The Perfect Theory

by Pedro G. Ferreira

  Subrahmanyan Chandrasekhar yearned to do great things, almost desperately. Born into an affluent Brahmin family in India, Chandra, as he became widely known, was an intense and committed student. He excelled at mathematics and was meticulous and fearless at calculating. While studying at the University of Madras, he was exposed to the new ideas coming over from Europe, starstruck by the great men who were building the new physics of the twentieth century. From a young age, and with a feverish passion, he set about trying to join the fray of modern physics. As he said, later in life, “Certainly one of the earliest motives that I had was to show the world what an Indian could do.”

  Chandra was entranced by the new quantum physics. He read all the new textbooks that came his way, among them Eddington’s recently published book, The Internal Constitution of the Stars. But what really won him over was a book on the quantum properties of matter by the German physicist Arnold Sommerfeld. Inspired by Sommerfeld’s work, he set to work making a name for himself by writing papers on the statistical properties of quantum systems and how they interact. One of the first papers he wrote was published in the Proceedings of the Royal Society when Chandra was not yet eighteen years old. Clearly capable of taking part in the great discoveries of the new quantum physics in Europe, Chandra chose England to pursue his calling and set off on the long trip to Cambridge for his PhD.

  It was during his long voyage on a ship of the Lloyd Triestino line that Chandra made the startling discovery that would transform his life. Obsessed with his work, he decided to spend his trip focusing on a paper written by Ralph Fowler, one of Eddington’s colleagues at Cambridge, which seemed to solve the problem of white dwarfs. Fowler had invoked two quantum concepts and dragged them into astrophysics. The first was Heisenberg’s uncertainty principle, the fact that you couldn’t pin down a particle and at the same time determine its state of motion or velocity. The second concept was the exclusion principle, which states that two electrons (or protons) within an atom cannot be in exactly the same physical state—the exotic matter wave that Schrödinger had proposed as the fundamental quantum description of a particle—at the same time. Indeed, it is as if there is a fundamental, inexorable repulsion between them, preventing them from occupying that same state.

  Fowler took the uncertainty and exclusion principles and set out to apply them to Sirius B. He reasoned that the material in a white dwarf such as Sirius B was so dense that he could think of it as a gas of electrons and protons being squeezed together. The electrons are so much lighter that they are allowed to roam more freely and jiggle about much more vigorously. The exclusion principle means that electrons have to be careful not to encroach on one another’s space, and as the density builds up, each electron has less and less space to move in. As each electron is pinned down more and more, the uncertainty principle kicks in and the velocities and motions get higher and higher, forcing the electrons against each other. These fast-moving, jiggling electrons lead to an outward push, a quantum pressure inside the white dwarf, that can counteract the pull of gravity. In a certain state, the gravity exactly balances the quantum pressure and the white dwarf can sit placidly, hardly glowing but resisting a catastrophic fate. Fowler’s explanation cleared up Eddington’s problem. It seemed that stars could end up as white dwarfs. It closed the narrative of stellar evolution and solved the cliffhanger in The Internal Constitution of the Stars—or so it seemed.

  Chandra took another look at Fowler’s result and did something very simple. He put in the numbers he expected for the density of the electron gas in the white dwarfs. The number he came out with was immense but unsurprising, exactly as Fowler had claimed in his paper. What Fowler had failed to do was work out how large the velocities of the electrons would actually be. When Chandra did this simple calculation, he was shocked: the electrons would have to be zipping around close to the speed of light. And this is where Fowler’s argument fell apart, for he had completely ignored the rules of special relativity that are so important when things start moving at the speed of light. Fowler made the mistake of assuming that the electrons inside the white dwarf could move as fast as they wanted, even if that meant they were zipping around faster than the speed of light.

  Chandra set out to fix Fowler’s mistake. He followed Fowler’s reasoning all the way up until the electrons were moving close to the speed of light. If the white dwarf was too dense, and the particles were indeed moving close to or at the speed of light, he used Einstein’s special theory of relativity, which posited that they couldn’t move any faster. The result he obtained was intriguing. He found that if the white dwarf became too heavy, it would also become too dense and the electrons would be unable to sustain the gravitational pull. In other words, there was a maximum mass for a white dwarf. In his calculation, Chandra found that it couldn’t be larger than about 90 percent of the mass of the sun. (Years later it would be shown that the correct value is more like 140 percent of the mass of the sun.) If a star ended its life as a white dwarf heavier than this maximum mass, it would be unable to support itself. Gravity would win out and inexorable collapse would ensue.

  When he arrived in Cambridge, Chandra gave Eddington and Fowler a draft of his calculation, but they ignored it. There was something deeply unsettling about the instability, which would wreck the edifice Eddington had so promisingly put forward and to which Fowler had added, and so the Cambridge men kept their distance. Over a period of four years, Chandra perfected his argument, and his confidence in his result grew. In 1933 Chandra finished his PhD and, at age twenty-two, was made a fellow of Trinity College. By 1935 Chandra had finessed his calculation still further and was prepared to present his results at one of the monthly meetings of the Royal Astronomical Society.

  On January 11, 1935, Chandra stood up in front of a crowd of distinguished astronomers at the Royal Astronomical Society, at Burlington House in London. Carefully and meticulously Chandra worked through his results, presenting the details of his nineteen-page paper, which was about to be published by the Monthly Notices of the society. He finished by saying, “A star of large mass cannot pass into the white dwarf stage, and one is left speculating on other possibilities.” This strange result was there in the mathematics and physics that they all believed and had to be taken seriously. When Chandra finished, there was polite applause and a smattering of questions. It was done.

  The president of the RAS then turned to Eddington and invited him to step up to the podium to talk on his own paper, “Relativistic Degeneracy.” Eddington stood up to give his brief, fifteen-minute talk. He carefully went over Chandra’s claim that his calculation scuppered Fowler’s solution to the problem of white dwarfs. And then he summarily dismissed Chandra’s watertight argument. To Eddington, Chandra’s result was “a reductio ad absurdum of the relativistic degeneracy formula.” In fact, he firmly believed that “various accidents may intervene to save the star,” and furthermore, “I think there should be a law of nature to prevent a star from behaving in this absurd way!” Eddington’s authority was such that Chandra’s talk was immediately dismissed by most of the audience. If Eddington thought it was wrong, it must be wrong.

  Chandra had come up against the mighty Eddington and lost. He was sabotaging Eddington’s beautiful story of how stars lived and died, and Eddington didn’t like it. If gravitational collapse overcame everything, Schwarzschild’s strange solution would have to be faced head-on, with all its bizarre consequences. As Chandra himself said, many years later, “Now, that clearly shows that . . . Eddington realized that the existence of a limiting mass implies that black holes must occur in nature. But he did not accept that conclusion. . . . If he had accepted that, he would have been 40 years ahead of anybody else. In a way it is too bad.”

  Chandra returned to Cambridge devastated. His run-in with Eddington was to mark him for the rest of his life. A few years later he was invited to take up a post in the Yerkes Observatory in Chicago. He stopped working on white dwarfs and shied away from thinking of what would h
appen if indeed their masses were too large. Would they lead to the inexorable formation of Schwarzschild’s solution, or would something prevent that from happening along the way? Robert Oppenheimer would be the one to answer those questions.

  J. Robert Oppenheimer was a child of the quantum. Brought up in an affluent New York family with van Goghs hanging on their walls, Oppenheimer had a gilded education, first studying at Harvard and then, in 1925, moving to Cambridge. Oppenheimer’s Harvard mentor wrote in his letter of recommendation to Cambridge that Oppenheimer “was evidently much handicapped by his lack of familiarity with ordinary physical manipulations,” although he added, “You will seldom find a more interesting betting proposition.” Oppenheimer’s sojourn in Cambridge was a disaster and short-lived. After a nervous breakdown during which he assaulted one of his colleagues and confessed to trying to poison another, Oppenheimer decided to leave and try his luck in Göttingen.

  Göttingen, the land of David Hilbert, had embraced quantum physics, and Oppenheimer couldn’t have been at a better place to take part in the new revolution. Over the next two years he wrote a series of papers with his supervisor, Max Born, that would indelibly imprint his name in the history of quantum physics. Indeed, the Born-Oppenheimer approximation is still taught in universities today and is part of the paraphernalia used to calculate the quantum behavior of molecules. Oppenheimer finished his PhD in 1927 and a few years later returned to the United States to take up a position at the University of California at Berkeley.

  At Berkeley, Oppenheimer set up one of the beacons of theoretical physics in 1930s America. Oppie, as he was fondly called, seemed to be able to hold forth on any topic, from art and poetry to physics and sailing. Sharp and incredibly quick at picking up on difficult concepts, he hopped from project to project, intellectually raiding new fields and making quick contributions that, while not necessarily profound, were undoubtedly timely and clever. He was impatient and sometimes cruel if he didn’t agree with or understand an argument, but Oppenheimer’s sheer magnetism and energy made him a natural leader, and he excelled at supporting and inspiring his group. He slowly and surely recruited a coterie of brilliant and enthusiastic students and researchers with whom he would tackle many of the new problems that were being discussed in Europe. Wolfgang Pauli, noting that Oppenheimer in his enthusiasm had a habit of muttering, dubbed his group the “nim nim boys.” Berkeley was Oppenheimer’s Göttingen, his Copenhagen.

  And then, after nearly ten years of focusing almost exclusively on the quantum, in 1938, Oppenheimer became intrigued by Einstein’s general theory of relativity. Like Chandra, he approached the theory from the quantum end, looking at how the quantum effects of matter might play off against the gravitational implosion of space and time.

  Every summer Oppenheimer would head down to Southern California with his crowd of students and researchers and take up residence at Caltech, in sunny Pasadena. There he could talk to not only the other physicists but also the astronomers who had followed Hubble’s success and had witnessed Lemaître’s lectures on the primeval atom at first hand. There, they still held a flame for general relativity. It was in Pasadena that Oppenheimer first read a paper by the Russian physicist Lev Davidovich Landau on what would happen if the cores of stars were purely made of a compact mess of neutrons.

  Landau was one of the leading lights in Soviet physics, growing up during the Russian Revolution, a truly brilliant physicist who benefited from the wave of modernization sweeping through the new Russia. Like Oppenheimer, he had spent time abroad, studying in the great laboratories of Europe and witnessing the birth of quantum physics. At nineteen he had already written a paper applying the new physics to the behavior of atoms and molecules, and when he returned to Leningrad, at twenty-three, he had earned the admiration of his older colleagues and was rapidly embraced by the Soviet system.

  With his flair for solving difficult and complex physical systems with quantum physics, Landau had decided to look at a novel source for energy in stars: neutrons, the neutrally charged particles found in the nuclei of atoms. Over the previous decade, it had become clear that adding neutrons or protons to or removing them from nuclei could lead to a copious amount of nuclear energy. So Landau conjectured that if the cores of stars could be packed with neutrons, it might be possible to unleash enough nuclear energy to generate light. If the neutrons were packed together to a density that resembled that of the nucleus of an atom, they might just be the necessary fuel. This nuclear material would be impossibly heavy—a teaspoon of material would weigh tons. If an atom in the stars’ bulk fell into the core, it would be smashed to smithereens, partly absorbed, and partly released as radiation. According to Landau, the neutron core fueled a star’s brightness—it was what made the sun shine. Landau proceeded to work out how big the core had to be and that for such a core to be stable, it just had to be heavier than a thousandth of the weight of the sun. These cores could be tucked away at the center of stars, burning away and fueling starlight.

  But as Landau was writing up his idea, he was also getting caught up in the wave of political repression that was sweeping the country. Two months after Landau published his short paper on neutron cores, “Origin of Stellar Energy,” in Nature, he was arrested by the NKVD. He had been caught editing an anti-Stalinist pamphlet to be distributed at the 1938 May Day parade in Moscow in which Stalin was accused of being a Fascist “with his rabid hatred of genuine Socialism” who had “become like Hitler and Mussolini.” Landau was incarcerated for a year in the Lubyanka prison, just after his Nature paper was feted in Izvestia, one of the main Soviet newspapers, as a source of pride for Soviet physics.

  Oppenheimer was intrigued by the brevity of Landau’s paper and the simple idea being proposed, so he decided to redo Landau’s calculations himself. It took three collaborations with three gifted students, but he eventually got where he wanted to go. His first collaborator was Robert Serber. Together, they gently pulled apart Landau’s idea that the neutron core could be easily tucked away in the sun, shrouded by the hot gases that puff the stars up, and showed that it was wrong. Oppenheimer and Serber published their letter, almost as short as Landau’s, in October 1938 in the Physical Review, while Landau languished in the Lubyanka. Oppenheimer then took the next step with another student, George Volkoff. The pair studied the stability of neutron cores. Their calculation, published in January 1939, is a beautiful mix of mathematics using clever simplifications of Einstein’s theory, with insightful physical intuition and hard calculations. They showed that neutron cores were incredibly unstable configurations and hence couldn’t even be invoked to fuel the energy of very large stars, yet another blow for Landau’s idea.

  At the end of their paper, Oppenheimer and Volkoff pointed out that “a consideration of non-static solutions must be essential” to understand the long-term fate of the neutron cores. Then Oppenheimer set off to do the last piece with yet another student, Hartland Snyder, this time taking general relativity far beyond what anyone had ever attempted. Oppenheimer and Snyder calculated how space and time (and the neutron core) would evolve once the neutron star became unstable. To do so they used a clever idea to understand the results that they were getting: they placed a fictitious observer very far away from the implosion and another fictitious observer right on the surface of the neutron core and compared what those observers would see. They found that the two observers would see very different things.

  A distant observer would see the neutron core implode. But as the surface of the neutron core got closer and closer to the strange shroud that Schwarzschild had found, the collapse would seem to proceed more and more slowly. At some point the implosion would be so slow that it would almost have ground to a halt. The wavelength of any light beam trying to escape from the neutron core would be stretched, redshifting more and more the closer the surface of the neutron core contracted to the critical surface. It would be as if space and time had stopped evolving, and the star would cease to communicate with the outsid
e world. It was very similar to what Eddington himself had said more than a decade before in his book The Internal Constitution of the Stars: “The mass would produce so much curvature . . . that space would close up round the star, leaving us outside (i.e. nowhere).”

  An observer riding the surface of the star as it imploded would see something completely different. He or she would witness the inexorable collapse of the neutron core, see the surface of the neutron core actually cross the critical radius and fall into the inner region of Schwarzschild’s magic surface. And furthermore, this poor, doomed observer would see the formation of the dreaded surface that Schwarzschild had found, the point of no return from which nothing could exit. In other words, if you could sit at the right (or wrong) place, you could see the actual formation of Schwarzschild’s solution.

  Oppenheimer and Snyder had completed Eddington’s life story of stars by showing that, indeed, if they were massive enough, they would collapse to form Schwarzschild’s strange solution. It meant that Schwarzschild’s solution might not be just an interesting, exotic solution to the general theory of relativity. These strange objects might actually exist in nature and had to be included in astrophysics, just like the study of stars, planets, and comets. Once again, general relativity had potentially revealed something unexpected and wonderful about the universe.