Why String Theory? Page 4

  The approximation of fixed spacetime is equivalent to neglecting the effects of gravity. For most purposes in particle physics this works wonderfully, because gravity as a force is so, so weak: using only my hands I can pull myself up and defeat the gravitational pull of the entire earth. The Planck length is defined as the distance scale at which this approximation must finally fail. At these scales, the effects of gravity can no longer be neglected, and the quantum theories of the electromagnetic, strong and weak forces must be joined by a quantum theory of the gravitational force.

  There is also an important point to make about the Planck length. The Planck length tells us the smallest possible length that the theories we know and love could be correct down to. It does not tell us the length at which these theories actually do break down. It is entirely possible that these theories fail on lengths considerably larger than the Planck length. While everyone agrees that what we know must fail by the time we have reached the Planck length, no one knows exactly where, between the scales we have probed experimentally and the Planck length, the great engine of known physics judders to a halt.

  A theory of the Planck scale requires a correct quantum mechanical description of gravity. Obtaining a definitively correct quantum theory of gravity is possibly the biggest open problem in theoretical physics. Cometh the hour, cometh the solution. And cometh another, and another, and another. Just as national crises see more than one politician volunteering themselves as the saviour of the country, so the important and well-posed nature of the quantum gravity problem sees many ideas put forward as proposed solutions.

  However, something new must come in at the Planck scale. What will that something be? Modern theoretical physics says – if not by unanimous agreement then at least by a large majority – that something is most likely to be string theory. String theory states that somewhere between the scales of the Large Hadron Collider and the Planck scale new degrees of freedom will come in, and those degrees of freedom will be the higher harmonics of vibrating strings.

  It should be said here that consensus opinions are dangerous things. There is a consensus opinion that the world is not flat, but there also used to be consensus opinions that smoking did not cause health problems and that there was no need to sterilise medical equipment between separate operations. Consensus opinions about the Planck scale deserve a heightened level of scrutiny because of the great difficulties in evaluating any such ideas experimentally.

  The simplest way to describe string theory, then, is as one candidate solution of the quantum gravity problem, and so as a candidate theory of physics at the smallest possible length scales. As the book will show, this is a rather limiting definition of the subject. If all string theory had to offer was a possible idea for quantum gravity, it would receive far less attention than it does. However, that argument is one I intend to develop over the rest of the book, at the same time showing that the consensus opinion about the importance of string theory is held for good reasons.

  The purpose of this chapter has been to locate string theory within the scientific vista. In this picture, its natural home is among the physics of the smallest possible scales. The ultimate story is richer, and we will later see surprising connections to physics at much larger distance scales, but this is where string theory has arisen from. The techniques it uses are most closely related to those of particle physics, with which it shares much of its language and personnel. Before narrowing down onto string theory however, I want to first review some of the great truths of physics.

  1As there are different ways to measure it, this number should not be taken as overly precise.

  2There exists a lazy but widespread prejudice that this discovery was not prehistoric. The beliefs that the geography of the earth was thought not to be spherical and that Christopher Columbus was warned not to sail because he would fall off the edge of the earth: these myths descend from the Victorian fallacy that there was a mediaeval fallacy that the world was flat.

  3William of Occam, a mediaeval Franciscan friar, studied in Oxford around 1300, possibly within some of the same buildings that are still used by the university today.

  CHAPTER 3

  Big Lessons of Physics

  3.1 SPACE AND TIME ARE CUT FROM THE SAME CLOTH

  ‘In the beginning …’ History, culture and religion – as in the first words of the Bible – reflect a classical notion of time that is easy to understand and appears obviously correct. This notion says that there exists one, universal time that applies everywhere and to everyone. It has counted from the beginning to now, and it counts from now to the future. The lives and fortunes of mice and men are marked out by the beat of a celestial gong. This view was expressed by Isaac Newton in the Principia, the work describing his laws of motion:

  ‘Absolute, true, and mathematical time, of itself, and from its own nature flows equably without relation to anything external, and by another name is called duration; relative, apparent, and common time is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.’

  According to this view, we could in principle synchronise our watches with distant aliens visiting from a far-off galaxy. The aliens could return to their home galaxy, stay there for many years, and then once again come to Earth. On their arrival we could compare our watches once more, and we would find them to be still identical. In this picture, time sits above and apart from space, acting as a universal coordinate labelling events in space. This notion of time is so ingrained in us that it took a very special person to realise that it is in fact wrong.

  That person was Albert Einstein, and the realisation occurred in 1905 with the development of special relativity. Special relativity is sometimes crudely described as the addition of a fourth dimension, with that dimension being time. If this were all special relativity was, Einstein would not have gone beyond the inheritance of Newton. Special relativity is instead built around two crucial insights, one conceptual and one technical. The former generates a philosophical revolution, the latter a calculational one.

  The conceptual insight is that space and time are knit from the same cloth. There is not, contra Newton, any division into absolute space and absolute time, with ‘separation in space’ and ‘separation in time’ being two entirely disjoint concepts. In relativity this division is not absolute, but varies from observer to observer. I, stationary in my armchair of reflection, may see two events as simultaneous and with a purely spatial separation. You, passing by on a bicycle – or more realistically, a spaceship travelling close to the speed of light – would instead see one event as clearly earlier than the other.

  This insight does not imply that time has no meaning. One of the most important roles of time is to mark causation, and to say what was ‘before’ and what was ‘after’. The sequence of causation remains unaltered in relativity. Some events cause other events and must always precede them. Whoever you are, you will agree that the event ‘My son Alexander was born’ came later in time than the event ‘I was born’ – the former could not have occurred without the latter. However, different observers may disagree on whether the event ‘I was born’ comes before or after the event ‘The alien Zak on the planet Zorg was born’.

  The second great insight is more technical in nature. It was already known that the laws of physics do not depend on where you are, or which direction you are looking in. Experiments in Madrid, Mumbai and Moscow get the same results, and the laws of physics do not care whether experiments are located in a north-facing building or a south-facing building. There can be local environmental effects: June in Madrid and December in Moscow require different levels of thermal insulation. However, once these environmental effects are removed the underlying laws of physics have no preferred orientation, as the three spatial directions can be rotated into one another without changing the laws of physics.

  Einstein also realised that these rotation
s can be extended to include the time direction. There is a particular set of rotations that mix the space and time coordinates, under which the laws of physics are invariant. This rotation is called a Lorentz transformation and underlies all the mathematics of both special and general relativity.

  It is not the purpose of this book to give a detailed treatise on relativity or to describe its mathematics. The essence of relativity was neither the mutual identification nor the abolition of space and time, but instead the realisation that they are both cut from the same cloth and are part of a single object: spacetime. In the words of one of the founders of relativity, the German physicist Hermann Minkowski: ‘Space of itself and time of itself will disappear into mere shadows, and only a kind of union between the two will survive.’

  3.2 SPACETIME IS DYNAMICAL

  Special relativity took Newton’s vision of absolute space and absolute time and destroyed it. Physics according to Newton involved events in space occurring to the backdrop of a universal time measured on a divine clock. This vision vanished with special relativity, and Newton’s concept of absolute time died a complete and irreversible death. As the arena in which the laws of physics played out, space and time were replaced in special relativity by the single object spacetime. This new entity then became the unchangeable background against which the laws of physics are formulated.

  This insight belonged to Einstein. It is to his great credit that he also made the further discovery that the new picture is also wrong – special relativity must in turn be replaced by general relativity. General relativity, which Einstein developed in 1915, takes the immutable spacetime of special relativity and breathes life into it. The single most important point of general relativity is that spacetime itself is dynamical. The geometry of spacetime evolves in response to matter. Matter tells geometry how to bend, and geometry tells matter how to move. The fixed spacetime of special relativity is revealed to be no more fixed than the fixed stars of Ptolemaic astronomy.1

  The details of exactly how geometry responds to matter and how matter responds to geometry lie within the equations of general relativity. Superficially, these equations are highly complex. There are ten of them, and they all depend on each other and have to be treated at the same time. For all but the very simplest cases, they are hard to solve exactly.

  However, properly understood, these equations describe one of the most conceptually simple of all scientific ideas.2 The gravitational response of matter to geometry in the curved spacetime of general relativity is extremely easy to state: matter always continues in the direction it is already going, following the shortest, straightest path it can find. What could be simpler? Aircraft flight paths, which appear as funny curves on two-dimensional projections of the earth, become straight lines when viewed on an actual globe. In curved spacetimes, this principle of shortness and straightness already encompasses all the motion of bodies in Newtonian gravity. Within general relativity, the inverse square force law of Newton’s law of gravity is simply a consequence of the statement that bodies travel in straight lines in curved spacetime.

  General relativity is the paradigmatic example of good theoretical physics. It is mathematically beautiful and elegant. It is conceptually simple, with no free parameters and no adjustable knobs that need tuning. It is also physically true and gives a precise description of many aspects of the universe. Its scientific virtues are completed by the fact that it is also practically useful, being essential for the correct functioning of GPS devices.

  That spacetime is dynamical – and the geometry of the universe itself is evolving – is also the foundation of modern cosmology. As we will discuss further below, observations of distant galaxies require for consistency that as time goes on, the space between the galaxies must be increasing. As time goes on, there is simply more space between the galaxies than there used to be. The overall geometry has changed and space itself has stretched.

  We give a one-dimensional analogy for this expansion. We consider two small ants on an elastic band. To start with, the band is unstretched, and the ants can walk around the band from one to another. We can imagine them as separated by a particular finite distance – say five centimetres – such that if one ant wanted to walk across to its friend it would take a particular time, for example ten seconds. We now gradually start pulling the band so that it stretches. As we do so, the separation between the ants increases. Neither ant has moved by themselves, as they have in no way exerted themselves. However, as we continue pulling on the band the ants progressively move further and further apart. The amount of time it would take one ant to travel to the other also grows: from ten seconds to thirty seconds to five minutes and more – the intermediate space has grown and expanded, and its geometry has changed.

  In this example, the change in the geometry of the band is imposed by an external agent (the person pulling on its ends). The band does not stretch dynamically by itself, but only through applied external forces. In general relativity applied to cosmology, the growth of the universe and the change in its geometry is dynamical – it is determined entirely by the sources of matter and energy within the universe.

  The dynamical nature of spacetime is also crucial to the existence of black holes. In normal circumstances the curvature of space is small. Regular matter – the earth, or even the sun – produces only small local distortions on a flat background spacetime. Why do I say small? In physics, ‘small’ and ‘large’ are always relative concepts. The sun’s gravitational force may be large by human standards, but it is perfectly described by Newton’s theory of gravitation, and ‘small’ in this context means ‘well described by Newton’. The sun is neither sufficiently dense nor sufficiently massive to affect spacetime. As an analogue for these distortions, imagine a tennis ball placed on a tight trampoline. The trampoline bends slightly but retains its shape. However, if enough matter is put into a small enough region, these distortions are no longer small. The analogy is no longer a tennis ball on a trampoline, but a beach ball filled with concrete. Spacetime is no longer similar but slightly curved, but bends in on itself beyond the point of return. Given enough matter, spacetime curves so much and so badly that it produces a region from which nothing can escape – a black hole.

  Black holes are a consequence of the dynamics of spacetime. Spacetime curves in response to matter, and if enough matter is present, spacetime curves enough to form a black hole. The densities required are large. A black hole with the mass of the earth would require all its matter to be squashed into a region nine millimetres in radius. The ‘squash radius’ grows linearly with the mass. The sun, one hundred and fifty thousand times heavier than Earth, would need to be compressed into a region around three kilometres in radius – which is indeed the typical size of the black holes that are formed as an endpoint of stellar evolution.

  Einstein’s theory of general relativity represents the highest development of classical physics. It made subtle modifications to Einstein’s previous theory of special relativity, modifications that would require several decades for decisive experimental test.

  However, it was only a few years after Einstein had constructed this theory that its classical foundations would be swept away.

  3.3 THE WORLD IS DESCRIBED BY QUANTUM MECHANICS

  I now turn to what was by a long way the most important discovery of twentieth century science. This is the fact that the world is described by quantum mechanics. No other scientific discovery, either before or since, has combined to the same degree foundational insight, philosophical significance and technological impact – not the discovery of DNA, not the laws of electromagnetism and not even the theory of evolution. A scientific outlook unenlightened by any contact with quantum mechanics is partial and limited. Such an outlook produces a view of nature that is foreshortened and darkened, attached to an understanding haunted by premodern incubi.

  The importance of quantum mechanics is threefold. First, it overthrew classical Newtonian mechanics, a theory so successful and influen
tial that the German philosopher Immanuel Kant saw it as an absolute truth, even almost a necessary one. The discovery of quantum mechanics showed that these fundamental laws of dynamics were wrong, both calculationally and conceptually – and not just for objects moving at close to the speed of light. Second, quantum mechanics has through its uncertainty principle permanently modified our ideas of what can be known, and even what it means to know something. Last and by no means least, quantum mechanics has had immense technological application. It is not only the foundation on which all of chemistry is built, but is also essential for the construction of devices ranging from lasers to semiconductors.

  Newton’s laws provided a deterministic outlook for how matter behaves. In Newton’s vision, bodies move because forces act on them. In this world objects have both a position and a velocity. After a small instant of time, the positions have changed as the bodies are moving, and the velocities have changed because there are forces acting. All the quantities one associates with an object – its position, its velocity and its energy – are also continuous quantities that can change smoothly and can take on any value.

  This vision is wrong in many ways. The ‘quantum’ in quantum mechanics refers to the fact that the smoothness no longer holds. Energies can become discrete, taking only specific, fixed values. The possible energies an electron can have within a hydrogen atom cannot change continuously – there is a discrete set of possibilities. These discrete jumps of energy are called quanta of energy. At small distances, positions and velocities cease to be good concepts: an electron in orbit around an atom cannot be said to have a position, but only a probabilistic distribution of possible positions.

  Newtonian mechanics also offered not just a dynamics but a philosophy – Newtonianism. This philosophy was admittedly not that of Newton himself, that deeply religious, semi-magical genius. However, the influence of Newton’s ideas spread widely. Newtonian mechanics is entirely deterministic. Given the initial values of the positions and velocities of particles, their future evolution is fixed. Arbitrarily precise measurements now lead to arbitrarily precise predictions for the future. If you know the positions and velocities of all particles, all aspects of the future – war, love, death – are predetermined for all time. At the point when the first humans rubbed flints together to make fire, the scorer of the winning goal in the 2018 football World Cup was already set. As Newton’s ideas were popularised in France (by among others Voltaire and his mistress Émilie du Châtelet), they produced a vision of a determinist future, where sufficiently precise measurements would allow sufficiently brilliant French mathematicians to calculate the future to arbitrary accuracy. This proud vision was eventually followed by the proverbial fall, as hubris led to nemesis. Newtonian determinism is not the way world works: in quantum mechanics, the uncertainty principle tells us that we can never know positions and velocities simultaneously. It is not just that we do not know; it is that we can never know.