Three Roads to Quantum Gravity

by Lee Smolin

  It may be that each science has one main thing to teach humanity, to help us shape our story of who we are and what we are doing here. Biology’s lesson is natural selection, as its exponents such as Richard Dawkins and Lynn Margulis have so eloquently taught us. I believe that the main lesson of relativity and quantum theory is that the world is nothing but an evolving network of relationships. I have not the eloquence to be the Dawkins or Margulis of relativity, but I do hope that after reading this book you will have come to understand that the relational picture of space and time has implications that are as radical as those of natural selection, not only for science but for our perspective on who we are and how we came to exist in this evolving universe of relations.

  Charles Darwin’s theory tells us that our existence was not inevitable, that there is no eternal order to the universe that necessarily brought us into being. We are the result of processes much more complicated and unpredictable than the small aspects of our lives and societies over which we have some control. The lesson that the world is at root a network of evolving relationships tells us that this is true to a lesser or greater extent of all things. There is no fixed, eternal frame to the universe to define what may or may not exist. There is nothing beyond the world except what we see, no background to it except its particular history.

  This relational view of space has been around as an idea for a long time. Early in the eighteenth century, the philosopher Gottfried Wilhelm Leibniz argued strongly that Newton’s physics was fatally flawed because it was based on a logically imperfect absolute view of space and time. Other philosophers and scientists, such as Ernst Mach, working in Vienna at the end of the nineteenth century, were its champions. Einstein’s theory of general relativity is a direct descendent of these views.

  A confusing aspect of this is that Einstein’s theory of general relativity can consistently describe universes that contain no matter. This might lead one to believe that the theory is not relational, because there is space but there is no matter, and there are no relationships between the matter that serve to define space. But this is wrong. The mistake is in thinking that the relationships that define space must be between material particles. We have known since the middle of the nineteenth century that the world is not composed only of particles. A contrary view, which shaped twentieth-century physics, is that the world is also composed of fields. Fields are quantities that vary continuously over space, such as electric and magnetic fields.

  The electric field is often visualized as a network of lines of force surrounding the object generating the field, as shown in Figure 1. What makes this a field is that there is a line of force passing through every point (as with a contour map, only lines at certain intervals are depicted). If we were to put a charged particle at any point in the field, it would experience a force pushing it along the field line that goes through that point.

  FIGURE 1

  The electric field lines between a positively and a negatively charged electron.

  General relativity is a theory of fields. The field involved is called the gravitational field. It is more complicated than the electric field, and is visualized as a more complicated set of field lines. It requires three sets of lines, as shown in Figure 2. We may imagine them in different colours, say red, blue and green. Because there are three sets of field lines, the gravitational field defines a network of relationships having to do with how the three sets of lines link with one another. These relationships are described in terms of, for example, how many times one of the three kinds of line knot around those of another kind.

  In fact, these relationships are all there is to the gravitational field. Two sets of field lines that link and knot in the same way define the same set of relationships, and exactly the same physical situation (an example is shown in Figure 3). This is why we call general relativity a relational theory.

  FIGURE 2

  The gravitational field is like the electric field but requires three sets of field lines to describe it.

  Points of space have no existence in themselves - the only meaning a point can have is as a name we give to a particular feature in the network of relationships between the three sets of field lines.

  This is one of the important differences between general relativity and other theories such as electromagnetism. In the theory of electric fields it is assumed that points have meaning. It makes sense to ask in which direction the field lines pass at a given point. Consequently, two sets of electric field lines that differ only in that one is moved a metre to the left, as in Figure 4, are taken to describe different physical situations. Physicists using general relativity must work in the opposite way. They cannot speak of a point, except by naming some features of the field lines that will uniquely distinguish that point. All talk in general relativity is about relationships among the field lines.

  FIGURE 3

  In a relational theory all that matters is the relationships between the field lines. These four configurations are equivalent, as in each case the two curves link in the same way.

  One might ask why we do not just fix the network of field lines, and define everything with respect to them. The reason is that the network of relationships evolves in time. Except for a small number of idealized examples which have nothing to do with the real world, in all the worlds that general relativity describes the networks of field lines are constantly changing.

  This is enough for the moment about space. Let us turn now to time. There the same lesson holds. In Newton’s theory time is assumed to have an absolute meaning. It flows, from the infinite past to the infinite future, the same everywhere in the universe, without any relation to things that actually happen. Change is measured in units of time, but time is assumed to have a meaning and existence that transcends any particular process of change in the universe.

  FIGURE 4

  In a non-relational theory it matters also where the field lines are in absolute space.

  In the twentieth century we learned that this view of time is as incorrect as Newton’s view of absolute space. We now know that time also has no absolute meaning. There is no time apart from change. There is no such thing as a clock outside the network of changing relationships. So one cannot ask a question such as how fast, in an absolute sense, something is changing: one can only compare how fast one thing is happening with the rate of some other process. Time is described only in terms of change in the network of relationships that describes space.

  This means that it is absurd in general relativity to speak of a universe in which nothing happens. Time is nothing but a measure of change - it has no other meaning. Neither space nor time has any existence outside the system of evolving relationships that comprises the universe. Physicists refer to this feature of general relativity as background independence. By this we mean that there is no fixed background, or stage, that remains fixed for all time. In contrast, a theory such as Newtonian mechanics or electromagnetism is background dependent because it assumes that there exists a fixed, unchanging background that provides the ultimate answer to all questions about where and when.

  One reason why it has taken so long to construct a quantum theory of gravity is that all previous quantum theories were background dependent. It proved rather challenging to construct a background independent quantum theory, in which the mathematical structure of the quantum theory made no mention of points, except when identified through networks of relationships. The problem of how to construct a quantum theoretic description of a world in which space and time are nothing but networks of relationships was solved over the last 15 years of the twentieth century. The theory that resulted is loop quantum gravity, which is one of our three roads. I shall describe what it has taught us in Chapter 10. Before we get there, we shall have to explore other implications of the principle that there is nothing outside the universe.

  CHAPTER 2

  IN THE FUTURE WE SHALL KNOW MORE

  One of the things that cannot exist outside the universe is ourselves. Thi
s is obviously true, but let us consider the consequences. In science we are used to the idea that the observers must remove themselves from the system they study, otherwise they are part of it and cannot have a completely objective point of view. Also, their actions and the choices they make are likely to affect the system itself, which means that their presence may contaminate their understanding of the system.

  For this reason we try as often as we can to study systems in which a clean boundary can be drawn separating the system under study from the observer. That we can do this in physics and astronomy is one of the reasons why those sciences are said to be ‘harder’. They are held to be more objective and more reliable than the social sciences because in physics and astronomy there seems to be no difficulty with removing the observer from the system. In the ‘softer’ social sciences there is no way around the fact that the scientists themselves are participants in the societies they study. Of course, it is possible to try to minimize the effects of this and, for better or worse, much of the methodology of the social sciences is based on the belief that the more one can remove the observer from the system, the more scientific one is being.

  This is all well and good when the system in question can be isolated, say in a vacuum chamber or a test tube. But what if the system we want to understand is the whole universe? We do live in the universe, so we need to ask whether the fact that cosmologists are part of the system they are studying is going to cause problems. It turns out that it does, and this leads to what is probably the most challenging and confusing aspect of the quantum theory of gravity.

  Actually, part of the problem has nothing to do with quantum theory, but comes from putting together two of the most important discoveries of the early twentieth century. The first is that nothing can travel faster than light; the second is that the universe seems to have been created a finite time ago. Current estimates put this time at about 14 billion years, but the exact number is not important. Together, the two things mean that we cannot see the whole universe. We can see only the contents of a region that extends around us to about 14 billion light years - the distance light could travel in this time. This means that science cannot, in principle, provide the answer to any question we might ask. There is no way to find out, for example, how many cats there are in the universe, or even how many galaxies there are. The problem is very simple: no observer inside the universe can see all of what is in the universe. We on Earth cannot receive light from any galaxy, or any cat, more than 14 billion or so light years from us. So if someone asserts that there are exactly 212,400,000,043 more cats in the universe than can be seen from Earth, no investigation we can do can prove them right or wrong.

  However, the universe is quite likely to be much larger than 14 billion light years across. Why this is so would take us too far afield, but let me say simply that we have yet to find any evidence of the universe either ending or closing in on itself. There is no feature in what we can see that suggests that it is not just a small fraction of what exists. But if this is so, then even with perfect telescopes we would be able to see only a small part of all that exists.

  Since the time of Aristotle, mathematicians and philosophers have investigated the subject of logic. Their aim has been to establish the laws by which we reason. And ever since its beginnings, logic has assumed that every statement is either true or false. Once this is assumed, it is possible to deduce true statements from other true statements. Unfortunately, this kind of logic is completely inapplicable when it comes to making deductions about the whole universe. Suppose we count all the cats in the region of the universe that we can see, and the number comes to one trillion. This is a statement whose truth we can establish. But what of a statement such as, ‘Fourteen billion years after the big bang, there are a hundred trillion cats in the whole universe’? This may be true or false, but we observers on Earth have absolutely no way of determining which. There may be no cats farther than 14 billion light years from us, there may be 99 trillion or there may be an infinite number. Although these are all assertions that we can state, we cannot decide whether they are true or false. Nor can any other observer establish the truth of any claim as to the number of cats in the universe. Since it takes only about four billion years for cats to evolve on a planet, no observer could know whether cats have evolved in some region of space so far away from her that light reflected from their mysterious eyes could not have yet reached her.

  However, classical logic demands that every statement be either true or false. Classical logic is therefore not a description of how we reason. Classical logic could be applied only by some being outside the universe, a being who could see the whole cosmos and count all its cats. But, if we insist on our principle that there is nothing outside the universe, there is no such being. To do cosmology, then, we need a different form of logic - one that does not assume that every statement can be judged true or false. In this kind of logic, the statements an observer can make about the universe are divided into at least three groups: those that we can judge to be true, those that we can judge to be false and those whose truth we cannot decide upon at the present time.

  According to the classical view of logic, the question of whether a statement can be judged to be true or false is something absolute - it depends only on the statement and not on the observer doing the judging. But it is easy to see that this is not true in our universe, and the reason is closely related to what we have just said. Not only can an individual observer only see light from one part of the universe; which part they can see depends on where they find themselves in the history of the universe. We can judge the truth or falseness of statements about the Spice Girls. But observers who live more than 14 billion years from us cannot because they will not have received any information that would even let them suspect the existence of such a phenomenon. So we must conclude that the ability to judge whether a statement is true or false depends to some extent on the relationship between the observer and the subject of the statement.

  FIGURE 5

  Observers in the future will be able to see more of the universe than we can see now. The diagonal lines represent the paths of light rays travelling from the past to us. Since nothing can travel faster than light, anything in our past that we can see or experience any effect of must lie within the triangle completed by the two diagonal lines. In the future we shall be able to receive light from farther away, and therefore see farther.

  Furthermore, an observer who lives on Earth a billion years from now will be able to see much more of the universe, for they will be able to see 15 billion light years out into the universe rather than the 14 billion light years we can see. They will see everything we can see, but they will see much more because they will see farther (Figure 5). They may be able to see many more cats. So, the list of statements they can judge to be true or false includes all that we can judge, but it is longer. Or consider an observer who lives 14 billion years after the big bang, as we do, but is 100 billion light years from us. Many cosmologists argue that the universe is at least 100 billion light years across; if they are right there is no reason for there not to be intelligent observers at that distance from us. But the part of the universe that they see has no overlap with the part of the universe that we see. The list of statements they can judge to be true or false is thus completely different from the list of statements that we here on Earth can judge to be true or false. If there is a logic that applies to cosmology, it must therefore be constructed so that which statements can be judged to be true or false depends on the observer. Unlike classical logic, which assumes that all observers can decide the truth or falsity of all statements, this logic must be observer-dependent.

  In the history of physics it has often happened that by the time the physicists have been able to understand the need for a new mathematics, they found that the mathematicians had got there first and had already invented it. This is what happened with the mathematics needed for quantum theory and relativity and it has happened here as well. For re
asons of their own, during the twentieth century mathematicians investigated a whole collection of alternatives to the standard logic we learned in school. Among them is a form of logic which we may call ‘logic for the working cosmologist’, for it incorporates all the features we have just described. It acknowledges the fact that reasoning about the world is done by observers inside the world, each of whom has limited and partial information about the world, gained from what they can observe by looking around them. The result is that statements can be not only true or false; they can also carry labels such as ‘we can’t tell now whether it’s true, but we might be able to in the future’. This cosmological logic is also intrinsically observer-dependent, for it acknowledges that each observer in the world sees a different part of it.

  The mathematicians, it seems, were not aware that they were inventing the right form of logic for cosmology, so they called it other names. In its first forms it was called ‘intuitionistic logic’. More sophisticated versions which have been studied more recently are known collectively as ‘topos theory’. As a mathematical formalism, topos theory is not easy. It is perhaps the hardest mathematical subject I’ve yet encountered. All of what I know of it comes from Fotini Markopoulou-Kalamara, who discovered that cosmology requires non-standard logic and found that topos theory was right for it. But the basic themes of it are obvious, for they describe our real situation in the world, and not only as cosmologists. Here in the real world, we almost always reason with incomplete information. Each day we encounter statements whose truth or falsity cannot be decided on the basis of what we know. And in the forms of our social and political life we recognize, often explicitly, that different observers have access to different information. We also deal every day with the fact that the truth or falsity of statements about the future may be affected by what we choose to do.