Reality Is Not What It Seems Part 2

Special relativity is a subtle and conceptually difficult theory. It is more difficult to digest than general relativity. Reader, don't become demoralized if the next few pages sound a bit abstruse. The theory shows, for the first time, that in the Newtonian vision of the world there isn't just something missing: rather, it must be radically modified in a way that goes completely against common sense. It is the first real leap into the revision of our most intuitive understanding of the world.

The extended present.

The theories of Newton and of Maxwell appear to contradict each other in a subtle way. Maxwell's equations determine a velocity: the velocity of light. But Newton's mechanics is not compatible with the existence of a fundamental velocity, because what enters Newton's equations is acceleration, not velocity. In Newton's physics, velocity can only be velocity of something with respect to something else. Galileo had underlined the fact that the Earth moves with respect to the Sun, even if we do not perceive this movement, because what we usually term 'velocity' is velocity 'with respect to Earth'. Velocity, we say, is a relative concept, that is, there is no meaning to the velocity of an object by itself: the only velocity which exists is the velocity of an object with respect to another object. This is what physics students learned in the nineteenth century, and what they learn today. But if this is so, then the speed of light determined by Maxwell's equations is velocity with respect to what?

One possibility is that there is a kind of universal substratum in relation to which light moves and has its speed. But the predictions of Maxwell's theory seem to be independent of this substratum. The experimental attempts to measure the speed of the Earth with respect to this hypothetical substratum tried at the end of the twentieth century all failed.

Einstein has claimed that he was not put on the right track by any experiments but only by reflecting on the apparent contradiction between Maxwell's equations and Newton's mechanics. He asked himself whether there was a way of rendering Newton's and Galileo's core discoveries and Maxwell's theory consistent.

In doing so, Einstein arrives at a stupefying discovery. To understand it, think of all the past, present and future events (with respect to the moment in which you are reading) and imagine them distributed as in figure 3.1.

Well, Einstein's discovery is that this diagram is incorrect. In reality, things are actually as they are depicted in figure 3.2.

Between the past and the future of an event (for example, between the past and the future for you, where you are, and in the precise moment in which you are reading) there exists an 'intermediate zone', an 'expanded present'; a zone that is neither past nor future. This is the discovery made with special relativity.

Figure 3.1 s.p.a.ce and time before Einstein.

Figure 3.2 The structure of 's.p.a.cetime'. For every observer, the 'extended present' is the intermediate zone between the past and the future.

The duration of this intermediate zone,fn10 which is neither in your past nor in your future, is very small and depends on where an event takes place relative to you, as ill.u.s.trated in figure 3.2: the greater the distance of the event from you, the longer the duration of the extended present. At a distance of a few metres from your nose, dear reader, the duration of what for you is the intermediate zone, neither past nor future, is no more than a few nanoseconds: next to nothing (the number of nanoseconds in a second is the same as the number of seconds in thirty years). This is much less than we could possibly notice. On the other side of the ocean, the duration of this intermediate zone is a thousandth of a second, still well below the threshold of our perception of time the minimum amount of time we perceive with our senses which is somewhere in the order of a tenth of a second. But on the Moon the duration of the expanded present is a few seconds, and on Mars it is a quarter of an hour. This means we can say that, on Mars, there are events that in this precise moment have already happened, events that are yet to happen, but also a quarter of an hour during which things occur that are neither in our past nor in our future.

They are elsewhere. We had never before been aware of this 'elsewhere' because, next to us, this 'elsewhere' is too brief; we are not quick enough to notice it. But it exists, and it is real.

This is why it is impossible to hold a smooth conversation between here and Mars. Say I am on Mars and you are here. I ask you a question and you reply as soon as you've heard what I said; your reply reaches me a quarter of an hour after I posed the question. This quarter of an hour is time that is neither past nor future to the moment in which you've replied to me. The key fact about nature that Einstein understood is that this quarter of an hour is inevitable: there is no way of reducing it. It is woven into the texture of the events of s.p.a.ce and of time: we cannot abbreviate it, any more than we can send a letter to the past.

Figure 3.3 The relativity of simultaneity.

It's strange, but this is how the world happens to be. As strange as the fact that in Sydney people live upside down: strange, but true. One gets accustomed to the fact, which then becomes normal and reasonable. It is the structure of s.p.a.ce and time that is made like this.

This implies that it makes no sense to say of an event on Mars that it is taking place 'just now', because 'just now' does not exist (figure 3.3).fn11 In technical terms, we say that Einstein has understood that 'absolute simultaneity' does not exist: there is no collection of events in the universe which exist 'now'. The collection of all the events in the universe cannot be described as a succession of 'now's, of presents, one following the other; it has a more complex structure, ill.u.s.trated in figure 3.2. The figure describes that which in physics is called s.p.a.cetime: the set of all past and future events, but also those that are 'neither-past-nor-future'; these do not form a single instant: they have themselves a duration.

In the Andromeda Galaxy, the duration of this expanded present is (with respect to us) 2 million years. Everything that happens during these 2 million years is neither past nor future with respect to ourselves. If a friendly advanced Andromeda civilization decided to send a fleet of s.p.a.cecraft to visit us, it would make no sense to ask whether 'now' the fleet has already left, or not yet. The only meaningful question is when we receive the first signal from the fleet: from that moment on not earlier the departure of the fleet is in our past.

The discovery of the structure of s.p.a.cetime made by the young Einstein in 1905 has concrete consequences. The fact that s.p.a.ce and time are intimately connected, as in figure 3.2, implies a subtle restructuring of Newton's mechanics, which Einstein rapidly completes in 1905 and 1906. A first result of this restructuring is that, as s.p.a.ce and time fuse together in a single concept of s.p.a.cetime, so the electric field and the magnetic fields fuse together in the same way, merging into a single ent.i.ty which today we call the electromagnetic field. The complicated equations written by Maxwell for the two fields become simple when written in this new language.

There is another implication of the theory, freighted with heavy consequences. The concepts of 'energy' and 'ma.s.s' become combined in the same way as time and s.p.a.ce, and electric and magnetic fields, are fused together in the new mechanics. Before 1905 two general principles appeared certain: conservation of ma.s.s, and conservation of energy. The first had been extensively verified by chemists: ma.s.s never changes in a chemical reaction. The second conservation of energy followed directly from Newton's equations and was considered one of the most incontrovertible laws. But Einstein realizes that energy and ma.s.s are two facets of the same ent.i.ty, just as the electric and magnetic fields are two facets of the same field, and as s.p.a.ce and time are two facets of the one thing: s.p.a.cetime. This implies that ma.s.s, by itself, is not conserved; and energy as it was conceived at the time is not independently conserved either. One may be transformed into the other: only one single law of conservation exists, not two. What is conserved is the sum of ma.s.s and energy, not each separately. Processes must exist that transform energy into ma.s.s, or ma.s.s into energy.

A rapid calculation teaches Einstein how much energy is obtained by transforming one gram of ma.s.s. The result is the celebrated formula E = mc. Since the speed of light c is a very large number, and c an even greater number, the energy obtained from transforming one gram of ma.s.s is enormous; it is the energy of millions of bombs exploding at the same time enough energy to illuminate a city and power the industries of a country for months or, conversely, capable of destroying in a second hundreds of thousands of human beings, in a city such as Hiroshima.

The theoretical speculations of the young Einstein had transported humanity into a new era: the era of nuclear power, an era of new possibilities, and new dangers. Today, thanks to the intelligence of a rebellious young man who would not abide rules, we have the instruments to bring light to the homes of the 10 billion human beings who will soon inhabit the planet, to travel in s.p.a.ce towards other stars, or to destroy each other and devastate the planet. It depends on our choices; on which leaders we call upon to decide for us.

Today the structure of s.p.a.cetime proposed by Einstein is well understood and repeatedly tested in laboratories; it is considered conclusively established. Time and s.p.a.ce are different from the way they had been conceived since Newton. s.p.a.ce does not exist independently from time. In the expanded s.p.a.ce of figure 3.2 there is no particular slice having a better claim than others to be called 's.p.a.ce now'. Our intuitive idea of the present the ensemble of all events happening 'now' in the universe is an effect of our blindness: our inability to recognize small temporal intervals. It is an illegitimate extrapolation from our parochial experience.

The present is like the flatness of the Earth: an illusion. We imagined a flat Earth because of the limitations of our senses, because we cannot see much beyond our own noses. Had we lived on an asteroid of a few kilometres in diameter, like the Little Prince, we would have easily realized we were on a sphere. Had our brain and our senses been more precise, had we easily perceived time in nanoseconds, we would never have made up the idea of a 'present' extending everywhere. We would have easily recognized the existence of the intermediate zone between past and future. We would have realized that saying 'here and now' makes sense, but that saying 'now' to designate events 'happening now' throughout the universe makes no sense. It is like asking whether our galaxy is 'above or below' the galaxy of Andromeda: a question that makes no sense, because 'above' or 'below' has meaning on the surface of the Earth, not in the universe. There isn't an 'up' or a 'down' in the universe. Similarly, there isn't either always a 'before' and an 'after' between two events in the universe. The resulting knitted structure that s.p.a.ce and time form together, depicted in figures like 3.2 and 3.3, is what physicists call 's.p.a.cetime' (figure 3.4).

When the Annalen der Physik published the article by Einstein in which all this was suddenly clarified, the impact upon the world of physics was momentous. The apparent contradiction between the equations of Maxwell and Newtonian physics were well known, and no one knew how to resolve them. Einstein's solution, astonishing and extremely elegant, took everyone by surprise. The story goes that in the dimly lit old halls of Cracow University, an austere professor of physics came out of his study, waving around Einstein's article, screaming, 'The new Archimedes is born!'

But despite the outcry provoked by the step forwards made by Einstein in 1905, we are not yet at his masterpiece. Einstein's triumph is the second theory of relativity, the theory of general relativity, published ten years later, when he was thirty-five.

Figure 3.4 What is the world made of?

The theory of 'general relativity' is the most beautiful theory produced by physics, and the first of the pillars of quantum gravity. It is at the heart of the narrative of this book. Here, the real magic of twentieth-century physics begins.

The most beautiful of theories

After publishing the theory of special relativity, Einstein becomes a renowned physicist and receives offers of work from numerous universities. But something troubles him: special relativity does not square with what was known about gravity. He realizes this while writing a review on his theory, and wonders whether the venerable theory of the 'universal gravity' of the father of physics, Newton, should not be reconsidered as well, to make it compatible with his relativity.

The origin of the problem is easy to understand. Newton had tried to explain why things fall and planets revolve. He had imagined a 'force' that draws all bodies towards one another: the 'force of gravity'. How this force managed to draw distant things together without anything between them was not understood. Newton himself, as we have seen, had suspected that in the idea of a force acting between distant bodies that do not touch there was something missing; and that in order for the Earth to attract the Moon something that could transmit this force had to be there between the two. Two hundred years later, Faraday had found the solution not for the force of gravity, but for the electric and magnetic forces: the field. Electric and magnetic fields 'carry around' the electric and magnetic force.

It's clear, at this stage, to any reasonable person, that the force of gravity must have its Faraday lines as well. It's clear also, by a.n.a.logy, that the force of attraction between the Sun and the Earth, or between the Earth and falling objects, must be attributed to a field in this case, a gravitational field. The solution discovered by Faraday and Maxwell to the question as to what carries the force must reasonably be applied not only to electricity but also to gravity. There must be a gravitational field, and some equations a.n.a.logous to Maxwell's, capable of describing how Faraday's gravitational lines move. In the first years of the twentieth century this is clear to any sufficiently reasonable person; that is to say, only to Albert Einstein.

Einstein, fascinated since adolescence by the electromagnetic field that pushed the rotors in his father's power stations, begins to look into this gravitational field and search for what kind of maths could describe it. He immerses himself in the problem. It would take ten years to resolve. Ten years of manic studies, attempts, mistakes, confusion, brilliant ideas, wrong ideas, a long series of articles published with incorrect equations, further mistakes and stress. Finally, in 1915, he commits to print an article containing the complete solution, which he names the General Theory of Relativity: his masterpiece. It is Lev Landau, the most outstanding theoretical physicist of the Soviet Union, who called it 'the most beautiful of theories'.

The reason for the beauty of the theory is not hard to see. Instead of simply inventing the mathematical form of the gravitational field and seeking to devise the equations for it, Einstein fishes out the other unresolved question in the furthest depths of Newton's theory and combines the two questions.

Newton had returned to Democritus's idea, according to which bodies move in s.p.a.ce. This s.p.a.ce had to be a large, empty container, a rigid box for the universe; an immense scaffolding in which objects run in straight lines, until a force causes them to curve. But what is this 's.p.a.ce' which contains the world made of? What is s.p.a.ce?

To us, the idea of s.p.a.ce seems natural, but it is our familiarity with Newtonian physics that makes it so. If you think about it, empty s.p.a.ce is not part of our experience. From Aristotle to Descartes, that is to say, for two millennia, the Democritean idea of s.p.a.ce as a peculiar ent.i.ty, distinct from things, had never been seen as reasonable. For Aristotle, as for Descartes, things have extension: extension is a property of things; extension does not exist without something being extended. I can take away the water from a gla.s.s, but air will fill it. Have you ever seen a really empty gla.s.s?

If between two things there is nothing, Aristotle reasoned, then there is nothing. How can there be at the same time something (s.p.a.ce) and nothing? What is this empty s.p.a.ce within which particles move? Is it something, or is it nothing? If it is nothing, it doesn't exist, and we can do without it. If it is something, can it be true that its only property is to be there, doing nothing?

Since antiquity, the idea of empty s.p.a.ce, halfway between a thing and a non-thing, had troubled thinkers. Democritus himself, who had placed empty s.p.a.ce at the basis of his world where atoms course, certainly wasn't crystal clear on the issue: he wrote that empty s.p.a.ce is something 'between being and non-being': 'Democritus postulated the full and the empty, calling one "Being", and the other "Non-Being",' says Simplicius.1 Atoms are being. s.p.a.ce is non-being a 'non-being' that, nevertheless, exists. It is difficult to be more obscure than this.

Newton, who resuscitated the Democritean idea of s.p.a.ce, had tried to patch things up by arguing that s.p.a.ce was G.o.d's sensorium. No one has ever understood what Newton meant by 'G.o.d's sensorium', perhaps not even Newton himself. Certainly, Einstein, who gave little credit to the idea of a G.o.d (with or without a sensorium), except as a playful rhetorical device, found Newton's explanation of the nature of s.p.a.ce utterly unconvincing.

Newton struggled considerably to overcome the scientists' and philosophers' resistance to his reviving the Democritean concept of s.p.a.ce; at first n.o.body took him seriously. Only the extraordinary efficacy of his equations, which turned out to predict always the correct outcome, ended up silencing criticism. But doubts concerning the plausibility of the Newtonian concept of s.p.a.ce persisted, and Einstein, who read philosophers, was well aware of them. Ernst Mach, whose influence Einstein readily acknowledged, was the philosopher who highlighted the conceptual difficulties of the Newtonian idea of s.p.a.ce the same Mach who did not believe in the existence of atoms. (A good example, incidentally, of how the same person can be short-sighted in one respect and far-seeing in another.) Thus, Einstein addresses not one but two problems. First, how can we describe the gravitational field? Second, what is Newton's s.p.a.ce?

And it's here that Einstein's extraordinary stroke of genius occurs, one of the greatest flights in the history of human thinking: what if the gravitational field turned out actually to be Newton's mysterious s.p.a.ce? What if Newton's s.p.a.ce was nothing more than the gravitational field? This extremely simple, beautiful, brilliant idea is the theory of general relativity.

Figure 3.5 What is the world made of?

The world is not made up of s.p.a.ce + particles + electromagnetic field + gravitational field. The world is made up of particles + fields, and nothing else; there is no need to add s.p.a.ce as an extra ingredient. Newton's s.p.a.ce is the gravitational field. Or vice versa, which amounts to saying the same thing: the gravitational field is s.p.a.ce (figure 3.5).

But, unlike Newton's s.p.a.ce, which is flat and fixed, the gravitational field, by virtue of being a field, is something which moves and undulates, subject to equations like Maxwell's field, like Faraday's lines.

It is a momentous simplification of the world. s.p.a.ce is no longer different from matter. It is one of the 'material' components of the world, akin to the electromagnetic field. It is a real ent.i.ty which undulates, fluctuates, bends and contorts.

We are not contained within an invisible, rigid scaffolding: we are immersed in a gigantic, flexible mollusc (the metaphor is Einstein's). The Sun bends s.p.a.ce around itself, and the Earth does not circle around it drawn by a mysterious distant force but runs straight in a s.p.a.ce that inclines. It's like a bead which rolls in a funnel: there are no mysterious forces generated by the centre of the funnel, it is the curved nature of the funnel wall which guides the rotation of the bead. Planets circle around the Sun, and things fall, because s.p.a.ce around them is curved (figure 3.6).

Figure 3.6 The Earth turns around the Sun because s.p.a.cetime around the Sun is curved, rather like a bead which rolls on the curved wall of a funnel.

A little more precisely, what curves is not s.p.a.ce but s.p.a.cetime that s.p.a.cetime which, ten years previously, Einstein himself had shown to be a structured whole rather than a succession of instants.

This is the idea. Einstein's only problem was to find the equations to make it concrete. How to describe this bending of s.p.a.cetime? And here Einstein is lucky: the problem had already been solved by the mathematicians.

The greatest mathematician of the nineteenth century, Carl Friedrich Gauss, the 'prince of mathematicians', had written maths to describe curved surfaces, such as the surfaces of hills, or such as the one portrayed in figure 3.7.

Figure 3.7 A curved (bidimensional) surface.

Then he had asked a talented student of his to generalize this maths to curved s.p.a.ces in three or more dimensions. The student, Bernhard Riemann, produced a ponderous doctoral thesis of the kind that seems completely useless.

Riemann's result was that the properties of a curved s.p.a.ce (or s.p.a.cetime) in any dimension are described by a particular mathematical object, which we now call Riemann curvature and indicate with the letter 'R'. If you think of a landscape of plains, hills and mountains, the curvature R of the surface is zero in the plains, which are flat 'without curvature' and different from zero where there are valleys and hills; it is at its maximum where there are pointed peaks of mountains, that is to say, where the ground is least flat, or most curved. Using Riemann's theory, it is possible to describe the shape of curved s.p.a.ces in three or four dimensions.

With a great deal of effort, seeking help from friends better versed in mathematics than himself, Einstein learns Riemann's maths and writes an equation where R is proportional to the energy of matter. In words: s.p.a.cetime curves more where there is matter. That is it. The equation is the a.n.a.logue of the Maxwell equations, but for gravity rather than electricity. The equation fits into half a line, and there is nothing more. A vision that s.p.a.ce curves becomes an equation.

But within this equation there is a teeming universe. And here the magical richness of the theory opens up into a phantasmagorical succession of predictions that resemble the delirious ravings of a madman but which have all turned out to be true. Even up to the beginning of the 1980s, almost n.o.body took the majority of these fantastical predictions entirely seriously. And yet, one after another, they have all been verified by experience. Let's consider a few of them.

To begin with, Einstein recalculates the effect of a ma.s.s like the Sun on the curvature of the s.p.a.ce that surrounds it, and the effect of this curvature on the movements of the planets. He finds the movements of the planets as predicted by Kepler's and Newton's equations, but not exactly: in the vicinity of the Sun, the effect of the curvature of s.p.a.ce is stronger than the effect of Newton's force. Einstein computes the movement of Mercury, the planet closest to the Sun and hence the one for which the discrepancy between the predictions of his and Newton's theories is greatest. He finds a difference: the point of the orbit of Mercury closest to the Sun moves every year 0.43 seconds of arc more than that predicted by Newton's theory. It is a small difference, but, within the scope of what astronomers were able to measure, and comparing the predictions with the observations of astronomers, the verdict is unequivocal: Mercury follows the trajectory predicted by Einstein, not the one predicted by Newton. Mercury, the fleet-footed messenger of the G.o.ds, the G.o.d of the winged sandals, follows Einstein, not Newton.

Einstein's equation, then, describes how s.p.a.ce curves very close to a star. Due to this curvature, light deviates. Einstein predicts that the Sun causes light to curve around it. In 1919 the measurement is achieved; a deviation of light is measured which turns out to be exactly in accordance with the prediction.

But it is not only s.p.a.ce that curves: time does, too. Einstein predicts that time on Earth pa.s.ses more quickly at higher alt.i.tude, and more slowly at lower alt.i.tude. This is measured, and also proves to be the case. Today we have extremely precise clocks, in many laboratories, and it is possible to measure this strange effect even for a difference in alt.i.tude of just a few centimetres. Place a watch on the floor and another on a table: the one on the floor registers less pa.s.sing of time than the one on the table. Why? Because time is not universal and fixed, it is something which expands and shrinks, according to the vicinity of ma.s.ses: the Earth, like all ma.s.ses, distorts s.p.a.cetime, slowing time down in its vicinity. Only slightly but two twins who have lived respectively at sea-level and in the mountains will find that, when they meet up again, one will have aged more than the other (figure 3.8).

This effect offers an interesting explanation as to why things fall. If you look at a map of the world and the route taken by an aeroplane flying from Rome to New York, it does not seem to be straight: the aeroplane makes an arc towards the north. Why? Because, the Earth being curved, crossing northwards is shorter than keeping to the same parallel. The distances between meridians are shorter the more northerly you are; therefore, it is better to head northwards, to shorten the route (figure 3.9).

Well, believe it or not, a ball thrown upwards falls downwards for the same reason: it 'gains time' moving higher up, because time pa.s.ses at a different speed up there. In both cases, aeroplane and ball follow a straight trajectory in a s.p.a.ce (or s.p.a.cetime) that is curved (figure 3.10).fn12 Figure 3.8 Two twins spend their time one at sea-level and the other in the mountains. When they meet up again, the twin who lived in the mountains is older. This is the gravitational dilation of time.

But the predictions of the theory go well beyond these minute effects. Stars burn as long as they have available hydrogen their fuel then die out. The remaining material is no longer supported by the pressure of the heat and collapses under its own weight. When this happens to a large enough star, the weight is so strong that matter is squashed down to an enormous degree and s.p.a.ce curves so intensely as to plunge down into an actual hole. A black hole.

Figure 3.9 The further north you go, the smaller the distance between two meridians.

Figure 3.10 The higher up something is, the more quickly time pa.s.ses for it.

When I was a university student, black holes were regarded as a scarcely credible implication of an esoteric theory. Today they are observed in their hundreds and studied in detail by astronomers. One of these black holes, with a ma.s.s a million times greater than the Sun, is located at the centre of our galaxy we can observe stars...o...b..ting around it. Some, pa.s.sing too close, are destroyed by its violent gravity.

Further still, the theory predicts that s.p.a.ce ripples like the surface of the sea, and that these ripples are waves similar to the electromagnetic ones which make television possible. The effects of these 'gravitational waves' can be observed in the sky on binary stars: they radiate such waves, losing energy and slowly falling towards each other.fn13 Gravitational waves produced by two black holes falling into one another were directly observed by an antenna on Earth in late 2015, and the announcement, given in early 2016, has once again left the world speechless. Once more, the seemingly mad predictions of Einstein's theory turn out to be precisely true.

And further still, the theory predicts that the universe is expanding and emerged from a cosmic explosion 14 billion years ago a subject I will discuss in more detail shortly.

This rich and complex range of phenomena bending of rays of light, modification of Newton's force, slowing down of clocks, black holes, gravitational waves, expansion of the universe, the Big Bang follow from understanding that s.p.a.ce is not a dull, fixed container but possesses its own dynamic, its own 'physics', just like the matter and the other fields it contains. Democritus himself would have smiled with pleasure, had he been able to see that his idea of s.p.a.ce would turn out to have such an impressive future. It is true that he termed it non-being, but what he meant by being () was matter; and he wrote that his non-being, the void, nevertheless 'has a certain physics () and a substantiality of its own'.fn14 How right he was.

Without the notion of fields introduced by Faraday, without the spectacular power of mathematics, without the geometry of Gauss and Riemann, this 'certain physics' would have remained incomprehensible. Empowered by new conceptual tools and by mathematics, Einstein writes the equations which describe Democritus's void and finds for its 'certain physics' a colourful and amazing world where universes explode, s.p.a.ce collapses into bottomless holes, time slows down in the vicinity of a planet, and the boundless expanses of interstellar s.p.a.ce ripple and sway like the surface of the sea ...

All of this sounds like a tale told by an idiot, full of sound and fury, signifying nothing. And yet, instead, it is a glance towards reality. Or better, a glimpse of reality, a little less veiled than our blurred and ba.n.a.l everyday view of it. A reality which seems to be made of the same stuff our dreams are made of, but which is nevertheless more real than our clouded daily dreaming.

And all this is the result only of an elementary intuition that s.p.a.cetime and the gravitational field are one and the same thing and a simple equation which I can't resist copying out here, even if most of my readers will certainly not be able to decipher it. I do so, anyway, in the hope that they might be able to catch a glimpse of its beautiful simplicity: In 1915 the equation was simpler still, because the term + gab, which Einstein added two years later (and which I discuss below) did not yet exist.fn15 Rab depends on Riemann's curvature, and together with Rgab represents the curvature of s.p.a.cetime; Tab stands for the energy of matter; G is the same constant that Newton found: the constant that determines the strength of the force of gravity.

That's it. A vision and an equation.

Mathematics or physics?

I would like to pause, before continuing with physics, to make a few observations about mathematics. Einstein was no great mathematician. He struggled with maths. He says this himself. In 1943 he replied in the following way to a nine-year-old child with the name of Barbara who wrote to him about her difficulties with the subject: 'Don't worry about experiencing difficulties with maths, I can a.s.sure you that my own problems are even more serious!'2 It seems like a joke, but Einstein was not kidding. With mathematics, he needed help: he had it explained to him by patient fellow students and friends, such as Marcel Grossman. It was his intuition as a physicist that was prodigious.

During the last year in which he was completing the construction of his theory, Einstein found himself competing with David Hilbert, one of the greatest mathematicians of all time. Einstein had given a lecture, attended by Hilbert, in Gttingen. Hilbert immediately understood that Einstein was in the process of making a major discovery, grasped the idea and tried to overtake Einstein and be the first to write the correct equations of the new theory Einstein was slowly building. The sprint to the finish line between the two giants was a nail-biting affair, eventually decided by a matter of just a few days. Einstein, in Berlin, ended up giving a public lecture almost every week, each time presenting a different equation, anxious that Hilbert would not get to the solution before him. The equation was incorrect every time. Until, that is, by a hair's breadth just marginally ahead of Hilbert Einstein found the right one. He had won the race.

Hilbert, a gentleman, never questioned Einstein's victory, even though he was working on very similar equations at the time. In fact, he left a gentle and beautiful phrase which captures perfectly Einstein's difficult relationship with mathematics, and, perhaps, the difficult relationship which exists generally between the whole of physics and mathematics. The maths that was necessary to formulate the theory was geometry in four dimensions, and Hilbert writes: Any youngster on the streets of Gttingenfn16 understands geometry in four dimensions better than Einstein. And yet, it was Einstein who completed the task.

Why? Because Einstein had a unique capacity to imagine how the world might be constructed, to 'see' it in his mind. The equations, for him, came afterwards; they were the language with which to make concrete his visions of reality. For Einstein, the theory of general relativity is not a collection of equations: it is a mental image of the world arduously translated into equations.

The idea behind the theory is that s.p.a.cetime curves. If s.p.a.cetime had only two dimensions, and we lived on a sort of plane, it would be easy to imagine what it means to say that 'physical s.p.a.ce curves'. It would mean that the physical s.p.a.ce in which we live is not like a flat table but resembles instead a surface with mountains and valleys. But the world we inhabit does not have only two dimensions, it has three. Four, in fact, when time is included. To imagine a curved s.p.a.ce in four dimensions is more complicated, because in our habitual perception we do not have an intuition of a 'larger s.p.a.ce' within which s.p.a.cetime can curve. But Einstein's imagination had no difficulty in intuiting the cosmic mollusc in which we are immersed, which can be squashed, stretched and twisted and that const.i.tutes the s.p.a.cetime around us. It is thanks to this visionary clarity that Einstein managed to be the first to write the theory.

In the end, a degree of tension between Hilbert and Einstein did develop. A few days before Einstein made his successful equation public, Hilbert had sent an article to a periodical which shows just how close he had come to the same solution and even today historians of science are faced with doubts when trying to evaluate the respective contributions of these two giants. At some point their relations cooled, and Einstein feared that Hilbert, more senior and powerful than him, would seek to attribute to himself too much of the merit for the construction of the theory. But Hilbert never claimed to be the first to discover general relativity and in a world such as that of science, where often, too frequently, disputes over precedence become poisonous the two gave a truly wonderful example of wisdom, clearing the field of all negative tension.

Einstein writes a marvellous letter to Hilbert, summarizing the profound sense of the shared course they had taken: There was a moment in which something like an irritation came between us, the origin of which I no longer want to a.n.a.lyse. I have fought against the bitterness which it provoked in me, and have succeeded completely in doing so. I again think of you with unclouded friendship, and I ask you to do the same for me. It is really a pity if companions such as we are, who have managed to forge a path aside from the pettiness of this world, could find anything other than joy in each other's company.3

The cosmos

Two years after the publication of his equation, Einstein decides to use it to describe the s.p.a.ce of the entire universe, considered at the largest scale. And here he has another of his amazing ideas.

For thousands of years, men had asked themselves whether the universe was infinite, or had a limit. Both hypotheses entail th.o.r.n.y problems. An infinite universe does not seem to stand to reason: if it is infinite, for example, there must exist somewhere a reader just like you who is reading the very same book (infinity is truly vast, and there are not sufficient combinations of atoms to fill it with things always different from each other). In fact, there must be not only one but an infinite series of readers identical to yourself ... But if there is a limit to the universe, what is that boundary? What sense is there in a border with nothing on the other side? Already in the fourth century CE, in Taranto, the Pythagorean philosopher Archytas had written: If I found myself in the furthest sky, that of the fixed stars, would I be able to stretch my hand, or a rod, out beyond it or not? That I should not be able to is absurd; but if I am able to, then an outside exists, be it of matter, or s.p.a.ce. In this way one could proceed ever further, towards the end, from time to time asking the same question, as to whether there will always be something into which to extend the rod.4 These two absurd alternatives the absurdity of an infinite s.p.a.ce, and the absurdity of a universe with a fixed border didn't seem to leave any reasonable choice between them.

But Einstein finds a third way: the universe can be finite and at the same time have no boundary. How? Just as the surface of the Earth is not infinite but does not have a boundary either, where it 'ends'. This can happen, naturally enough, if something is curved: the surface of the Earth is curved. And in the theory of general relativity, of course, three-dimensional s.p.a.ce can also be curved. Consequently, our universe can be finite but borderless.

On the surface of the Earth, if I were to keep walking in a straight line, I would not advance ad infinitum: I would eventually get back to the point I started from. Our universe could be made in the same way: if I leave in a s.p.a.cecraft and journey always in the same direction, I fly around the universe and eventually end up back on Earth. A three-dimensional s.p.a.ce of this kind, finite but without boundary, is called a 3-sphere.

Figure 3.11 A sphere can be represented as two discs which in reality are smoothly joined all along their edges.

To understand the geometry of a '3-sphere', let us return to the ordinary sphere; the surface of a ball, or the Earth. To represent the surface of the Earth on a plane, we can draw two discs, as is customary when drawing the continents (figure 3.11).