Julian Barbour - The End of Time - The Next Revolution in Physics

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.

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.