The End of Time: The Next Revolution in Physics

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.