The End of Time: The Next Revolution in Physics

CHAPTER 10

The Discovery of General Relativity

FUNNY GEOMETRY

This chapter is about how Einstein progressed from special relativity, which does not incorporate gravity, to general relativity, which does. Einstein believed that he was simultaneously incorporating Mach’s principle as its deepest foundation, but later, as I said, he changed his mind and left this topic in a great muddle. My view is that, nevertheless, without being aware of it, Einstein did incorporate the principle. This has important implications for time. We start with a bit more about Minkowski’s discoveries, which is necessary if we are to understand the way Einstein set about things.

One of the most important concepts in physics and geometry is distance, which is measured with rods. Distances can be measured in a space of any number of dimensions. You can measure them along a line or curve, on a flat or curved surface, or in space. In Part 2 we saw how an abstract ‘distance’, the action, can be introduced in multidimensional configuration spaces like Platonia. Minkowski showed that a remarkable kind of four-dimensional distance exists in space-time. Its existence is a consequence of the experimental facts that underlie special relativity. These things are most easily explained if we assume that space has just one dimension, not three; space-time then has two dimensions. Such a space-time is shown in Figure 27. We must first of all learn about past, present, and future in space-time.

One of the distinguished coordinate systems that exists in space-time is shown in Figure 27, in which the x axis is for space and the t axis for time, which increases upward. This is the Lorentz frame of Alice in Figure 25. Her world line is the vertical t axis. The units of time and distance are chosen to make the speed of light unity. Light pulses that pass through event O at t = 0 in opposite directions in space travel in space-time along the two lines marked future light cone. Their continuations backward (the light’s motion before it reaches O) define the past light cone.

Figure 27 Past and future light cones and the division of space-time in time-like and space-like regions, as described in the text.

Each event has a light cone, but only O’s is shown. Relativity differs from Newtonian theory mainly through the light cone and its associated distinguished speed c, which is a limiting speed for all processes. Light plays a distinguished role in relativity simply because it has that speed. No material object can travel at or faster than it. If a material object passes through O, its world line must lie somewhere inside the light cone, for example OA in Figure 27.

The light cone divides space-time into qualitatively different regions. An event like A can be reached from O by a material object travelling slower than light. Two such events are time-like with respect to each other. For two such events there exists a Lorentz frame in which they have the same space coordinates but different time coordinates. For the points O and A this frame is shown in the upper right of Figure 28.

Next we consider events like B and C in Figure 27, outside the light cone of O. They are space-like with respect to O. No material body can reach them from O, since to do so it would have to travel faster than light. For two events that are mutually space-like there exists a Lorentz frame in which they have the same time coordinate but different space coordinates. For two space-like events, it is impossible to say which is the earlier in any absolute sense. In some Lorentz frames one will be earlier than the other (thus O is earlier than both B and C in Alice’s frame in Figure 27), but in others the temporal order will be reversed.

Figure 28 Past, present and future in a space-time with two dimensions of space. The object that moves along OA (bottom left) is at rest in the starred frame (top right). Its world line is O*A (O and O* are the same event).

Finally, two events that can be connected by a light ray have a light-like relationship. All points on the light cone of event 0, for example the point F, are light-like with respect to O.

These three basic relationships between events – being time-like, spacelike or light-like – are the same in all Lorentz frames. This is because the three types are determined by the light cones, which are real features in space-time, just as rivers are real features of a continent. In contrast, the coordinate axes are like lines ‘painted’ on space-time – they are no more real than the grid lines on a map. Moreover, in a change from one frame to another, the coordinate axes never cross the light cones. The time axis moves but stays within the light cone, while the space axes stay within the ‘present’ as defined above. This is illustrated in Figure 28 for space with two dimensions, which shows how the light cone gets its name. It also highlights the great difference between the Newtonian and Einsteinian worlds. In the former, past, present and future are defined throughout the universe, and the present is a single simultaneity hyperplane. In the latter, they are defined separately for each event in space-time, and the present is much larger.

Now we can talk about distance. In ordinary space it is always positive. The distance relationships are reflected in Pythagoras’ theorem: the square of the hypotenuse in any right-angled triangle is equal to the sum of the squares of the other two sides: H² = A²+B².

Minkowski was led to introduce a ‘distance’ in space-time by noting a curious fact. For observers who use the xy frame in Figures 27 and 28, event A is separated from O by the space-like interval EA and by the time-like interval DA. For observers who use the starred frame, however, O and A are at the same space point and are merely separated by the time-like interval OA. The xy observers measure EA with a rod and DA with a clock, obtaining results X and T, respectively. With their clock, observers in the starred frame can measure only the time-like interval OA. Now, their clock runs at a different rate to the xy clock, so they will find that OA is not T but T_starred. Using Einstein’s results, Minkowski found that (T_starred)² = T²−X². This is just like Pythagoras’ theorem, except for the minus sign.

There are several important things about this result. Einstein had shown that observers moving relative to each other would not agree about distances and times between pairs of events. However, Minkowski found something on which they will always agree. Measurements of the space-like separation (by a rod) and the time-like separation (by a clock) of the same two events O and A can be made by observers moving at any speed. They will all disagree about the results of the separate measurements, but they will all find the same value for the square of the time-like separation minus the square of the space-like separation. It will always be equal to the square of the time-like separation, called the proper time, of the unique observer for whom O and A are at the same space position. This result created a sensation. Space and time, like rods and clocks, seem to have completely different natures, but Einstein and Minkowski showed that they are inseparably linked.

What is more, Minkowski showed that it is very natural to regard space and time together as a kind of four-dimensional country in which any two points (events in space-time) are separated by a ‘distance’. This ‘distance’, found by measurements with both rods and clocks, is to be regarded as perfectly real because everyone will agree on its value. In fact, Minkowski argued that it is more real than ordinary distances or times, since different observers disagree on them. Only the ‘distance’ in space-time is always found to be the same. But it is a novel distance – positive for the time-like OA in Figure 27, zero for the light-like OF and negative for the space-like OC. (It is a convention, often reversed, to make time-like separations positive and space-like ones negative. What counts is that they have opposite signs. Also, if the units of space and time are not chosen to make the speed of light c equal to 1, the square of the space-time ‘distance’ becomes (cT)² – X2.)

Almost everything mysterious and exciting about special relativity arises from the enigmatic minus sign in the space-time ‘distance’. It causes the ‘skewing’ of both axes of the starred frame of the starred twins in Figure 25, and leads to the single most startling prediction – that it is possible, in a real sense, to travel into the future, or at least into the future of someone else, since the future as such is not uniquely defined in special relativity. What we call space and time simply result from the way observers choose to ‘paint coordinate systems’ on space-time, which is the true reality. Minkowski’s diagrams made all these mysteries transparent – and intoxicatingly exciting for physicists. However, this is not the place to discuss time travel and the other surprises of relativity, which are dealt with extensively in innumerable other books.

EINSTEIN’S WAY TO GENERAL RELATIVITY

For physicists, ‘relativity’ has two different meanings. The more common is the one employed by Einstein when he created relativity. He related it to the empirical fact, first clearly noted by Galileo in 1632, that all observations made within an enclosed cabin on a ship sailing with uniform speed are identical to observations made when the ship is at rest. Einstein illustrated this fact with experiments on trains. The lesson he drew from it was that uniform motion as such could not be detected by any experiment. The laws of nature could therefore not be expressed in a unique frame of reference known to be at rest. They could be expressed only in any one of a family of distinguished frames in uniform motion relative to one another. The relativity principle states that the laws of nature have the identical form in all such frames. For reasons shortly to be explained, this later became known as the restricted or special relativity principle.

This meaning of relativity is tied to a special feature of the world – the existence of the distinguished frames and their equivalence for expressing the laws of nature. The other meaning of relativity is more primitive and less specific. It simply recognizes that space and time are invisible: all we ever see are objects and their relative motions. We can speak meaningfully of the position and motion of an object only if we say how far it is from other objects. Position and motion are relative to other objects. This is often called kinematic relativity, to distinguish it from Galilean relativity.

Both relativity principles have played important – often decisive – roles in physics. Copernicus and Kepler used kinematic relativity to great effect in the revolution they brought about. Galileo used the other relativity principle to explain how we can live on the Earth without feeling its motion. That was almost as wonderful a piece of work as Einstein’s, nearly three hundred years later. A natural question is this: what is the connection between the two relativity principles? Any satisfactory answer must grapple with and resolve the issue of the distinguished frames of reference. How are they determined? What is their origin? As we have seen, neither Einstein nor Minkowski addressed these questions when they created special relativity, and they have been curiously neglected ever since. This is a pity, since they touch upon the nature of time. We cannot say what time is – and whether it even exists – until we know what motion is.

Poincaré sought to unite the two relativity principles in a single condition on the structure of dynamics, as formulated in the two-snapshots idea. Had he succeeded, he would have derived the empirical fact of Galilean relativity solely on the basis of a natural criterion derived from kinematic relativity. He died without taking this idea any further, but in any case it is doubtful whether the two relativity principles can be fully fused into one. Poincaré formulated his idea in 1902, before the relativistic intermingling of space and time became apparent, and it is hard to see how that can ever be derived from the bare fact of kinematic relativity. It is, however, of great interest to see how far Poincaré’s idea can be taken. We shall come to this when we have seen how Einstein thought about and developed his own relativity principle and thereby created general relativity.

It is important not to be overawed by the genius of Einstein. He did have blind spots. One was his lack of concern about the determination in practice of the distinguished frames that play such a vital role in special relativity – he simply took them for granted. It is true that they are realized approximately on the reassuringly solid Earth in skilfully engineered railway carriages. But how does one find them in the vast reaches of space? This is not a trivial question. Matching this lack of practical interest, we find an absence of theoretical concern. Einstein asked only what the laws of nature look like in given frames of reference. He never asked himself whether there are laws that determine the frames themselves. At best, he sought an indirect answer and got into a muddle – but a most creative muddle.

To see why, it is helpful to trace the development of his thinking – a fascinating story in its own right. As an extremely ambitious student, he read Mach’s critique of Newton’s absolute space. This made him very sceptical about its existence. Simultaneously, he was exposed to all the issues related to the aether in electrodynamics. Lorentz, in particular, had effectively identified absolute space with the aether, in the form of an unambiguous state of rest. But, writing to his future wife Mileva in August 1899, Einstein was already questioning whether motion relative to the aether had any physical meaning. This would develop into one of the key ideas of special relativity. If it is impossible to detect motion relative to it, the aether cannot exist. It was natural for Einstein to apply the same thought to absolute space.

His 1905 paper killed the idea that uniform motion relative to any kind of absolute space or aether could be detected. But Newton had based his case for absolute space on the detection not of uniform motion, but of acceleration. In 1933, Einstein admitted that in 1905 he had wanted to extend the relativity principle to accelerated as well as uniform motion, but could not see how to. The great inspiration – ‘the happiest thought of my life’ – came in 1907 when he started to consider how Newtonian gravity might be adapted to the framework of special relativity. He suddenly realized the potential significance of the fact, noted by Galileo and confirmed with impressive accuracy by Newton, that all bodies fall with exactly the same acceleration in a gravitational field.

Most physicists saw this as a quirk of nature, but Einstein immediately decided to elevate it to another great principle and exploit it as he had the relativity principle. Unable to divine new laws of gravitation straight off, he formulated the equivalence principle, according to which processes must unfold in a uniform gravitational field in exactly the same way as in a frame of reference accelerated uniformly in a space free of gravity. He argued that pure acceleration could not be distinguished from uniform gravitation. Suppose that you awoke from a deep narcotic sleep in a dark bedroom to find that gravity was mysteriously stronger. There could be two different causes. You might have been transported, bedroom and all, to another planet with stronger gravity. But you might still be on the Earth but in an elevator accelerating uniformly upward. No experiments you could perform in your bedroom would enable you to distinguish between these alternatives.

Einstein saw here a striking parallel with the relativity principle. The relativity principle prevented an observer from detecting uniform motion. In its turn, the equivalence principle prevented an observer from detecting uniform acceleration – observed acceleration could be attributed either to acceleration in gravity-free space or to a gravitational field. Einstein recognized the immediate short-term potential of his new principle. He knew how processes unfolded in gravity-free space. Mere mathematics showed how they would appear in an accelerated frame, but by the equivalence principle it was possible to deduce that these same processes must occur in a uniform gravitational field. Once again, Einstein’s inspired selection of a simple universal principle – all bodies fall in the same way – enabled him to perform a startling conjuring trick. He showed that the rate of clocks must depend on their position in a gravitational field. Clocks closer to gravitating bodies must run slow relative to clocks farther away.

This fact is often said to show that ‘time passes more slowly’ near a gravitating body. However, objective facts within relativity can seem utterly mysterious and logically impossible if we imagine time as a river. Such a time does not exist. Relativity makes statements about actual clocks, not time in the abstract. It is easy to imagine – and physicists now find it comparatively easy to verify – that otherwise identical clocks run at different rates at the top and bottom of a higher tower. Incidentally, the ‘time dilation’ effect in gravity is much easier to accept than the similar effect associated with motion. There is no reciprocal slowing down. Thus, observers at the top and bottom of the tower both agree that the clock at the top runs faster.

By 1907 Einstein was also able to show that gravity must deflect light. Both his early predictions, made precise in his fully developed theory, have been confirmed with most impressive accuracy in recent decades. However, Einstein saw his early predictions merely as stepping stones to something far grander. The equivalence principle persuaded him that inertia (i.e. the tendency of bodies to persist in a state of rest or uniform motion) and gravity, which Newton and all other physicists had regarded as distinct, must actually be identical in nature. He started to look for a conceptual framework in which to locate this conviction. At the same time, he saw a great opportunity to abolish not only the aether but also all vestiges of absolute space. So far he had managed to achieve two steps in this process by showing that uniform motion and uniform acceleration could not correspond to anything physically real in the world. However, much more general motions could be imagined. Einstein aimed to show that the laws of nature could be expressed in identical form whatever the motion of the frame of reference.

The relativity he had so far established was very special. What he wanted was complete general relativity. This idea, nurtured and developed over eight years and involving intense and often agonizing work during the last four, explains the name he gave to his unified theory of gravitation and inertia that finally emerged in 1915. Viewed in the light of the ancient debate about absolute and relative motion, Einstein’s approach was very distinctive and somewhat surprising since he made no attempt to build kinematic relativity directly into the foundations of his theory. Unlike Mach and many other contemporaries, he did not insist that only relative quantities should appear in dynamics. He went at things in a roundabout way, mostly because of his preference for general principles. However, I think it was also a result of the way he thought about space and time.

As far as I can make out, Einstein did conceive of space-time as real and as the container of material things – fields and particles. However, he recognized that all its points were invisible and that they could be distinguished and identified only by observable matter present at them. Since space-time was made ‘visible’ by such matter, he supposed he could lay out coordinate grid lines on space-time and express the laws of nature with respect to them.

Now came the decisive issue. Einstein saw space-time without any matter in it as a blank canvas. Nothing about it could suggest why the coordinate grid lines should be drawn in one way rather than another. Any choice would be arbitrary and violate the principle of sufficient reason. Einstein found this intolerable. That is no exaggeration: his faith in rationality of nature – as opposed to human beings – was intense. The only satisfactory resolution was general relativity. In truth, there can be no distinguished coordinate systems. It must be possible to express the laws of nature in all systems in exactly the same form.

The only justification for the distinguished systems that appeared in Newtonian dynamics and special relativity was the law of inertia. But the equivalence principle had opened up the possibility of unifying inertia and gravity. This insight sustained Einstein in his long search for general relativity. His contemporaries would all have been content simply to find a new law of gravity. He was after something sublime.

It is suggestive that both Poincaré and Einstein – the old and young giants – began their attack on absolute space from the principle of sufficient reason. The difference between their approaches is interesting. Working within the traditional dynamical framework, Poincaré said that only directly observable quantities – the relative separations of bodies and their rates of change – should be allowed as initial data for dynamics. In such a theory, we may say that perfect Laplacian determinism holds (it doesn’t hold in Newtonian theory, which uses invisible absolute space and time). Einstein had a more general approach. He merely insisted that there should be no arbitrary choice of the coordinate systems used to express the laws of nature.

THE MAIN ADVANCES

The desire to express the laws of nature in progressively more general coordinates led to all Einstein’s major breakthroughs. Newton had argued that centrifugal forces proved the existence of absolute space. The laws of nature looked different in rotating systems. Einstein wanted to attack this problem head on. Could he perhaps show that, if expressed properly, the laws of nature did after all have the same form in rotating and non-rotating coordinates? The principle of equivalence suggested that what Newton had taken to be absolute inertial effects in a rotating system might be the gravitational effects of distant matter. The point is that in a rotating system the distant stars would themselves appear to be rotating. Since rotating electric charges generate electric and magnetic fields, rotating masses might generate new kinds of gravitational field. Nearly thirty years earlier, Mach had suggested that rotating matter ‘many leagues thick’ might generate measurable centrifugal forces within it. Einstein now conjectured that the gravitational field was the mechanism through which such forces could arise.

He therefore started to consider what form the laws of nature would take in a rotating system. This immediately led him to a startling conclusion: the ordinary laws of Euclidean geometry could not hold in such a system! His argument was based on the contraction of measuring rods in motion which he had proved in special relativity. First, imagine observers at rest on a surface who measure the circumference and diameter of a circle painted on it. They will find that their ratio is π. That agrees with Euclidean geometry – a recognized law of nature. Now imagine other observers on a disk above the painted circle and rotating about its centre. Their rods will undergo Fitzgerald-Lorentz contraction when laid out in the direction of motion, around the circumference. However, when laid out along the diameter, the rods will not contract. (The contraction occurs only in the direction of motion.) Therefore, the rotating observers will not find π when they measure the ratio of the circumference to the diameter. For them, Euclidean geometry will not hold.

Because Einstein wanted so passionately to generalize the relativity principle, he took this result seriously. According to the hint from the equivalence principle, novel effects in accelerated coordinate systems (as a rotating one is) could be attributed to gravitational effects. He concluded that geometry would not be Euclidean in a gravitational field. This happened during 1911/12, when he was working in Prague. Through either the suggestion of a colleague or the recollection of lectures on non-Euclidean geometry he had heard as a student, Einstein’s attention was drawn to a classic study in the 1820s by the German mathematician Carl Friedrich Gauss.

Gauss had studied the curvature of surfaces in Euclidean space. As a rule, material surfaces in space are not flat but curved. Think of the surface of the Earth or any human body. Gauss’s most important insight was that a surface in three-dimensional space is characterized by two distinct yet not entirely independent kinds of curvature. He called them intrinsic and extrinsic curvature. The intrinsic curvature depends solely on the distance relationships that hold within the surface, whereas the extrinsic curvature measures the bending of the surface in space. A surface can be flat in itself – with no intrinsic curvature – but still be bent in space and therefore have extrinsic curvature. The best illustration of this is provided by a flat piece of paper, which has no intrinsic curvature. As it lies on a desk it has no extrinsic curvature either: it is not bent in space. However, it can be rolled into a tube. It is then bent – but not stretched – and acquires extrinsic curvature.

In contrast to a sheet of paper, the surface of a sphere, like the earth, has genuine intrinsic curvature. Gauss realized that important information about it could be deduced from distance measurements made entirely within the surface. Imagine that you can pace distances very accurately, and that you walk due south from the north pole until you reach latitude 85° north. Then you turn left and walk due east all the way round the Earth at that latitude. All the time you will have remained the same distance R from the north pole. If you believed the Earth to be flat, you would expect to have to walk the distance 2πR before returning to the point of your left turn. However, you will find that you get there having walked a somewhat shorter distance. This shows you that the surface of the Earth is curved.

To describe these things mathematically for all smooth surfaces, Gauss found it convenient to imagine ‘painting’ curved coordinate lines on the surface. On a flat surface it is possible to introduce rectangular coordinate grids, but not if the surface is curved in an arbitrary way. So Gauss did the next best thing, which is to allow the coordinate lines to be curved, like the lines of latitude and longitude on the surface of the Earth. He showed how the distance between any two neighbouring points on a curved surface could be expressed by means of the distances along coordinate lines, and also how exactly the same distance relations could be expressed by means of a different system of coordinates on the same surface. About thirty years later, another great German mathematician, Bernhard Riemann, showed that not only two-dimensional surfaces but also three-dimensional and even higher-dimensional spaces can have intrinsic curvature. This is hard to visualize, but mathematically it is perfectly possible. Just as on the Earth, in a curved space of higher dimensions, you can, travelling always in the same direction, come back to the point you started from. These more general spaces with curvature are now called Riemannian spaces.

Einstein realized that he had to learn about all this work thoroughly, and it was very fortunate that he moved at that time to Zurich, where Marcel Grossmann, an old friend from student days, was working. Grossmann gave him a crash course in all the mathematics he needed. When he had fully familiarized himself with it, Einstein became extremely excited for two reasons.

First, Minkowski had shown that space-time could be regarded as a four-dimensional space with a ‘distance’ defined in it between any two points. Except that the ‘distance’ was sometimes positive and sometimes negative, whereas Riemann had assumed the distance to be always positive and had never envisaged time as a dimension, considered mathematically Minkowski’s space-time was just like one of Riemann’s spaces. But it was special in lacking curvature – it was like a sheet of paper rather than the Earth’s surface. Einstein had meanwhile become convinced that gravity curves space-time. This led to one of his most beautiful ideas: in special relativity, the world line (path) of a body moving inertially is a straight line in space-time. This is a special example of a ‘shortest curve’, or geodesic. The corresponding path in a space with curvature would be a geodesic, like a great circle on a sphere.

Einstein assumed that the world line of a body subject to inertia and gravity would be a geodesic. In this way he could achieve his dream of showing that inertia and gravity were simply different manifestations of the same thing – an innate tendency to follow a shortest path. This will be a straight line if no gravity is present, so that space-time has no curvature, but in general it will be a curved (but ‘straightest’) line in a genuinely curved space-time. Since matter causes gravity, Einstein assumed that matter must curve space-time in accordance with some law, for which he immediately started to look. Bodies moving in such a space-time would follow the geodesics corresponding to the curvature produced by the matter, so the gravitational effect of the matter would be expressed through the curvature it produces. Another important insight was that in small regions the effect of curvature would be barely noticeable, just as the Earth seems flat in a small region, so that in those small regions physical phenomena would appear to unfold just as in special relativity without gravity. This gave full expression to the equivalence principle.

The second reason why Einstein became so excited was that Gauss’s method matched his own idea of general relativity. He disliked the distinguished frames of special relativity because they corresponded to special ways of ‘painting’ coordinate systems onto space-time. He felt that this was the same as having absolute space and time. They would be eliminated only if the coordinate systems could be painted on space-time in an arbitrary way. But this was what Gauss’s method amounted to. In fact, in a curved space it is mathematically impossible to introduce rectangular coordinates. Mathematicians call the possibility of using completely arbitrary coordinate systems general covariance. Specifically, laws are said to be generally covariant if they take exactly the same form in all coordinate systems. Einstein identified this with his requirement of general relativity.

To summarize this part of the story, in 1912 Einstein became aware of the possibilities opened up by non-Euclidean geometry and the work initiated by Gauss. He had begun to suspect that gravitational fields would make geometry non-Euclidean. He was also almost desperate to find a formalism that did not presuppose distinguished frames of reference. He found that Gauss’s method of arbitrary coordinates was tailor-made for his ambitions. He also saw that, space and time having been so thoroughly fused by Minkowski, the only natural thing to do was to make space-time into a kind of Riemannian space. The ideas of Gauss and Riemann must be applied, not to space alone, but to space and time. This is the incredibly beautiful idea that Minkowski made possible: gravity was to be explained by curvature in space and time. Einstein thus conjectured that space-time is curved by gravity, and that bodies subject only to gravity and inertia follow geodesics determined by the distance properties of space-time, which encapsulate all its geometrical properties. Einstein’s conjecture has been brilliantly confirmed to great accuracy in recent decades.

THE FINAL HURDLE

Finding the law of motion of bodies in a gravitational field was only part of Einstein’s problem. He also had to find how matter created a gravitational field. He needed to find equations for the gravitational field somewhat like those that Maxwell had found for the electromagnetic field. They would establish how matter interacted with the gravitational field, and also how the field itself varied in regions of space-time free of matter (matching the way electromagnetic radiation propagated as light through space-time). This part of the problem created immense difficulties for Einstein, mostly through very bad luck.

Much as I would like to tell the complete story, which is fascinating and full of ironies, I shall have to content myself with saying that, after three nerve-wracking years, Einstein finally found a generally covariant law that described how matter determined the curvature of space-time. It involves mathematical structures called tensors, all the properties of which had already been studied by mathematicians. In particular, for space-time free of matter, Einstein was able to show that a tensor known as the Ricci tensor (because it had been studied by the Italian mathematician Gregorio Ricci-Curbastro) must be equal to zero. Ironically, Grossmann had already suggested to Einstein in 1912 that in empty space the vanishing of the Ricci tensor might be the generally covariant law he was seeking. However, some understandable mistakes prevented them from recognizing the truth at that time.

It is a striking fact that all the mathematics Einstein needed already existed. In fact, I believe it is significant that he did not have to invent any of it. In 1915, he was immediately able to show that, to the best accuracy astronomers could achieve at that time, his theory gave identical predictions to Newtonian gravity except for a very small correction to the motion of Mercury. All planetary orbits are ellipses. A planet’s elliptical orbit itself very slowly rotates, under the gravitational influence of the other planets. This is known as the advance of the perihelion, the perihelion being the point at which the planet is closest to the Sun, marking one end of the ellipse’s longest diameter. According to Einstein’s theory, Mercury’s perihelion should advance by 43 seconds of arc per century more than was predicted by Newtonian theory. This very small effect shows up for Mercury because it is closer to the Sun than the other planets, and also has a large orbital eccentricity. For many years, the sole discrepancy in the observed motions of the planets had been precisely such a perihelion advance for Mercury of exactly that magnitude. All attempts to explain it had hitherto failed. Einstein’s theory explained it straight off.

GENERAL RELATIVITY AND TIME

Many more things could be said about general relativity and its discovery. However, what I want to do now is identify the aspects of the theory and the manner of its discovery that have the most bearing on time.

First, the classical (non-quantum) theory as it stands seems to make nonsense of my claim that time does not exist. The space-time of general relativity really is just like a curved surface except that it has four and not two dimensions. A two-dimensional surface you can literally see: it is a thing extended in two dimensions. In their mind’s eye, mathematicians can see four-dimensional space-time, one dimension of which is time, just as clearly. It is true that time-like directions differ in some respects from space-like directions, but that no more undermines the reality of the time dimension than the difference between the east-west and north-south directions on the rotating Earth makes latitude less real than longitude. However, the qualification ‘as it stands’ at the start of this paragraph is important. In the next chapter we shall see that there is an alternative, timeless interpretation of general relativity.

Next, there is the matter of the distinguished coordinate systems. In one sense, Einstein did abolish them. Picture yourself in some beautiful countryside with many varied topographic features. They are the things that guide your eye as you survey the scene. The real features in space-time are made of curvature, and hills and valleys are very good analogies of them. Imagined grid lines are quite alien to such a landscape. In general relativity, the coordinated lines truly are merely ‘painted’ onto an underlying reality, and the coordinates themselves are nothing but names by which to identify the points of space-time.

For all that, space-time does have a special, sinewy structure that needs to be taken into account. Distinguished coordinate systems still feature in the theory. This is because the theory of measurement and the connection between theory and experiment is very largely taken over from special relativity. In fact, much of the content of general relativity is contained in the meaning of the ‘distance’ that exists in space-time. This is where the analogy between space-time and a landscape is misleading. We can imagine wandering around in a landscape with a ruler in our pocket. Whenever we want to measure some distance, we just fish out the ruler and apply it to the chosen interval. But measurement in special relativity is a much more subtle and sophisticated business than that. In general, we need both a rod and a clock to measure an interval in space-time. Both must be moving inertially in one of the frames of reference distinguished by that theory, otherwise the measurements mean nothing. The theory of measurement in general relativity simply repeats in small regions of space-time what is done in the whole of Minkowski space-time in special relativity. No measurements can be contemplated in general relativity until the special structure of distinguished frames that is the basis of special relativity has been identified in the small region in which the measurements are to be made.

This is something that is often not appreciated, even by experts. It comes about largely because of the historical circumstances of the discovery of general relativity and the absence of an explicit theory of rods and clocks. There is also the stability of our environment on the Earth and the ready availability in our age of clocks. It is easy for us to stand at rest on the Earth, watch in hand, and perform a measurement of a purely timelike distance. But nature has given us the inertial frame of reference for nothing, and skilful engineers made the watch. Finally, because we can and very often do see a three-dimensional landscape spread out before our eyes, it is very easy to imagine four-dimensional space-time displayed in the same way. All textbooks and popular accounts of the subject positively encourage us to do so. They all contain ‘pictures’ of space-time. Now the picture is indeed there, and very wonderful it is too. But it arises in an immensely sophisticated manner hidden away within the mathematical structure of the Ricci tensor. The story of time as it is told by general relativity unfolds within the Ricci tensor. It performs the miracle – the construction of the cathedral of space-time by intricate laying and interweaving of the bricks of time. I shall try to explain this in qualitative terms in the next chapter. Let me conclude this one by highlighting again the importance of the historical development. It made possible the discovery of a theory without full appreciation of its content.

At the end of November 1915, Einstein wrote an ecstatic letter to his lifelong friend Michele Besso, telling him that his wildest dreams had come true: ‘General covariance. Mercury’s perihelion with wonderful accuracy.’ These two verbless sentences say it all. Einstein was convinced that general covariance had deep physical consequences and had led him to one of the greatest triumphs of all time. Yet, barely two and a half years later, he admitted, in response to a quite penetrating criticism from a mathematician called Erich Kretschmann, that general covariance had no physical significance at all.

In a way, this is obvious. Space-time is a beautiful sculpture. What makes it beautiful is the way in which its parts are put together. The fact that one can paint coordinate lines on the finished product and measure distance on the sculpture between points on it labelled by the arbitrary coordinates clearly leaves the sculpture exactly the same. All this changing of coordinates is purely formal. It tells you nothing about the true rules that make the sculpture.

Belatedly, Einstein came to see that his whole drive to achieve general covariance as a deep physical principle had no foundation in fact. It was just a formal mathematical necessity. Ever determined to find new and even more beautiful laws of nature, he never felt the need to go back and see exactly how his sculpture was actually created. In a book I wrote some years ago on the discovery of dynamics, I commented on the fact that Kepler (so very like Einstein in his dogged holding on to an idea that eventually transformed physics) never realized quite what a wonderful discovery he had made. I likened him to

a boy who finds for the first time a ripe horse-chestnut with the outer shell intact. Cherishing the golden and curiously shaped object, he might take it home, quite unaware of the shiny brown and perfectly smooth conker ready to spring from the shell on application of a little directed pressure. That was Kepler’s fate: he died without an inkling of what his nut really contained.

The same thing happened to Einstein. Realizing while still at Prague the sort of thing he needed, he hurried to a shop called ‘Mathematics’ owned by his friend Grossmann in Zurich. Straight off the shelf, at a bargain price, he bought a wonderful device called the Ricci tensor. Three years later, after agonizing struggles, he learned how to turn the handles properly, and out popped the advance of Mercury’s perihelion and the exact light deflection at eclipses.

But it never entered his head to ask how the device actually worked. He died only half aware of the miracle he had created.