Three Roads to Quantum Gravity

by Lee Smolin

  What we already know about quantum gravity suggests that the second possibility is right. The apparent smoothness of space and time are illusions; behind them is a world composed of discrete sets of events, which can be counted. Different approaches give us different pieces of evidence for this conclusion, but they all agree that if we looks finely enough at our world the continuity of space and time will dissolve as surely as the smoothness of material gives way to the discrete world of molecules and atoms.

  The different approaches also agree about how far down we have to probe the world before we come to the elementary events. The scales of time and distance on which the discrete structure of the world becomes manifest is called the Planck scale. It is defined as the scale at which the effects of gravity and quantum phenomena will be equally important. For larger things, we can happily forget about quantum theory and relativity. But when we get down to the Planck scale we have no choice but to take it all into account. To describe the universe at this scale we need the quantum theory of gravity.

  The Planck scale can be established in terms of known fundamental principles. It is calculated by putting together in appropriate combinations the constants that come into the fundamental laws. These are Planck’s constant, from quantum theory; the speed of light, from special relativity; and the gravitational constant, from Newton’s law of gravitation. In terms of the Planck scale, we are absolutely huge. The Planck length is 10-33 centimetres, which is 20 orders of magnitude smaller than an atomic nucleus. On the scale of the fundamental time, everything we experience is incredibly slow. The Planck time, which must be roughly the time it takes for something truly fundamental to happen, is 10-43 of a second. That is, the quickest thing we can experience still takes more than 1040 fundamental moments. A blink of an eye has more fundamental moments than there are atoms in Mount Everest. Even the fastest collision ever observed between two elementary particles fills more elementary moments than there are neurons in the brains of all the people now alive. It is hard to avoid the conclusion that everything we observe may still be incredibly complicated on the fundamental Planck scale.

  We can go on like this. There is a fundamental Planck temperature, which is likely to be the hottest anything can get. Compared with it, everything in our experience, even the interiors of stars, is barely above absolute zero. This means that, in terms of fundamental things the universe we observe is frozen. We begin to get the feeling that we know as much about nature and its potential phenomena as a penguin knows of the effects of forest fire, or of nuclear fusion. This is not just an analogy - it is our real situation. We know that all materials melt when raised to a high enough temperature. If a region of the world were raised to the Planck temperature, the very structure of the geometry of space would melt. The only hope we have of experiencing such an event is by peering into our past, for what is usually called the big bang is, in fundamental terms, the big freeze. What caused our world to exist was probably not so much an explosion as an event that caused a region of the universe to cool drastically and freeze. To understand space and time in their natural terms, we have to imagine what was there before everything around us froze.

  So, our world is incredibly big, slow and cold compared with the fundamental world. Our job is to remove the prejudices and blinkers imposed by our parochial perspective and imagine space and time in their own terms, on their natural scale. We do have a very powerful toolkit that enables us to do this, consisting of the theories we have so far developed. We must take the theories that we trust the most, and tune them as best we can to give us a picture of the Planck scale. The story I am telling in this book is based on what we have learned by doing this.

  In the earlier chapters I argued that our world cannot be understood as a collection of independent entities living in a fixed, static background of space and time. Instead, it is a network of relationships the properties of every part of which are determined by its relationships to the other parts. In this chapter we have learned that the relations that make up the world are causal relations. This means that the world is not made of stuff, but of processes by which things happen. Elementary particles are not static objects just sitting there, but processes carrying little bits of information between events at which they interact, giving rise to new processes. They are much more like the elementary operations in a computer than the traditional picture of an eternal atom.

  We are very used to imagining that we see a three-dimensional world when we look around ourselves. But is this really true? If we keep in mind that what we see is the result of photons impinging on our eyes, it is possible to imagine our view of the world in a quite different way. Look around and imagine that you see each object as a consequence of photons having just travelled from it to you. Each object you see is the result of a process by which information travelled to you in the shape of a collection of photons. The farther away the object is, the longer it took the photons to travel to you. So when you look around you do not see space - instead, you are looking back through the history of the universe. What you are seeing is a slice through the history of the world. Everything you see is a bit of information brought to you by a process which is a small part of that history.

  The whole history of the world is then nothing but the story of huge numbers of these processes, whose relationships are continually evolving. We cannot understand the world we see around us as something static. We must see it as something created, and under continual recreation, by an enormous number of processes acting together. The world we see around us is the collective result of all those processes. I hope this doesn’t seem too mystical. If I have written this book well then, by the end of it, you may see that the analogy between the history of the universe and the flow of information in a computer is the most rational, scientific analogy I could make. What is mystical is the picture of the world as existing in an eternal three-dimensional space, extending in all directions as far as the mind can imagine. The idea of space going on and on for ever has nothing to do with what we see. When we look out, we are looking back in time through the history of the universe, and after not too long we come to the big bang. Before that there may be nothing to see - or, at the very least, if there is something it will most likely look nothing like a world suspended in a static three-dimensional space. When we imagine we are seeing into an infinite three-dimensional space, we are falling for a fallacy in which we substitute what we actually see for an intellectual construct. This is not only a mystical vision, it is wrong.

  II

  WHAT WE HAVE LEARNED

  CHAPTER 5

  BLACK HOLES AND HIDDEN REGIONS

  In the cultural iconography of our time, black holes have become mythic objects. In science fiction novels and films they often evoke images of death and transcendence, recalling the irreversibility of certain passages and the promise of our eventual emergence into a new universe. I am not a very good actor, but I was once asked by a friend, the director Madeline Schwartzman, to act in one of her films. Luckily I got to play a physics professor giving a lecture on black holes. In the film, called Soma Sema, the myth of Orpheus is merged with two major scientific and technological themes of our time: total nuclear war and black holes. Orpheus, my student, seeks through her music to be an exception to all three versions of the irreversible.

  Among those of us who think about space and time professionally, black holes play a central role. A whole subculture of astronomers is devoted to understanding how they form and how to find them. By now, dozens of candidate black holes have been observed. But what is most exciting is that there are probably vast numbers of them out there. Many if not most galaxies, including our own, seem to have an enormous black hole at their centre, with a mass millions of times that of our Sun. And there is evidence, both observational and theoretical, that a small fraction of stars end their lives as black holes. A typical galaxy such as ours could well contain tens or even hundreds of millions of these stellar black holes. So black holes are out there, and interstella
r travellers of the far future will have to be careful to avoid them. But beyond the fascination they hold for astronomers, black holes are important to science for other reasons. They are a central object of study for those of us who work on quantum gravity. In a sense, black holes are microscopes of infinite power which make it possible for us to see the physics that operates on the Planck scale.

  Because they feature prominently in popular culture, almost everyone knows roughly what a black hole is. It is a place where gravity is so strong that the velocity required to escape from it is greater than the speed of light. So no light can emerge from it, and neither can anything else. We can understand this in terms of the notion of causal structure we introduced in the last chapter. A black hole contains a great concentration of mass that causes the light cones to tip over so far that the light moving away from the black hole actually gets no farther from it (Figure 13). So the surface of a black hole is like a one-way mirror: light moving towards it can pass into it, but no light can escape from it. For this reason the surface of a black hole is called the horizon. It is the limit of what observers outside the black hole can see.

  I should emphasize that the horizon is not the surface of the object that formed the black hole. Rather it is the boundary of the region that is capable of sending light out into the universe. Light emitted by any body inside the horizon is trapped and cannot get any farther than the horizon. The object that formed the black hole is rapidly compressed, and according to general relativity it quickly reaches infinite density.

  Behind the horizon of a black hole is a part of the universe made up of causal processes that go on, in spite of the fact that we receive no information from them. Such a region is called a hidden region. There are at least a billion billion black holes in the universe, so there are quite a lot of hidden regions that are invisible to us, or to any other observer. Whether a region is hidden or not depends in part on the observer. An observer who falls into a black hole will see things that her friends who stay outside will never see. In Chapter 2 we found that different observers may see different parts of the universe in their past. The existence of black holes means that this is not just a question of waiting long enough for light from a distant region to reach us. We could be right next to a black hole, yet never be able to see what observers inside it can see, however long we waited.

  FIGURE 13

  Light cones in the vicinity of a black hole. The solid black line is the singularity where the gravitational field is infinitely strong. The dotted lines are the horizons, consisting of light rays that stay the same distance from the singularity. Light cones just at the horizon are tilted to show that a light ray trying to move away from the black hole just stays at the same distance and travels along the horizon. A light cone inside the horizon is tilted so far that any motion into the future brings one closer to the singularity.

  All observers have their own hidden region. The hidden region of each observer consists of all those events that they will not be able to receive information from, no matter how long they wait. Each hidden region will include the interiors of all the black holes in the universe, but there may be other regions hidden as well. For example, if the rate at which the universe expands increases with time, there will be regions of the universe from which we shall never receive light signals, no matter how long we wait. A photon from such a region may be travelling in our direction at the speed of light, but because of the increase in the rate of the expansion of the universe it will always have more distance to travel towards us than it has travelled so far. As long as the expansion continues to accelerate, the photon will never reach us. Unlike black holes, the hidden regions produced by the acceleration of the expansion of the universe depend on the history of each observer. For each observer there is a hidden region, but they are different for different observers.

  This raises an interesting philosophical point, because objectivity is usually assumed to be connected with observer independence. It is commonly assumed that anything that is observer dependent is subjective, meaning that it is not quite real. But the belief that observer dependence rules out objectivity is a residue of an older philosophy, usually associated with the name of Plato, according to which truth resides not in our world but in an imaginary world consisting of all ideas which are eternally true. According to this philosophy, anybody could have access to any truth about the world, because the process of finding truth was held to be akin to a process of remembering, rather than observing. This philosophy is hard to square with Einstein’s general theory of relativity because, in a universe defined by that theory, something may be both objectively true and at the same time knowable only by some observers and not others. So ‘objectivity’ is not the same as ‘knowable by all’. A weaker, less stringent interpretation is required: that all those observers who are in a position to ascertain the truth or falsity of some observation should agree with one another.

  The hidden region of any observer has a boundary that divides the part of the universe they can see from the part they cannot. As with a black hole, this boundary is called the horizon. Like the invisible regions, horizons are observer dependent concepts. For any observer who remains outside it, a black hole has a horizon - the surface that divides the region from which light cannot escape from the rest of the universe. Light leaving a point just inside the horizon of the black hole will be pulled inexorably into the interior; light just outside the horizon will be able to escape (Figures 13 and 14). Although the horizon of a black hole is an observer dependent concept, there are a large number of observers who share that horizon: all those who are outside that black hole. So the horizon of a black hole is an objective property. But it is not a horizon for all observers, because any observer who falls through it will be able to see inside. And an observer who crosses the horizon of a black hole will become invisible to observers who remain outside.

  FIGURE 14

  The paths of three light rays that move away from the singularity. They start just inside, outside and just at the horizon.

  It helps to know that horizons are themselves surfaces of light. They are made up of those light rays that just fail to reach the observer (Figure 14). The horizon of a black hole is a surface of light that has begun to move outwards from the black hole but, because of the black hole’s gravitational field, fails to get any farther from its centre. Think of the horizon as a curtain made of photons. Photons leaving from any point just inside the horizon are drawn inwards, even if they were initially moving away from the centre of the black hole.

  On the other hand, a photon that starts just outside the horizon of a black hole will reach us, but it will be delayed because light cones near the horizon are tilted almost so far that no light can escape. The closer to the horizon the photon starts, the longer will be the delay. The horizon is the point where the delay becomes infinite - a photon released there never reaches us.

  This has the following interesting consequence. Imagine that we are floating some distance from a black hole. We drop a clock into the black hole, which sends us a pulse of light every thousandth of a second. We receive the signal and convert it to sound. At first we hear the signal as a high-pitched tone, as we receive the signals at a frequency of a thousand times a second. But as the clock nears the horizon of the black hole, each signal is delayed more and more by the fact that it takes a little more time for each successive pulse to arrive. So the tone we hear falls in pitch as the clock nears the horizon. Just as the clock crosses the horizon, the pitch falls to zero, and after that we hear nothing.

  This means that the frequency of light is decreased by its having to climb out from the region near the horizon. This can also be understood from quantum theory, as the frequency of light is proportional to its energy, and, just as it takes us energy to climb a flight of stairs, it takes a certain amount of energy for the photon to climb up to us from its starting point just outside the black hole. The closer to the horizon the photon begins its flight, the more energy it must give up
as it travels to us. So the closer to the horizon it starts, the more its frequency will have decreased by the time it reaches us. Another consequence is that the wavelength of the light is lengthened as the frequency is decreased. This is because the wavelength of light is always inversely proportional to its frequency. As a result, if the frequency is diminished, the wavelength must be increased by the same factor.

  But this means that the black hole is acting as a kind of microscope. It is not an ordinary microscope, as it does not act by enlarging images of objects. Rather, it acts by stretching wavelengths of light. But nevertheless, this is very useful to us. For suppose that at very short distances space has a different nature than the space we see looking around at ordinary scales. Space would then look very different from the simple three-dimensional Euclidean geometry that seems to suffice to describe the immediately perceptible world. There are various possibilities, and we shall be discussing these in later chapters. Space may be discrete, which means that geometry comes in bits of a certain absolute size. Or there may be quantum uncertainty in the very geometry of space. Just as electrons cannot be localized at precise points in the atom, but are forever dancing around the nucleus, the geometry of space may itself be dancing and fluctuating.

  Ordinarily we cannot see what is happening on very small length scales. The reason is that we cannot use light to look at something which is smaller than the wavelength of that light. If we use ordinary light, even the best microscope will not resolve any object smaller than a few thousand times the diameter of an atom, which is the wavelength of the visible part of the spectrum of light. To see smaller objects we can use ultraviolet light, but no microscope in existence, not even one that uses electrons or protons in place of light, can come anywhere near the resolution required to see the quantum structure of space.