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

It was the desire to quantize general relativity that led to the work described in this chapter. One important approach, called canonical quantization , is based on analysis of the dynamical structure of the classical theory. This is how general relativity came to be studied in detail as a dynamical theory nearly half a century after its creation as a geometrical space-time theory. The ‘hidden dynamical core’, or deep structure, of the theory was revealed. The decisive analysis was made in the late 1950s by Paul Dirac and the American physicists Richard Arnowitt, Stanley Deser and Charles Misner. They created a particularly elegant theory, now known universally as the ADM formalism. (Because it is regarded as controversial by some, the initials are occasionally reshuffled as MAD or DAM.)

The dynamical form of general relativity is often called geometrodynamics . The term, like ‘black hole’ and several others, was coined by John Wheeler, who, together with his many students at Princeton, did much to popularize this form of the theory. The interpretation of it proposed in this chapter is very close to one put forward by Wheeler in the early 1960s. However, I believe it brings out the essentially timeless nature of general relativity rather more strongly than Wheeler’s well-known writings of that period. What is at stake here is the plan of general relativity. What are its ultimate elements when it is considered as a dynamical theory, and how are they put together?

This is what Dirac and ADM set out to establish. The answer was manifestly a surprise for Dirac at least, since it led him to make the remarkable statement quoted in the Preface. They found that if general relativity is to be cast into a dynamical form, then the ‘thing that changes’ is not, as people had instinctively assumed, the four-dimensional distances within space-time, but the distances within three-dimensional spaces nested in space-time. The dynamics of general relativity is about three-dimensional things: Riemannian spaces.

To connect this with the topics of Part 2, let me tell you about the work that Bruno Bertotti and I did after the work described there. We began to wonder whether we could be more ambitious and construct not merely a non-relativistic, Machian mechanics, but perhaps an alternative to general relativity. At the time, we believed that Einstein’s theory did not accord with genuine Machian principles. Experimental support for it was beginning to seem rather convincing, but tiny effects have often led to the replacement of a seemingly perfect theory by another with a very different structure. We were aware of quite a lot of the work of Wheeler and ADM, and various arguments persuaded us that the geometry of three-dimensional space might well be Riemannian, possess curvature and evolve in accordance with Machian principles. We wanted to find a Machian geometrodynamics, which we did not think would be general relativity. The first task was to select the basic elements of such a theory. What structures should represent instants of time and be the points of the theory’s Platonia?

This question was easily answered. Any class of objects that differ intrinsically but are all constructed according to the same rule can form a Platonia. So far, we have considered relative configurations of particles in Euclidean space. There is nothing to stop us considering three-dimensional Riemannian spaces, especially if they are finite because they close up on themselves. This is difficult for a non-mathematician to grasp, but the corresponding things in two dimensions are simply closed, curved surfaces like the surface of the Earth or an egg. The points of Platonia for this case are worth describing. The surface of any perfect sphere is one point; each sphere with a different radius is a different point. Now imagine deforming a sphere by creating puckers on its surface. This can be done in infinitely many ways. There can be all sorts of ‘hills’ and ‘valleys’ on the surface of a sphere, just as there are on the Earth and the Moon. And there is no reason why the surface should remain more or less spherical: it can be distorted into innumerable different shapes to resemble an egg, a sausage or a dumbbell. On all of these there can be hills and valleys. Each different shape is just one point in Platonia, and could be a model instant of time. In this case you can form a very concrete image of what each point in Platonia looks like. These are things you could pick up and handle. Note that only the geometrical relationships within the surface count. Surfaces that can be bent into each other without stretching, like the sheet of paper rolled into a tube, count as the same. However, this is a mere technicality. The important thing is that the points of any Platonia are real structured things, all different from one another.

Imagining the points that constitute this Platonia is easy enough. It is much harder to form a picture of Platonia itself because it is so vast and has infinitely many dimensions. Triangle Land has three dimensions, and we can give a picture of it (Figures 3 and 4). But Tetrahedron Land already has six dimensions, and is impossible to visualize. When there are infinitely many dimensions, all attempts at visualization break down, but as mathematical concepts such Platonias do exist and play important roles in both mathematics and physics.

Riemannian spaces are actually empty worlds since they contain nothing that we should recognize as matter. You might wonder in what sense they exist. They certainly exist as mathematical possibilities, and the proof of this was one of the great triumphs of mathematics in the nineteenth century. But they can also contain matter, just like flat familiar Euclidean space. Its properties and existence were originally suggested by the behaviour of matter within it, and evidence for curved space can be deduced through matter as well, as the experimental confirmations of general relativity show. I hope that this disposes of any worries you might have. In fact, the Platonia of three-dimensional Riemannian spaces is well known in the ADM formalism as superspace (another Wheeler coining, and not to be confused with a different superspace in superstring theory).

The Platonia that models the actual universe certainly cannot consist of only empty spaces, since we see matter in the world. To get an idea of what is needed, imagine surfaces with marks or ‘painted patterns’ on them to represent configurations of matter or electric, magnetic or other fields in space. This will hugely increase the number of points in Platonia, since now they can differ in both geometry and the matter distributions. Any two configurations that differ intrinsically in any way count as different possible instants of time and different points of Platonia.

Within classical general relativity, the concept of superspace is not without difficulties, which could undermine my entire programme. Since the issues are decidedly technical, I have put the discussion of them in the Notes. However, I can say here that marrying general relativity and quantum mechanics is certain to require modification of the patterns of thought that have been established in the two separate theories. Superspace certainly arises as a natural concept in the framework of general relativity. The question is whether it is appropriate in all circumstances.

I feel that, when everything has been taken into account, superspace is the appropriate concept, though its precise definition and the kinds of Nows it contains are bound to be very delicate issues. Now, making the assumption they can be sorted out, what can we do with the new model Platonia?

The key idea in Part 2 is the ‘distance’ between neighbouring points in Platonia based solely on the intrinsic difference between them. It was obvious to Bruno and me that if we were to make any progress with our more ambitious goal, we should have to find an analogous distance in the new Platonia. We had to look for some form of best matching appropriate in the new arena.