The End of Time: The Next Revolution in Physics

CHAPTER 6

The Two Great Clocks in the Sky

WHERE IS TIME?

Newton’s mysterious ‘timepiece’ and speeds measured relative to it figured prominently in the last chapter. But is it really there, and how can we ever read its time if it is invisible? This chapter is about these two questions.

A simple but famous experiment of Galileo provides strong evidence for something very like Newton’s absolute time. He rolled a ball across a table and off its edge. His analysis of its fall was a major step in mechanics. First he noted the ball’s innate tendency to carry on forward in the direction it had followed on the table. It also started to fall under gravity, picking up speed. Galileo conjectured that two processes were at work independently, and that each could be analysed separately. The total effect would be found by simply adding the two processes together.

Galileo’s recognition of the tendency to keep moving forward anticipated Newton’s law of inertia. He did not recognize it as a universal law, but he did make it precise in some special cases. For the example of the ball, he conjectured that but for gravity (and air resistance) the ball would move for ever forward with uniform speed. (He actually thought that the motion would be around the Earth – Galileo’s inertia was circular. Luckily, the difference was far too small to affect his analysis.)

As for the second process, Galileo had already found that if an object is dropped from rest and in the first unit of time falls one unit of distance, then in the next it will fall a further three, in the next five, and so on. He was entranced by this, and called it the odd numbers rule. Now consider the sequence:

at t = 1, distance fallen = 1,

at t = 2, distance fallen = 1 + 3 = 4,

at t = 3, distance fallen = 1 + 3 + 5 = 9,

at t = 4, distance fallen = 1 + 3 + 5 + 7 = 16,…

The distance fallen increases as the square of the time: 1² = 1, 2² = 4, 3² = 9, 4² = 16,... . Galileo’s originality was to seek for a deeper meaning in this pattern.

Many teenagers can now do in seconds a calculation that took Galileo a year or more – it was so novel. He asked: if the distance fallen increases as the square of the time, how does the speed increase with time? He eventually found that it must increase uniformly with time. If after the first unit of time the object has acquired a certain speed, then after the second it will have twice that speed, after the third three times, and so on. Galileo’s work showed that, in the absence of air resistance, a falling body always has a constant acceleration. It never ceases to amaze me what consequences flowed from Galileo’s simple but precise question. It taught his successors how to read the ‘great book of nature’ (Galileo’s expression). From a striking empirical pattern, he had found his way to a simpler and deeper law.

To analyse the falling ball, Galileo simply combined the two processes – inertia and falling – under the assumption that each acts independently. He obtained the famous parabolic motion (Figure 18). In each unit of time, the ball moves through the same horizontal distance, but in the vertical dimension the distance fallen grows as the square of the time. The resulting curve traced by the ball is part of a parabola. Newton applied Galileo’s method for terrestrial motions to the heavens, and showed that the laws of motion had universal validity. This was the first great unification in physics. There may be a lesson for us here in our present quest – the search for time. We may have to look for it in the sky.

A search is needed. It is striking that all the elements in Galileo’s analysis are readily visualized. You can easily call up a table and the parabola traced by the ball in your imagination. Yet one key player seems reluctant to appear on the stage. Where is time? This is the question I have so far dodged. It presents a severe challenge to the idea that configurations are all that exist. Suppose that we take snapshots of the ball as it rolls across Galileo’s table in Padua, where he experimented. These snapshots can show everything in his studio. However, only the ball is moving. We take lots of snapshots, at random time intervals, until the ball is just about to fall over the edge. We put the snapshots, all mixed up, in a bag and, supposing time travel is possible, present it to Galileo and ask him whether, by examining the snapshots, he can tell where the ball will land.

He cannot. Had we rolled the ball twice as fast, it would have passed through the identical sequence of positions to the table’s edge, and they are all the snapshots capture. The speed is not recorded. But the ball’s speed determines the shape of the parabola, and hence where the ball lands. In fact Galileo will not even know in which direction the ball is going. Perhaps it will fall off the right-hand side of the table. More clearly than with the three-body evolutions, which mix the effects of time and spin, we see here the entire evidence for absolute time. The speed determines the shape of the parabola. There is manifestly more to the world than the snapshots reveal. What and where is it? Galileo himself provides an answer of sorts. He tells us that he measured time by a water-clock – a large water tank with a small hole in the bottom. His assistant would remove a finger from the hole and let the water flow into a measuring flask until the timed interval ended. The amount of water measured the time.

Figure 18 Galileo’s own diagram of parabolic motion. The ball comes from the right and then starts to fall. Incidentally, this diagram illustrates how conventions get established and become rigid – a modern version of it would certainly show the ball coming from the left and falling off on the right.) The uniformity of the horizontal inertial motion is shown by the equality of the intervals be, cd, de, ..., while the odd-numbers rule is reflected in the increasing vertical descents bo, og, gl, ....

We have only to include the water tank and assistant in the snapshots, and everything is changed. Galileo can tell us where the ball will land because he can now deduce its speed. There are some important lessons we can learn from this. First, it is water, not time, that flows. Speed is not distance divided by time but distance divided by some real change elsewhere in the world. What we call time will never be understood unless this fact is grasped. Second, we must ask what change is allowed as a measure of time. Galileo measured the water carefully and made sure that it escaped steadily from the tank – otherwise his measure of time would surely have been useless. But the innocent word ‘steadily’ itself presupposes a measure of time. Where does that come from? It looks as if we can get into an unending search all too easily. No sooner do we present some measure that is supposed to be uniform than we are challenged to prove that it is uniform.

It is an indication of how slowly basic issues are resolved – and how easily they are put aside – that Newton highlighted the issue of the ultimate source of time nearly two hundred years before serious attempts were made to find it. Even then, the attempts remained rather rudimentary and few scientists became aware of them. It is interesting that Galileo had already anticipated the first useful attempt. This was actually forced upon him by the brevity of free fall in the ball experiment: it was all over much too quickly for the water-clock to be of any use. (It came into its own when Galileo rolled balls down very gentle inclines.) To analyse the parabola, he found a handy substitute. He noted that if the horizontal motion of the falling ball does persist unchanged, then the horizontal distance traversed becomes a direct measure of time. He therefore used the horizontal motion as a clock to time the vertical motion. His famous law of free fall was then coded in the shape of the parabola. Its defining property is that the distance down from the apex (where the ball falls off the table) increases as the square of the horizontal distance from the axis. But this measures time.

Thus, time is hidden in the picture. The horizontal distance measures time. It would be nice if one could say ‘the horizontal distance is time’. This is the goal I am working towards: time will become a distance through which things have moved. Then we shall truly see time as it flows, because time will be seen for what it is – the change of things. However, there are many different motions in the universe. Are they all equally suitable for measuring time? A second question is this: what causal connections are at work here? Galileo measured time by the flow of water, but it is hard to believe that a little water flowing out of a tank in the corner of his studio directly caused the balls to trace those beautiful parabolas through the air. If time derives from motion and change – and it is quite certain that all time measures do – what motion or change, in the last resort, is telling the ball which parabola to trace? The first question is more readily answered.

THE FIRST GREAT CLOCK

Nearly two thousand years ago, astronomers knew that some motions are better than others as measures of time. This they discovered experimentally. For the early astronomers, there were two obvious and, on the face of it, equally good candidates for telling time. Both were up in the sky and both had impressive credentials. The stars made the first clock, the Sun the second.

The stars remain fixed relative to each other and define sidereal time. Any star can be chosen as the ‘hand’ of the stellar clock: one merely has to note when it is due south. The stellar clock then ticks whenever that star is due south (i.e. when it crosses the meridian). Fractions of the ‘tick unit’ are measured by its distance from the meridian. A mere glance at the night sky could tell the ancient astronomers the time to within a quarter of an hour. With some care, times could be told to a minute. There is something wonderful about this great clock in the sky. It was a unique gift to the astronomers. The discoveries that culminated with Kepler’s laws of planetary motion, and many more made until well into the twentieth century, are unthinkable without it. No other phenomenon in nature could match it for convenience and accuracy. In millennia it has lost a few hours.

But there is a rival – the Sun. It defines solar time. This is the clock by which humanity and all other animals have always lived. The principle is the same: it is noon when the sun crosses the meridian. You don’t even have to be an astronomer to tell the time by this clock; a sundial will do.

Merely describing the clocks shows that speed is not distance divided by time, but distance divided by some other real change, most conveniently another distance. Roger Bannister ran one mile in four minutes; normal mortals can usually walk four miles in one hour. What does that mean? It means that as you or I walk four miles, the sun moves 15° across the sky. But this is not quite the complete story of speed and time, because there is a subtle difference between the two clocks in the sky – they do not march in perfect step. One and the same motion will have a different speed depending on which clock is used. One difference between the clocks is trivial: the solar day is longer than the sidereal. The Sun, tracking eastwards round the ecliptic, takes on average four minutes longer to return to the meridian than the stars do. This difference, being constant, is no problem. However, there are also two variable differences (Box 6).

BOX 6 The Equation of Time

The first difference between sidereal and solar time arises from one of the three laws discovered by Kepler that describe the motion of the planets. The Sun’s apparent motion round the ecliptic is, of course, the reflection of the Earth’s motion. But, as Kepler demonstrated with his second law, that motion is not uniform. For this reason, the Sun’s daily eastward motion varies slightly during the year from its average. The differences build up to about ten minutes at some times of the year.

The second difference arises because the ecliptic is north of the celestial equator in the (northern) summer and south in the winter. The Sun’s motion is nearly uniform round the ecliptic. However, it is purely eastward in high summer and deep winter, but between, especially near the equinoxes, there is a north-south component and the eastward motion is slower. This can lead to an accumulated difference of up to seven minutes.

The effects peak at different times, and the net effect is represented by an asymmetric curve called the equation of time (it ‘equalizes’ the times). In November the Sun is ahead of the stars by 16 minutes, but three months later it lags by 14 minutes. This is why the evenings get dark rather early in November, but get light equally early in January. The stars, not the sun, set civil time.

Since the Sun is much more important for most human affairs than the stars, how did the astronomers persuade governments to rule by the stars? What makes the one clock better than the other? The first answer came from the Moon and eclipses. Astronomers have always used eclipse prediction to impress governments. By around 140 BC, Hipparchus, the first great Greek astronomer, had already devised a very respectable theory of the Moon’s motion, and could predict eclipses quite well.

Now, in the timing and predicting of eclipses, half an hour makes a difference. They can occur only when the Moon crosses the ecliptic – hence the name – and the Moon moves through its own diameter in an hour. There is not much margin for error. By about AD 150, when Claudius Ptolemy wrote the Almagest, it was clear that eclipses came out right if sidereal, not solar time was used. No simple harmonious theory of the Moon’s motion could be devised using the Sun as a clock. But the stars did the trick.

What Hipparchus and Ptolemy took to be rotation of the stars we now recognize as rotation of the Earth. It is strikingly correlated with the Moon’s motion. Even more striking is the correlation established by Kepler’s second law, according to which a line from the Sun to a planet sweeps out equal areas in equal intervals of sidereal time. Whenever astronomers and physicists look carefully, they find correlations between motions. Some are simple and direct, as between the water running out of Galileo’s water-clock and the horizontal distance in his parabolas; others, especially those found by the astronomers, are not nearly so transparent. But all are remarkable.

If two things are invariably correlated, it is natural to assume that one is the cause of the other or both have a common cause. It is inconceivable, as I said, that water running from a tank in Padua can cause inertial motion of balls in northern Italy. It is just as inconceivable that the spinning Earth causes the planets to satisfy Kepler’s second law. Kepler, in fact, thought that it arose because all the planets were driven in their orbits by a spinning Sun, but we must look further now for a common cause. We shall find it in a second great clock in the sky. This will be the ultimate clock. The first step to it is the inertial clock.

THE INERTIAL CLOCK

The German mathematician Carl Neumann took this first step to a proper theory of time in 1870. He asked how one could make sense of Newton’s claim, expressed in the law of inertia, that a body free of all disturbances would continue at rest or in straight uniform motion for ever. He concluded that for a single body by itself such a statement could have no meaning. In particular, even if it could be established that the body was moving in a straight line, uniformity without some comparison was meaningless. It would then be necessary to consider at least two bodies. He introduced the idea of an inertial clock. He supposed that one body was known to be free of forces, so that equal intervals of its motion could then be taken to define equal intervals of time. With this definition, it would be possible to see if the other body, also known to be free of forces, moved uniformly. If so, then in this sense Newton’s first law would be verified.

Neumann’s idea illustrates the truth that time is told by matter – something has to move if we are to speak of time. Unfortunately, it left unanswered at least three important questions. How can we say that a body is moving in a straight line? How can we tell that it is not subject to forces? How are we to tell time if we cannot find any bodies free of forces?

The answers to these questions will tell us the meaning of duration. If we leave aside for the moment issues related to Einstein’s relativity theories and quantum mechanics, time as we experience it has two essential properties: its instants come in a linear sequence, and there does seem to be a length of time, or duration. I have tried to capture the first property by means of model instants. A random collection of such model instants would correspond to points scattered over Platonia. They would not lie on a single curve, and the fact that they do is, if verified, an experimental fact of the utmost importance. It enables us to talk about history.

But what enables us to talk so confidently of seconds, minutes, hours? What justification is there for saying that a minute today has the same length as a minute tomorrow? What do astronomers mean when they say the universe began fifteen billion years ago? Conditions soon after the Big Bang were utterly unlike the conditions we experience now. How can hours then be compared with hours now? To answer this question, I shall first assume that there are no forces in the world and that the only kind of motion is inertial. This simplification already enables us to get very close to the essence of time, duration and clocks. Then we shall see what forces do.

Suppose Newton claims that three particles, 1, 2 and 3, are moving purely inertially and that someone takes snapshots of them. These snapshots show the distances between the particles but nothing else (except for marks that identify the particles). We know neither the times at which the snapshots were taken nor any of the particles’ positions in absolute space. How can we test Newton’s assertion? We shall be handed a bag containing triangles and told to check whether they correspond to the inertial motion of three particles at the corners of the triangles. The Scottish mathematician Peter Tait solved this problem in 1883 (Box 7).

BOX 7 Tait’s Inertial Clock

Tait used the relativity principle (Box 5) to simplify things. If the particles are moving inertially, one can always suppose that particle 1 is at rest; it is shown in Figure 19 as the black diamond where the three coordinate axes x, y, z in absolute space meet. Now, unless the particles collide at some time – and we need not bother about this exceptional case – there must come a time at which particle 2 passes 1 at some least distance a. The coordinate axes can be chosen so that the line of its motion is as shown by the string of black diamonds. We can choose the unit of time so that particle 2 has unit speed. It will be Neumann’s clock, and each unit of distance it goes will mark one unit of time. Several positions of particle 2 are shown. They are the ‘ticks’ of the clock. At time t = 0, let particle 2 be at the point closest to particle 1 (at the black diamond on the x axis). At this time, particle 3 can be anywhere (three unknown coordinates) and have any velocity (three more unknowns). Thus, seven numbers are unknown: six for particle 3 and the distance a.

Figure 19 The arrangement of coordinates in Tait’s problem.

Now, each snapshot contains three independent data – the three sides of the triangle at the instant of the snapshot. It would seem that three snapshots give nine data – more than enough for the accomplished Tait to solve the problem. But since we know none of the times at which the snapshots are taken, each gives only two useful data. Thus, four snapshots will give eight useful data, seven of which will establish the Newtonian frame into which the triangles fit, while the remaining one will verify that they are indeed obeying Newton’s law. Figure 20 shows a typical solution.

Figure 20 A solution of Tait’s problem. The four ‘snapshots’ of the triangles (the given data) are shown in plan, with an indication of their positions in the ‘sculpture’ of four triangles created by the solution.

Tait’s solution creates nothing less than Newton’s invisible framework. It is both absolute space and the positions of the triangles in it at different instants of ‘time’. We start with triangles. They are all that we have. We are told that they are not random triangles but have arisen through a law. We test this assertion and succeed in creating a simple ‘sculpture’ of three straight lines. The order created is dramatic. Apart from exceptional cases, the distances of the particles from one another do not change in a mutually uniform manner. In fact, they vary with respect to one another in a quite complicated way. Moreover, the triangles in themselves give no hint that there is any space in which their corners lie on straight lines. Yet such lines can be found.

It is equally remarkable that the motions along these lines are mutually uniform. Each particle is the ‘hand’ of a clock for the motion of the other two. The sculpture is a clock with many hands (two in this case, since particle 1 defines the origin). And, now that we have the rigid structure, we see that absolute space is redundant. The sculpture holds together on its own. The ‘room’ was never there until it was created from the triangles and rules. It is they that give an almost bodily tangibility to space and time.

They also explain the meaning of duration and the statement that a second today is the same as a second tomorrow. Duration is reduced to distance. If today or tomorrow any one of the ‘hands’ of the inertial clock moves through the same distance, then we can say that the ‘same amount of time’ has passed. The extra time dimension is redundant: everything we need to know about time can be read off from distances. But note how special is the distance that leads to a meaningful definition of duration. Any change of distance ‘labels’ the instants of time. In statements like ‘Particle A hits B when C is five metres from D’, ‘five metres’ identifies an instant of time – it labels the instant. That is sufficient for history. However, the obvious changes of distance – those between particles – do not lead to a sensible definition of duration. The secrets of time are rather well hidden.

A similar construction can be repeated for any number of particles – a hundred, a billion, quite enough to fill the sky and make a galaxy or even a universe. It is important that if there are more than five particles then three snapshots are already sufficient to solve Tait’s problem, but two are never enough. This is very odd. For a thousand bodies, three snapshots contain far more information than we need, but two never give quite enough. The problem is exactly the one we encountered earlier. Two snapshots tell nothing about the relative orientations or the separation in time. We lack four pieces of information, and all the mystery of absolute space and time resides in them. We cannot make a clock until we get our hands on them. But when we have them, the properties that are revealed are very striking.

For example, Tait’s construction is a good model for the motions of many thousands of stars that are relatively close neighbours of our Sun. They all ‘fall’ in the gravitational field of the Galaxy in much the same way. That motion hardly shows up in the relative separations. But also, because the stars are so far apart there is very little gravitational interaction between them. Their motions are thus well described by the construction. What is more, any three stars can be chosen to make a ‘Tait clock’ and tell the time. Any other three will make another. Thousands, millions of such clocks can be made. All these clocks, light years apart, keep time with one another.

I mentioned earlier the importance of not being misled by the special circumstances of our existence. One of them is the Earth. Only the tiniest fraction of the matter in the universe is in solid form. Indeed, only a small fraction of the Earth – its crust, on which we live, and the innermost core – is solid. This is our home, and we take it for the normal run of things. The ground, trees, buildings, hills and mountains make a framework, which is so like absolute space. It does seem quite natural that a body should move in a straight line in such a space. But we need to think what the universe in its totality is like. Take a billion particles and let them swarm in confusion – that is the reality of ‘home’ almost everywhere in the universe. The stars do seem to swarm, so do the atoms in the stars. To understand the real issues of timekeeping, we must imagine trying to do it in typical circumstances. We must master celestial timekeeping and not be content with the short cuts that can be taken on the Earth, for they hide the essence of the problem.

THE SECOND GREAT CLOCK

We have seen how to check whether bodies are moving inertially without prior access to absolute space and time. But all matter in the universe interacts. Interactions make things more complicated, and not only because the calculations are hard. If objects are moving inertially, any three will suffice to construct an inertial clock. But in a system of interacting bodies it is not possible to treat any of them separately because each is affected by the others. In addition, we can find no framework at all in which the bodies move uniformly in straight lines.

There are three parts to a clock: a mechanism, a clock-face and hands. The main problem of celestial timekeeping arises because the clock-face is invisible. A further problem is that the hands run at varying rates. Imagine an isolated system of three gravitationally interacting stars. We are again given only their relative positions, from which we are to construct a clock. Because of their interactions, no framework exists in which the three stars move along straight lines. The best we can achieve is some ‘spaghetti sculpture’ (Figure 13). This is found by telling a computer that there does exist a framework of absolute space and time in which the stars obey Newton’s laws. However, the computer is given only the successive relative positions, not the positions in the framework at given times. But this is real information, and if the computer is given a sufficient number of snapshots it can search for an arrangement of them in a spaghetti sculpture in which the stars do obey Newton’s laws. The positions in the framework and the separations in time are found by trial and error.

Suppose we are given ten snapshots. We can mark the positions of the triangles formed by the three bodies in Triangle Land (Figures 3 and 4). We can then tell the computer four of the positions. If the snapshots have indeed been generated by bodies that satisfy Newton’s laws, the computer will find a curve that passes through them and the other six. We obtain a curve like those in Shape Space in Figures 9 and 10. We have to use both representations, in Triangle Land and in absolute space, because the raw data come to us in the former, but it is in the latter that we can make sense of them. Once we have solved the problem in absolute space, a timing of the evolution has been established. It is that timing of the events for which Newton’s laws do hold. If the computer tried to assign other timings, they would not. The timing that does work can then be transferred back to the raw data: the curve in Triangle Land. We can make marks along it corresponding to the passage of the time found by the computer.

The same thing can be done for any number of bodies. Their relative configurations will correspond to different points along a curve in the corresponding Platonia. To lay out ‘marks of equal intervals of time’ on it, we have to go through the same procedure with the computer, telling it to find a framework and a time in which the bodies do satisfy Newton’s laws. Only two facts about this process are significant. First, because all the bodies interact, all their positions must be used if the ‘time marks’ are to be found. To tell the time by such a clock, we need to know where all its bodies are. Time cannot be deduced from a small number of them, unlike inertial time; the clock has as many hands as the system has bodies. Second, no matter how many bodies there are in a system, the data in just two snapshots are never enough to find the spaghetti sculpture in absolute space and construct a clock. We always need at least some data from a third snapshot. As we have seen, this ‘two-and-a-bit puzzle’ is the main – indeed the only – evidence that absolute space and not Platonia is the arena of the universe.

You might think that this is all far removed from practical considerations. It is true that scientists have learned to make extremely accurate clocks using atomic phenomena. But this is a comparatively recent development. Before then, astronomers faced a tricky situation, which is worth recounting.

For millennia, the Earth’s rotation provided a clock sufficiently reliable and accurate for all astronomical purposes. It was unique – the astronomers had access to no other comparable clock. However, about a hundred years ago, astronomical observations had become so accurate that deficiencies in it began to show up. Tidal forces of the Moon acting on the Earth sometimes give rise to unpredictable changes of the mass distribution in its interior. As my accident in Oxford demonstrated, such changes in a rotating body must change its rotation rate. The clock was beginning to fail the astronomers’ growing needs for greater accuracy. Such crises highlight fundamental facts. What could the astronomers do?

They managed to find a natural clock more accurate than the Earth: the solar system. To make this into a clock, they assumed that Newton’s laws governed it. (After the discovery of general relativity, small corrections had to be made to them, but this did not change the basic idea.) However, the astronomers had no direct access to any measure of time. Instead, they had to assume the existence of a time measure for which the laws were true. Making this assumption and using the laws, they could then deduce how all the dynamically significant bodies in the solar system should behave. Although they had no access to it, they then knew where the various bodies should be at different instants of the assumed time. Monitoring one body – the Moon, in fact – they could check when it reached positions predicted in the assumed time and verify that the other bodies in the solar system reached the positions predicted for them at the corresponding times. The astronomers were thus forced into the exercise just described, and they used the Moon as the hand of a clock formed by the solar system.

They originally called the time defined in this manner Newtonian time. It is now called ephemeris time. (An ephemeris is a publication which gives positions of celestial objects at given times.) For a decade or more it was actually the official time standard for civil and astronomical purposes. More recently, atomic time, which relies on quantum effects, has been adopted. There are several important things about ephemeris time. First, it is unthinkable without the laws that govern the solar system. Second, it is a property of the complete solar system (because all its bodies interact, all co-determine one another’s positions). Third, it exists only because the solar system is well isolated as a dynamical system from the rest of the universe.

Ephemeris time may be called the unique simplifier. This is an important idea. If, as Mach argued, only configurations exist and there is no invisible substance of time, what is it that we call time? When we hold the configurations apart in time and put a duration between them, this something we put there is a kind of imagined space, a fourth dimension. The spacing is chosen so that the happenings of the world unfold in accordance with simple laws (Newton’s or Einstein’s). This is a consequence of the desire to represent things in space and time, and our inability hitherto to find laws of a simple form in any other framework.

Ephemeris time is the only standard we can use if clocks are to march in step. If we could not construct such clocks, we could never keep appointments and clocks would be useless. To see that there is only one sensible definition of duration, imagine that two teams of astronomers were sent to two similar but nevertheless different isolated three-body stellar systems. All they can do is observe their motions. From them they must generate time signals. Each team works separately, but the signals they generate must march in step – one clock may run faster than the other, but the relative rate must stay constant. There is only one measure of duration they can choose. In general, no motion in one system marches in step with any motion in the other. Only ephemeris time, deduced from the system as a whole, does the trick. A clock is any mechanical device constructed so that it marches in step with ephemeris time, the unique simplifier.

We can now see that there is only one ultimate clock: the universe. Although it would not be practicable, if we wanted to obtain time of ultra-high accuracy from the solar system, we would, sooner or later, have to take into account the disturbances exerted by bodies farther away. Since there are no perfectly isolated systems within the universe, this process can only stop, if ever, when the entire universe has been made into a clock. The universe is its own clock.

In the light of this, let us think again about Galileo’s ball rolling across the table in Padua. Snapshots of the ball alone were not sufficient to tell what would happen when it rolled over the edge. It seemed inconceivable that the ball’s path could be determined by the little bit of water flowing from a tank used to tell the time. For such reasons as this, Newton rejected speed relative to any one motion as a fundamental concept and invoked instead speed relative to an abstract time. However, if we conceive the universe as a single dynamical entity, the abstract time becomes redundant. The speed of Galileo’s ball that determines which parabola it will trace is its speed as measured by the totality of motions in the universe. This explains why some motions are distinguished from others for timekeeping. They are those that march in step with the cosmic clock, the unique true measure of time. This time is the distillation of all change. High noon is a configuration of the universe.

But the two-and-a-bit puzzle persists. We still have no simple direct way to measure time in Platonia, we always have to go through the intermediary of absolute space. This reflects the hybrid nature of energy. Kinetic energy is defined in absolute space, whereas potential energy is determined by instantaneous configurations and is thus independent of Newton’s invisible framework. We shall be able to claim that Platonia is the arena of the world only if we can dispense with absolute space in the definition of kinetic energy. That is the next topic.