The Magicians

by Marcus Chown

  Einstein’s struggle to find the elusive theory of gravity culminated in his presentation of the ‘general theory of relativity’ in four papers to the Prussian Academy of Sciences in November 1915. It painted a picture of a new and unexpected world.

  According to Newton, a ‘force’ of gravity exists between the Sun and the Earth, much like an invisible tether that connects the two bodies and keeps the Earth perpetually trapped in orbit around the Sun. Einstein showed this is wrong; what a mass like the Sun actually does is create a valley in the space–time around it.11 The Earth then traverses the upper slopes of this valley like a ball in a roulette wheel. Though no one would use such words for another half a century, what the general theory of relativity says in essence is: ‘Matter tells space–time how to warp, and warped space–time tells matter how to move.’12

  Warped space–time, according to Einstein, is gravity. However, since space–time is a four-dimensional thing and we are three-dimensional beings, we are utterly unaware of the hills and valleys of space–time. To explain the motion of a body like the Earth around the Sun, we have, therefore, invented a ‘force’ called gravity.

  Copies of Einstein’s papers on his new theory reached Schwarzschild on the Eastern Front within days of their presentation in Berlin. He had instantly fallen in love with the general theory of relativity.13 Its beauty and daring took his breath away. But more importantly, it took him to another place, far away from the death, destruction and the pounding of the guns. Incredibly, he found time between his calculations of artillery trajectories to absorb the complex mathematics and think deeply about the consequences of the theory.

  Einstein had used his theory to explain the baffling motion of the planet closest to the Sun. Mercury, like all the planets, is tugged by the gravity not only of the Sun but also of the other planets in the solar system. They cause its elliptical orbit to gradually change its orientation in space, or ‘precess’. But even when this effect is taken into account, there remains an unexplained bit left over: the ‘anomalous precession of the perihelion of Mercury’.

  Einstein realised that Mercury’s proximity to the biggest mass in the solar system meant that it was orbiting in the most warped space–time of any planet and that its path through space would therefore be affected by this. He used his theory of gravity to predict the path and found that it was exactly what was observed by astronomers. It was a triumph. His theory perfectly explained the anomalous motion of Mercury.

  However, Einstein’s calculations were scrappy and inelegant. The problem was that the machinery of his theory of gravity was complex. It used the mathematics of curved space–time, which had been developed in the nineteenth century by a number of mathematicians, most notably Carl Friedrich Gauss and Bernhard Riemann. Whereas one equation is sufficient to describe gravity in Newton’s theory, Einstein’s requires a total of ten.14 Consequently, finding the shape of the space–time for a given distribution of matter – known technically as a ‘solution’ of Einstein’s equations of the gravitational ‘field’ – is hard. Einstein himself had believed it impossible, so, in explaining the anomalous motion of Mercury, he had resorted to an approximate expression for the curved space–time around the Sun.

  Schwarzschild was familiar with ‘Riemannian geometry’: the mathematics of curved space. While the guns boomed around him, he wondered whether he could do better than Einstein. Could he find an exact formula for the curvature of space around a localised mass like the Sun?

  He started by making some basic assumptions. First, that the Sun – or indeed any star – is perfectly spherical. Secondly, that the curvature of the space–time around it does not change with time. And thirdly, that the curvature of space–time does not depend on direction, but only on the ‘radial’ distance from the Sun. Remarkably, these insights allowed Schwarzschild to hugely simplify Einstein’s equations, reducing them from ten to just one. He then employed a little mathematical wizardry, and – miracle of miracles – discovered that the lone equation had a unique solution.

  Schwarzschild had achieved the impossible: he had out-Einsteined Einstein. Rather than an approximate expression for the curvature of space–time surrounding the Sun, he had found a precise description, which was the first exact solution of Einstein’s theory of gravity ever found. In years to come, in recognition of the difficulty of finding a solution to Einstein’s equations, physicists would refer to such solutions by the names of their discoverers. His would be immortalised as the Schwarzschild solution, or, more precisely, the ‘Schwarzschild metric’.

  Using his exact solution, Schwarzschild quickly confirmed Einstein’s claim that his theory explained the anomalous motion of Mercury. ‘It is quite a wonderful thing that from such an abstract idea the explanation of the Mercury anomaly emerges so inevitably,’ he observed.15

  Schwarzschild then wrote up his calculations as a paper, and on 22 December 1915, wrote a covering letter to Einstein. It concluded, ‘As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas.’16 As yet, he had not realised the seriousness of the blisters which had begun to form in his mouth and would soon see him invalided out of the army to a field hospital.

  *

  It was a surprise to receive a letter from the Eastern Front. Einstein was aware – because it was such an extraordinary thing – that the director of the Berlin Observatory, despite being forty years old, had volunteered for the Kaiser’s army at the outbreak of war, but what could he be writing to him about?

  On reading, he was astonished to find a calculation using his own theory. He ran his finger along the lines of algebra, nodding emphatically as he did so. He had presented his theory of gravity to the Prussian Academy of Sciences little more than a month ago, yet Schwarzschild had not only mastered it but driven it forward into new territory. Here was the first exact solution of his general theory of relativity, something Einstein himself had considered impossible.

  Immediately, he replied to Schwarzschild, ‘I have read your paper with the utmost interest. I had not expected that one could formulate the solution to the problem in such a simple way. I liked very much your mathematical treatment of the subject.’17

  Einstein promised to present the work to the Prussian Academy on the following Thursday, along with a few words of explanation. He was as good as his word and delivered a summary of Schwarzschild’s paper on 13 January 1916. But Schwarzschild, lying in a hospital bed, had not finished with Einstein’s theory. He had examined the case of an idealised star – a spherical mass – and found an exact description of the curvature of space–time on the outside, but what about the inside? That was the subject of his second paper, whose calculations he was now checking, and which he was on the verge of sending to Einstein.

  The topic had captivated him for several days, and most importantly it had taken away his pain. He was oblivious to everything, a man lost in the dream. ‘Professor Schwarzschild!’ he remembered them shaking him. ‘We need to change your dressing … your bedclothes … You need to go for a walk …’

  What he had discovered by perusing his miraculous solution was something incredible. If a celestial body were ever to be compressed within a certain critical radius, space–time would become so grossly warped that it would no longer be a mere valley.18 Instead, it would morph into a bottomless pit from which nothing, not even a beam of light, could ever climb out. The star would become cut off from the universe forever and would appear like a hole in space. He had no name for such a region of grossly warped space–time, but one day there would be almost nobody on Earth who had not heard of the term ‘black hole’.

  The threshold radius was ridiculously small. Just like Schwarzschild’s space–time solution itself, it would one day bear the name of its discoverer. For the Sun, the ‘Schwarzschild radius’ was 1.47 kilometres, and for the Earth it was a mere five millimetres. If the Sun and the Earth were squeezed this small, they would wink out
of existence, disappearing from view forever.

  But the Sun is more than a million kilometres across, so this would mean compressing its material to an enormous, mind-boggling density. Schwarzschild’s first reaction was that this was ‘very weird and perhaps just a mathematical curiosity’, but he did not dismiss it out of hand.19 ‘History tells us that the mathematical solutions are often realised in nature, as if there were some kind of pre-established harmony between mathematics and physics,’ he wrote. This idea had been quite strongly present at the University of Göttingen, where he had worked before Berlin, and he confessed to being a ‘believer’. Maybe the monster described by his equation might actually exist.

  Schwarzschild folded the letter containing his new paper, slipped it in an envelope and sealed it. When an orderly came by, he gave it to him to post.

  *

  Einstein presented Schwarzschild’s black hole solution to the Prussian Academy of Sciences on 13 February 1916. In March, Schwarzschild, whose condition had worsened, was moved to Berlin, and he died on 11 May 1916. He was just forty-two. But one thing did not die with him: his solution of Einstein’s equations for a black hole.

  Herstmonceux, Sussex, Winter 1971

  After the meeting with Louise Webster, Paul Murdin found it impossible to settle: it was the adrenalin coursing through him. Webster, as reserved and unperturbable as ever, worked at her desk, immune to distraction, while Murdin paced up and down in the turret room, going over the logic of the remarkable conclusion they had come to.

  The X-rays in Cygnus X-1 came from matter torn off the blue supergiant star and heated to incandescence by internal friction as it swirled down into a black hole. If the two astronomers were right, they had made a truly momentous astronomical discovery. Nevertheless, it was still hard to believe that such a wild theoretical prediction could come true. ‘The surprising thing is that black holes turn out to be real objects,’ remarked Murdin. ‘Incredibly, they actually exist!’

  Murdin hoped that finding the first black hole out in space would make his name in astronomy and that, even more importantly, it might lead to a permanent job. With a young family to support, that was more than a trivial consideration.

  Murdin and Webster wrote a 500-word paper on their discovery, but they had a surprising amount of difficulty in getting permission to send it to Nature for publication. The Royal Observatory’s director, Richard Woolley, did not believe in black holes, thinking they were some sort of ‘new-age-ish’ magic. ‘In one conversation he even asked me why, exactly, I believed that Cygnus X-1 contained a “black box”,’ says Murdin.

  Part of the reason for Woolley’s foot-dragging was that he had been the student of Arthur Eddington, who had not believed in black holes, but another reason was that the Royal Greenwich Observatory at Herstmonceux had until recently been run by the Royal Navy. Public perception was extremely important to the navy, and Woolley was fearful that the Observatory might make a claim that would open it up to derision and ridicule.

  But the observations Murdin and Webster had accumulated were nothing if not convincing: all the evidence pointed to HDE 226868 being in orbit around something invisible and massive, and the only conceivable object that fit the bill was a black hole. Finally, after consulting with other senior members of the RGO, Woolley caved in and gave permission to Murdin and Webster to submit their paper for publication.

  A lot was at stake for Murdin and it was a nail-biting time. There was no guarantee that someone else would not come to the same conclusion about HDE 226868 and beat him and Webster into print. To guard against this, Murdin decided to deliver the paper personally to the Nature offices in central London, making sure it was stamped with the date. However, while driving to Hastings Station to catch the train, he half heard a news item on his car radio that seemed to be about a very energetic event in the stars. He immediately thought, ‘Oh no, someone else has got our fantastic result! We’ve been scooped!’

  All day in London, Murdin was worried sick. It was only when he returned to Hastings that evening that he heard the news item repeated; to his enormous relief it turned out to be about a storm on Mars.

  The paper appeared in Nature on 7 January 1972.20 Murdin got his permanent job and his family got a bigger house. In fact, he was the first person in history to have his mortgage paid by a black hole.21

  The day the paper was published, Murdin and his wife Lesley celebrated by taking their two small boys to a café on Hastings seafront and treating them to ‘knickerbocker glories’. As the boys, aged three and seven, dug their long spoons into the layers of ice cream, fruit and syrup, it was not difficult to guess what was going through their minds. ‘I think they were hoping their dad would find some more black holes,’ says Murdin.22

  It would be hard to imagine a greater contrast between the world of the man who had co-discovered black holes and that of the magician who had predicted the existence of black holes from his hospital bed on the Eastern Front fifty-six years earlier.

  *

  By the time Murdin and Webster discovered the first black hole in Cygnus X-1, the theory of such objects had moved on from Schwarzschild’s exact solution to Einstein’s theory of gravity.23 Einstein never believed in the possibility of black holes, and most others who considered the solution shared his view – though not Schwarzschild – that a body from which nothing, not even light, can escape is simply too weird for words. When a massive star shrinks at the end of its life, they reasoned, some as-yet-unknown force must intervene to prevent the formation of such a monstrosity. Such a force appeared to be provided by a revolutionary new description of the world of atoms and their constituents, which was formulated in the 1920s.

  ‘Quantum theory’ recognises that the fundamental building blocks of the world such as atoms, electrons and photons have a weird ‘dual’ nature.24 They can behave both as particles – like tiny billiard balls – and waves, like ripples on a pond. Because such quantum waves are spread out and so need a lot of room, the particles with which they are associated are hard to squeeze into a small volume. Or, to put it another way, when they are compressed, they push back.

  It turns out that the smaller the particle, the bigger the quantum wave. The smallest familiar particle, with the biggest quantum wave, is the electron, so when the matter of a star is squeezed into a small volume, the electrons that orbit inside its atoms push back. This electron ‘degeneracy pressure’ is the force that intervenes to stop a star shrinking to form a black hole. Or so people thought.

  In 1930, a nineteen-year-old physics student, travelling by sea from India to England, showed that everything is changed by Einstein’s special theory of relativity. Subrahmanyan Chandrasekhar imagined things from the particle viewpoint. Electrons push back when squeezed into a small volume because they buzz about ever faster, like a swarm of angry bees. However, Einstein’s theory recognises that nothing can travel faster than light, so there is a limit to how fast electrons can go and how hard they can push back. If a dying star is less than about 1.4 times the mass of the Sun, electron degeneracy pressure can indeed hold gravity at bay, resulting in a highly compact ‘white dwarf’, but for a star above this ‘Chandrasekhar limit’, things are different. Its gravity is strong enough to overcome the buzzing of its electrons, so nothing can prevent its runaway shrinkage to form a black hole.

  A twist to the story was added with the discovery of the neutron by James Chadwick in 1932. Together with protons, neutrons compose an atom’s central ‘nucleus’, around which electrons whirl like planets around the Sun. If a star is squeezed into a small enough volume, its electrons are squeezed into its protons, creating a dense ball of neutrons. The neutrons of such a ‘neutron star’, like electrons, have a quantum wave associated with them and resist being squeezed. But like electrons, there is a limit to how fast they can buzz about and how hard they can push back. The effect is more complicated to calculate than for electrons because it involves the ‘strong nuclear force’, which is hard for theorists to
model. But for a star above about three times the mass of the Sun, gravity is strong enough to overcome the buzzing of neutrons and nothing can stop the formation of a black hole. Black holes are unavoidable. And so too are the problems they create for physics.

  The main reason black holes have been considered such monstrosities is that, when a star undergoes runaway shrinkage to form a black hole, it eventually ends up squeezed into an infinitesimal point, with its density sky-rocketing to infinity. Such a ‘singularity’ signals the breakdown of space and time – indeed of physics itself.

  ‘Black holes are very exotic objects,’ says Andrea Ghez of the University of California at Los Angeles. ‘Technically, a black hole puts a huge amount of mass inside of zero volume. So our understanding of the centre of black holes doesn’t make sense, which is a big clue to physicists that we don’t have our physics quite right.’25

  The American physicist John Wheeler put it more poetically: ‘The black hole teaches us that space can be crumpled like a piece of paper into an infinitesimal dot, that time can be extinguished like a blown-out flame, and that the laws of physics that we regard as “sacred”, as immutable, are anything but.’26

  No wonder Einstein abhorred the fact that his theory of gravity predicted the existence of such monsters – it contains within it the seeds of its own destruction. To understand what really happens to space and time at the heart of a black hole, it will be necessary to find a deeper, singularity-free theory of gravity. Einstein’s theory of gravity is expected to be an approximation of this deeper theory, just as Newton’s theory of gravity turned out to be an approximation of Einstein’s.