial Kaluza’s idea with the quantum mechanical apparatus and found out that the additional dimension should have the Planck size – or 10 to the negative 35 metres – whose investigation requires energies which are way beyond our current and even imaginable limits. Things are getting less clear when String Theory comes into play though. We shall come to considering the additional dimensions in String Theory shortly.
Examining a Wire Universe
The examples given above help us understand why there is a reason to believe that our Universe might contain additional spatial dimensions besides those three that we are familiar with. But even for those who study this subject intensively, with mathematics and all the rest, it is very hard – if possible at all – to clearly imagine the picture of a universe having more than three spatial dimensions. For this reason, physicists usually try to derive an intuitive perception of the question by taking a step back and analysing how we would see three-dimensional space if we lived in a two- or even one-dimensional universe. For example, what would happen if the entire Universe was like a wire which we considered in the earlier section? This “Wire Universe” should be considered, literally, as an entire universe, which means that our wire is all there is, so that we should forget about looking at the wire as a remote observer. Instead, we shall imagine ourselves being some kind of a living organism in that universe. Now, for a more complete analogy, let us start with an extreme example, for which we shall assume that the circumferential dimension in this universe is compactified, and that we have no idea about its presence. Now we are left with only one dimension, so we will call this universe the “Lineland”, because a line is a one-dimensional object as well (this example is taken from Brian Greene’s book, so I have no credit for it).
Life in this universe differs dramatically from what we are familiar with. For example, the bodies from our Universe, such as ants, just cannot exist in the Lineland for quite a simple reason: there is no place for them to be there, since there is only one dimension in the Lineland. Being a creature within the Lineland, you have to find room inside it. Try to imagine that and you’ll immediately find out that you can live in the Lineland only if your body has neither width nor height, but only length.
Now imagine that you, like a mammal from our Universe, have two eyes. But being a one-dimensional creature, the only reasonable location of your eyes would be at each end of the body because otherwise you would not be able to see anything except some point inside of your body, which is far from being practical! Moreover, instead of being capable of revolving in the eye pits like in a human body, your eyes are eternally focused in a single possible direction: straight ahead. This is not an anatomic limitation, but rather the only solution open to evolution! And there is something more. If you think about it, you’ll realise that all you see in the Lineland is an object – or creature – right next you! And this is true for both your eyes. There is just no way for you to see anything except that object because, once again, the entire universe is a straight line.
We could continue reasoning through possible implications of living in the Lineland, but would very quickly conclude that such a life is not very rich in various possibilities. In the Lineland, you cannot walk past your friend “standing” in front of you. Moreover, you don’t even see their body, but only one eye! That is, after the inhabitants of the Lineland have settled in, they would have no chance to swap positions between each other. As we can see, the life in such a universe is far from being interesting.
But suppose one day two friends from this universe with the names Lineluza and Kline appeared to have a brilliant idea. What if – these two suggested – the Lineland is not actually one-dimensional, but instead has an additional circumferential dimension that has remained unseen due to its extraordinarily small size? They started to envisage all those magical implications which would take place if only the size of that additional dimension were to be made larger – the possibility which could not be ruled out according to the earlier works of their colleague Linestein. Lineluza and Kline described a magical place where the inhabitants could easily bypass each other by means of the second spatial dimension; they would also be able to see each others’ bodies at different angles, and their lives would obtain a whole range of new colours. This idea, no matter how radical it sounds, brings a great deal of hope into the heart of the Lineland inhabitants.
Now, if such a thing really happened – scientists from the Lineland magnified the size of the compactified circumferential dimension – the Lineland becomes nothing but The Wire Universe, or the “Wireland”. Not only did the inhabitants acquire the opportunity to walk through the universe in two dimensions (as shown in the figure 3 above), and to easily bypass each other, but also an entire new field is now open to evolution: the bodies of our inhabitants will become two-dimensional in some time. So that if we look at the Wireland in the next several thousand or several million years, we will find a whole range of different shapes there, including those that you used to study in the secondary school under the subject of Euclidean geometry. With time the inhabitants of the Wireland will inevitably become two-dimensional flat creatures just like in the famous Edwin A. Abbott’s novel “Flatland” published in 1884. Abbot’s Flatland is rich with culture and even some kind of a caste system which is based on the geometric form of inhabitants’ bodies.
Although it is hard to imagine anything interesting in the one-dimensional Lineland, the life in the Wireland is very rich! The shift from one to two dimensions transforms life in the universe radically.
But why stop there? Our Wireland might contain yet another additional dimension curled up in its structure. If this additional dimension were to unfold, the inhabitants of the Wireland would happen to experience another set of radical changes. In fact, if this third dimension becomes sufficiently large, this would be the universe such as ours. After some time has passed, you will find there a whole range of complex creatures such as human beings!
But wait, we can now ask the same question again: why do we need to stop there? This brings us to the Kaluza-Klein theory, according to which our Universe might contain additional spatial dimension. In the later sections of this chapter we will come to consider even more radical changes required by String Theory. In the early versions of String Theory the number of additional dimensions given by mathematical analysis was equal to 6 (the critical number of dimensions in bosonic string theory grows up to 26, and in superstring theory to 10 – i.e. 4 space-time dimensions familiar to us plus 6 additional dimensions required for the equations to work – but the details are too technical, so we won’t dive into them). If some – or all – of these additional dimensions really exist, and if they unfolded up to macroscopic size, life as we know it would experience dramatic changes. But what’s interesting here is that even if these additional dimensions remain compactified indefinitely, just their existence already leads to some profound consequences which we come to consider later.
Gravity and Electromagnetism Go Hand In Hand in the Kaluza-Klein World
Although Kaluza’s proposal about the existence of the fourth spatial dimension rolled up into the fabric of space-time is tantalising in its own right, physicists’ interest in this idea comes from a slightly different aspect. Einstein developed his General theory of Relativity in a particular way – with three spatial and one temporal dimension – because he had initially started up with such a model in mind. But if he wanted to, it seems like he could have developed his theory for two, five, or any other number of dimensions. And indeed, the mathematical apparatus of his theory can be extended to incorporate additional dimensions of space. And Kaluza did just that. He performed mathematical analysis and derived additional equations for a universe with four spatial dimensions instead of three.
What he found was rather astounding. He found that in this mathematical formulation the equations related to three spatial dimensions were basically similar to Einstein’s field equations, but in the case of one more dimension additional equations showed up. These additional equations were nothing but the equations derived by James Clerk Maxwell to explain electromagnetism! So, as you now see, by adding one additional spatial dimension Kaluza combined gravity and electromagnetism into one mathematical framework.
The gravitational and the electromagnetic interactions had hitherto been considered to be two separate and distinct forms of interaction. There seemed to be no connection between the two apart from the very similar formulae given by Newton’s law of universal gravitation and Coulomb’s law of electrostatic force. For those interested in this similarity I provide the two: Newton’s law of gravity is given by F = Gm1m2/r2 and Coulomb’s law of electrostatic force by F = ke|q1q2|/r2. Anyway, this similarity represented no connection in physical reality at that time. By making a radical assumption about the existence of a never before seen spatial dimension, however, Kaluza showed that these two forces might actually be intrinsically connected. His theory suggests that both of these interactions are related to waves propagating through the fabric of space-time. Gravity is mediated by waves propagating in three dimensions of space familiar to us, whereas electromagnetism is mediated by waves in an additional rolled up dimension.
Kaluza sent the article explaining the results of his work to Einstein. Initially, Einstein became very interested in these results. In his response to Kaluza in 1919 he wrote that he had never even thought that such unification might be achieved by looking at a five-dimensional world. A week later, however, in his other letter Einstein wrote that he finds Kaluza’s idea very interesting and doesn’t see any obvious flaws in there, but on the other hand he has to admit that he doesn’t find the arguments really convincing either. Two more years had passed and Einstein wrote yet another letter to Kaluza where he said that he had reassessed Kaluza’s work and is finally ready to present it in academy.
Unfortunately, when physicists started to explore the implications of the idea they found out that it had some serious contradictions with experimental data. Physicists’ attempt to include the electron into the theory led to the prediction about the ratio of its mass to its charge that clearly violated the experimental data. Since there was no clear way of resolving this problem, many physicists quickly lost interest in the model. Although Einstein and several other scientists continued exploring the idea, it did not get much attention among physicists to say the least.
Nevertheless, it now seems that Kaluza’s and Klein’s idea was just ahead of its time. In 1920s, when it came about, the entire new field of quantum mechanics was being developed intensively. New principles of the quantum world were catching theoretical physicists’ attention, which resulted in the development of quantum field theories. Experimentalists were deriving new ways of testing the predictions of the theory and performed experiments, which, in turn, adjusted the theories. The theory directed the experiment, and the experiment refined the theory. All of this resulted in deriving and developing the Standard Model of particle physics, whose relevance can hardly be overestimated. It does not seem surprising that at that time the proposal about an additional dimension hidden deep in the structure of the Universe didn’t bear fruit.
However, by the early 1970s the principles of the Standard Model had been established, and by the early 1980s the majority of its predictions had been confirmed. Even though some of those predictions – for example the existence of the Higgs boson – would remain unconfirmed for decades, the majority of physicists working on these questions did not have much doubt that those predictions will later be confirmed by experiment. So eventually it became clear that the time had come to explore and resolve the major conflict of XX century theoretical physics: the inconsistency between General Relativity and Quantum Mechanics. The success in the formulation of quantum field theories for the three fundamental interactions has inspired physicists to look for a similar theory for the gravitational interaction: a theory of quantum gravity. After numerous hypotheses have failed, the physics community became more appreciative of more radical approaches. The Kaluza-Klein hypothesis, which was left to die by itself in the late 1920s, was now revitalised.
The Kaluza-Klein Theory Resurrection
Since the time of the Kaluza-Klein proposal’s appearance the understanding of the physical world surrounding us has experienced a dramatic boost. The laws of quantum theory had been pretty much established and gained very strong experimental support by the 1970s. Two new kinds of fundamental forces unknown in 1920s were now observed and included into quantum theory, so that they were now on equal footings with electromagnetism and gravity. Many physicists at that time started to believe that Kaluza’s hypothesis initially failed due to the very conservative approach used by the author. He tried to unify two fundamental interactions and didn’t even consider the possibility of there being more. In the 1970s physicists showed that even though one additional spatial dimension could have led to the unification of gravity and electromagnetism, it was just insufficient.
By the mid 1970s there had appeared a whole bunch of various theoretical research aimed at developing theories with greater dimensionality in order to explain the world around us. In the figure 5 below you can see an example with two additional dimensions curled up into spherical shape. Similar to the case where we had one additional dimension, these two exist at each point of space-time in our Universe with three extended spatial dimensions. These two additional dimensions may also be rolled up into different shapes. For instance, they can have a toroidal shape shown in the figure 6.
Next we can imagine more complex structures having more than 2 additional dimensions, albeit these can hardly be represented on a two-dimensional screen. Mathematically, we can have any number of additional dimensions curled up in different shapes, and there are whole fields of mathematics – linear algebra, abstract algebra, complex analysis, topology, and others – whose purpose is partially to explore these ideas of higher dimensionality. But as long as there has been no experimental evidence of the existence of additional dimensions, they have to be compactified strongly enough to remain inaccessible for our modern-day experimental apparatuses (in later articles we’ll explore another interesting idea regarding additional dimensions, but for now we shall assume that they are indeed compactified).
The most promising theories concerned with extra-dimensions have been those which included supersymmetry as well. Physicists have hoped that the partial cancelling of intense quantum fluctuations provided by the pairs of superpartners would mitigate the discrepancies between gravitation and quantum fields. These theories were given a name Supergravity.
As with the original Kaluza’s hypothesis, supergravity theories superficially looked very promising. As a result of adding more dimensions they brought up additional equations which are very similar to those used to explain electromagnetic, strong and weak nuclear forces. Further careful analysis, however, showed that old problems were still remaining. The catastrophic quantum fluctuations that had been occurring in former theories were mitigated by supersymmetry, but insufficiently to make the theory consistent. Moreover, when working on the theories of supergravity physicists ran into trouble trying to include in them the concept of chirality which represents an essential part of the Standard Model. The concept of chirality is too abstract and is hard to understand conceptually, but for our purposes it would be enough if I give a very rough explanation. In the mid 1900s experimenters showed that there are some phenomena in our Universe which are not identical to their mirror images, i.e. our Universe is chiral. Essentially, they showed that some phenomena related to the weak nuclear interaction cannot have their mirror analogues. That is, those mirror analogues cannot exist in our Universe. So if you watch a movie showing some physical phenomenon and you notice that it contains a process violating this rule, you can be certain that what you are watching is a mirror version, not a real phenomenon. And this rule seemed almost impossible to include in a theory of supergravity.
As we now see, although the pieces of a final unified theory started to find their places in the puzzle, a central element capable of binding these pieces together was still absent. In 1984 superstring theory was shown to be the main candidate for taking the place of this central element.
Additional Dimensions in Superstring Theory
For now you should have got the idea of why our Universe might have additional spatial dimensions. Indeed, unless we have sufficiently advanced technology to test the structure of space on extraordinarily small scales, we cannot prove they do not exist. Anyway, these additional dimensions might seem to be just a mathematical focus pocus. As long as we have no way to experimentally test the presence of those dimensions, we can speculate that there might be whole civilizations of some extravagant creatures at Planck-scales. Even though this speculation does not have any mathematical support, postulating any of these currently untestable ideas might seem to be equally arbitrary.
For the people working with String Theory, however, things change dramatically. The mathematical framework of this theory successfully unifies Quantum Theory and General Relativity, hence it solves the main contradiction of modern-day theoretical physics. Furthermore, it unifies our understanding of the basic components of matter and the four fundamental forces. And what’s important here is that String Theory demands the existence of extra dimensions!
Let’s see why this is so. As we saw in chapter 4, the laws of quantum mechanics state that it is impossible to measure some parameters more precisely than Heisenberg’s uncertainty principle allows. The results of mathematical calculations based on the rules of Quantum Theory are represented by probabilities having certain values. As you probably know, probabilities can have values ranging either between 0 and 1 or between 0% and 100% (these, essentially, are equal representations: one can easily be converted to the other). Any probability with a value outside of this range has no meaning whatsoever. As physicists figured out in 1900s, in some situations the rules of QT break down because the calculated probabilities happen to have values outside of this range. As we also saw in chapter 5, the conflict between GR and QT – with its model based on dimensionless particles – occurs when the calculated probabilities reach an infinite value. As was discussed in chapter 6, String Theory solves this conundrum and gives us answers lying within the admissible range. What we haven’t touched upon, however, is that in the first versions of String Theory physicists found some probabilities which had negative values. These are again outside of the allowable range for a probability. Thus, at the first glance, String Theory also seemed inconsistent.
Nevertheless, scientists did not give up and looked for the cause of these unacceptable results. Eventually they found that cause, and I’m going to try to build a conceptual sense of it for you. Let’s start by assuming that strings exist in the Wireland which we considered in earlier sections (remember that it was a 2-dimensional universe). In such a universe our strings would have only two independent directions to vibrate in: left-right and back-forth (i.e. they would have only two degrees of freedom). If there were a third spatial dimension in the Wireland, however, the number of independent directions for our strings to vibrate would increase to three, i.e. the direction up-down becomes available. Further extension of this idea may be troublesome to imagine, but the overall scheme remains the same: larger number of spatial dimensions allows additional vibrational patterns for strings.
Now you might probably guess the source of the meaningless results contained in the early versions of String Theory. These results occurred because physicists considered a version which had too little a number of degrees of freedom. The cause of negative probabilities was in the inconsistency between what reality seemed to dictate and the requirements of the theory. The performed calculations clearly showed that if strings could vibrate in 9 dimensions of space, then the negative probabilities would go away. Anyway, this is important for the theory, but who cares? If String Theory seeks a way to explain the world with three spatial dimensions but works consistently only in a world with nine dimensions, then we are still in trouble.
Or are we? The idea put forward by Kaluza and Klein gives us a way around this trouble. Since strings are so small they can vibrate not only in large unfolded spatial dimensions, but also in those miniscule rolled up dimensions which we do not have access to. Thus we can satisfy the requirements of String Theory by assuming – like Kaluza and Klein did – that apart from three dimensions that we all know of, our Universe contains six additional compactified dimensions. Moreover, instead of postulating the existence of additional dimensions, String Theory demands it. For String Theory to be mathematically consistent, the Universe has to be 10 dimensional: 9 dimensions for space, and 1 for time. (In later articles we shall see that in 1995 this number grew up to 11, starting the second superstring revolution) Thereby, the descendant of Kaluza’s and Klein’s initial proposal has steadily come to the scene.
Some Questions Remained
After all this revolution, which String Theory brought up, a few important questions immediately showed up. Firstly, why does String Theory demand that number of additional spatial dimensions to be mathematically consistent instead of any other? Unfortunately, this question is one which is extremely hard to answer without resorting to the mathematical apparatus of the theory. Strings need 10 degrees of freedom to avoid getting inconsistent results with negative probabilities and such. The calculations performed on the theory’s equations lead to this result, but nobody could give an explanation having a clear conceptual picture without technical details involved. This is partly due to the fact that String Theory is still work in progress, and a lot of its details aren’t known as yet. This happened before, though. For example, the Standard Model of particle physics, which is based entirely on quantum theory, was derived and finalised more than 5 decades after Quantum Theory itself had been developed. And Quantum Theory still has its own questions that need to be addressed. Here I must remind you that although later work by E. Witten showed that String Theory actually demands 10 spatial dimensions, we shall be considering 9 unless we’ve got to the second superstring revolution in later articles.
Secondly, if the equations of String Theory show us that the Universe contains 9 spatial and 1 temporal dimensions, why did only 3 spatial dimensions unfold at some point in time, while the others have remained curled up? Why aren’t all of them unfolded, or why did all of them not remain compactified? Why was some other configuration of those dimensions not realised in our Universe? Currently we do not have an answer to this question. If String Theory is correct, then, hopefully, someday we will find the answer, but for now we don’t have such. This doesn’t mean though that nobody has even tried to find the explanation in the existing framework of String Theory. For instance, works on cosmological implications of the theory have shown that it is possible that initially all of the dimensions were rolled up, but moments after the Big Bang three of them unfolded while others remained compactified. In later articles we shall consider these possibilities, but I should say that those conjectures that we will touch upon are still exist in the form of speculation, and a lot of further work needs to be done in order to either support them or rule them out. For now we shall be taking it as a given: three spatial dimensions are unfolded, while others are compactified.
Thirdly, if String Theory demands the existence of extra spatial dimensions, couldn’t it be the case that its solutions might imply the existence of additional temporal dimensions as well? Think about it a little bit and you’ll see how weird a possibility that would be. We live in the Universe with a few spatial dimensions, so we could at least imagine the existence of others, right? But what could it even mean to have more than one time? If one of those would be the one familiar to us, then what would be the others? Could some of them run backwards? This idea becomes even more bizarre if those extra temporal dimensions could be compactified the way additional spatial dimensions are thought to be. For example, if a Planck-size ant were able to run across extra space-dimensions, it would be able to return to its original location every time it’s completed a full ‘cycle’. Nothing fanciful here since you are already used to returning to your apartment every workday. But if we had an additional temporal dimension, and our ant were to travel across it, then it would return to its initial position in time! You’ve got it right: no time has passed for the ant when it ran across the additional temporal dimension in this case. This, of course, goes far beyond our everyday experience. Time for us seems to run only in one possible direction: forward. Never do we return to a moment that has already passed. Surely enough, such additional curled up temporal dimensions could have different characteristics than ‘our’ time. As opposed to the extra spatial dimensions, though, those temporal dimensions – if only they exist – might alter our perception of the physical world surrounding us far more drastically. Some theorists are exploring the possibilities of including additional temporal dimensions into the framework of the theory, but these attempts are far from being in any way conclusive. In our further considerations we shall be concerned with one time dimension, though we should not forget about the intriguing possibilities which some researchers are playing around with.
Physical Implications of the Additional Dimensions
Although the size of the extra dimensions of space has to be extraordinarily small in order to explain why we haven’t detected their presence, according to String Theory they have an important indirect influence on some physical phenomena. In order to understand this let us remind ourselves that the mass and electric charge of a particle is determined by the vibrational patterns of the corresponding strings. It should be obvious to us that those vibrational patterns, in turn, are influenced by the space around them. Think about an ocean wave. Given the entire surface of the ocean, a single isolated wave might take any form and move in any direction because of the many degrees of freedom it would enjoy. If, on the other hand, our wave were to move through a very narrow canal, the number of degrees of freedom would substantially drop down because the form of the wave and the number of possible directions in which it can move would be largely determined by the width, depth and form of the canal. The compactified dimensions influence the number of degrees of freedom for fundamental strings in a similar way. As long as fundamental strings vibrate in all accessible spatial dimensions including additional ones, their vibrational patterns are largely determined by the shape of these extra dimensions. The shape, in turn, is determined by the geometry of the extra dimensions. We can summarise thus: the geometry of additional spatial dimensions determines the fundamental physical characteristics of a particle, such as its mass, charge and spin which we detect in our experiments.
This result is one of the most important implications of String Theory. And since this is so important researchers have spent decades trying to figure out which shape of the extra dimensions might lead to the fundamental characteristics of the particles we know of. So far, however, no one has been able to do this. We’ll come to this question in later articles but for now let us see what string theorists have been able to find on this avenue.
The Form of Extra Dimensions
The extra dimensions in String Theory cannot be compactified in an arbitrary way. That is, the equations of the theory restrict the number of possible shapes which those dimensions can take. In 1984 string theorists Philip Candelas, Andrew Strominger and Gary Horowitz alongside Edward Witten introduced the idea of compactification to String Theory. In their work these physicists showed that the mathematical apparatus of the theory requires that additional dimensions must be curled up into a form called Calabi-Yau shape (or Calabi-Yau manifold). It is named after two mathematicians Eugenio Calabi and Shing-Tung Yau, whose work, completed before the development of String Theory started, has played a crucial role in understanding the properties of these shapes. The mathematical description of Calabi-Yau manifolds is very complicated, but, thankfully, we don’t need it in order to get a sense of what they might look like.
The example of a Calabi-Yau shape can be seen in the Figure 7 above. You should remember though that this picture has some limitations, because a 6-dimensional manifold here is represented on a 2-dimensional surface, which inevitably leads to some sort of distortion. Nevertheless, the image shows a rough sketch of what this shape would look like for an external observer. In this image we see just one example of virtually infinite number of shapes allowed by String Theory, so such a restriction might seem not a big deal. However, if you consider the situation from a mathematical perspective, you quickly find out that without such a restriction this number would literally be infinite. In mathematics such a difference might lead to profound consequences, and some string theorists still hope that further development in the theory might provide some insights which will eventually help to rule out the majority of possible Calabi-Yau shapes and deal with a manageable number.
Now, remember the figure 5 where we depicted a 2-dimensional sphere at each point of space. To extend this idea with our new information we need to replace each sphere with a Calabi-Yau manifold that would also exist at each point of our familiar 3-dimensional space.
In the figure 9 you can see this picture more vividly.
In other words, according to String Theory six additional spatial dimensions rolled up into this kind of peculiar shape should exist everywhere around you and me, as well as at completely different corner of the Universe. Despite this, though, these dimensions are curled up so tightly that when you move through three unfolded ones you cannot perceive that your body does an amazing journey through these additional dimensions.
This is an astonishing prediction of String Theory, but it’s certainly not enough for an idea to be just mathematically consistent in order to be proven valid. We want any of our ideas to be tested somehow in order to either get confidence in them, or rule them out as just a mathematical artefact. Since those extra dimensions are so extraordinarily small, we want to find some traces of the influence which those dimensions would leave on things we can access, such as particles and the like. In the next article we shall consider possible ways to test the predictions of String Theory, but before we go on I’d like to give an answer to another very important question.
Are these Dimensions Really Necessary?
Regarding this question I can say that there is no definite answer as yet. Some physicists feel uncomfortable with the idea of extra space-dimensions which have never been seen. Because of that some theorists are trying to get round this idea by making use of some exotic mathematical constructions whose scope is far beyond this series of articles. However, to conclude this chapter I just wanted to mention a few models which try to build a 4-dimensional String Theory by introducing the additional degrees of freedom by different means. Those include Asymmetric Orbifold models, Four-Dimensional Covariant Lattices, Non-geometric Calabi-Yau Compactifications, 4d N=2 strings (what do all these names even mean?) and others. I’ve given you the links to some of the arxiv papers, but I’d like to warn you: these models are exceedingly complex even by the String Theory standards, which is why the majority of string theorists remain in the other camp. In our later articles we will be avoiding these 4-dimensional models because they are still in their infancy.
So as you now see, the mathematical apparatus of String Theory is so vast that even the theory’s essential assumptions can be questioned, which some physicists see as the theory’s serious drawback, while others regard this as its major benefit.
Thanks everyone for taking your time to read the article, and I hope to see you all next time.
Previous articles in this series can be found at:
Part 1: Following Einstein’s Dream
Part 2: Special Relativity – the Picture of Space and Time
Part 3: General Relativity – The Heart Of Gravity
Part 4: Quantum Mechanics – the World of Weirdness
Part 5: General Relativity vs Quantum Mechanics
Part 6: The Basic Principles
Part 7: Supersymmetry