The End of Time: The Next Revolution in Physics

CHAPTER 11

General Relativity: The Timeless Picture

THE GOLDEN AGE OF GENERAL RELATIVITY

Strange as it may seem, general relativity was little studied for about forty years. This was not for want of admiration, for it was soon recognized as a supreme achievement. Confirmation of the predicted bending of starlight near the Sun by Arthur Eddington’s eclipse expedition in 1919, communicated by telegram to The Times, made Einstein into a world celebrity overnight. The problem was that there seemed to be little one could do except wonder at the miracle of the theory he had created.

The main difficulty was the extreme weakness of all readily accessible gravitational fields. Apart from three small differences from Newtonian theory, which were all reasonably well confirmed, no further experimental tests seemed possible. A further problem was the mathematical complexity of the theory. Its solutions contained fascinating structures, above all black holes, but it was decades before these were discovered and fully understood. Finally, interest in general relativity was overshadowed by the discovery in 1925/6 of quantum mechanics. In fact, truly active research in general relativity commenced only in 1955, ironically the year Einstein died, with a conference held in Bern (where Einstein had worked as a patent clerk in 1905) to mark the fiftieth anniversary of special relativity.

Since then, research has concentrated in three main fields. First, there have been tremendous experimental advances, made possible above all by technological developments, including space exploration. The foundations and some detailed predictions of the theory have been tested to a very high accuracy. Particularly important was the discovery a quarter of a century ago of the first binary pulsar, observations of which have provided strong evidence for the existence of the gravitational waves predicted by the theory. General relativity has also played a crucial role in observational astronomy and cosmology.

There have been two broad avenues of theoretical research. First, general relativity has been studied as a classical four-dimensional geometrical theory of space-time, a systematical and beautiful development of Minkowski’s pioneering work. Roger Penrose has probably done more than anyone else in this field, though many others, including Stephen Hawking, have made very important contributions. Second, the desire to understand the connection between general relativity and quantum mechanics (Box 2) has stimulated much work. Here it is necessary to distinguish two programmes. The less ambitious one accepts space-time as a classical background and seeks to establish how quantum fields behave in it. This work culminated in the amazing discovery by Hawking that black holes have a temperature and emit radiation. In Black Holes and Time Warps, Kip Thorne has given a gripping account of this story. Although the full significance of Hawking’s discovery is still far from understood, nobody doubts its importance for the more ambitious programme, which is to transform general relativity itself into a quantum theory (Box 2). This transformation, which has not yet been achieved, is called the quantization of general relativity.

In fact, many researchers believe that it is a mistake to try to quantize general relativity directly before gravity has been unified with the other forces of nature. This they hope to achieve through superstring theory. However, a substantial minority believe that general relativity contains fundamental features likely to survive in any future theory, and that a direct attempt at its quantization is therefore warranted. This is my standpoint. In particular, I regard general relativity as a classical theory of time. It must surely be worth trying to establish its quantum form. Even if we have to await a future theory for the final details, the quantization of general relativity should give us important hints about the quantum theory of time.

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.

PLATONIA FOR RELATIVITY

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?

BEST MATCHING IN THE NEW 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.

This made it possible to determine a provisional difference. For example, I could compare the two fungi using the darkness of their brown surfaces. Alternatively, and much closer to what happens in general relativity, I could compare the curvatures at matching points. The essential point is that some intrinsic property is compared at each pair of matched points, and an average of all the resulting differences is then determined. This average, one number, is the provisional difference. I leave out the mathematical details, which are intricate even though the underlying idea is simple.

This provisional difference is clearly arbitrary since the pairing on which it is based could have been made differently. To find an intrinsic difference that can have real physical meaning, we must now embark – in imagination at least – on an immensely laborious task. Keeping the pins on Tristan fixed, we need to rearrange the pins (reasonably continuously so that the mathematics works) on Isolde in every conceivable way. For each trial pairing of all points on Isolde to all points on Tristan, we must find the provisional difference. We shall know that we have found the best-matching pairing and corresponding intrinsic difference when the provisional difference remains unchanged if we go from the given pairing to any other pairing that differs from it ever so slightly. (In mathematics, the fulfilment of this condition indicates that one has found a maximum, a minimum or a so-called stationary point of the quantity being considered. It turns out that a stationary point is what is found in this case, but that is a mere technicality.) Since there is an immense – indeed infinite – number of ways of changing the pairings, the best-matching requirement imposes a very strong condition. It is impossible to conceive of a more refined and delicate comparison of two things that are different but of the same kind. However, as Bruno and I realized, it is made necessary by the nature of the compared things.

It leads immediately to the ne plus ultra of best matching – and rationality.

CATCHING UP WITH EINSTEIN

It was around 1979 that Bruno and I developed the new best-matching idea. We did quite a lot of technical work, and were beginning to get quite hopeful. We knew that we could construct various forms of Machian geometrodynamics, and we began to think that one of them might be a serious rival to general relativity. But it is not easy to beat Einstein, as we were soon to find. This came about through the intervention of another friend, Karel Kuchař, whom I had got to know in 1972, when we had several discussions. Karel is Czech and studied physics at the Charles University in Prague, specializing in relativity. In 1968 he won an award to study at Princeton with John Wheeler, where he quickly established himself as a leading expert in the canonical quantization of gravity (the most straightforward quantization procedure (Box 2) that can be used in the attempt to quantize gravity), in which Dirac and ADM had been the pioneers. Some years later he became a professor of physics at the University of Utah in Salt Lake City, where he still works. Over the years I have profited greatly from discussions with Karel, and certainly would not have been in the position to write this book without assistance from him at some crucial points. However, I hasten to add that Karel is sceptical about my idea that time does not exist at all. As we shall see, general relativity presents a great dilemma. Karel gives more weight to one Born of this dilemma, I to the other.

The issue came into clear focus for me in 1980. In April of that year, Karel gave a memorable review talk at an international conference in Oxford, during which I had an opportunity to discuss with him the ideas that Bruno and I were developing. He invited me to come to Salt Lake City, which I did in the late fall, just in time to see the pale gold of the aspens in the Wasatch mountains. Getting to know Utah and the magnificent deserts of the western United States has been a great bonus from the study of physics for me and my family. But as this is a book about physics, not travel, I had better not digress.

To come straight to the point, it soon became clear in the discussions with Karel that the idea of best matching and the whole way of thinking about duration as a measure of difference were already both contained within the mathematics of general relativity, though not in a transparent form. These facts are still not widely known, mainly, I think, because of a certain inertia. General relativity was discovered as a theory of four-dimensional space-time, and that is still essentially the way it is presented. The fact that it is simultaneously a dynamical theory describing the changes of three-dimensional things is given much less weight. This is why so few people are aware that there is such a deep issue and crisis about the nature of time at the heart of general relativity.

I think that the nature of the problem can be explained to a non-scientist. Here, at least, is my attempt. Figure 29 is a very schematic representation of the three different kinds of four-dimensional space-time that have been considered in this book. As usual, only one of the three dimensions of space is shown. It and its material contents are represented by the horizontal direction, while time runs vertically. Thus, the more or less horizontal lines and curves in the three parts of the diagram represent space and its material contents at different ‘times’. They are each Nows in my sense. As we have seen, Newtonian space-time is like a pack of ordinary cards. Each card is a Now, and they are all horizontal. I called Minkowski space-time a magical pack of cards because its Nows, or hyperplanes of simultaneity, can be drawn in different ways. Depending on the Lorentz frame that is chosen, different families of parallel Nows are obtained. Time has become relative to the frame. In general relativity, this relativity of time is taken much further: provided the Nows do not cut through the light cone, they can be drawn in an immense number of different ways. It is the complete absence of uniqueness in the way this is done that led Einstein to comment that the concept of Now does not exist in modern physics. However, this reflects the space-time viewpoint. The dynamical viewpoint puts things in a different perspective.

Figure 29 The three different kinds of space-time: on the left, Newtonian space-time, with ‘horizontal’ Nows; in the middle, Minkowski space-time, with alternative ‘tilted’ Nows; on the right, the space-time of general relativity, with Nows running in arbitrary directions.

To see this, suppose we consider two neighbouring Nows, as shown in Figure 30, in a space-time that satisfies the equations of general relativity. Each Now is a 3-space with its own intrinsic three-dimensional geometry and material contents embedded within space-time. This four-dimensional space-time has its own geometry too, and permits the construction of ‘struts’ between the two Nows. The struts are the world lines of bodies that follow geodesics in space-time, leaving the earlier Now along the space-time direction that is perpendicular to it at the point of departure. Each ‘strut’ is, so to speak, erected on the first Now. It will pierce the second Now at some point. Taken altogether, such struts uniquely determine a pairing of each point of the first Now with a point of the second Now. They do something else, too. If a clock travels along each strut between its two ends, it will measure the proper time between them as it goes. Because the two Nows have been chosen arbitrarily, the proper time will in general be different for each strut.

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.

A SUMMARY AND THE DILEMMA

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.

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.