You can see directly how absolute space and time are created out of timelessness. Take some point on one of the Machian geodesics in Platonia; it is a configuration of masses. Take another point a little way along the geodesic; it is a slightly different configuration. Without any use of absolute space and time, using just the two configurations, you can bring the second into the position of best matching relative to the first. You can then take a third configuration, a bit farther along the path, and bring it into its best matching position relative to the second configuration. You can go along the whole path in this way. The entire string of configurations is oriented in a definite position relative to the first configuration. What looks like a framework is created, but it is not a pre-existing framework into which the configurations of the universe are slotted: it is brought into being by matching the configurations. Nevertheless, we get something like the Newtonian picture in Figure 1, except that we do not as yet have the ‘spacings in time’.
But this too emerges from the Machian theory. In the equations that describe how the objects move in the framework built up by best matching, it is very convenient to measure how far each body moves by making a comparison with a certain average of all the bodies in the universe. The choice of the average is obvious, and simplifies the equations dramatically. No other choice does the trick. For this reason it needs a special name; I shall call it the Machian distinguished simplifier. It is directly related to the quantities used to determine the geodesic paths in Platonia. To find how much it changes as the universe passes from one configuration to another slightly different one, it is necessary only to divide their intrinsic difference by the square root of minus the potential. (The action, by contrast, is found by multiplying it by the same quantity.) When this distinguished simplifier is used as ‘time’, it turns out that each object in the universe moves in the Machian framework described above exactly as Newton’s laws prescribe. Newton’s laws and his framework both arise from a single law of the universe that does not presuppose them.
In such a universe, the ultimate standard of time that determines which curve is traced by Galileo’s ball when it falls off his table in Padua is unambiguous. It is the average of all the changes in the universe that defines the Machian distinguished simplifier. Time is change, nothing more, nothing less.
The difference between the Newtonian and Machian theories can be summarized as follows. If we do not know the energy and angular momentum of a Newtonian system, we always need at least three snapshots of its configurations in order to reconstruct the framework of space and time in which they obey Newton’s laws. The task is complicated, to say the least. If, however, the system is Machian, the framework can be found with just two snapshots and the task is vastly simpler. It simply requires best matching of the two configurations.
When, later, I suggest that the quantum universe is timeless in a deeper sense than the classical Machian universe just described, that will be a conjecture. But it is made plausible by the results of this chapter. They are not speculation but mathematical truths. Every phenomenon explained by Newton’s laws, including the beautiful rings of Saturn and the spectacular structure of spiral galaxies, can be explained without absolute space and time. They follow from a simpler, timeless theory in Platonia.
Nature and Exploration (p. 109) For physicists and mathematicians who do not know the book, a wonderful account of the variational principles of mechanics, together with much historical material, is given by Lanczos (1986).
Developing Machian Ideas (p. 115) Translations of the papers by Hofrnann, Reissner and Schrödinger, along with other historical and technical papers on Mach’s principle, can be found in Barbour and Pfister (1995).
Exploring Platonia (p. 115) The special properties of Newtonian motions with vanishing angular momentum were discovered independently of the work of Bertotti and myself by A. Guichardet in the theory of molecular motions and by A. Shapere and E Wilczek in the theory of how micro-organisms swim in viscous fluids! A rich mathematical theory has meanwhile developed, and is excellently reviewed in the article by Littlejohn and Reinsch (1997), which contains references to the original work mentioned above. All mathematical details, as well as references to the earlier work by Bertotti and myself, can be found in Barbour (1994a).
PART 3
The Deep Structure of General Relativity
Now we come to relativity. My aim is not to give an extended account, only to show how its fundamental features relate to the book’s theme. But I have a tough nut to crack. My subject is the non-existence of time, whereas time is almost everything in relativity as it is usually presented. Is relativity Hamlet without the Prince of Denmark?
In fact, the evidence for the non-existence of time in relativity has long been hidden by accidents of historical development, and is far stronger than many people realize. Yet the case is not quite conclusive. We have seen how the space and time of Newton’s theory can be constructed from instants of time as defined in this book. Taking them to be the true atoms of existence, we have shown that no external framework is needed. Einstein’s space-time can also be put together from instants in a strikingly similar way. However, in the finished product they are knit together far more tightly than in Newtonian theory. Explaining the wonderful way in which this happens is the goal now. If the world were classical, no one would try to pull space-time apart into instants. But quantum theory will probably shatter space-time. It is therefore sensible to consider the constituents into which it might shatter. This is what I shall do in Part 3.
I begin by looking at the special theory of relativity, in which gravity plays no role. I then go on to the general theory, in which Einstein found a most brilliantly original way to describe gravity. In both relativity theories time seems to be very real and to behave in baffling ways. But, as became clear only after Einstein’s death, his theory has a deep structure which is revealed only by an analysis of how it works as a dynamical theory. It is this deep structure that is timeless. Quite a large proportion of Part 3 will explain the purely historical accidents that obscured the deep structure of general relativity for so long.