Why String Theory?

by Joseph Conlon

  There is an advanced side comment here. In certain limits, these extra dimensions cease to behave as a classical space within which familiar interpretations of dimension and geometry can be sustained. In these limits, the behaviour of the extra dimensions is best described as quantum geometry. This idea will reappear in the next chapter, but for now I will simply say that this behaviour has a well-defined mathematical description and can be smoothly joined up with the more familiar language of classical geometry. In these limits, string theory can be said to require extra quantum dimensions rather than extra classical dimensions.

  Be that as it may, the necessary existence of extra dimensions is at both first and second sight a serious problem. For an idea to be right about this world, it first of all has to be not wrong, and ‘wrong’ is exactly what an additional six dimensions of space appears to be. Before discussing minutiae, we first need to explain why the existence of extra dimensions, in the real universe, is not in flagrant contradiction with observations. Having done this, we can then explore the idea further, and examine what the positive consequences of extra dimensions are for the physics of this world.

  So why is the existence of extra dimensions not immediately marked as inconsistent? We know about two ways of moving across and one way of moving up. There is no ‘hyper-up’ direction: where is this extra dimension?

  To address this, it is helpful to think about what we mean by a dimension and how we distinguish between one, two and three spatial dimensions. The easiest way to think about this is in terms of labels. In a space with two dimensions, we need two labels – two numbers – to uniquely specify the position of an object. Whether we call these labels x and y, or latitude and longitude, or distance and angle, the key feature of a space with two dimensions is that it involves two labels. The analogous property is also true for three or one dimensions: in a three-dimensional space, we need three coordinates to label where something is and in one dimension you just need a single number. We also label positions with distances, and the better our rulers are, the more we can refine our notion of a position. It does not make sense to talk about objects being a micrometre apart if the only ruler we have has gradations of millimetres.

  If extra spatial dimensions are present, there must be additional coordinates we can use to describe position. These extra spatial dimensions are consistent provided that all distance scales involved are smaller than the distances our rulers can resolve: the extra dimensions are curled up. There is an analogy for this as good as it is standard: an ant walking along a piece of string (the ‘string’ here is the regular type one buys from the supermarket!). The walking ant experiences the string as a one-dimensional object. The ant can go forward and it can go backwards. There is only one direction it can move along, and the world of the ant is effectively one-dimensional.

  The string does however have another direction: the transverse direction around itself. For a bacterium deposited by the ant’s feet, the surface of the string is two-dimensional. The bacterium is so small that even the direction around the string still counts as large, and because it is so much smaller than the ant, its world is two-dimensional.

  In a similar way, extra dimensions that are small enough are literally invisible to us. The wavelength of visible light is around one ten-thousandth of a centimetre: any distances smaller than this cannot be resolved by means of visible light. Provided these extra dimensions are smaller than the resolution of any probe available to us, their possible existence is consistent with all observations and experiments performed to date. We can no more tell that they are present than use our fingers to feel viruses. In practice, this requires the sizes of extra dimensions to be less than around one billionth of a nanometre.

  What then are the implications of extra dimensions? For if string theory is true, the existence of extra-dimensional gravity is a genuinely true feature of the world – at the smallest possible scales, there are more than three spatial dimensions. If this statement is true, it should have consequences. What are these consequences? As ten-dimensional gravity is the Texan version of Einstein’s gravity – the same theory, except bigger and better – the behaviour of ten-dimensional gravity both includes and generalises the behaviour of four-dimensional gravity.

  What does this imply? There are many paths one can follow here and I am going to choose one, guided by my own research and interests. I shall explore one aspect of four-dimensional gravity and explain what its generalisation to ten dimensions implies. This path starts with the fact that Einstein’s theory tells us that spacetime is dynamical. It is not the rigid globe of the Ptolemaic heavens, unchanged and imperturbable. It is responsive, as the air is responsive to the click of fingers through the production of sound waves. Spacetime itself, and its geometry, sends out ripples in response to a disturbance, just as the surface of a pond ripples with water waves in response to the entry of a stone or a hippopotamus. These ripples in spacetime are called gravitational waves, and their size and strength depend on the magnitude of the disturbance. In the same way that a belly-flopping hippo makes larger water waves than a dropped pebble, mergers of black holes make larger gravitational waves than colliding ping-pong balls. These waves all represent changes – whether macro or micro – in the geometry of spacetime, propagating outwards from their point of origin.

  Such gravitational waves have never been directly detected. This is because they are intrinsically very weak and so their observation requires exquisitely precise technology. When direct detection does first occur, it will come from extreme astrophysical events, such as collisions between black holes or massive stars, capable of generating the largest ripples in spacetime. As with any other kind of wave, these ripples become weaker as they spread out. By the time they reach us, the geometric perturbations the ripples represent manifest themselves as distance wobbles as small as one nuclear width across a length of several kilometres. It sounds fantastical that one could attain the technological precision required to measure such a change, but this is precisely the goal of the current and planned gravitational wave experiments that hope to detect gravitational waves within the next decade.

  Gravitational waves are a clear prediction of Einstein’s theory. Despite the lack of a direct detection, there is extremely strong indirect evidence for their existence, beyond simply the accumulated evidence for the correctness of general relativity.

  As with any other form of wave, gravitational waves carry energy. A system currently emitting gravitational waves has less energy after the emission than before, and this change in energy can be measured. This precise effect can be seen in stellar binary systems of a pulsar in a close mutual gravitational orbit with another star. A pulsar is a magnetised, rotating neutron star that emits a beam of radiation at a precise frequency. Every time the star rotates, the beam passes across earth like the beam of a lighthouse. As the pulsar orbits its companion star, gravitational waves are emitted which carry off energy. The energy that is carried off reduces the orbital energy of the system, bringing the two stars slightly closer together.

  The special properties of binary pulsars allow this reduction to be measured directly. While this is a rare configuration of stars, such a system was first discovered by Russell Hulse and Joseph Taylor in 1974. The Hulse-Taylor binary has now been observed for over forty years, and the observations match perfectly with the predictions of general relativity. The system loses energy at precisely the rate it should if gravitational waves are given off at the level predicted by general relativity. For the discovery of this binary pulsar system, Hulse and Taylor were awarded the 1993 Nobel Prize for Physics.

  Gravitational waves are the response of spacetime to a disturbance. How many different types of gravitational wave are there? In three spatial dimensions, there are precisely two types. This is – accidentally, it turns out – the same as the number of different light waves. There are also two types of light wave, distinguished by their polarisation. General light is a random admixture of both types of polarisation. B
y inserting filters that transmit only one type, the polarised components can be extracted separately. By inserting crossed filters that require first one and then the other type of polarisation, almost all light can be blocked, and this effect is used in sunglasses to reduce the glare and intensity of light.

  Why does visible light have two polarisations? The answer is because there are three dimensions’. A light wave in two spatial dimensions would have only one kind of polarisation, and a light wave moving in four spatial dimensions would have three kinds of polarisation. In general, a light wave moving in D spatial dimensions has a total of (D − 1) possible ways that it can be polarised. The underlying reason for this is that a light wave is an oscillation in the electromagnetic field. It is a fact that the oscillation can occur in any direction – except along the direction the wave is travelling. For a wave moving in D spatial dimensions, there are exactly (D − 1) independent ways for the field to point, accounting precisely for the possible number of polarisation states.

  The mathematics generates a similar but slightly different relationship for gravity waves. It is a fact that gravity waves require the specification of not one but two directions orthogonal to the line of travel. The mathematics then results in gravity waves in D spatial dimensions having possible ‘polarisations’. In three spatial dimensions, this formula allows for two possible types of gravity wave.2 This is the same as the number of possible light waves, although this appears to be coincidence and no deep reason is known for this similarity. It is worth noting that for two spatial dimensions, the above formula gives the answer of zero. The ‘zero’ is surprising, but correct. The gravitational force is actually not a dynamical force in less than three spatial dimensions, and gravitational waves are unable to propagate.

  In a world with two spatial dimensions, there are no gravitational ripples. However as the dimensions increase the possibilities proliferate, and once we reach nine spatial dimensions, there are a total of thirty-five possible forms of gravitational wave.

  Three spatial dimensions give two forms of gravitational waves. Nine spatial dimensions give thirty-five forms of gravitational waves. We started with the question: how does physics with nine spatial dimensions differ from physics with three spatial dimensions? These statements show that one partial answer is that with additional dimensions there are both additional forms of light wave and additional forms of gravitational wave. This represents a general truth about theories with additional dimensions. The more dimensions there are, the more ways there are for waves to propagate and the more types of polarisation there are.

  What are these polarisation types and what do they correspond to? Polarised light waves correspond to oscillations of the electromagnetic field along directions transverse to the direction of motion. With extra spatial dimensions, the extra polarisation possibilities of light correspond to internal electromagnetic oscillations inside the extra dimensions. While there is still a wave transporting energy from A to B, the field oscillations now live in the extra dimensions.

  The same is true of gravitational waves. If any extra dimensions exist, we cannot resolve them, and the gravitational oscillations are entirely confined within the extra dimensions. Whatever it might be on the smallest possible scales, the effective number of spatial dimensions is three on all distance scales we have measured. All our physics is described in terms of laws involving three spatial dimensions. How would extra-dimensional physics manifest itself if we are forced to express it in terms of laws that involve three, and only three, spatial dimensions? What has happened to all these extra gravitational polarisations?

  The answer is that living in our world, we see these extra polarisations as new particles, all of whose interactions are of gravitational strength. Mathematically, this interaction strength follows from the origin of these particles in the fundamental theory as gravitational polarisation modes. The traces left within our three-dimensional world by the extra dimensions are additional new particles, which interact by neither the electromagnetic, weak or strong forces. Instead, all their interactions are only by forces of gravitational strength.3

  One effect of extra dimensions on the observable world is then through the existence of new, additional particles with gravitational-strength interactions, and only gravitational strength interactions. This feature is not the only effect of extra dimensions. It is, however, the most universal consequence of extra dimensions, which is the reason I focus on it here. To generate this feature, it does not matter how many extra dimensions there are, it does not matter how they are curved, it does not matter what their topology is, it does not matter whether they resemble a rainbow or a rugby ball, and it does not matter whether they are described by classical or quantum geometry. It is a universal feature. If extra dimensions are present, they invariably lead to the existence of new additional particles whose interactions are at levels vastly weaker than any of the particles present in the Standard Model of particle physics.

  Such particles are called ‘moduli’. The physics of moduli is the main topic of this chapter. I have so far given one route to the existence of moduli by viewing them as extra-dimensional polarisations. There is also another way to motivate the existence of moduli, and to understand their nature. This is to understand moduli as counting the number of ways one can change the extra-dimensional geometry without expending large amounts of energy. This draws on an analogy – an analogy that in the mathematics is in fact extremely close – between the physics of extra dimensions and the physics of oscillations of surfaces.

  We know that different surfaces make different sounds because they vibrate in different ways. A drum is not the same as a bell, and a bell is not the same as a violin: the oscillations of each are sui generis, with each instrument having its own unique patterns of vibration. Furthermore, each instrument can also ring in many different ways – when a violin is bowed, its sound comes from vibrations at many different frequencies.

  What in the name of Guarneri has this to do with moduli? The mathematics shows that you can view moduli as the oscillatory modes of the extra dimensions. This is not a poetic metaphor or some loose platitudinous analogy. It is as direct and as true a statement as one could hope to make without equations. The reason for this is that in Einstein’s theory, geometry is dynamical. If you strike it, it rings. One ‘strikes’ geometry through a ‘violent’ nearby event: for example, through the gravitational collapse of a star to form a black hole. In theories with extra dimensions, a sufficiently violent hit will also cause the extra dimensions to vibrate. The energetics required for this to happen are large. Nonetheless, if the hit is large enough, the extra dimensions will vibrate. Just as for a violin, the manner of vibration will be as a superposition of many different oscillatory modes.

  The mathematical statement is that every possible oscillatory mode of the extra dimensions appears in the lower-dimensional theory as a different particle. In a theory with extra dimensions, the types of particles reflect the structure and geometry of the extra dimensions. The superposition of many different frequencies of oscillation corresponds to the creation of many different types of particle. In this language moduli correspond to the deformations that are in a sense easiest to make – they cost the least amount of energy, and require the minimal amount of bang’.

  These pages may have come across as hard to follow and mathematically heady. It is worth restating the key (and true) point. In theories with extra dimensions, one of the traces left by the extra dimensions in our three-dimensional world is the existence of additional particles whose interactions are exceedingly weak, and indeed only of gravitational strength. These particles are called moduli, and their existence is an unavoidable prediction of any theory including extra dimensions. If string theory is true, the existence of moduli is also a true statement about nature. Evidence for the existence of moduli would also be evidence for the existence of extra dimensions.

  10.2 WHAT MAKES MODULI AND WHAT MODULI MAKE

  This looks like big progre
ss. Extra dimensions generically predict the existence of a new type of particle that has not yet been observed! The way to make the concept of extra dimensions scientific now appears clear. All we need to do is to devise an experiment that tells us whether moduli do or do not exist. If this experiment turns out positive, we have positive evidence for the reality of extra dimensions. If it fails, then we can falsify the existence of both moduli and extra dimensions. The history of particle physics is all about devising experiments to look for specific particles – we just need to think of an experiment to look for moduli.

  Unfortunately, however appealing this sounds, it is difficult to do. It is extraordinarily hard to devise any laboratory-based experiment that can look for moduli. The reason for this comes from the most distinctive feature of moduli: all their interactions are of gravitational strength, and gravitational interactions are by far the weakest. This fact has been discussed before, but it is worth re-stating. As I write these words, my puny humanoid hand and its puny humanoid muscles can lift the pen up against the gravitational pull of the entire earth. There are almost as many kilograms in the earth as there are atoms in a kilogram. The number of protons and neutrons in the earth is so large that it is more than the number of grains of sand that could fit inside the volume of Pluto’s orbit around the sun – and yet the combined gravitational pull of every single one of these particles, added together, is insufficient to restrain the pen against the electromagnetic tug generated by my muscles. If this is what so many known particles can fail to do via gravitational interactions, what hope do we have of making an experiment to detect a single unknown particle whose interactions are of similar strength?