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.
To explain the problem, let me first recall what best matching does and achieves in the Newtonian case of a large (but fixed) number of particles. Each instant of time, each Now, is defined by a relative configuration of them in Euclidean space. We modelled each Now as a ‘megamolecule’, and compared two such Nows, without reference to any external space or time, by moving one relative to the other until they were brought as close as possible to coincidence as measured by a suitable average. This is where the real physics resides, since the residual difference between the Nows in the best-matching position defines the ‘distance’ between them in Platonia. Once we possess all such ‘distances’ between neighbouring Nows, we can determine the geodesics in Platonia that correspond to classical Machian histories. Besides defining these ‘distances’, the best matching automatically brings the two Nows into the position they have in Newton’s absolute space, if we want to represent things in that way.
However, to complete that Newtonian-type picture, we have still to determine ‘how far apart in time’ the two Nows are. This is the problem of finding the distinguished simplifier, the time separation that unfolds the dynamical history in the simplest or most uniform way. As we saw in the final section of Chapter 6, in the discussion of ephemeris time, the choice of distinguished simplifier is unique if we want to construct clocks that will enable their users to keep appointments. Our ability to keep appointments is a wonderful property of the actual world in which we find ourselves, and we must have a proper theoretical understanding of its basis. This is achieved if we insist that a clock is any mechanism that measures, or ‘marches in step with’, the distinguished simplifier. This is the theory of duration and clocks that Einstein never addressed explicitly. However, the most important thing is that history itself is constructed in a timeless fashion. The distinguished simplifier is introduced after the event to make the final product look more harmonious. Duration is in the eye of the beholder.
In Newtonian best matching, the compared Nows are moved rigidly relative to one another. We could conceive of a more general procedure, but since the Nows are defined by particles in Euclidean space its flatness and uniformity make that an additional complication. We should always try to keep things simple.
However, if we adopt curved three-dimensional spaces, or 3-spaces as they are often called, as Nows, any best-matching procedure for them will have to use a more general pairing of points between Nows. For example, two 3-spaces (which may or may not contain matter) may have different sizes. It will then obviously be impossible to pair up all points as if they were sitting together in the same space. More generally, the mere fact that both spaces are curved – and curved in different ways – forces us to a much more general and flexible method for achieving best matching.
In a talk, I once illustrated what has to be done by means of two magnificent fungi of the type that grow on trees and become quite solid and firm. For reasons that will become apparent, I called them Tristan and Isolde. Tristan was a bit larger than Isolde, and both were a handsome rich brown, the darkness of which varied over their curved and convoluted surfaces. I wanted to explain how one could determine a ‘difference’ between the two by analogy with the best-matching for mass configurations in flat space. In some way, this would involve pairing each point on Tristan to a matching point on Isolde. A little reflection shows that the only way to do this is to consider absolutely all possible ways of making the matching.
I took lots of pins, numbered 1, 2, 3, ..., and stuck them in various positions into Tristan. I then took a second set, also numbered 1, 2, 3, ..., and stuck them into Isolde. Since they had similar shapes, I placed the pins in corresponding positions, as best as I could judge. I could then say that, provisionally, pin 1 on Tristan was ‘at the same position’ as pin 1 on Isolde. All the other points on them were imagined to be paired similarly in a trial pairing.