General Theory of Relativity explained: The beautiful simplicity (and weird time sags) of Einstein's 100-year-old masterpiece

In his youth, Albert Einstein spent a year loafing aimlessly. You don't get anywhere by not "wasting" time – something, unfortunately, which the parents of teenagers tend frequently to forget. He was in Pavia. He had joined his family having abandoned his studies in Germany, unable to endure the rigours of his high school there. It was the beginning of the 20th century, and in Italy the beginning of its industrial revolution. His father, an engineer, was installing the first electrical power plants in the Paduan plains.

Albert was reading Kant and attending occasional lectures at the University of Pavia – for pleasure, without being registered there or having to think about exams. It is thus that serious scientists are made.

After this, he registered at the University of Zurich, and immersed himself in the study of physics. A few years later, in 1905, he sent three articles to the most prestigious scientific journal of the period, the Annalen der Physik. Each of these is worthy of a Nobel Prize.

The first shows that atoms really exist. The second lays the first foundation for quantum mechanics. The third presents his first theory of relativity (known today as "special relativity"), the theory which elucidates how time does not pass identically for everyone: identical twins find that they are different in age, if one of them has travelled at speed.

Einstein became a renowned scientist overnight and received offers of employment from various universities. But something disturbed him: despite its immediate acclaim, his theory of relativity did not fit with what we know about gravity – namely, with how things fall. He came to realise this when writing an article summarising his theory, and began to wonder if the law of "universal gravity" as formulated by the father of physics himself, Isaac Newton, was in need of revision in order to make it compatible with the new concept of relativity. He immersed himself in the problem. It would take 10 years to resolve. Ten years of frenzied study, attempts, errors, confusion, mistaken articles, brilliant ideas, misconceived ideas.

Finally, in November 1915, he committed to print an article giving the complete solution: a new theory of gravity, which he called "the General Theory of Relativity", his masterpiece and the "most beautiful of theories", according to the great Russian physicist Lev Landau.

There are absolute masterpieces which move us intensely: Mozart's Requiem; Homer's Odyssey; the Sistine Chapel; King Lear. To fully appreciate their brilliance may require a long apprenticeship, but the reward is sheer beauty – and not only this, but the opening of our eyes to a new perspective upon the world. Einstein's jewel, the general theory of relativity, is a master-piece of this order.

I remember the excitement I felt when I began to understand something about it. It was summer. I was on a beach at Condofuri in Calabria, immersed in the sunshine of the Hellenic Mediterranean, and in the last year of my university studies. Undistracted by schooling, one studies best during vacations.

I was studying with the help of a book that had been gnawed at the edges by mice, because at night I'd used it to block the holes of these poor creatures in the rather dilapidated, hippie-ish house on an Umbrian hillside where I used to take refuge from the tedium of university classes in Bologna. Every so often I would raise my eyes from the book and look at the glittering sea: it seemed to me that I was actually seeing the curvature of space and time imagined by Einstein. It was as if a friend were whispering into my ear an extraordinary hidden truth, suddenly raising the veil of reality to disclose a simpler, deeper order. Ever since we discovered that the Earth is round and turns like a mad spinning top, we have understood that reality is not as it appears to us; every time we glimpse a new aspect of it, it is a deeply emotional experience. Another veil has fallen.

But among the numerous leaps forward in our understanding that have succeeded each other over the course of history, Einstein's is perhaps unequalled. Why? In the first place because, once you understand how it works, the theory has a breathtaking simplicity.

I'll summarise the idea. Newton had tried to explain why things fall and planets turn. He had imagined the existence of a "force" which draws all material bodies towards one another, and called it "the force of gravity".

How this force was exerted between things distant from each other, without there being anything between them, was unknown – and the great father of modern science was cautious of offering a hypothesis.

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Newton also imagined that bodies moved through space, and that space was a great empty container, a large box which enclosed the universe, an immense structure through which all objects ran true until a force obliged their trajectory to curve. What this "space" was made of, this container of the world he invented, Newton could not say. But a few years before the birth of Einstein, two great British physicists, Michael Faraday and James Maxwell, had added a key ingredient to Newton's cold world: the electromagnetic field. This field is a real entity which, diffused everywhere, carries radio waves, fills space, can vibrate and oscillate like the surface of a lake, and "transports" the electrical force.

Since his youth, Einstein had been fascinated by this electromagnetic field which turned the rotors in the power stations built by his father, and he soon came to understand that gravity, like electricity, must be conveyed by a field as well: a "gravitational field" analogous to the "electrical field" must exist. He aimed at understanding how this gravitational field worked, and how it could be described with equations.

And it is at this point that an extraordinary idea occurred to him, a stroke of pure genius: the gravitational field is not diffused through space; the gravitational field is that space itself. This is the idea of the theory of general relativity. Newton's "space", through which things move, and the "gravitational field" are one and the same.

It's a moment of enlightenment. A momentous simplification of the world: space is no longer something distinct from matter, it is one of the "material" components of the world. An entity that undulates, flexes, curves, twists. We are not contained within an invisible, rigid infrastructure: we are immersed in a gigantic, flexible snail shell. The sun bends space around itself, and the Earth does not turn around it because of a mysterious force but because it is racing directly in a space which inclines, like a marble that rolls in a funnel. There are no mysterious forces generated at the centre of the funnel; it is the curved nature of the walls which causes the marble to roll.

Planets circle around the sun, and things fall, because space curves.

How can we describe this curvature of space? The most outstanding mathematician of the 19th century, Carl Friedrich Gauss, the so-called "prince of mathematicians", had written mathematical formulae to describe two-dimensional undulating surfaces, such as the surfaces of hills. Then he had asked a gifted student of his to generalise the theory to encompass spaces in three or more dimensions. The student in question, Bernhard Riemann, had produced an impressive doctoral thesis of the kind that seems completely useless. The conclusion of Riemann's thesis was that the properties of a curved space are captured by a particular mathematical object which we know today as Riemann's curvature, and indicate with the letter "R". Einstein wrote an equation which says that R is equivalent to the energy of matter. That is to say: space curves where there is matter. That is it. The equation fits into half a line, and there is nothing more. A vision – that space curves – became 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.

To begin with, the equation describes how space bends around a star. Due to this curvature, not only do planets orbit the star, but light stops moving in a straight line and deviates. Einstein predicted that the sun causes light to deviate. In 1919, this deviation was measured, and the prediction verified.

But it isn't only space that curves; time does too. Einstein predicted that time passes more quickly high up than below, nearer to the Earth. This was measured and turned out to be the case. If a man who has lived at sea level meets up with his twin who has lived in the mountains, he will find that his sibling is slightly older than him. And this is just the beginning.

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When a large star has burnt up all of its combustible substance (hydrogen), it goes out. What remains is no longer supported by the heat of the combustion and collapses under its own weight, to a point where it bends space to such a degree that it plummets into an actual hole. These are the famous "black holes". When I was studying at university, they were considered to be the barely credible predictions of an esoteric theory. Today, they are observed in the sky in their hundreds, and studied in great detail by astronomers.

But this is still not all. The whole of space can expand and contract. Furthermore, Einstein's equation shows that space cannot stand still; it must be expanding. In 1930, the expansion of the universe was actually observed. The same equation predicts that the expansion ought to have been triggered by the explosion of a young, extremely small and extremely hot universe: by what we now know as the "Big Bang". Once again, no one believed this at first, but the proof mounted up until cosmic background radiation – the diffuse glare that remains from the heat generated by the original explosion – was actually observed in the sky. The prediction arising from Einstein's equation turned out to be correct.

And further still, the theory contends that space moves like the surface of the sea. The effects of these "gravitational waves" are observed in the sky on binary stars, and correspond to the predictions of the theory even to the astonishing precision of one part to one hundred billion. And so forth.

In short, the theory describes a colourful and amazing world where universes explode, space collapses into bottomless holes, time sags and slows near a planet, and the unbounded extensions of interstellar space ripple and sway like the surface of the sea – and all of this, which emerged gradually from my mice-gnawed book, was not a tale told by an idiot in a fit of lunacy, or a hallucination caused by Calabria's burning Mediterranean sun and its dazzling sea. It was reality.

Or better, a glimpse of reality, a little less veiled than our blurred and banal everyday view of it. A reality which seems to be made of the same stuff which our dreams are made of, but which is nevertheless more real than our clouded quotidian dreaming.

All of this is the result of an elementary intuition: that space and gravitational field are the same thing. And of a simple equation which I cannot resist giving here, even though you will almost certainly not be able to decipher it. Though perhaps anyone reading this will still be able to appreciate its wonderful simplicity: Rab − ½ R gab = Tab.

That's it.

You would, of course, need to study and digest Riemann's mathematics in order to master the technique to read and use this equation. It takes a little commitment and effort. But less than is necessary to come to appreciate the rarefied beauty of a late Beethoven string quartet. In both cases, the reward is sheer beauty, and new eyes with which to see the world.

Extracted from 'Seven Brief Lessons in Physics' by Carlo Rovelli, published by Allen Lane on 24 September, priced £9.99. Rovelli is head of the Equipe de Gravité Quantique at the Theoretical Physics Department of Aix-Marseille University