The End of Time: The Next Revolution in Physics

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.