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

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.

The Inertial Clock(p. 99) Tait’s work, which I feel is very important, passed almost completely unnoticed. This is probably because two years later the young German Ludwig Lange introduced an alternative construction for finding inertial frames of reference, coining the expression ‘inertial system’. Lange deserves great credit for bringing to the fore the issue of the determination of such systems from purely relative data, but Tait’s construction is far more illuminating. Lange’s work is discussed in detail in Barbour (1989) and Tait’s in Barbour (forthcoming).

The Second Great Clock(p. 107) A very nice account of the history of the introduction of ephemeris time was given by the American astronomer Gerald Clemence (1957).

The two-and-a-bit puzzle is the statement that two snapshots of a dynamical system are nearly but not quite sufficient to predict its entire history. We need to know not only two snapshots, but also their separation in time and their relative orientation in absolute space. These are exactly the things that determine the energy and angular momentum of any system, both of which, as we have seen, have a profound influence on its behaviour.

There are two different ways to approach this problem. Either we assume the known laws of nature are correct and simply ask how they can be verified, or we take a more ambitious stance and ask if they arise from some deeper level that we have not yet comprehended. The latter is the approach of this chapter. We shall forget absolute space and time and take Platonia for real. I have likened it to a country; countries are there to be explored. In exploring a country, one follows a path through it. Any continuous curve through Platonia is such a path. A natural question is whether some paths are distinguished compared with others. It leads directly to the idea of optimization.

Optimization problems arise naturally, and were already well known to mathematicians in antiquity. It seems they were also known and understood by Queen Dido, who when she came to North Africa was granted as much land as she could enclose within the hide of a cow. She cut it into thin strips, out of which she made a long string. Her task was then to enclose the maximum area of land within it. The solution to this problem of maximizing the area within a figure of given perimeter is a circle. However, Dido’s territory was to adjoin the coast, which did not count as part of the perimeter. For a straight coastline the solution to this problem is a semicircle, and this was said to be the origin of the territory of Carthage. A rich body of mathematical and physical theory has developed out of similar problems. It cannot explain why the universe is, but given that the universe does exist it goes a long way to explain why it is as it is and not otherwise.

In early modern times, Pierre de Fermat (of the famous last theorem) developed a particularly fruitful idea due to Hero of Alexandria, who had sought the path of a light ray that passes from one point to another and is reflected by a flat surface on its way. Hero solved this problem by assuming that light travels at a constant speed and chooses the path that minimizes the travel time. Fermat extended this least-time idea to refraction, when light passes from one medium to another, in which it may not travel at the same speed as in the first medium. When a ray of light passes from air into water, the ray is refracted (bent) downward, towards the normal (perpendicular) to the surface. If this behaviour is to be explained by the least-time idea, light must travel slower in water than in air. For a long time it was not known if this were so, so Fermat’s proposal was a prediction, which was eventually confirmed.

In 1696 John Bernoulli posed the famous ‘brachistochrone’ (shortesttime) problem. A bead, starting from rest, slides without friction under gravity on a curve joining two points at different heights. The bead’s speed at any instant is determined by how far it has descended. What is the form of the curve for which the time of descent between the two points is shortest? Newton solved the problem overnight, and submitted his solution anonymously, but Bernoulli, recognizing the masterly solution, commented that Newton was revealed ‘as is the lion by the claw print’. The solution is the cycloid, the curve traced by a point on the rim of a rolling wheel.

Soon there developed the idea that the laws of motion – and thus the behaviour of the entire universe – could be explained by an optimization principle. Leibniz, in particular, was impressed by Fermat’s principle and was always looking for a reason why one thing should happen rather than another. This was an application of his principle of sufficient reason: there must be a cause for every effect. Leibniz famously asked why, among all possible worlds, just one should be realized. He suggested, rather loosely, that God – the supremely rational being – could have no alternative but to create the best among all possible worlds. For this he was satirized as Dr Pangloss in Voltaire’s Candide . In fact, in his main philosophical work, the Monadology , Leibniz makes the more defensible claim that the actual world is distinguished from other possible worlds by possessing ‘as much variety as possible, but with the greatest order possible’. This, he says, would be the way to obtain ‘as much perfection as possible’.

Inspired by such ideas, the French mathematician and astronomer Pierre Maupertuis (another victim of Voltaire’s satire), advanced the principle of least action (1744). From shaky initial foundations (Maupertuis wanted to couple his idea with a proof for the existence of God) this principle grew in the hands of the mathematicians Leonhard Euler and Joseph-Louis de Lagrange into one of the truly great principles of physics. As formulated by Maupertuis, it expressed the idea that God achieved his aims with the greatest economy possible – that is, with supreme skill. In passing from one state at one time to another state at another time, any mechanical system should minimize its action , a certain quantity formed from the masses, speeds and separations of the bodies in the system. The quantities obtained at each instant were to be summed up for the course actually taken by the system between the two specified states. Maupertuis claimed that the resulting total action would be found to be the minimum possible compared with all other conceivable ways in which the system could pass between the two given states. The analogy with Fermat’s principle is obvious.

Unfortunately for Maupertuis’s theological aspirations, it was soon shown that in some cases the action would not have the smallest but greatest possible value. The claims for divine economy were made to look foolish. However, the principle prospered and was cast into its modern form by the Irish mathematician and physicist William Rowan Hamilton a little under a hundred years after Maupertuis’s original proposal. A wonderfully general technique for handling all manner of mechanical problems on the basis of such a principle had already been developed by Euler and, above all, Lagrange, whose Mécanique analytique of 1788 became a great landmark in dynamics.