The End of Time: The Next Revolution in Physics

CHAPTER 4

Alternative Frameworks

ABSOLUTE OR RELATIVE MOTION?

Both Copernicus and Kepler believed that the universe, with the solar system at its centre, was bounded by a huge and distant rigid shell on which the luminous stars were fixed. They did not speculate what lay beyond – perhaps it was simply nothing. They defined all motions relative to the shell, which thus constituted an unambiguous framework. Many factors, above all Galileo’s telescopic observations in 1609 and the revival of interest in the Greeks’ idea of atoms that move in the void, destroyed the old cosmology. New ideas crystallized in a book that Descartes wrote in 1632. He was the first person to put forward clearly an idea which, half a century later, Newton would make into the most basic law of nature: if nothing exerts a force on them, all bodies travel through space for ever in a straight line at a uniform speed. This is the law of inertia. Descartes never published his book because in 1633 the Inquisition condemned Galileo for teaching that the Earth moves. The Copernican system was central to Descartes’s ideas, and to avoid Galileo’s fate he suppressed his book.

He did publish his ideas in 1644, in his influential Principles of Philosophy, but with a very curious theory of relative motion as an insurance policy. He argued that a body can have motion only relative to some other body, chosen as a reference. Since any other body could play the role of reference, any one body could be regarded as having many different motions. However, he did allow a body to have one true ‘philosophical motion’, which was its motion relative to the matter immediately adjacent to it. (Descartes believed there was matter everywhere, so any body did always have matter adjacent to it.) This idea let him off the Inquisition’s hook, since he claimed that the Earth was carried around the Sun in a huge vortex, as in a whirlpool. Since the Earth did not move relative to the immediately adjacent matter of the vortex, he argued that it did not move!

However, he then formulated the law of inertia, just as in 1632. When, sometime around 1670, long after Descartes’s death in 1650, Newton came to study his work, he immediately saw the flaw. To say that a body moves in a straight line presupposes a fixed frame of reference, which Descartes had denied. Since Newton could see the great potential of the law of inertia, to exploit it he came up with the concept of an immovable space in which all motion takes place. He was very scornful of Descartes’s inconsistency, and when he published his own laws in 1687 he decided to make it a big issue, without, however, mentioning Descartes by name. He introduced the notion of absolute space and, with it, absolute time.

Newton granted that space and time are invisible and that one could directly observe only relative motions, not the absolute motions in invisible space. He claimed that the absolute motions could nevertheless be deduced from the relative motions. He never gave a full demonstration of this, only an argument designed to show that motion could not be relative. He was making a very serious point, but at the same time he wanted to make a fool of Descartes. This had strange and remarkable consequences.

Descartes had sought to show that all the phenomena of nature could be explained mechanically by the motion of innumerable, tiny, invisible particles. Vital to his scheme was the centrifugal force felt as tension in a string that retains a swung object. The object seems to be trying to escape, to flee from the centre of rotation. In Newtonian terms, it is actually trying to shoot off along the tangent to the circle, but that is still a motion that would take it away from the centre and create the tension. Descartes claimed that light was pressure transmitted from the Sun to the Earth by centrifugal tension set up in the vortex that he pictured swirling around the Sun. Because centrifugal force was so important to Descartes, Newton used it to show that motion cannot be relative. Newton’s intention was to hoist Descartes by his own petard.

Newton imagined a bucket filled with water and suspended by a rope from the ceiling. The bucket is turned round many times, twisting the rope, and is then held still until the water settles. When the bucket is released, the rope unwinds, twisting the bucket. Initially the surface of the water remains flat, but slowly the motion of the bucket is transmitted to the water, which starts to spin, feels a centrifugal force and starts to rise up the side of the bucket. After a while, the water and bucket spin together without relative motion, and the water surface reaches its greatest curvature.

Newton asked what it was that caused the water’s surface to curve. Was it the water’s motion relative to the side of the bucket (Descartes’s claimed true philosophical motion relative to the immediately adjacent matter) or motion relative to absolute space? Surely the latter, since when the relative motion is greatest, at the start, there is no curvature of the water’s surface, but when the relative motion has stopped (and the water and bucket spin together) the curvature is greatest. This was Newton’s main argument for absolute space. It was strong and it ridiculed Descartes.

In Newton’s lifetime, his notion of absolute space, to which he gave such prominence, attracted strong criticism. If space were invisible, how could you say an object moves in a straight line through a space you cannot see? Newton never satisfactorily answered this question. Many people felt, as Descartes did, that motion must be relative to other matter, though not necessarily adjacent matter. Bishop Berkeley argued that, as in Copernican astronomy, motion must ultimately be relative to the distant stars, but he failed to get to grips with the problem that the stars too must be assumed to move in many different ways and thus could not define a single fixed framework, as Copernicus and Kepler had believed.

Newton’s most famous critic was the great German mathematician and philosopher Wilhelm Gottfried Leibniz, who had been involved in a very unpleasant dispute with Newton about which of them had first discovered the calculus, the revolutionary new form of mathematics that made so many things in science much easier, including the development of mechanics. In 1715, Leibniz began a famous correspondence on Newton’s ideas with Samuel Clarke, who was advised by Newton. The Leibniz-Clarke Correspondence has become a classic philosophy text. Many undergraduates study it, and philosophers of science often discuss it.

The exchange had an inconclusive outcome. It is generally agreed that Leibniz advanced effective philosophical arguments, but he never addressed the detailed issues in mechanics. Typically, he argued like this. Suppose that absolute space does exist and is like Newton claimed, with every point of space identical to every other. Now consider the dilemma God would have faced when he created the world. Since all places in absolute space are identical, God would face an impossible choice. Where would he put the matter? God, being supremely good and rational, must always have a genuine reason for doing something – Leibniz called this the ‘principle of sufficient reason’ (I have already appealed to this when I discussed brain function and consciousness, by requiring an observable effect to have an observable cause) – and because absolute space offered no distinguished locations, God would never be able to decide where to put the matter. Absolute time, on the assumption that it existed, presented the same difficulty. Newton had said that all its instants were identical. But then what reason could God have for deciding to create the world at some instant rather than another? Again, he would lack a sufficient reason. For reasons like these, not all of them so theological, Leibniz argued that absolute space and time could not exist.

A century and a half passed before the issue became a hot topic again. This raises an important issue: how could mechanics have dubious foundations and yet flourish? That it flourished nevertheless was due to fortunate circumstances that are very relevant to the theme of this book. First, although the stars do move, they are so far away that they provide an effectively rigid framework for defining motions as observed from the Earth. It was found that in this framework Newton’s laws do hold. It is hard to overestimate the importance of this fortunate effective fixity of the distant stars. It presented Newton with a wonderful backdrop and convenient framework. Had the astronomers been able to observe only the Sun, Moon and planets but not the stars (had they been obscured by interstellar dust), Newton could never have established his laws. Thus, scientists were able to accept Newton’s absolute space as the true foundation of mechanics, using the stars as a substitute for the real thing – that is, a true absolute frame of reference. They also found that Newton’s uniformly flowing time must march in step with the Earth’s rotation, since when that was used to measure time (in astronomical observations spanning centuries, and even millennia) Newton’s laws were found to hold. Once again, a substitute for the ‘real thing’ was at hand. One did not have to worry about the foundations. Fortunate circumstances like these are undoubtedly the reason why it is only recently that physicists have been forced to address the issue of the true nature of time.

The person who above all brought the issue of foundations back to the fore was the Austrian physicist Ernst Mach, whose brilliant studies in the nineteenth century of supersonic projectiles and their sonic boom are the reason why the Mach numbers are named after him. Mach was interested in many subjects, especially the nature and methods of science. His philosophical standpoint had points in common with Bishop Berkeley, but even more with the ideas of the great eighteenth-century Scottish empiricist David Hume. Mach insisted that science must deal with genuinely observable things, and this made him deeply suspicious of the concepts of invisible absolute space and time. In 1883 he published a famous history of mechanics containing a trenchant and celebrated critique of these concepts. One suggestion he made was particularly influential.

It arose as a curious consequence of the covert way Newton had attacked Descartes. Considering Newton’s bucket argument, Mach concluded that, if motion is relative, it was ridiculous to suppose that the thin wall of the bucket was of any relevance. Mach had no idea that Newton was attacking Descartes’s notion of the one true philosophical motion, just as Newton had not seen that Descartes had invented it only to avoid the wrath of the Inquisition. Newton had used the bucket argument to show that relative motion could not generate centrifugal force, but Mach argued that the relative motions that count are the ones relative to the bulk of the matter in the universe, not the puny bucket. And where is the bulk of the matter in the universe? In the stars.

This led Mach to the revolutionary suggestion that it is not space but all the matter in the universe, exerting a genuine physical effect, that creates centrifugal force. Since this is just a manifestation of inertial motion, which Newton claimed took place in absolute space, Mach’s proposal boiled down to the idea that the law of inertia is indeed, as Bishop Berkeley believed, a motion relative to the stars, not space. Mach’s important novelty was that there must be proper physical laws that govern the way distant matter controls the motions around us. Each body in the universe must be exerting an effect that depends on its mass and distance. The law of inertia will turn out to be a motion relative to some average of all the masses in the universe. For this basic idea, Einstein coined the expression Mach’s principle, by which it is now universally known (though attempts at precise definition vary quite widely).

Mach’s idea suggests that the Newtonian way of thinking about the workings of the universe, which is still deep-rooted, is fundamentally wrong. The Newtonian scheme describes an ‘atomized’ universe. The most fundamental thing is the containing framework of space and time: that exists before anything else. Matter exists as atoms, tiny unchanging masses that move in space and time, which govern their motion. Except when close enough to interact, the atoms move with complete indifference to one another, each following a straight and lonely path through the infinite reaches of absolute space. The Machian idea takes the power from space and time and gives it to the actual contents of the universe, which all dance in their motions relative to one another. It is an organic, holistic view that knits the universe together. Very characteristic is this remark of Mach in his The Science of Mechanics (pp.287-8):

Nature does not begin with elements, as we are obliged to begin with them. It is certainly fortunate for us that we can, from time to time, turn aside our eyes from the overpowering unity of the All and allow them to rest on individual details. But we should not omit, ultimately to complete and correct our views by a thorough consideration of the things which for the time being we have left out of consideration.

Mach himself made only tentative suggestions for a new relative mechanics, but his remarks caught the imagination of many people, above all Einstein, who said that Hume and Mach were the philosophers who had influenced him most deeply. Einstein spent many years trying to create a theory that would embody Mach’s principle, and initially believed that he had succeeded in his general theory of relativity. That is why he gave it that name. However, after a few years he came to have doubts. Eventually he concluded that Mach’s idea had been made obsolete by developments in physics, especially the theory of electro-magnetism developed by Faraday and Maxwell, which had introduced new concepts not present in Newton’s scheme.

Throughout the twentieth century, physicists and philosophers discussed Mach’s principle at great length, without coming to any conclusion. It is my belief that the problem lies in Einstein’s highly original but indirect approach. Mach had not made a really clear proposal, and Einstein never really stopped and asked himself just what should be achieved by Mach’s principle. I shall consider this in Part 3, but I need to anticipate a small part of the story in order to justify Part 2. Einstein’s theory is rather complicated and achieves several things at once. It is not easy to separate the parts and see the ‘Machian’ structure. In my opinion, general relativity is actually as Machian as it could be. What is more, it is the Machian structure that has such dramatic consequences when one tries to reconcile the theory with quantum mechanics. If, as I believe, the quantum universe is timeless, it is so because of the Machian structure of general relativity. To explain the core issues, I need a simplified model that captures the essentials. This Part 2 will provide. It will also provide a direct link between the great early debate about the foundations of mechanics and the present crisis of quantum cosmology. Two key issues are still the same: what is motion, and what is time? It will also enable me to explain the main work in physics with which I have been involved, and make it easier for you to see why I have come to doubt the existence of time.

Science advances in curious ways, and scientists are often curiously unconcerned with foundations. Descartes was one of the greatest philosophers, yet in that first book in 1632 he never gave a moment’s thought to the definition of motion. We are so used to living on the solid Earth that it seems unproblematic to say that a body moves in a straight line. If the Inquisition had not condemned Galileo, Descartes would never have argued for the relativity of motion. But for the inconsistency of his system, Newton would not have made an issue out of absolute space and time. He would not have devised the bucket argument, Mach might never have had his novel idea, and Einstein would not have been inspired to his greatest creation.

Had the Inquisition condemned Galileo a few months later, Descartes would have published his ideas in their original form – and general relativity might never have been found.

AN ALTERNATIVE ARENA

I would like to say a bit more about my own personal development, which as the book progresses will help you to understand why I am so deeply convinced of the need to have a new concept of time. In the very first days after my trip to the Bavarian Alps, while thinking hard about time, I came across Mach’s book. Like so many others, I was captivated by his idea about inertia. His comments on time also encouraged me greatly: ‘It is utterly beyond our power’, he said, ‘to measure the changes of things by time. Quite the contrary, time is an abstraction, at which we arrive by means of the changes of things.’ This was just the conclusion I had reached. A year or so later, after I had decided to study the foundations of physics, I started to read the papers Einstein had written when he was creating general relativity. Comparing them with what Mach had written, I came to the conclusion that Einstein had simply not set about the problem in the right way: he had not attacked it directly. It seemed to me necessary to go back to first principles.

It was six or seven years before I came to form really clear ideas. I eventually concluded that what was needed above all was a new arena in which to describe the universe. I arrived at the notion of Platonia (or, as I originally called it, the relative configuration space of the universe). The argument was quite simple. First, it is a fact that we orient ourselves in real life by objects we actually see, not by invisible space (see the Notes on the previous chapter). Things are the signposts that tell us where we are. There is also the fortunate fact that we live on the nearly rigid Earth. We can orient ourselves by means of just a few objects fixed on its surface, say church spires when hiking in the English countryside. Always there, the Earth provides a natural background. Motion seems to take place in a framework. But imagine what life would be like if we lived on a jellyfish!

The fact is that we live in a very special location. Only the tiniest fraction of matter in the solar system, let alone the universe, is in solid form. Imagine that we lived in an environment much more typical of the universe – in space. To simplify things, let there be only a finite number of objects, all in motion relative to one another. At any instant there are certain distances between these other objects and us. There is nothing else. In these circumstances, what would be the natural way to answer what is always a fundamental question: where are we? We have no other means of saying where we are except in terms of our distances to other objects. What is more, it would be artificial to choose just a few of them to locate ourselves. Why these rather than those? It would be much more natural to specify our distances to all objects. They define our position. This conclusion is very natural once we become aware that nothing is fixed. Everything moves relative to everything else.

Taking this further, thinking about the position and motion of one object is artificial. We are part of Mach’s All, and any motion we call our own is just part of a change in the complete universe. What is the reality of the universe? It is that in any instant the objects in it have some relative arrangement. If just three objects exist, they form a triangle. In one instant the universe forms one triangle, in a different instant another. What is to be gained by supposing that either triangle is placed in invisible space? The proper way to think about motion is that the universe as a whole moves from one ‘place’ to another ‘place’, where ‘place’ means a relative arrangement, or configuration, of the complete universe.

An arena is the totality of places where one can go in some game. But who is playing the game and where? In Newton’s game, individual objects play in absolute space. In Mach’s game, there is only one player – the universe. It does not move in absolute space, it moves from one configuration to another. The totality of these places is its relative configuration space: Platonia. As the universe moves, it therefore traces out a path in Platonia. This captures, without any redundant structure, the idea of history. History is the passage of the universe through a unique sequence of states. In its history, the universe traces a path through Platonia.

However, such language makes it sound as though time exists. I may have inadvertently conjured up an image in your mind of the universe as a lone hiker walking the fells in northern Platonia. Properly understood, the Machian programme is much more radical. For no Sun rises or sets over that landscape to mark the walker’s progress. The Sun, like the moving parts of any clock, is part of the universe. It is part of the walker. Of course, to say that time has passed, we must have some evidence for that. Something must move. That is the most primitive fact of all. In the Newtonian picture, as in Feynman’s quip, time can pass without anything happening. If we deny that, the grandstand clock must go. There is nothing outside the universe to time it as it goes from one place to another in Platonia – only some internal change can do that. But just as all markers are on an equal footing for defining position, so are all changes for the purposes of timing. We must reckon time by the totality of changes. But changes are just what takes the universe from one place in Platonia to another. Any and all changes do that. We must not think of the history of the universe in terms of some walker on a path who can move along it at different speeds. The history of the universe is the path. Each point on the path is a configuration of the universe. For a three-body universe, each configuration is a triangle. The path is just the triangles – nothing more, nothing less.

With time gone, motion is gone. If you saw a jumbled heap of triangles, it would not enter your head that anything moved, or that one triangle changed into another. When Newton’s superstructure is removed, Newtonian history is like that jumbled heap of triangles, except that it is a special heap. If you picked up each triangle – I call that picking up an instant of time – and marked its position in Triangle Land, you would find that the marks of the triangles form a continuous curve.

This was the decisive picture that crystallized in my mind about 1971. At that stage I had no thought of applications to quantum mechanics, and no inkling that it might lead to the replacement of one clearly delineated path through Platonia by a mist that hovers over the same timeless landscape. We had a blackboard in our kitchen in College Farm, and I wrote at the top it it: The history of the universe is a continuous curve in its relative configuration space.’ My wife, perhaps understandably, was rather sceptical about the progress I was making. After all, fourteen words were not much to show for seven years of thought. But the clear formulation of the concept of Platonia was the important thing. It shifts attention from the parts of the universe to the universe itself. It shows that time is not needed as an extra element, the Great Timekeeper outside the universe. The universe keeps track of itself. In one instant it is where it is, in another it is somewhere else. That is what a different instant of time is: it is just a different place in Platonia. Instants of time and positions of the objects within the universe are all subsumed into the single notion of place in Platonia. If the place is different, the time is different. If the place is the same, time has not changed. This change of viewpoint is made possible only because the universe is treated as a single whole and time is reduced to change.

I think the reason why I take the possibility of a completely timeless universe more seriously than almost all other physicists is this background that came from thinking about Mach’s principle. As we shall see, Platonia is the natural arena for the realization of that idea. Many years after I had first recognized that Platonia would provide the basis for the solution to the Machian problem, I began to see that it had deep relevance in the quantum domain too. The problems of the origin of inertia and of quantum cosmology form a seamless whole.