The Aims of Machian Mechanics (1) (p. 71) In creating the beautiful diagrams that form such an important part of this section, Dierck Liebscher was able to draw on initial data devised by Douglas Heggie (University of Edinburgh), using software written by Piet Hut (Institute for Advanced Study), Steve McMillan (Drexel University) and Jun Makino (University of Tokyo). Dierck has written a very interesting book (alas, as yet published only in German) on the connection between different possible geometries and Einstein’s relativity theory (Liebscher 1999). It contains many striking computer-generated diagrams.
(2) Poincaré’s discussion is contained in his Science and Hypothesis, which, along with the writings of his contemporary, Mach, became a popular-science best-seller. In fact, in this book I am actually revisiting many of the themes discussed by Poincaré and Mach, but with the advantage of hindsight. How are the great issues they raised changed by the discovery of general relativity and quantum mechanics? I have adapted Poincaré’s discussion somewhat to match the requirements of a timeless theory (he considered only the possibility of eliminating absolute space).
(3) Since writing Box 3, which draws attention to the present unsatisfactory use of absolute dislance in physics, I have discovered a way to create dynamical theories in which distance is not absolute. This is achieved by a very natural extension of the best-matching idea described later in the book. The new insights that I mention in the Preface are in part connected with this development. One of the most exciting is that, if such theories do indeed describe the world, gravitation and the other forces of nature are precisely the mechanism by means of which absolute distance is made irrelevant. Since this work is still in progress, I shall make no attempt to describe it in detail, but I shall keep my website (www.julianbarbour.com) up to date with any progress (see also p.358).
CHAPTER 6
The Two Great Clocks in the Sky
Newton’s mysterious ‘timepiece’ and speeds measured relative to it figured prominently in the last chapter. But is it really there, and how can we ever read its time if it is invisible? This chapter is about these two questions.
A simple but famous experiment of Galileo provides strong evidence for something very like Newton’s absolute time. He rolled a ball across a table and off its edge. His analysis of its fall was a major step in mechanics. First he noted the ball’s innate tendency to carry on forward in the direction it had followed on the table. It also started to fall under gravity, picking up speed. Galileo conjectured that two processes were at work independently, and that each could be analysed separately. The total effect would be found by simply adding the two processes together.
Galileo’s recognition of the tendency to keep moving forward anticipated Newton’s law of inertia. He did not recognize it as a universal law, but he did make it precise in some special cases. For the example of the ball, he conjectured that but for gravity (and air resistance) the ball would move for ever forward with uniform speed. (He actually thought that the motion would be around the Earth – Galileo’s inertia was circular. Luckily, the difference was far too small to affect his analysis.)
As for the second process, Galileo had already found that if an object is dropped from rest and in the first unit of time falls one unit of distance, then in the next it will fall a further three, in the next five, and so on. He was entranced by this, and called it the odd numbers rule. Now consider the sequence:
at t = 1, distance fallen = 1,
at t = 2, distance fallen = 1 + 3 = 4,
at t = 3, distance fallen = 1 + 3 + 5 = 9,
at t = 4, distance fallen = 1 + 3 + 5 + 7 = 16,…
The distance fallen increases as the square of the time: 12 = 1, 22 = 4, 32 = 9, 42 = 16,... . Galileo’s originality was to seek for a deeper meaning in this pattern.
Many teenagers can now do in seconds a calculation that took Galileo a year or more – it was so novel. He asked: if the distance fallen increases as the square of the time, how does the speed increase with time? He eventually found that it must increase uniformly with time. If after the first unit of time the object has acquired a certain speed, then after the second it will have twice that speed, after the third three times, and so on. Galileo’s work showed that, in the absence of air resistance, a falling body always has a constant acceleration. It never ceases to amaze me what consequences flowed from Galileo’s simple but precise question. It taught his successors how to read the ‘great book of nature’ (Galileo’s expression). From a striking empirical pattern, he had found his way to a simpler and deeper law.
To analyse the falling ball, Galileo simply combined the two processes – inertia and falling – under the assumption that each acts independently. He obtained the famous parabolic motion (Figure 18). In each unit of time, the ball moves through the same horizontal distance, but in the vertical dimension the distance fallen grows as the square of the time. The resulting curve traced by the ball is part of a parabola. Newton applied Galileo’s method for terrestrial motions to the heavens, and showed that the laws of motion had universal validity. This was the first great unification in physics. There may be a lesson for us here in our present quest – the search for time. We may have to look for it in the sky.
A search is needed. It is striking that all the elements in Galileo’s analysis are readily visualized. You can easily call up a table and the parabola traced by the ball in your imagination. Yet one key player seems reluctant to appear on the stage. Where is time? This is the question I have so far dodged. It presents a severe challenge to the idea that configurations are all that exist. Suppose that we take snapshots of the ball as it rolls across Galileo’s table in Padua, where he experimented. These snapshots can show everything in his studio. However, only the ball is moving. We take lots of snapshots, at random time intervals, until the ball is just about to fall over the edge. We put the snapshots, all mixed up, in a bag and, supposing time travel is possible, present it to Galileo and ask him whether, by examining the snapshots, he can tell where the ball will land.
He cannot. Had we rolled the ball twice as fast, it would have passed through the identical sequence of positions to the table’s edge, and they are all the snapshots capture. The speed is not recorded. But the ball’s speed determines the shape of the parabola, and hence where the ball lands. In fact Galileo will not even know in which direction the ball is going. Perhaps it will fall off the right-hand side of the table. More clearly than with the three-body evolutions, which mix the effects of time and spin, we see here the entire evidence for absolute time. The speed determines the shape of the parabola. There is manifestly more to the world than the snapshots reveal. What and where is it? Galileo himself provides an answer of sorts. He tells us that he measured time by a water-clock – a large water tank with a small hole in the bottom. His assistant would remove a finger from the hole and let the water flow into a measuring flask until the timed interval ended. The amount of water measured the time.