Figure 30. The two continuous curves represent (in one dimension) the two slightly different 3-spaces mentioned in the text; the more or less vertical lines are the ‘struts’.
What has this to do with best matching? Everything. Imagine mean-minded mathematicians who stick ‘pins’ like those that I stuck into Tristan and Isolde into the two 3-spaces to identify the two ends of all the struts in Figure 30. The pins carry little flags with the ‘lengths’ of the corresponding struts – the proper time – along them. However, all this information, which tells us exactly how the two 3-spaces are positioned relative to each other in space-time, is made invisible to other mathematicians who are ‘given’ just the two 3-spaces, the Nows with their intrinsic geometries and matter distributions, and set the task of finding the struts’ positions and lengths. Will they succeed?
Despite niggling qualifications, the answer is yes. When you unpack the mathematics of Einstein’s theory and see how it works from the point of view of geometrodynamics, it appears to have been tailor-made to solve this problem. This was shown in 1962 in a remarkable, but not very widely known paper of just two pages by Ralph Baierlein, David Sharp and John Wheeler (the first two were students of Wheeler at Princeton). I shall refer to these authors, whose paper has the somewhat enigmatic title ‘Three-dimensional geometry as a carrier of information about time’, as BSW. Initials can become a menace, but the BSW paper is so central to my story that I think they are warranted.
It is the implications of the BSW paper that I discussed with Karel in 1980. They can be quickly summarized. The basic problem that BSW considered was what kind of information, and how much, must be specified if a complete space-time is to be determined uniquely. This is exactly analogous to the question that Poincaré asked in connection with Newtonian dynamics, and then showed that the information in three Nows was needed. As we have seen, a theory will be Machian if two Nows are sufficient. What BSW showed is that the basic structure of general relativity meets this requirement.
In fact, the all-important Einstein equation that does the work is precisely a statement that a best-matching condition between the two 3-spaces does hold. The pairing of points established by it is exactly the pairing established by the orthogonal struts. In fact, the key geometrical property of space-times that satisfy Einstein’s equations reflects an underlying principle of best matching built into the foundations of the theory. I think that Einstein, with his deep conviction that nature is supremely rational, would have been most impressed had he lived to learn about it.
Equally beautiful and interesting is the condition that determines ‘how far apart in time’ the 3-spaces are. It is closely analogous to the rule by which duration can be introduced as a distinguished simplifier in Machian dynamics and the method by which the astronomers introduced ephemeris time. There is, however, an important difference. In the simple Machian case, the distinguished simplifier creates the same ‘time separation’ across the whole of space. In Einstein’s geometrodynamics, the separation between the 3-spaces varies from point to point, but the principle that determines it is a generalization, now applied locally, of the principle that works in the Newtonian case and explains how people can keep appointments. This is why I say that, quite unbeknown to him, Einstein put a theory of Mach’s principle and duration at the heart of his theory.
I go further. The equivalence principle too is very largely explained by best matching. To model the real universe, the 3-spaces must have matter distributions within them. The analogue in two dimensions is markings on bodies or paintings on curved surfaces. When we go through the best-matching procedure, sticking pins into Isolde, it is not only points on her skin that are matched to points on Tristan, but also any tattoos or other decorative markings. All these decorations – matter in the real universe – contribute with the geometry in determining the best-matching position and the distinguished simplifier that holds the 3-spaces apart and creates proper time between them. When this idea is combined with the relativity requirement, the equivalence principle comes out more or less automatically.
Since the equivalence principle is essentially the condition that the law of inertia holds in small regions of space-time, and all clocks rely in one way or another on inertia, this is the ultimate explanation of why it is relatively easy (nowadays at least) to build clocks that all march in step. They all tick to the ephemeris time created by the universe through the best matching that fits it together.
We have reached a crucial stage, and a summary is called for. In all three forms of classical physics – in Newtonian theory, and in the special and general theories of relativity – the most basic concept is a framework of space and time. The objects in the world stand lower in the hierarchy of being than the framework in which they move. We have been exploring Leibniz’s idea that only things exist and that the supposed framework of space and time is a derived concept, a construction from the things.
If it is to succeed, the only possible candidates for the fundamental ‘things’ from which the framework is to be constructed are configurations of the universe: Nows or ‘instants of time’. They can exist in their own right: we do not have to presuppose a framework in which they are embedded. In this view, the true arena of the world is timeless and frameless – it is the collection of all possible Nows. Dynamics has been interpreted as a rule that creates histories, four-dimensional structures built up from the three-dimensional Nows. The acid test for the timeless alternative is the number of Nows needed in the exercise. If two suffice, perfect Laplacian determinism holds sway in the classical world. It will have a fully rational basis. There will be a reason for everything, found by examination and comparison of any two neighbouring Nows that are realized. There is perfection in such dynamics: every last piece of structure in either Now plays its part and contributes, but nothing more is needed.
In non-relativistic dynamics, Newton’s seemingly incontrovertible evidence for a primordial framework and the secondary status of things can be explained if the universe is Machian. Then the roles will be reversed, things will come first, and the local framework defined by inertial motion will be explained. However, without access to the complete universe such a theory cannot be properly tested. In any case, the Newtonian picture is now obsolete even if it did clarify the issues. In general relativity the situation is much more favourable and impressive, since the best matching is infinitely refined and its effects permeate the entire universe. We can test for them locally. Finding that they are satisfied at some point in space-time is like finding a visiting card: ‘Ernst Mach was here’. The strong evidence that Einstein’s equations do hold suggests that physics is indeed timeless and frameless.
For all that, the manner in which space-time holds together as a four-dimensional construct is most striking. It is highlighted by the fact that there is no sense in which the Nows follow one another in a unique sequence. This is what, in the Newtonian case, gives rise to the beautifully simple image of history as a curve in Platonia. But in special relativity and, much more strikingly, in general relativity such a unique curve of history is lost. One and the same space-time can be represented by many different curves in Platonia. Even though no extra structure beyond what already exists in Platonia is needed to construct space-time, the way it holds together convinces most physicists that space-time (with the matter it contains) is the only thing that should be regarded as truly existing. They are very loath to accord fundamental status to 3-spaces in the way the dynamical approaches of Dirac, ADM and BSW require. Even though most of them grant that quantum theory will almost certainly modify drastically the notion of space-time, they are still very anxious to maintain the spirit of Minkowski’s great 1908 lecture. They are convinced that space and time hang together, and they want to preserve that unity at all costs. Within the purely classical theory, it seems to me that the argument is finely balanced. Perhaps an unconventional image of space-time will show how delicate this issue – space-time as against dynamics – is.