by Lee Smolin
Argument for the Bekenstein bound
Let us suppose that The Thing is big enough to be described both in terms of an exact quantum description and in terms of an averaged, macroscopic description. We shall argue by contradiction, which means that we first assume the opposite of what we are trying to show. Thus we assume that the amount of information required to describe The Thing is much larger than the area of The Screen. For simplicity, we assume that The Screen is spherical.
We know that The Thing is not a black hole, because we know that the entropy of any black hole that can fit into The Screen must be equivalent to an area less than that of the screen. But in this case its entropy must be lower than the area of the screen, in Planck units. If we assume that the entropy of a black hole counts the number of its possible quantum states, this is much less than the information contained in The Thing.
It then follows (from a theorem of classical general relativity) that The Thing has less energy than a black hole that would just fit inside The Screen. Now, we can slowly add energy to The Thing by dripping it slowly through the screen. We shall reach some point by which we shall have given it so much energy that, by that same theorem, it must collapse to a black hole. But then we know that its entropy is given by one-quarter of the area of the screen. Since that is lower than the entropy of The Thing initially, we have managed to lower the entropy of a system. This contradicts the second law of thermodynamics.
We dripped the energy in slowly to ensure that nothing surprising happens outside The Screen which might increase the entropy strongly somewhere else. There seem to be no loopholes in this argument. Therefore, if we believe the second law of thermodynamics, we must believe that the most entropy that we, outside the Screen, can attribute to The Thing is one-quarter of the area of The Screen. And because entropy is a count of answers to yes/no questions, this implies the Bekenstein bound as we have stated it.
Another reason to believe the Bekenstein bound is that it can be derived directly from loop quantum gravity. To do this one only has to study the problem of how a screen is described by the quantum theory. As shown in Figure 40, in loop quantum gravity a screen will be pierced by edges of a spin network. Each edge that intersects the screen contributes to the total area of the screen. It turns out that each edge that is added also increases the amount of information that can be stored in a quantum theoretic description of the screen. We can add more edges, but the information a screen can store cannot increase faster than its area. This is just what is required by the Bekenstein bound.
Perhaps the first person to realize the radical implications of the Bekenstein bound was Louis Crane. He deduced from it that quantum cosmology must be a theory of the information exchanged between subsystems of the universe, rather than a theory of how the universe would look to an outside observer. This was the first step towards the relational theories of quantum cosmologies later developed by Carlo Rovelli, Fotini Markopoulou and myself. Gerard ’t Hooft later began to think about the horizon of a black hole as something like a computer, along the lines I have described. He proposed the first version of the holographic principle and gave it its name. It was then quickly championed by Leonard Susskind, who showed how it could be applied to string theory. Since then at least two other versions of the holographic principle have been proposed. So far there is no consensus on which is right. I shall explain two of the versions, which are called the strong and weak holographic principles.
The idea of the strong holographic principle is very simple. Since the observer is restricted to examining The Thing by making observations through The Screen, all of what is observed could be accounted for if one imagined that, instead of The Thing, there was some physical system defined on the screen itself (Figure 41). This system would be described by a theory which involved only The Screen. This ‘screen theory’ might describe The Screen as something like a quantum computer, with one bit of memory for every pixel, each pixel being 2 Planck lengths on each side. Now suppose that the observer sends some signal through The Screen, which interacts with The Thing. The result is a signal which comes back out through The Screen. As far as the observer is concerned, the same thing would happen if the light interacted with the quantum computer on The Screen and returned a suitable signal. The point is that there is no way for the observer to tell if they were interacting with The Thing itself, or merely with its image, represented as a state of the screen theory. If the screen theory were suitably chosen, or the computer representing the information on the screen suitably programmed, the laws of physics holding inside the screen could equally well be represented by the response of The Screen to the observer.
FIGURE 41
The Screen is like a television set with pixels measuring 2 Planck lengths on each side. One can only see as much information about the world beyond The Screen as can be represented on it.
In this form, the holographic principle states that the most succinct description that can be given of the part of the world that lies on the other side of any surface is actually a description of how its image evolves on that surface. This might seem weird, but the important thing is the way it relies on the Bekenstein bound. The Screen description is adequate because no more information can be gained about The Thing than can ever be represented by the state of the pixels on The Screen. The strong form of the holographic principle says that the world is such that the physical description of any object in nature can equally well be represented by the state of such a computer, imagined to exist on a surface surrounding it. That is, for every set of true laws that might hold inside The Screen, there is a way to program the computer representing the screen theory so that it reproduces all true predictions of those laws.
This is weird enough, but it does not go as far as it might. The problem is that it describes the world in terms of things. But remember, in Chapter 4 I argued that when we get down to the fundamental theory there will be no things, only processes. If we believe this, we cannot believe in any principle which expresses the world in terms of things. We should reformulate the principle so that it makes references only to processes. This is what the weak holographic principle does. It states that we are mistaken to think that the world consists of Things that occupy regions of space. Instead, all that there exists in the world are Screens, on which the world is represented. That is, it does not posit that there are two things, bulky things, and images or representations of them on their surfaces. It posits that there is only one kind of thing - representations by which one set of events in the history of the universe receives information about other parts of the world.
In such a world, nothing exists except processes by which information is conveyed from one part of the world to another. And the area of a screen - indeed, the area of any surface in space - is really nothing but the capacity of that surface as a channel for information. So, according to the weak holographic principle, space is nothing but a way of talking about all the different channels of communication that allow information to pass from observer to observer. And geometry, as measured in terms of area and volume, is nothing but a measure of the capacity of these screens to transmit information.
This more radical version of the holographic principle is based on the ideas introduced in Chapters 2 and 3. It relies strongly on the idea that the universe cannot be described from the point of view of an observer who exists somehow outside of it. Instead there are many partial viewpoints, where observers may receive information from their pasts. According to the holographic principle, geometrical quantities such as the areas of surfaces have their origins in measuring the flow of information to observers inside the universe.
Thus, it is not enough to say that the world is a hologram. The world must be a network of holograms, each of which contains coded within it information about the relationships between the others. In short, the holographic principle is the ultimate realization of the notion that the world is a network of relationships. Those relationships are revealed by this new principle to involve nothi
ng but information. Any element in this network is nothing but a partial realization of the relationships between the other elements. In the end, perhaps, the history of a universe is nothing but the flow of information.
The holographic principle is still a new and very controversial idea. But for the first time in the history of quantum gravity we have in our hands an idea which at first seems too crazy to be true, but which survives all our attempts to disprove it. Whatever version of it finally turns out to be the true one, it is an idea which seems to be required by what we understand so far about quantum gravity. But it is also the kind of idea which will make it quite impossible, if it is ever accepted, to go back to any previous theory that did without it. The uncertainty principle of quantum theory and Einstein’s equivalence principle were also ideas of this type. They contradicted the principles of older theories and, at first, just barely seemed to make sense. Just like them, the holographic principle is the kind of idea one hopes to run into just as one is turning the corner to a new universe.
CHAPTER 13
HOW TO WEAVE A STRING
Perhaps the main reason why some physicists do not get very excited about loop quantum gravity is that, although it succeeds very well in describing how the geometry of space must look on the Planck scale, it is basically pretty boring. There are no new principles involved. To set up the theory we just put in the basic principles of quantum theory and relativity. We get a lot out that is new and could even be tested experimentally. But it is perhaps not so surprising that when geometry is treated quantum theoretically it behaves like a quantum theoretic system. Things that used to be continuous, such as the range of possible volumes a space could have, now become discrete. The main lesson is that we really can treat space and time in a background independent way, and see them as a nothing but a network of relationships. This is good, but this is also what the principles we put in demanded. That it works is a good consistency check, but we should not consider it either surprising or revolutionary. The main strength of this approach, its simplicity and transparency, is perhaps also its main weakness.
String theory is just the opposite. We start not with basic principles, but by contradicting the thing we feel most certain about quantum gravity - that it must be a background independent theory. We ignore this, and search for a theory of gravitons and other particles moving against a background of empty space; and, by trial and error, we find it. Our guiding principle is to find something that works. To do this we have to change the rules, not once but over and over again. There are not particles, there are strings. There are not three dimensions of space but nine. There are extra symmetries. String theory is unique. Actually, it is not quite unique - it comes in an enormous number of versions. And in fact there are not just strings, but membranes of many different dimensions. And there are not nine dimensions, but ten. And so on. String theory has been nothing but a series of surprises, one after the other. We put in no principles - all we put in is the desire for a theory of gravitons that makes sense. And we get out a long list of unexpected facts, a whole new world to be explored.
For more than ten years, from about 1984 to 1996, these two theories of quantum gravity were developed by two different groups of people completely independently. Each group was successful in solving the problems it set for itself. Although we listened to each other’s talks, and maintained friendships formed before the split, it must be said that almost everyone thought that their group was on the right path and the others were misguided. To each group it was obvious why the other could not succeed. The loop guys (and gals) said to the string guys, ‘Your theory is not background independent, it cannot be a real quantum theory of space and time. Only we know how to make a successful background independent theory.’ The string guys said to the loop guys, ‘Your theory does not give a consistent description of the interactions between gravitons and other particles. Only our theory describes a consistent unification of gravity with the other interactions.’ I am ashamed to admit that few in either community rose to the challenge. During this whole period, for example, there was not a single person who worked on both theories. Many seemed to make the understandable mistake of confusing the solution of part of the problem of quantum gravity with the solution of the whole problem.
Many misunderstandings have resulted. I have had the experience more than once of sitting next to someone from one camp listening to a talk by someone from the other. The person next to me would get very agitated: “That young person is so arrogant, they claim they have solved everything!” In fact the speaker had given a very measured presentation full of careful qualifications and caveats and had not made a single claim that went beyond what they had done. The problem is that such qualifications have to be presented in the terminology specific to the theory, and the person next to me, from the opposing theory’s camp, was unable to follow it. This has happened to me in both directions. Even now, one can go to a conference and find that string theory and loop quantum gravity are the subjects of separate parallel sessions. The fact that the same problems are being addressed in the two sessions is noticed only by the small handful of us who do our best to be in both rooms.
There are many remarkable aspects of this situation, including the fact that almost every one of these people is quite sincere. Just as the existence of Moslems does not deter some Christians from the sincere conviction that theirs is the one true religion, and vice versa, there are many string theorists and many loop quantum gravity people who do not seem to be troubled by the existence of a whole community of equally sincere and smart people who pursue a different approach to the problem they are spending their lives attacking.
But this is a problem not of science but of the sociology of the academy. Sometimes, rushing from the loop room to the string room and back again, I have wondered what would have happened had physics in the seventeenth century been carried out in the same sociological context as present-day science. So let us wind back time and consider an alternative history of science. By 1630 there would have been two large groups of natural philosophers working on the successor to Aristotelian science. At conferences they would have divided into two parallel sessions with, as today, little overlap. In one room would be those who thought that falling bodies provided the key to the new physics. They would spend their time in profound reflections on the motion of bodies on the Earth. They would launch projectiles, experiment with pendulums and roll balls down inclined planes. Each of them would have their own personal version of the theory of falling bodies, but they would be united by the conviction that no theory could succeed that did not incorporate the deep principle discovered by Galileo that objects fall with a constant acceleration. They would be unconcerned with the motion of planets, because they would see nothing to disagree with the old and profoundly beautiful idea that planets move in circular orbits.
Two floors above them there would be a larger room where the ellipse theorists met. They would spend their time studying the orbits of planets, both in the real solar system and in imagined worlds of various dimensions. For them the key principle would be the great discovery by Kepler that planets move on elliptical orbits. They would be quite unconcerned with how bodies fall on Earth because they would share the view that only in the heavens could one see the true symmetries behind the world, uncontaminated by the complexities of the Earth, where so many bodies pushed on one another as they sought the centre. In any case they would be convinced that all motion, including that on Earth, must in the end reduce to complicated combinations of ellipses. They would assure sceptics that it was not yet time to study such problems, but when the time came they would have no problem explaining falling bodies in terms of the theory of ellipses.
Instead, they would focus their attention on the recent discovery of D-planets, which would have been found to follow parabolas rather than ellipses. So the definition of ellipse theory would be extended to include parabolas and other such curves such as hyperbolas. There would even be a conjecture that all the diff
erent orbits could be unified under one common theory, called C-theory. However, there was no agreed set of principles for C-theory, and most work on the subject required new mathematics that most physicists could not follow.
Meanwhile, another new form of mathematics was being invented by a brilliant mathematician and philosopher in Paris, René Descartes. He propounded a third theory, in which planetary orbits have to do with vortices.
It is true that while Galileo and Kepler did correspond, each seemed to show little interest in the key discoveries of the other. They wrote to each other about the telescope and what it revealed, but Galileo seems never to have mentioned ellipses, and to have gone to his grave believing the planetary orbits were circles. Nor is there any evidence that Kepler ever thought about falling bodies or believed them to be relevant to explaining the motions of the planets. It took a young scientist of a later generation, Isaac Newton, born the year of Galileo’s death, to wonder whether the same force that made apples fall drew the Moon to the Earth and the planets to the Sun. So, while my story is fanciful, it really did happen that scientists with the stature of Galileo and Kepler each contributed an essential ingredient to a scientific revolution while remaining almost ignorant of and apparently uninterested in each other’s discoveries.
We can hope that it will take less time to bring the different pieces of the quantum theory of gravity together than it did for someone to see the relationship between the work of Kepler and Galileo. The simple reason is that there are many more scientists working now than there were then. Whereas Kepler and Galileo might each have complained, if asked, that they were too busy to look at what the other was doing, there are now plenty of people to share the work. However, there is now the problem of making sure that young people have the freedom to wander across boundaries established by their elders without fear of jeopardizing their careers. It would be naive to say this is not a significant issue. In many areas of science we are paying for the consequences of an academic system that rewards narrowness of focus over exploration of new areas. This underlines the fact that good science is, and will always be, as much a question of judgement and character as it is a question of cleverness.