The End of Time: The Next Revolution in Physics

CHAPTER 7

Paths in Platonia

NATURE AND EXPLORATION

The two-and-a-bit puzzle is the statement that two snapshots of a dynamical system are nearly but not quite sufficient to predict its entire history. We need to know not only two snapshots, but also their separation in time and their relative orientation in absolute space. These are exactly the things that determine the energy and angular momentum of any system, both of which, as we have seen, have a profound influence on its behaviour.

There are two different ways to approach this problem. Either we assume the known laws of nature are correct and simply ask how they can be verified, or we take a more ambitious stance and ask if they arise from some deeper level that we have not yet comprehended. The latter is the approach of this chapter. We shall forget absolute space and time and take Platonia for real. I have likened it to a country; countries are there to be explored. In exploring a country, one follows a path through it. Any continuous curve through Platonia is such a path. A natural question is whether some paths are distinguished compared with others. It leads directly to the idea of optimization.

Optimization problems arise naturally, and were already well known to mathematicians in antiquity. It seems they were also known and understood by Queen Dido, who when she came to North Africa was granted as much land as she could enclose within the hide of a cow. She cut it into thin strips, out of which she made a long string. Her task was then to enclose the maximum area of land within it. The solution to this problem of maximizing the area within a figure of given perimeter is a circle. However, Dido’s territory was to adjoin the coast, which did not count as part of the perimeter. For a straight coastline the solution to this problem is a semicircle, and this was said to be the origin of the territory of Carthage. A rich body of mathematical and physical theory has developed out of similar problems. It cannot explain why the universe is, but given that the universe does exist it goes a long way to explain why it is as it is and not otherwise.

In early modern times, Pierre de Fermat (of the famous last theorem) developed a particularly fruitful idea due to Hero of Alexandria, who had sought the path of a light ray that passes from one point to another and is reflected by a flat surface on its way. Hero solved this problem by assuming that light travels at a constant speed and chooses the path that minimizes the travel time. Fermat extended this least-time idea to refraction, when light passes from one medium to another, in which it may not travel at the same speed as in the first medium. When a ray of light passes from air into water, the ray is refracted (bent) downward, towards the normal (perpendicular) to the surface. If this behaviour is to be explained by the least-time idea, light must travel slower in water than in air. For a long time it was not known if this were so, so Fermat’s proposal was a prediction, which was eventually confirmed.

In 1696 John Bernoulli posed the famous ‘brachistochrone’ (shortesttime) problem. A bead, starting from rest, slides without friction under gravity on a curve joining two points at different heights. The bead’s speed at any instant is determined by how far it has descended. What is the form of the curve for which the time of descent between the two points is shortest? Newton solved the problem overnight, and submitted his solution anonymously, but Bernoulli, recognizing the masterly solution, commented that Newton was revealed ‘as is the lion by the claw print’. The solution is the cycloid, the curve traced by a point on the rim of a rolling wheel.

Soon there developed the idea that the laws of motion – and thus the behaviour of the entire universe – could be explained by an optimization principle. Leibniz, in particular, was impressed by Fermat’s principle and was always looking for a reason why one thing should happen rather than another. This was an application of his principle of sufficient reason: there must be a cause for every effect. Leibniz famously asked why, among all possible worlds, just one should be realized. He suggested, rather loosely, that God – the supremely rational being – could have no alternative but to create the best among all possible worlds. For this he was satirized as Dr Pangloss in Voltaire’s Candide. In fact, in his main philosophical work, the Monadology, Leibniz makes the more defensible claim that the actual world is distinguished from other possible worlds by possessing ‘as much variety as possible, but with the greatest order possible’. This, he says, would be the way to obtain ‘as much perfection as possible’.

Inspired by such ideas, the French mathematician and astronomer Pierre Maupertuis (another victim of Voltaire’s satire), advanced the principle of least action (1744). From shaky initial foundations (Maupertuis wanted to couple his idea with a proof for the existence of God) this principle grew in the hands of the mathematicians Leonhard Euler and Joseph-Louis de Lagrange into one of the truly great principles of physics. As formulated by Maupertuis, it expressed the idea that God achieved his aims with the greatest economy possible – that is, with supreme skill. In passing from one state at one time to another state at another time, any mechanical system should minimize its action, a certain quantity formed from the masses, speeds and separations of the bodies in the system. The quantities obtained at each instant were to be summed up for the course actually taken by the system between the two specified states. Maupertuis claimed that the resulting total action would be found to be the minimum possible compared with all other conceivable ways in which the system could pass between the two given states. The analogy with Fermat’s principle is obvious.

Unfortunately for Maupertuis’s theological aspirations, it was soon shown that in some cases the action would not have the smallest but greatest possible value. The claims for divine economy were made to look foolish. However, the principle prospered and was cast into its modern form by the Irish mathematician and physicist William Rowan Hamilton a little under a hundred years after Maupertuis’s original proposal. A wonderfully general technique for handling all manner of mechanical problems on the basis of such a principle had already been developed by Euler and, above all, Lagrange, whose Mécanique analytique of 1788 became a great landmark in dynamics.

The essence of the principle of least action is illustrated by ‘shortest’ paths on a smoothly curved surface. In any small region, such a surface is effectively flat and the shortest connection between any two neighbouring points is a straight line. However, over extended areas there are no straight lines, only ‘straightest lines’, or geodesics, as they are called. As the idea of shortest paths is easy to grasp, let us consider how they can be found. Think of a smooth but hilly landscape and choose two points on it. Then imagine joining them by a smooth curve drawn on the surface. You can find its length by driving pegs into the ground with short intervals between them, measuring the length of each interval and adding up all the lengths. If the curve winds sharply, the intervals between the pegs must be short in order to get an accurate length; and as the intervals are made shorter and shorter, the measurement becomes more and more accurate. The key to finding the shortest path is exploration. Having found the length for one curve joining the chosen points, you choose another and find its length. In principle, you could examine systematically all paths that could link the two chosen points, and thus find the shortest.

This is indeed exploration, and it contains the seed of rational explanation. There is something appealing about Leibniz’s idea of God surveying all possibilities and choosing the best. However, we must be careful not to read too much into this. There does seem to be a sense in which Nature at least surveys all possibilities, but what is selected is subtler than shortest and more definite than ‘best’, which is difficult to define. Nothing much would be gained by going into the mathematical details, and it will be sufficient if you get the idea that Nature explores all possibilities and selects something like a shortest path. However, I do need to emphasize that Newton’s invisible framework plays a vital role in the definition of action.

Picture three particles in absolute space. At one instant they are at points A, B, C (initial configuration), and at some other time they are at points A*, B*, C* (final configuration). There are many different ways in which the particles can pass between these configurations: they can go along different routes, and at different speeds. The action is a quantity calculated at each instant from the velocities and positions that the particles have in that instant. Because the positions determine the potential energy, while the velocities determine the kinetic energy, the action is related to both. In fact, it is the difference between them. It is this quantity that plays a role like distance. We compare its values added up along all different ways in which the system could get from its initial configuration to its final configuration, which are the analogues of the initial and final points in the landscape I asked you to imagine. The history that is actually realized is one for which the action calculated in this way is a minimum. As you see, absolute space and time play an essential role in the principle of least action. It is the origin of the two-and-a-bit puzzle. Now let us see how it might be overcome.

DEVELOPING MACHIAN IDEAS

After it became clear to me that Platonia was the arena in which to formulate Mach’s ideas, I soon realized that it was necessary to find some analogue of action that could be defined using structure already present in Platonia. With such an action it would be possible to identify some paths in Platonia as being special and different from other paths. In Leibniz’s language, such paths could be actual histories of the universe, as opposed to merely possible ones. The problem with Hamilton’s action was that it included additional structure that was present if absolute space and time exist, but absent if you insist on doing everything in Platonia. In 1971, with a growing family and financial commitments, I was doing so much translating work I had little time for physics. As luck would have it, the postal workers in Britain went on an extended strike. No more work reached me (no one thought of using couriers in those days) – it was bliss. I got down to the physics and soon had a first idea. It still took quite a time to develop it adequately, but eventually I wrote it up in a paper published in Nature in 1974. Mach’s principle may be controversial, but it always attracts interest, and Nature also published quite a long editorial comment on the paper. Perhaps it was worth waiting ten years before getting my first paper published.

It was certainly a turning point in my life. Some months after it appeared, I received a letter with some comments on it from Bruno Bertotti, who was, and still is, a professor of physics at the University of Pavia in Italy. Bruno, who is a very competent mathematician, has worked in several fields in theoretical physics. In fact, he was one of the last students of the famous Erwin Schrödinger, the creator of wave mechanics (Box 1). But he has also been active in experimental gravitational physics, and he organized the first two – and very successful – international conferences in the field. Although I can never stop thinking about basic issues in physics, I am at best an indifferent mathematician, so I was very lucky that my correspondence with Bruno soon developed into active collaboration. Sometimes Bruno came to work at College Farm, but mostly I went to Pavia. For seven years I went there for about a month, every spring and autumn. It was a very fruitful and rewarding collaboration: my work on Mach’s principle would never have developed into a real theory without Bruno’s input. I cannot say that we discovered any really new physics, because in the end we had to recognize that Einstein had got there long before us. What I think was important was that in two papers, published in 1977 and 1982, we laid the foundations of a genuine Machian theory of the universe. To our surprise, we then found that this theory is already present within general relativity, though so well hidden that no one (not even Einstein) suspected it. We had found a quite new route to his theory, and had the consolation to know that Einstein had by no means fully grasped the significance of his own theory.

In this connection, a remarkable coincidence that happened to me on my first visit to Pavia is worth recounting. I arrived on a Friday night. I was going to spend the first weekend sightseeing, and after breakfast on Saturday morning I wandered off with no set aim through the streets of Pavia in the warm April sunshine. After about twenty minutes I chanced upon a grand medieval house. A plaque outside said that in the 1820s the poet Ugo Foscolo had lived there. One could walk into the courtyard, which I did. It was Italy as you dream of it. This, I thought, was the place to live. Six months later, quite by chance, I learned that for two years, in the 1890s, it had been Einstein’s home. In his teens, the electrical firm run by his father and uncle in Munich had failed, and they had moved to Pavia and started another firm (which also failed). Somehow that chance episode in Pavia seems symbolic of my efforts in physics. Einstein was there first, long ago, but it was still worth the journey to see the place from the inside. It yielded a perspective, quite different from Einstein’s, which persuades me that Platonia is the true arena of the universe. If it is, we shall have to think about time differently.

The first idea Bruno and I developed had several interesting and promising properties. Above all, it showed that a mechanics of the complete universe containing only relative quantities and no extra Newtonian framework could be constructed. Hitherto, most people had thought this to be impossible. Just as Mach had suspected, the phenomenon that Newton called inertial motion in absolute space could be shown to arise from motion relative to all the masses in the universe. We also showed that an external time is redundant. However, besides the desirable features we obtained effects which showed that the theory could not be right. While the universe as a whole could create the experimentally observed inertial effects that we wanted, the Galaxy would create additional effects, not observed by astronomers, that ruled out our approach.

The idea that Bruno and I first developed seemed so natural it surprised us that no one had thought of it earlier. However, I learned quite recently that something similar was proposed in 1904 (in an obscure booklet by one Wenzel Hofmann), and then rediscovered in 1914 by the physicist Hans Reissner and again in 1925 by none other than Schrödinger, just before he discovered wave mechanics. This was especially ironic since Bruno had been Schrödinger’s student. I think the main reason why these papers got overlooked was that they were completely overshadowed by Einstein’s general relativity and the excitement of the discovery of quantum mechanics in 1925/6. There is also an undoubted tendency for physicists to work within a so-called paradigm (the American philosopher of science Thomas Kuhn’s famous expression), and pay at best fleeting attention to ideas that do not fit within the existing established patterns of thought.

I mention these things because the next idea that Bruno and I tried seems to me just as natural as our first idea, if one approaches the problems of describing motion and change with an open mind. It does, however, seem very different from the present paradigm, which has become deep-rooted with the long hegemony of Newtonian ideas, which were only partly changed by Einstein. Although, as we shall see, our second idea is actually built into Einstein’s theory at its very heart, within the context of classical physics it merely provides a different perspective on that theory. However, for the study of quantum effects it does represent a genuine alternative, and the attempt to create a quantum theory of the universe may force its adoption, alien though it may appear to many working scientists.

EXPLORING PLATONIA

Let me now explain this second idea. So far, I have explained only what the points of Platonia are. Each is a possible relative arrangement, a configuration, of all the matter in the universe. If there are only three bodies in it, Platonia is Triangle Land, each point of which is a triangle (Figures 3 and 4). Can we somehow say ‘how far apart’ any two similar but distinct triangles are? If so, this will define a ‘distance’ between neighbouring points in Triangle Land, and just as mathematicians seek geodesics using ordinary distances on curved surfaces, we can start to look for geodesics in Platonia. If we can find them, they will be natural candidates for actual histories of the universe, which we have identified as paths in Platonia. They will be Machian histories if the ‘distance’ between any two neighbouring points in Platonia is determined by their structures and nothing else, and we shall not need to suppose that they are embedded in some extra structure like absolute space.

There is such a simple and natural solution to the problem of finding geodesics in Platonia that I would like to spell it out. The fact that it does seem to be used by nature is one of the two prime pieces of evidence I have for suggesting that the universe is timeless. (The second, equally simple in its way, comes from quantum mechanics.) How it works out for the simplest example of a universe of three bodies is described in Box 8.

BOX 8 Intrinsic Difference and Best Matching

In Figure 21, triangle ABC is one point in Triangle Land, and the slightly different triangle A*B*C* is a neighbouring point. A ‘distance’ between them can be found in many ways, but one of the simplest is the following. Imagine that ABC is held fixed, and A*B*C* is placed in any position relative to it. This creates ‘distances’ AA*, BB*, CC* between the corresponding vertices, at which we suppose there are bodies of masses a, b, c. We form a ‘trial distance’ d by taking each mass and multiplying it by the square of the corresponding distance, adding the results and taking the square root of the sum. Thus

This is an arbitrary quantity, since the relative positioning of the two triangles is arbitrary. It is, however, possible to consider all relative positionings and find the one for which d is minimized. This is a very natural quantity to find, and it is not arbitrary. Two different people setting out to find it for the same two triangles would always get the same result. It measures the intrinsic difference between the two matter distributions represented by the triangles. It is completely determined by them, and does not rely on any external structure like absolute space.

The intrinsic difference between two arbitrary matter distributions can be found similarly. One distribution is supposed fixed, and the other moved relative to it. In any trial position, the analogue of the above expression is calculated, and the position in which it is minimized is sought. Because this special position reduces the apparent difference between the two matter configurations to a minimum, it may be called the best-matching position.

Figure 21 A trial relative placing of the two triangles.

Using the intrinsic difference defined in Box 8, we can determine ‘shortest paths’ or ‘histories’ in Platonia as explained above. However, the intrinsic difference by itself does not lead to very interesting histories, and it is more illuminating to consider a related quantity. The potential energy of any matter distribution (Figure 17) is determined by its relative configuration, and is therefore already ‘Machian’. Each matter distribution has its own Newtonian gravitational potential energy. Two nearly identical matter distributions have almost the same potential. Now, the intrinsic difference is determined by two nearly identical configurations. To obtain more interesting histories we can simply multiply the intrinsic differences by the potential (strictly speaking, by the square root of minus the potential). This will change the definition of ‘distance’, but it will still enable us to determine ‘shortest distances’. You do not need to worry about these details, but I do want to give you a flavour of what is involved.

I think you will agree that finding shortest paths in an imagined timeless landscape bears little direct resemblance to our powerful sense of the passage of time. Yet the outcome turned out to be remarkably like what happens in Newtonian theory. Let me explain, taking again the example of a three-body universe, for which Platonia is Triangle Land.

Any continuous path in it corresponds to a sequence of triangles: they are the ‘points’ through which the path passes. But this is very similar to what comes out of Newtonian theory (Figure 1) – which, however, yields not only the triangles but also their positions in absolute space and separations in time. Remember that the triangles in Figure 1 are ‘lit up’ by flashes at each unit of absolute time, and that we see them, in perspective, in absolute space at those times. However, these are invisible aids. The astronomers see neither when they look through telescopes, all they see are stars. Thus, as far as observable things are concerned, both theories yield the same kind of thing – sequences of triangles. The question is, what kind of sequences do the two theories yield? In what respects do the theories differ when it comes down to what is actually observable?

The major difference is that the Machian theory makes more definite predictions. As a theory of geodesics, it determines the shortest path between any two fixed points in Platonia. It covers Platonia with such paths. These geodesics have the following important property: at any one point in Platonia, many of them pass through it. In fact, for every direction that you can go from a point, there is just one geodesic. This is the crucial difference. As Figures 13 and 14 highlighted, when the Newtonian histories are represented as paths in Platonia, it turns out that many can pass through the same point, and have at that point the same direction. However, these paths then ‘splay out’ and go to quite different places in Platonia. In Newtonian terms, they differ in energy and angular momentum. The difference is not apparent in the initial point and direction, but comes to light dramatically in the later evolution of the paths. This defect is absent in the Machian theory. For any given point and direction at that point, there is just one geodesic. Bruno and I constructed the theory precisely to achieve that aim.

What did quite surprise us was to discover that the unique Machian history with a given direction through a point is identical to one of the many Newtonian histories through the point with the same direction. It is, in fact, the Newtonian history for which the energy and angular momentum are both exactly zero. The small fraction of Newtonian solutions with this property are all the solutions of a simpler timeless and frameless theory.

This brought to light an unexpected reconciliation between the positions of Newton and Leibniz in their debate about absolute and relative motion. Both were right! The point is that in a universe which, like ours, contains many bodies, there can be innumerable subsystems that are effectively isolated from one another. This is true of the solar system within the Galaxy, and also for many of the galaxies scattered through the universe. Each subsystem, considered by itself, can have nonzero energy and angular momentum. However, if the universe is finite, the individual energies and angular momenta of its subsystems can add up to zero. In a universe governed by Newton’s laws this would be an implausible fluke. But if the universe is governed by the Machian law, it must be the case. It is a direct consequence of the law. What is more, the Machian law predicts that in a large universe all sufficiently isolated systems will behave exactly as Newton predicted. In particular, they can have nonzero energy and angular momentum, and therefore seem to be obeying Newton’s laws in absolute space and time. But what Newton took to be an unalterable absolute framework is shown in the Machian theory to be simply the effect of the universe as a whole and the one law that governs it. What physicists have long regarded as laws of nature and the framework of space and time in which they hold are, as I said in Chapter 1, both ‘local imprints’ of that one law of the universe.

You can see directly how absolute space and time are created out of timelessness. Take some point on one of the Machian geodesics in Platonia; it is a configuration of masses. Take another point a little way along the geodesic; it is a slightly different configuration. Without any use of absolute space and time, using just the two configurations, you can bring the second into the position of best matching relative to the first. You can then take a third configuration, a bit farther along the path, and bring it into its best matching position relative to the second configuration. You can go along the whole path in this way. The entire string of configurations is oriented in a definite position relative to the first configuration. What looks like a framework is created, but it is not a pre-existing framework into which the configurations of the universe are slotted: it is brought into being by matching the configurations. Nevertheless, we get something like the Newtonian picture in Figure 1, except that we do not as yet have the ‘spacings in time’.

But this too emerges from the Machian theory. In the equations that describe how the objects move in the framework built up by best matching, it is very convenient to measure how far each body moves by making a comparison with a certain average of all the bodies in the universe. The choice of the average is obvious, and simplifies the equations dramatically. No other choice does the trick. For this reason it needs a special name; I shall call it the Machian distinguished simplifier. It is directly related to the quantities used to determine the geodesic paths in Platonia. To find how much it changes as the universe passes from one configuration to another slightly different one, it is necessary only to divide their intrinsic difference by the square root of minus the potential. (The action, by contrast, is found by multiplying it by the same quantity.) When this distinguished simplifier is used as ‘time’, it turns out that each object in the universe moves in the Machian framework described above exactly as Newton’s laws prescribe. Newton’s laws and his framework both arise from a single law of the universe that does not presuppose them.

In such a universe, the ultimate standard of time that determines which curve is traced by Galileo’s ball when it falls off his table in Padua is unambiguous. It is the average of all the changes in the universe that defines the Machian distinguished simplifier. Time is change, nothing more, nothing less.

The difference between the Newtonian and Machian theories can be summarized as follows. If we do not know the energy and angular momentum of a Newtonian system, we always need at least three snapshots of its configurations in order to reconstruct the framework of space and time in which they obey Newton’s laws. The task is complicated, to say the least. If, however, the system is Machian, the framework can be found with just two snapshots and the task is vastly simpler. It simply requires best matching of the two configurations.

When, later, I suggest that the quantum universe is timeless in a deeper sense than the classical Machian universe just described, that will be a conjecture. But it is made plausible by the results of this chapter. They are not speculation but mathematical truths. Every phenomenon explained by Newton’s laws, including the beautiful rings of Saturn and the spectacular structure of spiral galaxies, can be explained without absolute space and time. They follow from a simpler, timeless theory in Platonia.