The End of Time: The Next Revolution in Physics

CHAPTER 5

Newton’s Evidence

THE AIMS OF MACHIAN MECHANICS

Merely changing the framework in which one conceives of the universe does nothing, but it is still very illuminating to look at some fundamental facts of mechanics in the alternative arenas of absolute space and Platonia. This exercise brings out the strengths of Newton’s position, and at the same time shows what a Machian approach must achieve. The following discussion is based on penetrating remarks made in 1902 by the great French mathematician Henri Poincaré. More clearly than Mach, he demonstrated what is required of a theory of relative motion. Unfortunately, his remarks were overshadowed by Einstein’s discovery of relativity and did not attract the attention they deserved – and still deserve.

You may find that this chapter requires more reflection than all the others. You certainly do not need to grasp it all, but I hope that you will be able to change from a way of thinking to which we have been conditioned by the fact that we evolved on the stable surface of the Earth to a more abstract way of thinking that would have been forced upon us had we evolved from creatures that roamed in space between objects moving through it in all directions. We have to learn how to find our bearings when the solid reassuring framework of the Earth is not there. This is the kind of mental preparation you need to understand the ideas Poincaré developed. In this respect, he was smarter than Einstein.

Poincaré simply asked, rather more precisely than anyone before him, what information is needed to predict the future. Another French mathematician, Pierre Laplace, had already imagined a divine intelligence that at one instant knows the positions and motions of all bodies in the universe. Using Newton’s laws, the divinity can then calculate all past and future motions – it can see, in its mind’s eye, all of history laid out for the minutest inspection. As an alternative to the standard representation in Newton’s absolute space, it will help to see this miracle performed in Triangle Land, the simplest Platonia. This will reveal a curious defect in Newtonian mechanics.

Figure 7 Here Alpha, the apex of Triangle Land, is at the bottom. The axes that were shown in Figures 3 and 4 have been removed since they would detract from the essence of this and the following figure. Two of the ribs of Triangle Land run upwards to the right and left. They are marked BC = 0 and CA = 0 to indicate that the triangles corresponding to points on these ribs have ‘collapsed’ because their sides BC and CA, respectively, are zero. The third rib recedes into the figure (this is the rib that in Figure 4 runs along the ‘floor’). The shaded planes cut the faces of the pyramid in the lines on the ‘sheets’ in Figure 4. As shown here, all points on any straight line from Alpha through Triangle Land represent different triangles, but they differ only in size. In fact, all the points on any one of the shaded sheets represent triangles that have the same perimeter. If we are interested only in the shapes of the triangles, these are represented by the points on just one of the planes (i.e. the different points on any one plane represent differently shaped triangles; this possibility is depicted more fully in Figure 8).

Figure 8 This shows one of the shaded planes in Figure 7. It can be called Shape Space, because each point in it represents a different possible shape of a triangle. The point at the centre represents an equilateral triangle (all three sides equal). Points on the three dashed lines correspond to isosceles triangle (two sides equal). All other possible triangles are scalene (all three sides unequal). Points on the three curved lines correspond to right-angled triangles (the central triangle at the top is the Pythagorean 3, 4, 5 triangle). All points inside the curved lines correspond to acute triangles (all angles less than 90°); all points outside the curved lines correspond to obtuse triangles (one angle greater than 90°). When I asked my friend Dierck Liebscher to create this diagram, I had no idea that it would turn out to be so beautiful. I do like the curved lines of the right-angled triangles! Let me remind you that all the points on the straight edges of Shape Space correspond to triangles that have become ‘flat’ because all three corners of each of these triangles are collinear (on one line). Each of the three vertices of Shape Space correspond to configurations in which two particles coincide, while the third is some distance from them.

You may like to refresh your memory by returning to Figures 3 and 4 before you examine Figures 7 and 8. Figures 3, 4, 7, 8 and 9 are very important. I am rather concerned that younger readers (those under forty, or even fifty!) may have some difficulty with them, since fundamental geometry is not taught nearly as thoroughly at school as it used to be. However, if you can spend a bit of time on these figures and begin to understand what they mean, you will certainly get a great deal more out of this book. In fact, you will also be absorbing some of the deepest and most fruitful concepts in mathematics and theoretical physics. Don’t worry – it can be done. Once the clutter of technical detail is removed, all the great ideas in mathematics and physics are in essence very simple and intuitive. But you need patience to absorb them. When Newton was asked how he had come to make his great discoveries, he answered: ‘By thinking about these things for a long time.’ Try lying in bed or a nice warm bath and thinking about Triangle Land!

Figure 8 brings out the rich topography of any Platonia (Shape Space is a possible Platonia). Wherever you go, you find something different. Each point is a different ‘world’ – and a different instant of time. There are even characteristically different regions (of acute and obtuse triangles), like provinces or counties, as well as internal and external frontiers (the right-angled and isosceles triangles). Any Platonia is quite unlike Newton’s absolute space, all points of which are identical. As I remark in the Notes to Chapter 4, there is something unreal about that property of absolute space. Real things have genuine attributes that distinguish them from other real things. Platonia is a land of real things. I find Figure 8 very suggestive. Leibniz always said that it is necessary to consider all possible worlds and find some reason why one rather than another occurs or is actually created. In Figure 8, we do see all the possible worlds of triangle shapes laid out before us. Box 3 contains a short digression on possible kinds of Platonias.

BOX 3 Possible Platonias

One of the big unsolved problems of physics is the origin of distance and whether it is absolute. Since we need a measuring rod to measure any distance, this suggests rather strongly that distance is relative (to the chosen rod). If we tried to double every distance in the universe, the length of the rod would be doubled too, and nothing actually observable would be changed. For reasons like this, physicists have a hunch that absolute scale should have no objective meaning. However, this is not confirmed by existing theories and experimental facts, which do suggest that distance is in a well-defined sense absolute. The hope remains that a physics completely without scale will be found. If so, Shape Space gives an idea of what the corresponding Platonia will be like. There is still a uniquely distinguished point in it – the central point in Figure 8. It takes the place of Alpha in Triangle Land. If you ‘touched’ the central point, the equilateral triangle would ‘light up’. This is the most symmetrical configuration the three-body universe can have. Symmetry is beautiful in one way but bland in another. The boundaries of Shape Space are somewhat unappealing too, because they represent improper triangles that flatten into a line (mathematicians call such configurations, collinear in this case, degenerate). The vertices of Shape Space correspond to collinear configurations in which one particle is infinitely far from the other two. The interesting structures in this case lie between the bland centre and the degeneracy of the frontiers.

In modern theoretical cosmology, distance is absolute and the universe expands. For reasons that are not yet understood, it simultaneously becomes more richly structured. In a cosmology without both time and scale, this would correspond, in a realistic scale-free Platonia, to going from the bland centre to the more interestingly structured ‘instants of time’ situated between it and the frontiers. That is where the mist I introduced in Chapter 3 must collect most thickly – have the highest intensity – at time capsules structured so that they seem to record evolution from the symmetric centre. This would be a cosmology of pure structure, an appealing thought. The scalene and obtuse triangles that inhabit the ‘favoured belt’ in Shape Space remind me of the line in Gerard Manley Hopkins’s poem ‘Pied Beauty’, in which he praises ‘All things counter, original, spare, strange’. In such a scheme, the bland centre and degenerate frontiers still have a vital role to play in the scheme of things. This is because a kind of resonance between all the instants of time determines where the mist settles. Any acoustician will recognize the importance of the walls and centre of a building in determining its harmonies. Platonia, shown here as Shape Space, is a ‘heavenly vault’ in which the music of the spheres is played. However, I should emphasize that the more realistic Shape Spaces corresponding to universes with more than three particles are most definitely ‘open ended’ and should not be thought of as enclosed spaces, as might appear from the simple example of Triangle Shape Space. Platonia is not a claustrophobic vault but an ‘echoing canyon’ open to the sky. What is more, there is a sense in which its echo, heard at any point within it, is what we call the past.

In completing this box, I note it has a nice unplanned symbolism. Box 3 considers Shape Space, which is represented by the perfect (equilateral) triangle, which itself, through each of its points, represents all triangles, all of which are unities in the sense mentioned on p.18. Shape Space illustrates Giordano Bruno’s monas monadum, the unity of the unities.

If Laplace’s divinity contemplates a three-particle universe, its history will be a curve in Triangle Land, which, omitting the triangle size, we can show as a curve in Shape Space. The example of real Newtonian three-body gravitational interaction shown in Figure 9 brings out very clearly the main fact that we associate with time, that its instants come in a unique succession. This translates beautifully into the winding path. (You can see why I am so indebted to Dierck Liebscher, who crafted this diagram.) However, the other two important attributes of time, duration and direction, are not yet reflected in Figure 9. There are no marks along the curve to indicate how much time elapses between any two points on it. It is also impossible to say in which direction along the curve time increases.

What information must we give to determine a history of the kind encoded in the path in Figure 9? According to Laplace, simply the positions and velocities of the bodies at one instant (together with their masses). Poincaré remarked, however, that the positions and motions of the bodies are defined in absolute space, and the speeds of the bodies are defined using absolute time. This is a subtle and important qualification.

If only relative quantities count, then Newton assumed too much structure. In a universe of just three particles, only the three distances between them (the triangle they form) should count. The triangle universe cannot have an overall position and orientation in some invisible containing space. Similarly, since the idea of an external ‘grandstand clock’ is absurd, we cannot say ‘how fast’ the universe travels along the curve in Figure 9. It simply occupies all the points along it.

If Mach is right, so that time is nothing but change and all that really counts in the world is relative distances, there should be a perfect analogue of Laplace’s scenario with a divine intelligence that contemplates Platonia. Machian dynamics in Platonia must be about the determination of paths in that timeless landscape. It should be possible to specify an initial point in Platonia and a direction at that point, and that should be sufficient to determine the entire path. Nothing less can satisfy a rational mind. The history in Figure 10 starts at the centre of Shape Space, so that there the particles form an equilateral triangle, and set off in a certain direction. In Machian dynamics, the initial position and initial direction (strictly in Triangle Land, not Shape Space) should determine the complete curve uniquely. Now we can test this idea in the real world. The heavens provide plenty of triple-star systems, and astronomers have been observing their behaviour for a long time. They certainly meet the Laplace-type condition when described in Newtonian terms. But are their motions comprehensible from a Machian point of view? This is the question Poincaré posed.

Figure 9 The (computer-generated) path traced in Shape Space by the triangles formed by three mutually gravitating particles. It is important to realize that the winding path shown is not traced by a single particle moving over the page, but that each point on the path represents the shape of a complete triangle. The curve shows the succession of triangle shapes. As explained in the text, it is impossible to say that time increases as you go in a particular direction along this curve. However, suppose we imagine it starts in the top left corner. This corresponds to particles A and C nearly colliding, while particle B is far away. Then they move to a configuration that is quite close to an equilateral triangle, after which A and B get very close together near the bottom of the diagram. Then the triangle shape evolves along the curve up to the top right. Where the curve nearly touches the top line, all three particles are almost on a line, with C between A and B. Finally the curve returns to the bottom of the figure, where the wiggles indicate that particles A and B are orbiting around each other, while C is far away. You see how history is all coded in one curve, but you just cannot tell in which direction it unfolds!

APPARENT FAILURE

The answer is very curious. The motions are nearly but not quite comprehensible. This can be highlighted by showing how different possible Newtonian motions look when represented as curves in Triangle Land, our model Platonia – or rather Shape Space, since this is much easier to represent. To create a vivid picture, let us imagine that we are holding two cardboard triangles that are slightly different. These can represent the relative configurations of three mutually gravitating bodies at two slightly different instants of Newton’s absolute time.

Figure 10 Another possible path traced by the same three particles as in Figure 9 (I refer to them now as bodies). This history starts (or ends if time is assumed to run the other way) at the configuration in which the three bodies form an equilateral triangle. The shape of the triangle changes in a definite way at all points along the curve. If it left the initial equilateral triangle along one of the dotted lines, that would mean two sides of the triangle remaining equal in length while the third changes – the equilateral triangle would become an isosceles triangle. In fact, for the example shown, the ratios of all three sides change. For readers used to thinking of motions in ordinary space, this example corresponds to particles that constantly orbit each other in a fixed plane. The positions at which the curve touches the dotted line that bound Shape Space correspond to eclipses, when one particle is between the other two and on the line joining them. Such a configuration is called a syzygy (that’s a nice word to show off with). In ordinary space, the one particle passes through the line joining the other two and comes out the other side. But the points on the curve in Figures 9 and 10 stand for the complete triangle, not one of the three particles. This is why the curve approaches the syzygy frontier and then returns into the interior of Shape Space. There are no triangles outside the syzygies!

Playing the role of Laplace’s divinity, we place the first triangle, at the instant when Newton’s grandstand clock says it is noon, at some position in absolute space. A second later, we place the second triangle somewhere near it in a slightly different position. The first triangle defines the initial positions of the three bodies. Given the position of the second triangle one second later, we can calculate the initial motions, since we know where the particles have gone and how long it took them. (Strictly, to calculate the instantaneous velocities we must take an infinitesimal time interval, not one second, but that is a minor detail). Imagine now that a strobe light illuminates the bodies with a flash once every second, corresponding to the seconds ticking on Newton’s clock, so that we can watch how the triangle formed by the bodies moves through absolute space. We have seen this already, in Figure 1. We can also plot the points corresponding to the triangles in either Triangle Land or Shape Space, obtaining a curve like those in Figures 9 and 10. This abstracts away the extra Newtonian information – the positions in absolute space and the time separations – that we possessed originally.

Now, wherever we place the two triangles, the resulting curves in either Triangle Land or Shape Space will all start at the same point, since we always begin with the same triangle, and that corresponds to just one fixed point in Platonia. The curve must also have the same initial direction, since that is determined by the position of the second triangle in Platonia, which is also fixed. This is explained in the caption to Figure 10. The question is, how does the curve run after that? What effect do the positioning of the first two triangles in absolute space and the time separation have on the subsequent evolution?

To answer this question, we need the notion of centre of mass (Box 4). For a given triangle, there are two different things to bear in mind when it comes to placing it in absolute space. First, we can place its centre of mass anywhere. Since space has three dimensions, this means that we can shift the centre of mass along three different directions. Physicists say that in such cases there are three degrees of freedom. Second, holding the centre of mass fixed, we can change the orientation of the triangle in space. This introduces three more degrees of freedom. To see this, picture an arrow passing through the centre of mass perpendicular to the triangle. It will point to somewhere on the two-dimensional sky, giving two degrees of freedom. The third arises because one can, keeping the arrow fixed, rotate the triangle around it as an axis.

BOX 4 Centre of Mass

The centre of mass of a system of bodies is the position of a fictitious mass equal to the sum of the masses of the system. For two unequal masses m and M, it lies on the line joining them at the position that divides the line in the ratio M/m – that is, closer to the heavier mass in that proportion. For any isolated system of bodies, the centre of mass either remains at rest or moves uniformly in a straight line through absolute space. The centre of mass for three bodies is shown in Figure 11.

Figure 11 Masses of 1, 2 and 3 units (indicated by their sizes) are shown at the vertices of the two slightly different triangles (which correspond to the two triangles discussed in the main text). The two centres of mass are shown by the big blob (mass 6 units). The position of the centre of mass is found by finding the centre of mass of any pair and then the centre of mass of it and the remaining third mass.

Now, wherever we place the centre of mass of the first triangle, and however we orient it, the sequence of triangles that then arises is always the same. The path traced out in Platonia is the same. The starting position in absolute space does not matter an iota. This is rather remarkable. It is as if you could grow identical carrots in your garden, at the bottom of the sea, and in outer space. Different locations in absolute space have a decidedly shadowy reality. Unlike real locations on the Earth, they do not have any observable effects.

The ‘location in time’ is equally difficult to pin down. We started our experiment at noon according to Newton’s clock. In fact, the starting time has no influence whatsoever: all the ‘carrots’ come out just the same. So, as far as the position of the first triangle in both absolute space and time is concerned, it has no influence whatsoever. We begin to wonder whether they play any role at all. This doubt is strengthened when we consider where to place the second triangle. It turns out that we can position its centre of mass anywhere in absolute space relative to the first triangle. This too has no effect at all on the sequence of triangles that then follow. This absence of effect is due to so-called Galilean relativity, which is one of the most fundamental principles of physics (Box 5).

BOX 5 The Galilean Relativity Principle

Galileo noted that all physical effects in the closed cabin of a ship sailing at uniform speed on a calm sea unfold in exactly the same way as in a ship at rest. Unless you look out of the porthole, you cannot tell whether the ship is moving. Quite generally, in Newtonian mechanics the uniform motion of an isolated system has no effect on the processes that take place within it. The left-hand diagram in Figure 12 shows (in perspective) the triangles formed by the three gravitating bodies in the history of Figure 10 at equal intervals of absolute time. The individual bodies move along the ‘spaghetti’ tubes. The centre of mass moves uniformly up the z axis. (Despite appearances, the triangles are always horizontal, i.e. parallel to the xy plane.) The right-hand diagram has two physically equivalent interpretations. First, it is how observers moving uniformly to the left past the system on the left would see that system receding behind them. Second, it is also how observers at rest relative to the system on the left would see a system identical to that system except for a uniform motion of the centre of mass to the right. This is how the happenings in the cabin of Galileo’s galley would be ‘sheared’ to the right for observers standing on the shore. Depending on the speed of the system, the centre of mass will be shifted in unit time by different amounts, but the actual sequence of triangles remains the same. This corresponds to the freedom mentioned in the text.

Because of the relativity principle, the laws of motion satisfied by bodies take exactly the same form in any frame of reference moving uniformly through absolute space as they do in absolute space itself. Although Newton did not like to admit it, this fact makes it impossible to say whether any such frame, which is called an inertial frame of reference, is at rest in absolute space or moves through it with some uniform velocity. Bodies with no forces acting on them move in a straight line with uniform speed in any inertial frame of reference (hence the name). It is impossible to say that you are at rest in absolute space, only that you are at rest in some inertial frame of reference. For historical reasons I use ‘absolute space’ in the text, but strictly I should be using ‘any inertial frame of reference’.

Figure 12 Unlike Figures 9 and 10 (and the later Figure 14), the lines followed by the spaghetti strands in this figure (and also Figure 13) show the tracks of the three individual particles in space. This is why there are three strands and not a single curve. It will help you a lot if you can get used to thinking about these two different ways of representing one and the same state of affairs. Here we see individual particles moving in absolute space. In Figures 9, 10 and 14 we ‘see’ (in our mind’s eye) the ‘world’ or ‘universe’ formed by the three particles moving in Platonia.

There are only four freedoms that remain. Having placed the centre of mass of the second triangle at some position, we can change its orientation (three freedoms). We can also change the amount of Newton’s absolute time that elapses between the instants at which the three bodies occupy the two positions (one freedom, the fourth). If the time difference is shortened, this means that the bodies travel farther in less of Newton’s time – that is, they are moving faster initially. In fact, since the motion of the centre of mass does not matter, we can keep it fixed and change only the orientation. Now, at last, we come to something that does matter. Both these changes – in the time difference and in the relative orientation – have dramatic consequences, which are illustrated in Figures 13 and 14.

Figures 13 and 14 express the entire mystery of absolute space and time. Both of Newton’s absolutes are invisible, yet their effects show up in the evolutions of the triangles, which are more or less directly visible. The astronomers do see stars and the spaces between them (admittedly in projection) when they look through telescopes. If time were merely change and only distances had dynamical effect, a decent Machian mechanics – one that would satisfy Laplace’s divine intelligence – should lead to exactly the same evolutions in all nine cases. This is manifestly not true for the real triple-star systems that astronomers observe. All the different kinds of evolution shown in Figures 13 and 14, and many more, are found. All the facts that enabled Newton to win his argument against Leibniz are contained in these diagrams, but it took about two centuries before Poincaré found the best way to demonstrate them. He concluded regretfully that a mechanics that uses only relative quantities, as Mach advocated, cannot get off the ground. It lacks perfect Laplacian determinism. Nevertheless, the failure is curious. Absolute space and time could have had an effect through all the freedoms allowed in the placing of the two triangles. There are fourteen degrees of freedom in total, of which ten have no effect whatsoever. This is just what the invisibility of space and time would lead us to expect. Yet four degrees of freedom do have a profound influence. Three are associated with twists in space, the fourth with the overall speed put into the system. These strange mismatches between expectation and reality have kept the philosophers arguing and the physicists puzzling for centuries.

The fact is that Newton’s absolute space and time play a decidedly odd role. The first problem is their invisibility. The more serious problem is what little part they play in the whole story, and how irrationally they enter the stage when they do participate in the action. Once we have chosen the relative orientation and time separation of the two triangles, we can take them anywhere in absolute space and time. They will always give rise to the same evolution. Absolute space and time seem to matter very little; only the relative orientation and time separation count.

Figure 13 These are ‘spaghetti diagrams’ of evolutions in absolute space like the left-hand one in Figure 12 (the one at the top left is the same evolution but with the triangles removed). The corresponding curves in Shape Space are shown in Figure 14. In each diagram the evolution commences with the three bodies forming an equilateral triangle, and all the corresponding curves start in the same direction in Triangle Land and Shape Space. This is because the second triangle is the same in both cases. The different evolutions are created by giving the bodies different initial speeds (they are different in the three rows) and by giving the triangles different orientational twists (different in the three columns).

Figure 14 These are the curves in Shape Space corresponding to the nine evolutions in absolute space shown in Figure 13. They all start from the same point with the same direction, but then diverge strongly. Remember, as I explained in the caption to Figure 9, that these curves represent not the motion of a single particle across the page, but the shapes of continuous sequences of triangles. If you ‘stuck a pin’ into any point on one of these curves, the triangle corresponding to it would ‘light up’. It is very important to appreciate that Figures 13 and 14 show identical happenings in two different ways. Since Newton’s time, nearly all physicists have believed the Newtonian representation, Figure 13, to be the physically correct way to think about these things.

Following Leibniz and Mach, I believe Figure 14 is the right way. However, this approach faces a severe difficulty explained in the text. It is only in Chapter 7 that I shall explain how it is overcome.

But these are our arbitrary choices. Once we have chosen two triangles, nothing about the triangles in themselves gives any hint as to how we should make the choices. Leibniz formulated two great principles of philosophy that most scientists would adhere to. The first is the identity of indiscernibles: if two things are identical in all their attributes, then they are actually one. They are the same thing. The second, which we have already met, is the principle of sufficient reason: every effect must have a cause. There must be some real observable difference that explains different outcomes.

Now we can see the problem. Considered in itself, the pair of triangles is just one thing. Each different relative orientation and time separation we give them depends on our whim. They should not have any effect. Yet each has momentous consequences: they create quite different universes. An exactly analogous problem arises if the universe consists of any number of particles. Two snapshots (the analogues of the two triangles) of the relative configuration of the universe are never quite enough to determine an entire history uniquely.

Before we look at the one possibility that can resolve this puzzle, it is worth considering how the four freedoms that do count show up in practice. We shall then be able to see what a great discovery Newton’s invisible framework was. We start with the twists.

SPACE AND SPIN

When I was a boy, there was only one sport at which I was any good: the high jump. One year I went on a training course at the athletics ground in Oxford. We were introduced to angular momentum and how it could be exploited to improve the jump. As tallest of the young hopefuls, I was chosen to give a demonstration. The instructor made me lie on my side, arms and legs outstretched, on a small bench turntable. He started to rotate it slowly and asked, ‘If you pull yourself into a crouched position, what will happen?’ I knew, I was studying physics: ‘Angular momentum will be conserved, and the turntable will spin faster.’ ‘Right,’ he said, ‘do it.’ Proudly, I pulled in my arms and legs with vigour. The effect was frightening. The turntable whizzed around so fast that I panicked, tried to get off, and was thrown onto the floor. I escaped with bruises. I am still kicking myself, not about the accident, but because I did not stay on another day. I would have seen Roger Bannister run the first four-minute mile.

Angular momentum is a kind of net spin about a fixed axis. To calculate it for the Earth, you multiply the mass of each piece of matter in the Earth by its perpendicular distance from the rotation axis and the speed of its circular motion about the axis. The Earth’s total angular momentum is the sum of the contributions of all the pieces. Clockwise and anticlockwise motions count oppositely. A jet plane flying round the world in the opposite sense to the daily rotation contributes with the opposite sign.

By Newton’s laws, this net spin cannot change for an isolated system. This universal law applies equally to humans and planets. When I pulled in my arms and legs in Oxford, I abruptly reduced the distance of much of my mass from the rotation axis. This inescapably enforced an equally abrupt increase of my rotational speed – with its unfortunate consequences. The same law explains why the Earth’s rotation axis stays fixed, pointing towards the pole star, and why the length of the day, the rotation period, does not change. The rotation speed could change only if the Earth could expand or contract, but, being rigid, it cannot. (Actually, both the axis and the day do change very slowly due to the external influence of the Sun and Moon.) For rigid bodies like the Earth and a top, the effects of angular momentum are rather obvious. However, its effects are far-reaching.

A globular cluster may contain a million stars. It has no rigidity – all its stars move individually in different directions, though gravity holds the cluster together. Its angular momentum is found by choosing three mutually perpendicular axes, and calculating the net spin around each of them. These correspond exactly to the three degrees of freedom to make twists, mentioned in the previous section. However, the three axes can always be chosen in such a way that the spin about two of them is zero, and all the net spin is thus about a single axis. This axis is a kind of arrow that points in a certain direction in space. It and the net spin remain completely unchanged as time passes. In astronomy, time passes in aeons. Since the stars all move in different directions, the bookkeeping exercise that nature performs is remarkable. A deep principle is at work.

The laws of nature are seldom seen to be operating in a pure form, and are hard to recognize. Air resistance and friction distort the basic laws of mechanics. But the greatest difficulty arises because the laws involve time, and we experience only one instant at a time. If only we could see all the instants of time stretched out before us, we could see the effects of the laws of motion directly, as in some of the diagrams earlier in the book.

However, a few phenomena reveal mechanics at work in a striking fashion. They are often associated with angular momentum. The humble top is one of the best examples. Riding a bicycle is another: the reassuring way in which balance is maintained as you speed down a hill with the air rushing past you is down to the angular momentum in the spinning wheels. Once the wheels are turning fast, they have a strong tendency to keep their axis of rotation horizontal. Indeed, a child’s hoop illustrates beautifully how the rotation axis maintains a fixed direction. So does the frisbee, spinning true as it floats through the air. Much grander examples occur naturally. I have already mentioned the earth’s rotation, which we see as the rising and setting of the Sun, Moon and stars and their ceaseless march across the sky. Many of our images of time come from this phenomenon, the child’s top writ large.

However, in all these examples there is a rigid body. The example of globular clusters tells of a mighty invisible framework behind the all too elusive phenomena. Newton knew it was there long before the astronomers found the grandest examples of its handiwork: spiral galaxies. In them, the initially invisible effects of the framework have become visible. Indeed, any isolated collection of matter, whatever its nature – a million stars in a globular cluster or a huge cloud of dust in space – has its associated fixed axis of net spin. Laplace called the plane perpendicular to it through the centre of mass the invariable plane, because its orientation can never change. Sometimes it can actually be seen. This is because some motions can be changed or even lost through mutual interactions, whereas others cannot. For example, objects moving parallel to the spin axis in opposite directions may collide and be deflected into the invariable plane. Over time, the matter in the system can ‘collect’ in or near it provided there is still the correct amount of circular motion about the axis. This has happened in spiral galaxies, in which the bright stars in the spiral arms are formed from such accumulated matter. They ‘light up’ the invariable plane, making it visible (Figure 15).

A similar effect has been at work in the solar system. About four and a half billion years ago the Sun and planets formed from a huge cloud of dust left over from a supernova explosion. The dust had some net spin, and an associated invariable plane. The Sun formed near the cloud’s centre of mass, and gathered up most of the mass in the cloud. More or less all of the solar system’s rotation now takes place in the plane of the ecliptic, in which the Earth moves around the Sun. Although the Sun got the bulk of the mass, Jupiter has most of the spin.

Figure 15 A spectacular spiral galaxy seen ‘from above’.

The fact that all the planets move in the same direction around the Sun in nearly coincident planes is thus a remote consequence of the relatively modest initial net spin of the primordial dust cloud. We see the result in the sky, since all the celestial wanderers – the Sun, Moon and planets – follow much the same track against the background of the stars. Ironically, Newton underestimated the power of his own laws. He could not bring himself to believe that the solar system had arisen naturally. ‘Mere mechanical causes’, he said, ‘could not give birth to so many regular motions.’ He asserted that ‘this most beautiful system’ could only have proceeded ‘from the counsel and dominion of an intelligent and powerful Being’. One wonders what Newton would have made of the modern pictures of Saturn and its rings (Figure 16). Of all the images created in the heavens by gravity and the invariable plane, this is surely the most perfect.

For three centuries, the best explanation for phenomena like the rings of Saturn has remained Newton’s: inertia, the inherent tendency of all objects to follow straight lines in the room-like arena of absolute space. If these are accepted, then the rings of Saturn, tops, frisbees and all the other manifestations of angular momentum can be explained. However, Newton’s account is not so much an explanation as a statement of facts in need of explanation. Since it is always matter that we actually see, should we not try to account for these things without the mysterious intermediaries of absolute space and time? Before we attack this problem, we need to consider energy and, in the next chapter, clocks and the measurement of time.

Figure 16 Saturn and its rings.

ENERGY

Energy is the most basic quantity in physics. It comes in two forms: kinetic energy measures the amount of motion in a system, while potential energy is determined by its instantaneous configuration. Like angular momentum, in an isolated system the sum of the two remains constant. If one decreases, the other must increase. For example, the potential energy of a falling body is proportional to its height and decreases as it falls. The speed of descent, and with it the kinetic energy, increases by an exactly compensating amount.

Energy, like the whole of mechanics, has a curious hybrid nature. Absolute space and time are needed to calculate kinetic but not potential energy. Each body of mass m and speed v in a system contributes a kinetic energy ½mv². The speed is measured in absolute space, which is why it is needed to calculate kinetic energy. By contrast, the potential energy of a system depends only on its relative configuration. For example, each pair of gravitating bodies in a system contributes to the system’s total potential energy an amount that is inversely proportional to their separation. If this is doubled, the potential energy of the pair is halved. Since each point in any Platonia corresponds to a different configuration of bodies, the potential energy changes from place to place in Platonia. This is illustrated for three bodies in Figure 17.

Figure 17 The gravitational potential energy of three bodies of different masses is shown as the height of a surface above Shape Space (Figure 8), each point on which corresponds to a different shape of the triangle formed by the three bodies. The overall scales of the configurations on the right are nine times greater, so the magnitude of the potential energy is much lower. Since potential energy is inversely proportional to separation, it increases sharply towards the corners of Shape Space, corresponding to two-particle coincidences, and becomes infinite at them. As this cannot be shown in the figure, the surfaces have been cut off at a certain height. The most distant corner of Shape Space corresponds in this figure to coincidence of the two most massive particles, so this is why the potential increases most strongly there.

Like angular momentum, the energy affects the appearance of systems and the behaviour of individual objects. For gravity the potential energy is negative, while the kinetic energy is positive. Thus the total energy E can be either positive, zero or negative. If a spacecraft is launched with sufficient speed, it can escape from the Earth’s gravity because its E is positive. If E is zero, the spacecraft has exactly the escape velocity, and escape is just possible. If E is negative, the spacecraft cannot escape from the Earth and will either orbit the Earth or fall back to ground. The planets can never escape from the Sun because they have negative E. Star clusters can remain concentrated in a relatively small region of space only if their energy is negative, otherwise they would rapidly disperse. This is why we do see such fine objects as the galaxies and star clusters in the sky. It is also largely the reason why the Sun and planets have their beautiful round shapes.

Thus, the shapes of almost all the objects astronomers observe in the sky reflect their energy and angular momentum. They, in turn, seem impossible to explain unless absolute space and time do exist and have a real influence, just as Newton claimed. The evidence for Newton’s invisible framework is written all over the sky. The evidence can be summarized as the two-snapshots problem. Suppose that snapshots of an isolated system taken at two closely spaced instants show only the separations of its bodies, not the overall orientations in absolute space. The separation in time between the snapshots is also unknown. If the system is a globular cluster, the snapshots contain millions of data. However, to determine the evolution of the system, four pieces of data are still lacking. They determine the kinetic energy (one piece of data) and the angular momentum (three pieces of data). Although they cannot be deduced from the two snapshots, they have a huge influence on the evolution, which can often be seen at a glance. A third snapshot will yield the data, but also much redundant information. The four missing pieces of data comprise the entire evidence for absolute space and time. Every system in the universe proclaims their existence. This seems to make nonsense of my claim that time does not exist. There appears to be more to the universe than its relative configurations. There is invisible structure, of which no trace can be found in Platonia.