by Marcus Chown
On Einstein’s first visit to London in 1921, he stayed in the home of the biologist J. B. S. Haldane. It was like having The Beatles in their screaming heyday come to stay. So overcome with excitement and hysteria was Haldane’s daughter at seeing Einstein walk through the front door that she promptly fainted.42
The morning before the lecture that he had come to London to deliver, Einstein left Haldane’s house and walked to Westminster Abbey. There, in the nave against the choir screen, he gazed on the marble tomb of his great predecessor: Isaac Newton.
Both Newton and Einstein had been inspired by falling bodies to create their theories of gravity. In the fall of an apple Newton had seen the fall of the Moon, and unified the earth with the heavens. In the fall of a man from a roof, Einstein had seen that the force of gravity is an illusion. Both men had known what it was like to ‘voyage through strange seas of thought alone’. ‘Nature to him was an open book, whose letters he could read without effort,’ said Einstein. What would he have given to have met Newton – a man who had died two centuries earlier but whose thought processes he understood better than any man alive?
With the general theory of relativity, Einstein now had in his hands one of the most powerful tools in the history of physics. But though he was a genius, he was not infallible. And, remarkably, he would miss some of the most important predictions of his own theory. Those predictions — black holes and the big bang – would reveal that Einstein’s theory of gravity, though a huge improvement on Newton’s, is itself flawed.
Further reading
Einstein, Albert, Relativity: The Special and General Theory, Folio Society, London, 2004.
Fölsing, Albrecht, Albert Einstein, Penguin, London, 1998. Levenson, Thomas, Einstein in Berlin, Bantam Books, New York, 2003.
Levenson, Thomas, The Hunt for Vulcan . . . And how Albert Einstein destroyed a planet, discovered relativity and deciphered the Universe, Head of Zeus, London, 2015.
Levin, Janna, Black Hole Blues, The Bodley Head, London, 2016.
Overbye, Dennis, Einstein in Love: A Scientific Romance, Viking, London, 2000.
Pais, Abraham, ‘Subtle is the Lord . . The Science and the Life of Albert Einstein, Oxford University Press, Oxford, 1983.
7
Where God divided by zero
How Einstein’s theory of gravity predicts daft things at the ‘singularity’ of a black hole and how a deeper theory is needed that doesn’t
For years, my early work with Roger Penrose seemed to be a disaster for science. It showed that the Universe must have begun with a singularity, if Einstein’s general theory of relativity is correct. That appeared to indicate that science could not predict how the Universe would begin.
Stephen Hawking1
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.
John Wheeler2
In February 1916, Einstein received a surprising package. It came from a soldier serving on the Eastern Front. Karl Schwarzschild had been the director of the Astrophysical Observatory in Potsdam, just outside Berlin. But at the outbreak of war in 1914 he had been overcome by extreme patriotic fervour and dropped everything to volunteer for military service. In his eighteen months in the Kaiser’s army, he had run a weather station in Belgium, calculated shell trajectories with an artillery battery in France, and now he was serving in Russia.
Despite being caught up in a vicious war, Schwarzschild found time to write several scientific papers, two of which were on Einstein’s theory of gravity, which he had learnt about soon after its publication at the end of 1915. What was notable about Schwarzschild’s work was that in such a short time he had taken a significant step beyond Einstein.
The equations of general relativity are complex. They substitute a total of ten equations for Newton’s single inverse-square law formula. Because of their complexity it is very hard to deduce the shape of the space-time in the vicinity of a realistic body. But Schwarzschild made a number of simplifying assumptions, which reduced Einstein’s equations to a more straightforward and manageable form, and enabled him to ‘solve them’.
Schwarzschild’s ‘solution’ described the shape of the warped space-time in the neighbourhood of a localised mass such as a star. Einstein was amazed. ‘I had not expected that one could formulate the exact solution of the problem in such a simple way,’ he wrote back to Schwarzschild.
Most remarkably, Schwarzschild showed that if enough mass were crammed into a small enough volume, space-time would become so extraordinarily warped as to become a bottomless well. So steep would be the sides of the well that a light beam trying to climb out would die of exhaustion, sapped of all its energy, before it could escape. With no light emerging, the region of space-time would appear blacker than night.
Schwarzschild had no word for what he had discovered. That would only be coined by the American physicist John Wheeler in 1967. But today there is hardly a person alive who does not know the term. Schwarzschild’s solution described a ‘black hole’.3,4
Schwarzschild’s story is a tragic one. While in Russia, he developed a rare and serious ‘autoimmune’ disease in which the body’s immune system malfunctions and attacks healthy tissue. Pemphigus vulgaris causes painful blisters on the skin as well as inside the mouth, nose, throat, anus and genitals. No one knows its cause – although it may be a combination of genetic and environmental factors – and there is no cure — although modern treatments with corticosteroids relieve the symptoms. If the blisters become infected, the infection can spread into the bloodstream and affect the whole body. This is what happened to Schwarzschild. He was invalided back to Berlin in March 1916 but died two months later on 11 May. He was forty-two.
Schwarzschild’s black hole was surrounded by an ‘event horizon’. Anything passing through this into the interior – whether light or matter – could never get out again. The event horizon provides a measure of the ‘size’ of a black hole. For the Sun to become a black hole, it would have to be crushed within a sphere only 3 kilometres in radius. For the Earth, the ‘Schwarzschild radius’ would be a mere 2 centimetres. Fortunately for us, the Sun and the Earth are not massive enough for their gravity ever to turn them into black holes.
But if a very massive star was crushed within its event horizon — causing it to wink out of existence as far as the rest of the Universe was concerned – its gravity would continue to crush the star all the way down to an infinitesimal point. With the star effectively snuffed out, all that would be left behind would be a bottomless well of space-time. ‘The black holes of nature are the most perfect macroscopic objects there are in the Universe,’ said the Indian Nobel prizewinner Subrahmanyan Chandrasekhar. ‘The only elements in their construction are our concepts of space and time.’5
At the centre of the black hole, where the matter of a star is crushed to infinite density, the curvature of space-time and the strength of gravity skyrocket to infinity.6 ‘Black holes are where God divided by zero,’ observed American actor and writer Stephen Wright. The appearance of such a nonsensical ‘singularity’ in any theory indicates that it no longer describes reality. It has broken down, become indistinguishable from gobbledygook.
Of Schwarzschild’s black hole paper, Einstein said: ‘If this result were real, it would be a true disaster.’ But not for an instant did Einstein – or even Schwarzschild – think the result was real. Neither did they entertain the thought that the black hole solution described an object that might actually exist in the Universe.
The few who did were not overly worried either. A star has a finite supply of energy and, when it exhausts it, its internal fires must go out. Since those fires push outwards and prevent gravity from crushing the star during its lifetime, the star will begin shrinking down to a singularity. But some new force was bound to come to the rescue and halt the shrinkage long
before that happened. It was inconceivable that nature would permit the formation of a monstrous singularity.
Actually, it appeared nature did indeed provide such a force. It was a consequence of ‘quantum theory’, the bizarre theory of the microscopic world of atoms and their constituents.7
Quantum stars
Quantum theory was stumbled on in the first decades of the twentieth century but given a firm mathematical foundation only in the mid-1920s. The theory recognised that the fundamental building blocks of matter behave both as localised particles -like tiny billiard balls – and spread-out waves – like ripples on a pond. This peculiar ‘wave-particle duality’ leads to a multitude of strange and unexpected phenomena – for instance, the ability of a single particle to be in two or more places at once. It also plays a crucial role when a star at the end of its life runs out of the fuel necessary to maintain its internal fires.8
Robbed of its ability to push outwards, the matter of a star is crushed by the iron fist of gravity until it fills a volume of about the size of the Earth. Such a ‘white dwarf’, 100 times smaller and about a million times denser than the Sun, is the endpoint of the evolution of all normal stars, including our own. In such superdense conditions – a sugar-cube-sized volume of white dwarf stuff weighs as much as a family car. — the electrons are forced very close together.
Squeezing a wave of any type into a small space causes it to become more choppy and violent. In the case of a quantum wave, more choppy and violent corresponds to a faster-moving particle (strictly speaking, one with greater ‘momentum’). This is the famous ‘Heisenberg Uncertainty Principle’. And it dictates that when electrons are squeezed tightly together inside a white dwarf, they attain extremely high speeds.
This is one quantum effect with important implications for white dwarfs. But there is a second one, which is a bit harder to explain.9 Take it on trust that another consequence of wave-particle duality is that the fundamental building blocks of matter come in two distinct tribes: ‘bosons’, which are gregarious; and ‘fermions’, which are antisocial. Fermions, which include the electron, are said to obey the ‘Pauli Exclusion Principle’, which states that no two fermions can occupy the same quantum ‘state’.10
For electrons in a white dwarf this means that two neighbouring particles must have distinctly different velocities. So, if one has a velocity dictated by the Heisenberg Uncertainty Principle, its neighbour must have an even higher velocity — in practice, double the velocity; its neighbour an even higher velocity – in practice, triple the velocity; and so on.
Picture a ladder, with each rung corresponding to a higher and higher velocity. According to the Pauli Exclusion Principle, there can be only one electron on each rung (actually, there can be two, but that is another story!).11 The Principle ensures that the electrons in a white dwarf have extraordinarily high velocities, boosted way beyond what the Heisenberg Uncertainty Principle might suggest. And it is these super-fast electrons buzzing about inside the stars that push back against gravity. Their so-called ‘electron degeneracy’ pressure keeps a white dwarf stable and prevents it from shrinking to a ball much smaller than the Earth.12
This was the state of play in the late-1920s. Quantum theory, miraculously, came to the rescue of a dying star. It staved off the runaway collapse down to a black hole with a nightmarish singularity in its heart. All was under control. Everything in the garden was rosy.
Or so it seemed.
The Chandrasekhar limit
In August 1930, a nineteen-year-old Indian embarked on a ship in Bombay bound for England and the University of Cambridge. I have already quoted his older self on the subject of the remarkable simplicity of black holes. His name was Subrahmanyan Chandrasekhar and he was something of a mathematical prodigy.
The voyage was initially assailed by bad weather and the ship had to steam at half-speed. But at Aden the sun came out. And as the ship made its way through the Suez Canal, Chandrasekhar was at last able to leave his cabin, where he had been largely imprisoned during the heavy seas.
I imagine him cutting an eccentric figure as he staggers out on deck carrying a teetering pile of books on quantum theory and astrophysics. Perspiring copiously, he dumps the books in one deckchair and collapses in another beside it. Other Indians promenading past shoot him odd looks. It is no more than he expects. He has made no effort to interact with any of his fellow countrymen and he is acutely aware that they consider him aloof, if not arrogant. But he does not care. At last, he has the peace and quiet to think, really think. And what he thinks about, incongruously, as the sands of the Sinai Peninsula sail past and the hot desert air scours his face, is white dwarfs.One question and one question alone occupied Chandrasekhar’s mind: were the electrons in a white dwarf relativistic? Flicking back and forth between his books and papers, he gathered together the formulae that described the interiors of stars and the quantum behaviour of electrons at ultra-high density. He put in the numbers he knew and cranked away until finally there emerged an answer. He checked and checked again. There was absolutely no doubt about it. The electrons inside a white dwarf would be moving at more than half the speed of light, a velocity at which the effects of Einstein’s special theory of relativity would be apparent. In the jargon, they were ‘relativistic’.
Such velocities were staggeringly huge: more than 150,000 kilometres per second. But more important to Chandrasekhar was the implication of such speeds. It meant that quantum theory alone was insufficient to understand white dwarfs. A correct theory must also incorporate Einstein’s special theory of relativity.
When night fell the sky was crowded with an impossible number of stars. Nobody guessed that the strange young man in the deckchair, so engrossed in his notebooks he often forgot to go to dinner, was calculating the properties of their interiors. His body may have been pinned to the deck of a ship but his mind ranged freely among the embers of dying suns.
It did not take long for Chandrasekhar to develop a properly relativistic theory of white dwarfs. And it did not take long for him to discover something unexpected and extraordinary, if not downright horrifying.
The more massive a white dwarf, the more its gravity squeezed the electrons in its interior and the faster they buzzed about. That much was true. Except that Einstein’s theory of relativity imposed a limit on how fast the electrons could go: the speed of light. As the electrons approached the cosmic speed limit, they became ever more massive and it became more and more difficult to boost their speed. But this created a problem. After all, it was the continual drumming of the electrons – like raindrops on a tin roof – that provided the outward force to oppose the gravity trying to crush a star. If, as they were squeezed harder and harder, their speeds were boosted by ever smaller amounts, their ability to oppose gravity was gradually drained away. The young Indian in the deckchair, with his head in the stars, saw the looming stellar catastrophe like a train bearing down on him in the night.
For a white dwarf, the stiffness of the electron gas holding back gravity was like the stiffness of a cricket ball resisting a bowler’s grip. But, above a certain stellar mass, everything changed. The cricket ball abruptly turned into marshmallow.
Chandrasekhar did the calculation, over and over again, checking and rechecking that he had made no mistake. But there was no doubt about it. If a star at the end of its life were more massive than 1.4 times the mass of the Sun, electron degeneracy pressure would not be enough to save it. Gravity would crush the star catastrophically. No known force in the Universe could stop it. The monstrous singularity was unavoidable.
Neutron stars
Two years later, in 1932, the English physicist James Chadwick found a particle as massive as the positively charged proton, but with no electric charge. With the discovery of the ‘neutron’, the picture of the atom was complete: negatively charged electrons orbit an ultra-compact nucleus which contains protons and neutrons and accounts for 99.9 per cent of an atom’s mass (the exception is an atom of the lightes
t element, hydrogen, which contains a lone proton in its nucleus).
Chadwick’s discovery had crucial implications for a star more massive than the ‘Chandrasekhar limit’ of 1.4 solar masses. Yes, its innards would be turned to marshmallow and it would be crushed ever smaller by the merciless grip of gravity. But this was not the whole story. The runaway shrinkage of the star would, inevitably, squeeze the electrons into the nuclei, where they would react with the protons to make neutrons.
Neutrons, like electrons, are fermions. And a neutron gas, just like an electron gas, would make the star stiff enough to resist gravity. But neutrons are much smaller than atoms. Instead of a white dwarf the size of the Earth, the result would be a ball of neutrons the size of Mount Everest. So dense would be such a ‘neutron star’ that a sugar-cube-sized volume would weigh as much as the entire human race.
In the 1940s, the British astronomer Fred Hoyle would suggest that the only possible power source of a ‘supernova’ — a type of stellar explosion so bright it can often outshine a galaxy of 100 billion stars – is the gravitational energy released in the catastrophic shrinkage of a star to form a neutron star. But it was not until 1967 that a Cambridge graduate student, Jocelyn Bell, discovered a neutron star, in the guise of a rapidly spinning ‘pulsar’.13
Although ‘neutron degeneracy pressure’ makes neutron stars stable against further gravitational collapse, such suns have the same Achilles’ heel as white dwarfs. They are ‘relativistic stars’ whose constituent particles are flying about at close to the speed of light. Consequently, above a certain threshold of mass, even the stuff of a neutron star turns to marshmallow.
The physics of neutrons, which are held together by nature’s ‘strong nuclear force’, is more complicated than the physics of electrons, which interact via the electromagnetic force. For this reason the threshold mass for a neutron star is not as precisely known as the Chandrasekhar limit. It was first calculated by the Russian physicist Lev Landau in 1932 and it is widely believed to be about three times the mass of the Sun. For stars above this mass, there is no known force that can stop their shrinkage down to a singularity.
