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Einstein’s ideas were so iconoclastic that it took time for mainstream scientists to welcome this deskbound civil servant into their community. Although he published his special theory of relativity in 1905, it was not until 1908 that he received his first junior academic post at Berne University. Between 1905 and 1908, Einstein continued to work at the patent office in Berne, where he was promoted to ‘technical expert, second class’ and given the time to push ahead with his effort to extend the power and remit of his theory of relativity.

The special theory of relativity is labelled special because it applies only to special situations, namely those in which objects are moving at constant velocity. In other words, it could deal with Bob observing Alice’s train travelling at a fixed speed on a straight track, but not with a train that was speeding up or slowing down. Consequently, Einstein attempted to reformulate his theory so that it would cope with situations involving acceleration and deceleration. This grand extension of special relativity would soon become known as general relativity, because it would apply to more general situations.

When Einstein made his first breakthrough in building general relativity in 1907, he called it ‘the happiest thought of my life’. What followed, however, was eight years of torment. He told a friend how his obsession with general relativity was forcing him to neglect every other aspect of his life:‘I cannot find the time to write because I am occupied with truly great things. Day and night I rack my brain in an effort to penetrate more deeply into the things that I gradually discovered in the past two years and that represent an unprecedented advance in the fundamental problems of physics.’

In speaking of ‘truly great things’ and ‘fundamental problems’, Einstein was referring to the fact that the general theory of relativity seemed to be leading him towards an entirely new theory of gravity. If Einstein was right, then physicists would be forced to question the work of Isaac Newton, one of the icons of physics.

Newton was born in tragic circumstances on Christmas Day 1642, his father having died just three months earlier. While Isaac was still an infant, his mother married a sixty-three-year-old rector, Barnabas Smith, who refused to accept Isaac into his home. It fell to Isaac’s grandparents to bring him up, and as each year passed he developed a growing hatred towards the mother and stepfather who had abandoned him. Indeed, as an undergraduate, he compiled a list of childhood sins that included the admission of ‘threatening my father and mother Smith to burne them and the house over them’.

Not surprisingly, Newton grew into an embittered, isolated and sometimes cruel man. For example, when he was appointed Warden of the Royal Mint in 1696, he implemented a harsh regime of capturing counterfeiters, making sure that those convicted were hung, drawn and quartered. Forgery had brought Britain to the brink of economic collapse, and Newton judged that his punishments were necessary. In addition to brutality, Newton also used his brains to save the nation’s currency. One of his most important innovations at the Mint was to introduce milled edges on coins to combat the practice of clipping, whereby counterfeiters would shave off the edges of coins and use the clippings to make new coins.

In recognition of Newton’s contribution, the British £2 coin issued in 1997 had the phrase STANDING ON THE SHOULDERS OF GIANTS around its milled edge. These words are taken from a letter that Newton sent to fellow scientist Robert Hooke, in which he wrote: ‘If I have seen further it is by standing on the shoulders of giants.’ This appears to be a statement of modesty, an admission that Newton’s own ideas were built upon those of illustrious predecessors such as Galileo and Pythagoras. In fact, the phrase was a veiled and spiteful reference to Hooke’s crooked back and severe stoop. In other words, Newton was pointing out that Hooke was neither a physical giant, nor, by implication, an intellectual giant.

Whatever his personal failings, Newton made an unparalleled contribution to seventeenth-century science. He laid the foundations for a new scientific era with a research blitz that lasted barely eighteen months, culminating in 1666, which is today known as Newton’s annus mirabilis. The term was originally the title of a John Dryden poem about other more sensational events that took place in 1666, namely London’s survival after the Great Fire and the victory of the British fleet over the Dutch. Scientists, however, judge Newton’s discoveries to be the true miracles of 1666. His annus mirabilis included major breakthroughs in calculus, optics and, most famously, gravity.

In essence, Newton’s law of gravity states that every object in the universe attracts every other object. More exactly, Newton defined the force of attraction between any two objects as

The force (F) between the two objects depends on the masses of the objects (m₁ and m₂)—the bigger the masses, the bigger the force. Also, the force is inversely proportional to the square of the distance between the objects (r²), which means that the force gets smaller as the objects move farther apart. The gravitational constant (G) is always equal to 6.67 × 10^-11 Nm²kg^-2, and reflects the strength of gravity compared with other forces such as magnetism.

The power of this formula is that it encapsulates everything that Copernicus, Kepler and Galileo had been trying to explain about the Solar System. For example, the fact that an apple falls towards the ground is not because it wants to get to the centre of the universe, but simply because the Earth and the apple both have mass, and so are naturally attracted towards each other by the force of gravity. The apple accelerates towards the Earth, and at the same time the Earth even accelerates up towards the apple, although the effect on the Earth is imperceptible because it is much more massive than the apple. Similarly, Newton’s gravity equation can be used to explain how the Earth orbits the Sun because both bodies have a mass and therefore there is a mutual attraction between them. Again, the Earth orbits the Sun and not vice versa because the Earth is much less massive than the Sun. In fact, Newton’s gravity formula can even be used to predict that moons and planets will follow elliptical paths, which is exactly what Kepler demonstrated after analysing Tycho Brahe’s observations.

For centuries after his death, Newton’s law of gravity ruled the cosmos. Scientists assumed that the problem of gravity had been solved and used Newton’s formula to explain everything from the flight of an arrow to the trajectory of a comet. Newton himself, however, suspected that his understanding of the universe was incomplete: ‘I do not know what I may appear to the world, but to myself I seem to have been only a little boy playing on the seashore, and diverting myself now and then in finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay undiscovered before me.’

And it was Albert Einstein who first realised that there might be more to gravity than Newton had imagined. After his own annus mirabilis in 1905, when Einstein published several historic papers, he concentrated on expanding his special theory of relativity into a general theory. This involved a radically different interpretation of gravity based on a fundamentally different vision of how planets, moons and apples attract one another.

At the heart of Einstein’s new approach was his discovery that both distance and time are flexible, which was a consequence of his special theory of relativity. Remember, Bob sees a clock slowing down and Alice getting thinner as they move towards him. So time is flexible, as are the three dimensions of space (width, height, depth). Furthermore, the flexibility of both space and time are inextricably linked, which led Einstein to consider a single flexible entity known as spacetime. And it turned out that this flexible spacetime was the underlying cause of gravity. This cavalcade of weird flexibility is undoubtedly mind-bending, but the following paragraph provides a reasonably easy way to visualise Einstein’s philosophy of gravity.

Spacetime consists of four dimensions, three of space and one of time, which is unimaginable for most mortals, so it is generally easier to consider just two dimensions of space, as shown in Figure 23. Fortunately, this rudimentary spacetime illustrates many of the key features of authentic spacetime, so this is a convenient simplification. Figure 23(a) shows that space (and indeed spacetime) is rather like a piece of stretchy fabric; the gridlines help to show that if nothing is occupying space, then its ‘fabric’ is flat and undisturbed. Figure 23(b) shows how two-dimensional space changes severely if an object is placed upon it. This second diagram could represent space being warped by the massive Sun, rather like a trampoline curving under the weight of a bowling ball.

In fact, the trampoline analogy can be extended. If the bowling ball represents the Sun, then a tennis ball representing the Earth could be launched into orbit around it, as shown in Figure 23(c). The tennis ball actually creates its own tiny dimple in the trampoline and it carries this dimple with it around the trampoline. If we wanted to model the Moon, then we could try to roll a marble in the tennis ball dimple and make it race around the tennis ball, while the tennis ball and its dimple raced around the hollow caused by the bowling ball.

Figure 23 These diagrams are two-dimensional representations of four-dimensional spacetime, ignoring time and one space dimension. Diagram (a) shows a flat, smooth, undisturbed grid, representing empty space. If a planet were to pass through this space, then it would follow a straight line.

Diagram (b) shows space warped by an object such as the Sun. The depth of the depression depends on the mass of the Sun.

Diagram (c) shows a planet orbiting the depression caused by the Sun. The planet causes its own little depression in space, but it is too small to be represented in this diagram because the planet is relatively light.

In practice, any attempt to model a complicated system on a trampoline soon breaks down, because the friction of the trampoline fabric disturbs the natural movement of the objects. Nevertheless, Einstein was arguing that exactly these sorts of trampoline effects were really happening in the fabric of spacetime. According to Einstein, whenever physicists and astronomers witnessed phenomena involving the force of gravitational attraction, they were actually seeing objects reacting to the curvature of spacetime. For example, Newton would have said that an apple fell to Earth because there was a mutual force of gravitational attraction, but Einstein now felt that he had a deeper understanding of what was driving this attraction: the apple fell to Earth because it was falling into the deep hollow in spacetime caused by the mass of the Earth.

The presence of objects in spacetime gives rise to a two-way relationship. The shape of spacetime influences the motion of objects, and at the same time those very objects determine the shape of spacetime. In other words, the dimples in spacetime that guide the Sun and the planets are caused by those selfsame Sun and planets. John Wheeler, one of the leading general relativists of the twentieth century, summed up the theory with the dictum ‘Matter tells space how to bend; space tells matter how to move.’ Although Wheeler sacrificed accuracy for snappiness (‘space’ should have been ‘spacetime’), this is still a neat summary of Einstein’s theory.

This notion of flexible spacetime may sound crazy, but Einstein was convinced that it was right. According to his own set of aesthetic criteria, the link between flexible spacetime and gravity had to be true, or as Einstein put it: ‘When I am judging a theory, I ask myself whether, if I were God, I would have arranged the world in such a way.’ However, if Einstein was to convince the rest of the world that he was right, he had to develop a formula that encapsulated his theory. His greatest challenge was to transform the rather vague notion of spacetime and gravity described above into a formal theory of general relativity, set in a rigorous mathematical framework.