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

What did quite surprise us was to discover that the unique Machian history with a given direction through a point is identical to one of the many Newtonian histories through the point with the same direction. It is, in fact, the Newtonian history for which the energy and angular momentum are both exactly zero. The small fraction of Newtonian solutions with this property are all the solutions of a simpler timeless and frameless theory.

This brought to light an unexpected reconciliation between the positions of Newton and Leibniz in their debate about absolute and relative motion. Both were right! The point is that in a universe which, like ours, contains many bodies, there can be innumerable subsystems that are effectively isolated from one another. This is true of the solar system within the Galaxy, and also for many of the galaxies scattered through the universe. Each subsystem, considered by itself, can have nonzero energy and angular momentum. However, if the universe is finite, the individual energies and angular momenta of its subsystems can add up to zero. In a universe governed by Newton’s laws this would be an implausible fluke. But if the universe is governed by the Machian law, it must be the case. It is a direct consequence of the law. What is more, the Machian law predicts that in a large universe all sufficiently isolated systems will behave exactly as Newton predicted. In particular, they can have nonzero energy and angular momentum, and therefore seem to be obeying Newton’s laws in absolute space and time. But what Newton took to be an unalterable absolute framework is shown in the Machian theory to be simply the effect of the universe as a whole and the one law that governs it. What physicists have long regarded as laws of nature and the framework of space and time in which they hold are, as I said in Chapter 1, both ‘local imprints’ of that one law of the universe.

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).