Julian Barbour - The End of Time - The Next Revolution in Physics

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 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).