The End of Time: The Next Revolution in Physics

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.