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 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.