The End of Time: The Next Revolution in Physics

CHAPTER 8

The Bolt from the Blue

HISTORICAL ACCIDENTS

In the whole of physics, nothing is more remarkable than the transformation wrought by a simple question that Einstein posed in 1905: what is the basis for saying that two events are simultaneous? Einstein was not the first to ask it. James Thomson, brother of Lord Kelvin, had in 1883. More significantly, so had Poincaré – a great figure in science – in 1898, in a paper that Abraham Pais, Einstein’s biographer, calls ‘utterly remarkable’. In connection with historical accidents, Poincaré’s paper is very interesting. He identified two problems in the definition of time.

First he considered duration: what does it mean to say that a second today is the same as a second tomorrow? He noted that this question had recently been widely discussed, and he outlined the astronomers’ solution, the ephemeris time described in Chapter 6. However, he then noted a second question, just as fundamental and in some ways more immediate, which had escaped close attention. How does one define simultaneity at spatially separated points? This was the question that Einstein posed and answered seven years later with such devastating effect. I read the subsequent history of relativity as follows. Einstein answered his question – Poincaré’s second – with such aplomb and originality that it eclipsed interest in the question of duration. It is not that duration plays no role in relativity – quite the opposite, it plays a central role. But duration is not derived from first principles. It appears indirectly.

One of the main aims of Part 3 is to redress the balance, to treat duration at the same level as simultaneity. There is, in fact, a beautiful theory of duration at the heart of general relativity, but it is hidden away in sophisticated mathematics. Einstein had no inkling of this. He said of his own theory that no one who had grasped its content could ‘escape its magic’. But the magic was more potent than even he realized. It can, it almost certainly will, destroy time.

BACKGROUND TO THE CRISIS

Much of nineteenth-century physics can be seen as meticulous preparation for the denouement over simultaneity. It had to come, but what a coup de théâtre Einstein made of it. Many readers will be familiar with the story, but since it introduces important ideas I shall briefly recall some key elements. It all started with an investigation of interference carried out in 1802 by the English polymath Thomas Young, famous among other things for his decipherment of the Egyptian hieroglyphs on the Rosetta Stone. In a sense, this was the start of both relativity and quantum theory. Young observed that if light from a single source is split into two beams that are subsequently recombined and projected onto a screen, then bright and dark fringes appear. He interpreted them in terms of a wave theory of light. If light is some kind of wave motion, there will be wave crests and troughs in both beams. When they are recombined, there will be places where the crests from one beam coincide with troughs in the other. They will cancel, giving dark fringes. But where crests coincide, they will enhance each other, giving bright fringes (Figure 22). Innumerable natural phenomena are explained by interference.

Young’s insight, which was developed more or less independently and much more thoroughly some years later by the Frenchman Augustin Jean Fresnel, soon gave rise to the notion that light waves must be vibrations of some elastic medium, which was called the aether. Meanwhile, the study of electricity and magnetism developed rapidly. In 1831, the English scientist Michael Faraday discovered electromagnetic induction, which not only showed that electricity and magnetism were related phenomena but rapidly became the basis of all electrical machinery. Deeply impressed by the patterns formed by iron filings sprinkled on paper held near a magnet (Figure 23), Faraday introduced the notion of lines of force and fields. A field can be thought of as a tension or excitation that exists throughout space and varies continuously (as demonstrated by induction) in both space and time. The field concept eventually changed physicists’ picture of what the world is ‘made of.

Figure 22 Thomas Young’s original explanation of the interference fringes in accordance with the wave theory of light, which he deduced by analogy with the behaviour of water waves. According to this interpretation, the beam reaches the barrier in the form of a plane wave, the successive parallel crests of which arrive simultaneously at the two slits A and B. The wave is diffracted at each slit, and spherical waves spread out from each point of the two slits towards the screen. At some points on the screen, the wave crests (or troughs) from the two slits arrive simultaneously, and the wave intensity is enhanced (bright regions). At other points, a wave crest from one slit arrives with a wave trough from the other. The wave intensity is cancelled at such a point (dark regions). This is the classical explanation of the fringes in terms of interference.

In the decade from 1855, the Scottish physicist James Clerk Maxwell took up Faraday’s qualitative field notion and cast it into mathematical form. His equations showed that electromagnetic effects should propagate through empty space as waves with a speed determined by the ratio of certain constants. It had already been noted that the ratio was equal to the known speed of light, leading to the strong suspicion that light was an electromagnetic effect. Maxwell’s equations proved this. Electromagnetic effects can propagate as waves of many different wavelengths: from radio waves (with wavelengths of around a metre to a kilometre), microwaves (wavelengths measured in centimetres), infrared waves (some ten to a thousand waves per centimetre), visible light (roughly ten thousand waves to the centimetre), ultraviolet light (up to around a million waves per centimetre), X-rays (of the order of ten million waves in every centimetre) and gamma rays (billions or even trillions of waves per centimetre). Hertz’s celebrated detection of waves from an electromagnetic source in 1888 was the first confirmation of this consequence of Maxwell’s theory.

Figure 23 Magnetic lines of force as revealed by placing iron filings in the magnetic field of a bar magnet.

Virtually all physicists were convinced that these electromagnetic excitations must be carried by some mechanical aether. This put a remarkable twist into the theory of motion and the status of Newton’s absolute space. Even in the framework of Newtonian theory, there had always been one serious problem with the notion. Newton was not entirely frank about it. In his guts, he certainly believed in a state of absolute rest. When he introduced absolute space, his words suggested the existence of one unique framework of motion. Either you moved with respect to it or you did not. However, in the body of the Principia he stated and used correctly the relativity principle, according to which the motions within a system are completely unaffected by any uniform overall motion it has (Box 5). This seriously undermined the idea of a unique state of rest – no criterion could establish whether one were in it or moving uniformly.

Figure 24 The pond argument. According to the spectator standing on the bank, the ripples move to the left and right with equal speed. But her partner, walking along the bank, sees things differently. He can almost keep up with the waves going to the right, while the left-moving waves recede from him almost twice as fast.

It was soon seen that the aether should introduce an experimentally verifiable standard of rest. The argument is simple and seems irrefutable (Figure 24). If you throw a stone onto the still surface of a pond, waves spread out in concentric rings. The water molecules do not move with the wave, they merely go up and down. The water plays the role of the conjectured aether: itself at rest, it is the material carrier of the waves. As seen by the woman standing on the bank, the waves spread out in all directions with the same speed. But for her partner walking along the bank, the wave process unfolds differently. The waves moving in the same direction as him have a different velocity relative to him compared with the waves moving in the opposite direction. A fast walker will even overtake some of the waves. The relativity principle cannot hold for such processes, and it was therefore expected that it would hold only for the mechanical processes described by Newton’s laws, but not for optical and electromagnetic phenomena. Moreover, as the Earth revolves in its orbit around the Sun it must be continually changing its speed through the aether. This ought to result in observable effects.

In fact, the argument is not quite so simple. Everyone agreed that light must be carried by an aether, but were all parts of the aether at rest relative to one another? As the Earth orbits the Sun, might it not carry some aether with it? It was also necessary to consider what would happen to light passing through water flowing on the Earth. Would the aether be carried along, partially or completely, by the water relative to the Earth? In fact, many issues had to be considered, including aberration, which makes the stars seem to shift slightly towards the point in the sky towards which the Earth is moving at any instant as it orbits the Sun. The arguments, some of which predated Maxwell’s work, developed over a period of eighty years, and many important experiments were made. By 1895, when the Dutch physicist Hendrik Anton Lorentz published an influential study, a consensus had more or less developed. It was that all known experimental results, with one crucial exception, could be explained by assuming the existence of a perfectly rigid aether.

The aether as proposed by Lorentz was actually devoid of all physical properties except rigidity. It was simply there to carry the excitations of Maxwell’s electromagnetic field and, in Lorentz’s words, to be the framework ‘relative to which all observable motions of the celestial bodies take place’. It therefore supplied a standard of rest like the water in the pond.

The one exception with which Lorentz had to contend was the famous Michelson-Morley experiment performed with great accuracy in 1887. Based on interference between light beams moving in the direction of the Earth’s motion and at right angles to it, it was designed to measure the change in the speed of the Earth’s motion through the aether over the course of a year. Its accuracy was sufficient to detect even only one-hundredth of the expected effect. But nothing was observed. It was a great surprise, and very puzzling.

Lorentz’s response was piecemeal. In particular, he suggested that, for some physical reason, the length of a body moving relative to the aether could be reduced in the direction of its motion by the amount needed to explain the Michelson-Morley result. Some years earlier, the Irish physicist George Fitzgerald had made the same proposal. Poincaré responded that some general principle should rule out all possibility of detecting motion relative to the aether. It should not be necessary to invoke ad hoc hypotheses. He began to think that the relativity principle might hold universally and not just for mechanical phenomena. Both he and Lorentz were working in this direction when Einstein appeared on the scene with a stunning solution.

EINSTEIN AND SIMULTANEITY

Two aspects of Einstein’s work ensured its triumphant success. First, he took the relativity principle utterly seriously. It was the bedrock, repeatedly exploited. Second, he took for real a ‘local time’ that Lorentz had introduced as a formal device to describe phenomena in a reference frame moving relative to the aether. Events simultaneous in the ‘local time’ were not so in the real time of the aether frame. But Einstein, committed to relativity, regarded one as just as real as the other. He made a virtue out of an apparent vice, and saw that the key to the entire mystery lay in the concept of simultaneity.

He deliberately highlighted an apparently irreconcilable paradox and then deftly presented its unique resolution: a radical proposal for saying when events are simultaneous. Hitherto this had seemed obvious, but Einstein showed that simultaneity was not a property of the world but a reflection of the way we describe it. By showing that the paradox could be resolved only by changing the notion of simultaneity – and with it time – he brought this issue to the fore.

The paradox was carefully prepared. He first defined the relativity principle. As in mechanics (Box 5), he postulated distinguished frames of reference in which all the laws of nature take their simplest form, and required this form to be the same in each frame. Any such frame, which constituted a kind of grid in space and time, should be in a state of uniform rectilinear (i.e. straight) motion relative to any other. He then postulated, in addition to this general principle, just one actual law of nature: that light propagates with the same speed c in all directions irrespective of the speed of the source. This was exactly what everybody had always assumed would hold in the unique frame of reference at rest in the aether. Einstein insisted it should happen in all frames.

The pond argument suggests that this is absurd. But Einstein realized that he had a hitherto unrecognized freedom: the grid lines defining simultaneity in space and time could be ‘drawn’ in a novel way. Simultaneity at spatially separated points must be defined in some way – but how? There must be a physical transmission of signals so as to synchronize clocks. The ideal would be infinitely fast signals. Then there would be no argument. This is effectively what happens in the pond experiment – the man and woman observe the water waves by light, which travels nearly a billion times faster than they do. We now see that the analogy between water waves in ponds and light waves in the aether is not perfect. For a full analogy, there would have to be signals that travel faster than light itself.

But such signals were unknown in Einstein’s time, and his theory would show that they could not exist. He therefore used the best substitute – light. This completely changed things. Light was to be analysed in a framework that light itself created, so the problem became self-referential. It might seem that Einstein cheated, making up the rules as he went along to ensure that he won the game. However, he was simply confronting a fact of life: laws of nature will be meaningful only if they relate things that can actually be observed. We live inside, not outside, the universe, and to synchronize distant clocks we have no alternative to the physical means available to us. Einstein’s hunch that we should use light because it would turn out to be the fastest medium available in nature has so far been totally vindicated.

The magical touch was that his choice was the most natural thing to do – in the theory of an aether and in the context of the relativity principle. Given their apparent irreconcilability, his subsequent demonstration of their compatibility was a coup. It also showed that there was something inevitable about the result.

For suppose there is a pond-like aether and that nothing is faster than light. It is natural to assume that it travels equally fast in all directions. Then how are we to define simultaneity throughout the aether? Einstein proposed setting up a master clock at a central reference point, sending a light signal to some distant identical clock at rest relative to it, and letting the signal be reflected back to the master clock. If it measures a time T for the round trip, we would obviously say that the light took ½T to reach the distant clock, which can be synchronized to read t + ½T on the arrival of a signal sent by the master clock when it reads t. In this way, clocks throughout the aether can be synchronized with the master clock. Standard measuring rods can be used to measure the distances between them. This is the obvious way to set up a space-time grid if the aether theory is correct.

However, it does assume that the aether is ‘visible’ and that we know when we are at rest in it. But this the relativity principle denies. Imagine a family of observers, equipped with clocks to measure time and rods to measure length, distributed in space and at rest relative to one another. Believing themselves at rest in the aether, they define simultaneity by Einstein’s prescription. There is also a second family, with identical rods and clocks, also at rest relative to one another but moving uniformly relative to the first. By the relativity principle, they can equally believe themselves at rest in the aether. So they too will use Einstein’s prescription to define simultaneity. Just as belief in the aether theory makes the prescription natural, belief in the relativity principle makes it natural for both families to adopt it. Nothing in nature privileges one family over the other. Whatever one family does, the other can do with equal right. In particular, each can use Einstein’s prescription.

The inescapable consequence is that the two families will disagree about which events are simultaneous. However, by accepting this, Einstein achieved his first goal – the demonstration that light propagation takes an identical form for both families (Box 9). This remarkable fact – that the relativity principle holds for light propagation and that simultaneity depends on the observer and on convention – is thus the great denouement towards which so much wonderful physics in the nineteenth century had been tending. It also showed that the aether is a redundant concept, since no experiment can establish whether we are moving relative to it.

Lack of simultaneity was only the beginning. Einstein went on to draw further amazing consequences from his iron insistence that all phenomena must unfold in exactly the same way for any two families of observers in uniform motion relative to each other. In particular, he was able to make some startling predictions about rods and clocks. The point is that the facts of light propagation are established by means of physical rods and clocks, but these tools are not immune to the relativity principle. Using simple equations and precise arguments, Einstein showed that two such families must each come to the conclusion that the clocks of the other family, moving relative to them, run slower than their own clocks. Each family also concludes that the rods of the other family are shorter than their own.

What is so remarkable about these results – and it seems so impossible that many quite intelligent people still refuse to accept it – is their mutual nature. How can it be that each family finds that the clocks of the other family run slower than their own? Box 10 explains.

BOX 9 Relativity in One Diagram and 211 Words

Figure 25 Alice (A) and Bob (B) believe they are at rest in the aether, and therefore draw a grid (dashes) with time vertical and space (only one dimension shown) horizontal. To synchronize their clocks, Alice sends Bob a light signal (solid line), which reaches him at X, where it is reflected back to Alice at T. Alice concludes that the signal reached Bob when she was at H. Their identical twins Alice* (A*) and Bob* (B*) are moving uniformly relative to them, but also believe they are at rest. Alice* sends a light signal, just as her twin does, at the moment they meet. It reaches Bob* at X*, and the reflection of it returns to Alice* at T*. She therefore concludes that her signal reached Bob* at H*, so she and Bob* have a grid (dots) inclined relative to their twins’ grid. The pairs do not agree on which events are simultaneous. Alice and Bob think H and X are, their twins think H* and X* are. However, they confirm the relativity principle, since both find that light travels along rays parallel to the diagonals (AX, XT and A*X*, X*T*) of their respective coordinate grids. Despite appearances, the situation is symmetric – in Alice* and Bob*’s grid their twins’ grid appears skew.

THE FORGOTTEN ASPECTS OF TIME

Fascinating as the results of Einstein’s two relativity theories are, many of them are not directly relevant to my main theme. Popular accounts that cover topics I omit are recommended in the section on Further Reading. My aim in Part 3 is to show how Einstein’s approach to relativity led him to an explicit theory of simultaneity but an implicit theory of duration. It is the latter that is important for this book, but it never got properly treated in relativity. The point is that remarkable facts about duration, as revealed through the clocks of different observers, follow inescapably from the definition of simultaneity and the relativity principle. Einstein did not need to create a theory of clocks and duration from first principles in order to learn some facts about them: they already followed from his two primary postulates.

BOX 10 The Impossible Becomes Possible

Figure 26 The horizontal and inclined strips, in which time increases vertically and the horizontal represents one space dimension, show the histories of two physically identical rods moving uniformly relative to each other. For Bob and Alice, points on the continuous line PPQQ are simultaneous and show the positions and lengths of the rods at the corresponding times. Their rod, PP, appears longer than Bob* and Alice*’s rod, QQ. But the starred twins think that points on the line P’P’Q’Q’ are simultaneous, and conclude that their rod Q’Q’ is longer than P’P’. A similar illustration could be given for clocks. Such diagrams do not explain this behaviour of rods and clocks, but do show that there is no outright logical contradiction. Einstein’s conclusions are as secure as his premises. His confidence in them has so far been totally vindicated.

The method Einstein used to create his relativity theories is an important factor. During the nineteenth century, mainly through the development of thermodynamics, physicists began to distinguish between, on the one hand, theories of the world in terms of truly basic laws and constituents (e.g. atoms and fields) and, on the other hand, so-called principle theories. In the latter, no attempt would be made to give an ultimate theory of things. Instead, the idea was to seek principles that seemed to hold with great generality and include them in the foundations of the description of phenomena. The repeated failures of all attempts to build perpetual-motion machines, of which two distinct types could be envisaged, became the basis of the first and second laws of thermodynamics. In the form in which it was developed on this basis, thermodynamics was a theory of the second kind – a principle theory.

In contrast, Lorentz’s combined theory of the electromagnetic field, electric charges and the aether was basically a theory of the first kind – it aspired to a fundamental description of the world in terms of its ultimate constituents. Einstein deliberately decided not to follow such a path in his own work on electrodynamics, from which the special theory of relativity emerged. He based it as far as possible on general principles. The fact is that Max Planck’s quantum discoveries (Box 1) and Einstein’s own development of them a few months before the relativity paper had persuaded Einstein that something very strange was afoot. Despite his admiration for Maxwell’s equations, he felt sure that they could not be the true laws of electromagnetism because they completely failed to explain the quantum effects. He had no confidence in his ability to find correct alternatives. Then, and to the end of his days, Einstein found the quantum baffling. He felt deeply that it was a huge mystery. By comparison, relativity (the special theory at least) was almost child’s play.

It was this attitude that largely shaped Einstein’s strategy in approaching the problem of the electromagnetic aether. He resolved to make no attempt at a detailed description of microscopic phenomena. Instead, he would rely on the relativity principle, for which there seemed to be strong experimental support, and make as few additional assumptions as possible. In the event, he was able to limit these to his assumption about the nature of light propagation. This was the one part of Maxwell’s scheme that he felt reasonably confident would survive the quantum revolution.

This had important consequences for the theory of time. Poincaré’s 1898 paper showed that it must answer two main questions: how simultaneity is to be defined, and what duration is. Associated with the second question is another, almost as important: what is a clock? Because of his approach, Einstein answered only the first question at a fundamental level. He gave at best only partial answers to the other two, and gave no explicit theories of either rods or clocks. Instead, he tacitly assumed the minimal properties they should possess. Otherwise, he relied to a very great extent on the relativity principle. It took him far. Few things in physics are more beautiful than the way he postulated the universal relativity principle and the one particular law of light propagation, and then deduced, from their combination, extraordinary properties of rods, clocks and time. If the premises were true, rods and clocks had to behave that way.

During his protracted creation of general relativity, Einstein used this trick several times. The strategy was always to avoid specific assumptions, and instead to seek principles. In this way he avoided ever having to address the physical working of rods and clocks: they were always treated separately as independent entities in both relativity theories. Their properties were not deduced from the inner structure of the theory, but were simply required to accord with the relativity principle. Einstein was well aware that this was ultimately unsatisfactory, and said so in a lecture delivered in 1923. He made similar comments again in 1948 in his Autobiographical Notes.

However, the tone of his comments does not suggest that he expected any great insight to spring from the rectification of this ‘sin’ (Einstein’s own expression). Only a ‘tidying up’ operation was needed. This gap in the theory of duration and clocks has still not been filled. I know of no study that addresses the question of what a clock is (and how crucially it depends on the determination of an inertial frame of reference) at the level of insight achieved in non-relativistic physics by James Thomson, Tait and Poincaré. Throughout relativity, both in its original, classical form and in the attempts to create a quantum form of it (which we come to in Part 5), clocks play a vital role, yet nobody really asks what they are. A distinguished relativist told me once that a clock is ‘a device that the National Bureau of Standards confirms keeps time to a good accuracy’. I felt that, as the theorist, he should be telling them, not the other way round.

The truth is that a chapter of physics somehow never got written. Despite his great admiration for Mach, Einstein was curiously insensitive to the issues highlighted by Mach and Poincaré. He did not directly address the nature and origin of the framework of dynamics. Despite an extensive search through his published papers and published and unpublished correspondence, I have found no indication that he ever thought really seriously about issues like those raised by Tait’s problem. This is rather surprising, since these were ‘hot topics’ during the very period in which Einstein created special relativity. He did not ask how the spatiotemporal framework (i.e. the framework of space and time used by physicists) arose; instead, he described the finished product and the processes that take place within the arena it creates.

In fact, Einstein and Hermann Minkowski, whose work will shortly be considered, brought about a marked change of emphasis in physics. To use an expression of John Wheeler, the ‘royal high road of physics’ from Galileo until Einstein was dynamics. Maxwell saw his own work as an extension of the principles developed by Galileo and Newton to new phenomena and to the field notion introduced by Faraday. At the same time, other scientists like Carl Neumann and Mach became aware of the need for new foundations of dynamics. In Poincaré’s writings of around 1900, one can see clear hints of how dynamics might have been developed further as the main stream of research. In particular, an explicit theory of the origin of the spatiotemporal framework might have emerged. That is more than evident from Poincaré’s 1898 paper on time and his 1902 comments, discussed in Chapter 5.

All this was changed by Einstein’s 1905 paper. Because of his quantum doubts, he distrusted explicit dynamical models. Within a few years a dualistic scheme appeared. Newton’s absolute space and time were replaced by space-time, but this was not the complete story. Actual physics emerged only with the statements about how rods and clocks behaved in space-time. This is where the scheme was dualistic. The behaviour of rods and clocks – and with it a theory of duration – never emerged organically from the structure of space-time, it was simply postulated. This is not to say the dualistic scheme is wrong in the statements it makes. Einstein’s theory is as secure as its foundations; there is no hint of failing there. However, insight into the nature of time and duration was lost.

For all that, general relativity does contain, hidden away in its mathematics (as I have already indicated), a theory of duration and the spatiotemporal framework. However, this did not come to light for many decades and even now is not properly appreciated. How this came about, and an account of the ‘hidden dynamical core’ of general relativity, are the subject of the next chapters.

It may help to end this chapter with a general remark on time. It is impossible to understand relativity if one thinks that time passes independently of the world. We come to that view only because change is so all-pervasive and so many different changes all seem to march in perfect step. Relativity is not about an abstract concept of time at all: it is about physical devices called clocks. Once we grasp that, many difficulties fall away. If light did not travel so much faster than normal objects, we would observe relativistic effects directly and they would not strike us as strange. There is nothing inherently implausible in the idea that clocks travelling past us at high speed should be observed to go slower than the watch on our wrist. Motion of the clock might well alter the rate at which it ticks. After all, when we swim through water, we feel the way our body responds. If there were an aether, clocks could well be affected by their motion through it. What is difficult to grasp is how observers travelling with the moving clocks think our wristwatch is running slow, while we think just the same about their clocks (this apparent logical impossibility has been dealt with in Box 10). However, the important thing is to get away from the idea that time is something. Time does not exist. All that exists are things that change. What we call time is – in classical physics at least – simply a complex of rules that govern the change.