Wagner’s opera Tristan und Isolde is widely regarded as a highpoint of the Romantic movement in music. General relativity is the ne plus ultra of dynamics. More explicitly, the way in which two 3-spaces are fitted together in its dynamical core is like two lovers seeking the closest possible embrace. This is the level of refinement at work in the principles that create the fabric of space-time. It is vastly more than just a four-dimensional block. Everywhere we look, it tells the same great story but in countless variations, all interwoven in a higher-dimensional tapestry. This is what Einstein made out of Minkowski’s magical pack of cards. Look at space-time one way, and we see Tristan and Isolde hanging, Chagall-like, in the sky. Look another way, and we see Romeo and Juliet, yet another way and it is Heloise and Abelard. All these pairs, each perfect in themselves, are all made out of each other. They and their stories stream through each other. They create a criss-cross fabric of space-time (Figure 31).
Figure 31. Space-time as a tapestry of interwoven lovers. Given just the ‘intrinsic structure’ of Tristan and Isolde, the BSW formalism determines in principle all the points on Tristan that will be paired with points on Isolde. The lengths of the struts (proper time between matched points) are obtained as a by-product of the basic problem – finding the ‘best position’ for the closest possible embrace. They are therefore shown as dashes. The lengths of the struts are local analogues of ephemeris time and, as they separate Tristan and Isolde, are simply the most transparent way of depicting the intrinsic difference between the two of them. The struts between the other pairs of lovers are determined similarly. We can see how the difference that keeps Tristan apart from Isolde is actually part of the body of Romeo (and Juliet). The struts between Romeo and Juliet are drawn with short dashes because they have a space-like separation. Einstein’s equations and the best-matching principle hold, however space-time is sliced.
It stretches to the limit the notion of substance. For the body of space-time, its fattening in time, is just the way we choose to hold things apart so that the story unfolds simply. At least, it is in Newtonian space-time. All the dynamics – what actually happens – is in the horizontal placing. We pull the cards apart in a vertical direction that we call time as a device for achieving simplicity of representation. Time is the distinguished simplifier. The substance is in the cards. They are the things; the rest is in our mind.
General relativity adds an amazing twist to this seemingly definitive theory of time. Considered alone, Tristan and Isolde are substance, and the separation between them is just the measure of their difference. They cannot come together completely simply because they are different. This difference we call time. But what is representation of difference between Wagner’s lovers is part of the very substance of Shakespeare’s lovers. Romeo and Juliet would not be what they are if Tristan and Isolde were not held apart by their difference. The time that holds Tristan apart from Isolde is the body of Romeo. This interstreaming of essence and difference all in one space-time is even more remarkable than Minkowski’s diagram containing two rods each shorter than the other.
Several profound ideas are unified and taken to the extreme in Figure 31: Einstein’s relativity of simultaneity, Minkowski’s fusion of time with space, Poincaré’s idea that the relativity principle should be realized through perfect Laplacian determinism, Poincaré’s idea that duration is defined so as to make the laws of nature take the simplest form possible, and the astronomers’ realization that it is measured by an average of everything that changes. Since best matching in general relativity holds throughout the universe in all conceivable directions, both time and space appear as the distillation of all differences everywhere in the universe. Machian relationships are manifestly part of the deep structure of general relativity. But are they the essential part?
If the world were purely classical, I think we would have to say no, and that the unity Minkowski proclaimed so confidently is the deepest truth of space-time. The 3-spaces out of which it can be built up in so many different ways are knitted together by extraordinarily taut interwoven bonds. This is where the deep dilemma lies. Four decades of research by some of the best minds in the world have failed to resolve it. On the one hand, dynamics presupposes – at the foundation of things – three-dimensional entities. Knowing nothing about general relativity, someone like Poincaré could easily have outlined a form of dynamics that was maximally predictive, flexible, refined and made no use of eternal space or time. Such dynamics, constrained only by the idea that there are distinct things, must have a certain general form. A whole family of theories can be created in the same Machian mould.
On the other hand, a truly inspired genius might just have hit on one further condition. Let dynamics do all those things with whatever three-dimensional entities it may care to start from. But let there be one supreme overarching principle, an even deeper unity. All the three-dimensional things are to be, simultaneously with all their dynamical properties, mere aspects of a higher four-dimensional unity and symmetry.
If certain simplicity conditions are imposed, only one theory out of the general family meets this condition. It is general relativity. It is this deeper unity that creates the criss-cross fabric of space-time and the great dilemma in the creation of quantum gravity. As we shall see, quantum mechanics needs to deal with three-dimensional things. The dynamical structure of general relativity suggests – and sufficiently strongly for Dirac to have made his ‘counter-revolutionary’ remark – that this may be possible. Yet general relativity sends ambivalent signals. Its dynamical structure says ‘Pull me apart’, but the four-dimensional symmetry revealed by Minkowski says ‘Leave me intact.’ Only a mighty supervening force can shatter space-time.
Note added for this printing. New work summarized on p. 358 could significantly change the situation discussed in this final section of the chapter. It suggests that the timeless Machian approach is capable of leading to a complete derivation of general relativity and that it is not necessary to presuppose ‘a higher tour-dimensional unity and symmetry.’ Since this new work has only just been published and has not yet been exposed to critical examination, I decided to leave the original text intact. However, as already indicated in the note at the end of the Preface, this new work does have the potential to strengthen considerably the arguments for the nonexistance of time.
Platonia for Relativity (p. 167) This is a technical note about the definition of superspace. The equations of general relativity lead to a great variety of different kinds of solution, including ones in which there are so-called closed time-like loops. These are solutions in which a kind of time travel seems to be possible. The question then arises of whether a given solution of general relativity—that is, a space-time that satisfies Einstein’s equations—can be represented as a path in superspace, in technical terms, as a unique succession of Riemannian three-geometries. If this is always so, then superspace does indeed seem a natural and appropriate concept. Unfortunately, it is definitely not so. There are two ways in which we can attempt to get round this difficulty. We could say that classical general relativity is not the fundamental theory of the universe, since it is not a quantum theory. This allows us to argue that superspace is the appropriate quantum concept and that it will allow only certain ‘well-behaved’ solutions of general relativity to emerge as approximate classical histories. For these, superspace will be an appropriate concept. Alternatively, we could extend the definition of super-space to include not only proper Riemannian 3-geometries (in which the geometry in small regions is always Euclidean), but also pscudo-Riemannian 3-geometries (in which the local geometry has a Minkowski type signature), and also geometries in which the signature changes within the space. For the reasons given in the long note starting on p. 348 below, I prefer the second option.