A singularity in space-time

TO achieve at least half a dozen major insights about the fundamental laws of nature in a single lifetime is rare, to say the least. That such insights are the subject of active research and puzzlement half a century after they were achieved is rarer still. And that they came from someone who was not expected to live beyond the age of 22 made Stephen Hawking truly one of a kind. His passing, on March 14, has left a void that no one can possibly fill.

Hawking achieved superstar status in his lifetime through a unique combination of factors. He was an outstanding theoretical physicist by any standards. Although his research involved particularly advanced mathematics and physics, he was masterful at describing it in intuitive and appealing language. To his public speeches, he brought an economical and wry turn of phrase that made everything seem simpler than it was. He was unfailingly optimistic about life and nature but attacked human folly with a gentle, yet devastating, wit. And he did not shy away from fame and public exposure but would cheerfully weigh in on any and every aspect of popular culture, speculative science, religion and politics.

And, of course, he was disabled in ways that most people had never seen before. For the latter half of his life, he was confined to a motorised wheelchair with virtually no movement left and could only communicate by speaking in a computerised voice. Hawking’s electronic speech became an integral part of his persona, and he used it with manifest enjoyment. The idea of an almost disembodied brain thinking deep thoughts about the cosmos and communicating them in oracular fashion via a voice synthesiser gripped the popular imagination in a unique way.

The thesis Hawking’s PhD thesis, “Properties of Expanding Universes”, completed by the age of 24, displays his penetrating intellect and the slightly acerbic wit that would later become a trademark. In the introduction to this young man’s thesis, one finds this eminently quotable, and often quoted, observation: “[S]ince the time of Copernicus we have been demoted to a medium sized planet going around a medium sized star somewhere near the edge of a fairly average galaxy.” He concludes that “we are now so humble that we would not claim to occupy any special position [in the universe]”.

The thesis started out by challenging the popular steady state cosmological model of Fred Hoyle and Jayant Narlikar, according to which the universe would have been expanding for an infinitely long time. With a simple and elegant argument, he showed that equations involving the steady creation of matter, as required by this model, contradicted the predictions of Einstein’s theory for an expanding universe. Hawking’s head-on challenge to a leading doctrine of his time, both in his thesis and in person during discussions at Cambridge, earned him a reputation for being brash but also extremely bright.

In the rest of the thesis, the young Hawking addressed several different issues in cosmology. One of his results regarding the role that gravitational perturbations play in galaxy formation is now thought to be incorrect. But the thesis is redeemed by its final chapter, which makes valuable progress on a key question: can the universe spring from a point or collapse to a point?

He would provide a definitive answer to this question in a collaborative paper with Roger Penrose—a leading theoretical physicist senior to him by a few years—just a few years later, but he took some essential steps in the thesis. The basic idea is intuitive: observations indicate that the universe is expanding, in the sense that all the galaxies are flying apart from each other. If one traces this process back in time, like a movie run backwards, one will, therefore, see the universe shrinking as matter distributions come closer together. This suggests the logical possibility that at some point in the distant past all the matter in the universe could have collapsed to a single point, the so-called “big bang”. The density at this point would be virtually infinite, so by Einstein’s theory there would correspondingly be a huge curvature in the fabric of the space-time continuum.

Mathematicians refer to points of infinite curvature as “singularities”. Hawking wanted to understand whether a space-time singularity could have existed in the far past. He also noted that such a singularity could arise in the distant future in the form of a “big crunch” if the universe stopped expanding and started to contract again. For the mathematical analysis of these questions, he drew on recent studies by Penrose. Penrose had observed that singularities could form when objects like stars collapsed under their own gravitational pull. Hawking’s new idea was to adapt these notions to the expanding/contracting universe as a whole. He found that given suitable conditions at a definite time, models of the universe would inevitably exhibit cosmological singularities in the far past or far future.

Decades later, Hawking recounted with characteristic humour how his proposal of a big crunch caused unexpected anxiety. “When I gave a lecture in Japan, I was asked not to mention the possible recollapse of the universe because it might affect the stock market. However, I can reassure anyone who is nervous about their investments that it is a bit early to sell: even if the universe does come to an end, it won’t be for at least 20 billion years. By that time, maybe the GATT trade agreement will have come into effect.”

Hawking’s thesis was made freely available online in 2017. The reader is immediately struck by his mastery of advanced physics and mathematics at a very young age. He absorbed both the fundamentals of Einstein’s theory of gravitation and later developments in the subject due to legendary figures such as Hermann Bondi, Hoyle, Narlikar, Kurt Godel and Amal Kumar Raychaudhuri. At the time, advanced general relativity was understood by very few people and had to be studied from original research papers rather than textbooks. Not only did the student Hawking absorb this knowledge, he quickly made major original contributions that enhanced it. Once out of graduate school, he was destined to outstrip many of the legends who had influenced his work.

The thesis also offers some insight into Hawking’s physical condition. It carries a handwritten statutory declaration “This dissertation is my own original work” in a shaky and irregular cursive, testifying to the onset of the degenerative disability—amyotrophic lateral sclerosis, also known as motor neurone disease—that had begun to affect his movements and would come to dominate his life.

Early career Hawking was now poised at the start of a prolific career that would give rise to over 200 publications: research papers, review articles and books. In the same year that his thesis was approved, he published half a dozen papers, many of them by himself. A series of original and important publications, both solo and collaborative, flowed from his pen in a continuous stream for the next few years.

The first major result was a joint work with Penrose in 1970 on the formation of singularities in space-time. This consolidated Hawking’s own previous studies in cosmology together with Penrose’s investigations on the gravitational collapse of material objects, now known as black holes. It also marked the first time Hawking ventured into the physics of black holes, but from then on he was hooked. Much of his life’s work focussed on these mysterious objects, and he was destined to make an unparalleled breakthrough in the field just five years later.

Although Arthur Eddington had conceived of stars so dense that light would be unable to escape from them and Subrahmanyan Chandrasekhar had proposed conditions under which a normal star could collapse under its own gravitational pull, the idea of actual black holes in the universe was considered very exotic in the 1960s and early 1970s. It was felt that such objects, from which nothing could escape, were somehow “unphysical” and would not form in practice. But from the outset, Hawking showed a deep conviction that they were real. His stand is completely vindicated by the current consensus that our galaxy alone contains as many as a hundred million black holes.

Curiously, his first solo effort in the field took the black hole concept one step further. Hawking proposed the existence of what are known today as “micro black holes”, conceptually similar to the giant objects formed by collapsed stars but completely different in scale. These hypothetical objects weigh as little as a hundredth of a microgram. Hawking’s motivation for this proposal was to explain the claimed detection—by a scientist called Joseph Weber—of gravitational waves. Unfortunately, Weber’s experiment later came to be completely discredited. Nonetheless, the idea of micro black holes lives on, and it is still conceivable that such objects exist and could be produced under the right conditions.

This paper also included an insight much ahead of its time: that tiny charged black holes and atomic nuclei, two apparently dissimilar objects, may actually behave quite similarly. Both will, for example, produce ionisation tracks when they travel through matter. Hawking wondered whether some of the observed tracks that had not found a conventional explanation could be due to microscopic black holes. While this has never been verified, the last few decades of theoretical research have led one to believe that small black holes have much in common with fundamental particles: both are essentially simple entities that carry mass, charge, angular momentum and not much else. This observation is at the heart of “dualities” between different descriptions of physical theories, an idea that has dominated theoretical physics for the last couple of decades.

Over the next three years, Hawking published a series of papers that illuminated many aspects of black holes and cosmology, touching upon gravitational radiation, the flow of energy and angular momentum into a black hole, the extraction of energy from rotating black holes, and the isotropy of the universe. In a two-page paper in 1971, he again attempted to explain the gravitational-wave experiments of Weber, which, as indicated earlier, turned out to be flawed. Nevertheless, the paper uncovered a fundamental mathematical property of black holes: the area of their bounding surface (“event horizon”) can never decrease in any physical process. Black holes can merge, enhancing their total area, but they can never split apart. This seemingly innocent observation was to have a profound impact in the next few years, after which physics truly would never be the same again.

Black hole thermodynamics In his 1972 PhD thesis, Jacob Bekenstein noted some analogies between the properties of black holes and the well-established field of thermodynamics. A system at some temperature, such as a gas, can be assigned an “entropy” that describes its degree of randomness or disorganisation. In any process, the net entropy of a closed system must increase; this is the second law of thermodynamics. Bekenstein was puzzled by the fact that one could apparently decrease the entropy of the universe by throwing some matter into a black hole. Since nothing could be emitted by a black hole, this matter would presumably disappear forever, and its own entropy would be erased. The result would be a more orderly universe, with lower entropy. This clearly contradicted the laws of thermodynamics.

Using Hawking’s result that the area of a black hole always increases, Bekenstein argued that when one decreased conventional entropy by throwing matter into a black hole, one simultaneously increased the area of the black hole. Hence, if one resorted to a new definition of entropy that included the area of the black hole, this generalised entropy might increase in any process. In this way the second law of thermodynamics, with the above generalisation, would continue to be valid.

A year later James Bardeen, Brandon Carter and Hawking took this idea further and developed a set of “laws of black hole mechanics” that had an appealing parallel with the four established laws of thermodynamics. While following Bekenstein’s relation of entropy to the area of a black hole, they additionally noted that a quantity called the “surface gravity” near a black hole played a role analogous to the temperature in thermodynamics. The stage was set for the synthesis of one of the newest fields of physics, the mechanics of black holes, with one of the oldest, thermodynamics. However, the authors of this work were cautious about taking their own results too seriously. They stressed that despite the close analogy, surface gravity should not be identified with temperature since the temperature of a black hole, which was believed not to radiate anything, must be exactly zero.

Hawking continued to reflect on this puzzle, and two years later, in 1975, he came out with his magnum opus, a paper titled “Particle Creation by Black Holes”. He had clearly been dissatisfied with the fact that surface gravity and temperature were so analogous and yet could not be identified with each other. So he sought, and found, a mechanism that allowed black holes to radiate. The bulk of the paper is a detailed mathematical analysis of quantised fields near the surface of a black hole, leading to the conclusion that black holes are not black nor are they cold as was always thought. Instead, they have a finite temperature and they emit a steady stream of thermal radiation just like any other body at that temperature. The paper provided a value for the temperature of a black hole in terms of its fundamental parameters.

Before presenting the detailed computation, Hawking provided an intuitive explanation for the process. Quantum mechanics implies that particle-antiparticle pairs spontaneously arise out of empty space for short periods. He reasoned that if this happened just outside the black hole and if then one member of the pair fell inside, the remaining particle would appear to have been emitted by the black hole. This reasoning is frequently cited as an explanation of Hawking radiation but has also led to much confusion. Hawking foreshadowed the confusion with this warning: “It should be emphasised that these pictures of the mechanism ...are heuristic only and should not be taken too literally.” His calculation spoke for itself. But he also admitted that “the strongest reason for believing that black holes can create and emit particles at a steady rate” was that it unified the laws of black hole mechanics and those of thermodynamics into a single generalised set of laws.

Hawking’s seminal discovery of radiation from black holes has never been confirmed in an experiment. The reason is that large black holes, of the kind known to populate the universe, have an extremely low temperature according to his formula. Indeed, they are colder than empty space itself and would actually absorb the background microwave radiation that pervades the universe. But micro black holes, if they exist, could be quite hot and even explosive towards the end stage of their evaporation. As Hawking dramatically observed in a follow-up paper, the energy released in the end stage of black hole decay would be “equivalent to about 1 million 1 Mton hydrogen bombs”.

But no micro black hole has yet been detected. Nonetheless, Hawking’s result is a direct consequence of Einstein’s well-tested theory of gravitation combined with the even more well-tested quantum theory, and its correctness is not generally doubted. Novel experiments may well detect Hawking radiation, and even perhaps micro black holes, some day.

Hawking’s proof that black holes emit thermal radiation solved one problem but created another. Thermal radiation carries no information beyond the value of its temperature. Thus, if one throws an object into a black hole, any detailed information encoded in that object appears to be completely lost forever. This contradicts a well-established law of quantum theory, which says that a state which encodes some precise information (technically a “pure state”) can never evolve into a state with less information (a “mixed state”). Thus, while inventing a generalised law of thermodynamics, Hawking appeared to have demolished a fundamental principle of quantum theory.

One cannot shrug one’s shoulders about such a possibility. Quantum theory has multiple ramifications in every aspect of the universe. Were one of its fundamental tenets found to be incorrect, the entire structure could fall apart. The hard-won understanding of atomic physics and subnuclear systems in the 20th century would be at risk. But Hawking did shrug his shoulders, at least metaphorically. He insisted that information was indeed lost inside black holes and that we should get used to it.

The next two decades of discourse in gravitational physics and string theory were significantly dominated by the “information loss paradox”, which even today is not completely resolved.

In 1997, Hawking famously took a bet with the Caltech physicist John Preskill that information would be lost. A few years later, he visited Mumbai for the Strings 2001 conference. At the press conference, a direct question was put to him by a colleague on whether he still believed in information loss. The answer, as I recall, was somewhat evasive and could best be interpreted as saying he had reconsidered the issue. In 2004, he formally conceded the bet to Preskill and, thereafter, accepted that information was not lost but would be extremely difficult and impractical to recover.

It is pointless to try and separate Hawking from his disability, and he himself never tried to do so. Afflicted with motor neurone disease from the age of 20, he faced a death sentence in a short time span. Hawking’s own website describes his undergraduate years, immediately preceding the diagnosis, in these words: “After three years and not very much work, he was awarded a first class honours degree in natural science.” It is hard to doubt that the imminent end transformed this lazy young man into a fiendishly hard-working, committed, even obsessive personality who would manipulate sophisticated equations in his head.

In the event, the end would not come until 54 years later, but Hawking did not have an easy life. His disability made it increasingly difficult to execute basic functions like talking, walking and eating. After an emergency surgery in 1985, he was completely unable to speak. He spent the rest of his life confined to a motorised wheelchair with a computer and a hand-held mouse. This ensemble allowed him to implement his autonomous movements, including ambulation and speech, until even the finger became non-functional. In his last few years, he used a cheek muscle to operate increasingly sophisticated devices.

Throughout this ordeal, his public persona was unfailingly cheerful. He constantly lauded the technological advances that allowed him to lead a functional life. In an advertisement for Intel, he said: “Medicine has not been able to cure me, so I rely on technology to help me communicate and live.” His pragmatic acceptance of his difficulties and public willingness to cooperate in resolving them will certainly have inspired many similarly affected people and society at large to think constructively about disability.

Science popularisation In our world there are brilliant scientists and there are gifted public speakers, but the two are rarely embodied in the same person. At best, some of the greatest achievers could be considered clear orators, but Hawking was in a different league altogether. His turn of phrase was precise, concise and memorable. Some of his epigrams are hard to get out of one’s mind, as when he answered a question about God at a press conference in Mumbai: “I use God as a metaphor for the laws of nature.”

He authored what is considered the bestselling popular science book of all time: A Brief History of Time . His reputation clearly had a lot to do with the sales, but the success of the book was due to the unique combination of Hawking’s traits. It had his oracular style, “our goal is nothing less than a complete description of the universe we live in”; his folksy popular touch, “in order to get married, I needed a job, and in order to get a job, I needed a PhD”; and, of course, his trademark humour: “When asked: ‘What did God do before he created the universe?’ Augustine didn’t reply: ‘He was preparing Hell for people who asked such questions.’” These flourishes were lightly superimposed on an accurate and lucid explanation of the most difficult questions of cosmology. The book sold over 10 million copies in its first two decades. There were many more books in this vein, in addition to compilations of great works in mathematics and science, children’s books (co-authored with his daughter, Lucy Hawking) and even an autobiography, appropriately titled My Brief History .

Hawking was ever willing to deliver public lectures, appear in TV serials and meet celebrities, and his voice even features in a Pink Floyd song. He argued that we must colonise other planets, warned that things would go badly if alien life were to meet us, and questioned whether God was necessary to understand the universe. He deflated those who think IQ tests are meaningful: “People who boast about their IQ are losers.” He used his remarkable popularity to propagate his diverse messages and always upheld the scientific spirit.

Now that he is no more, one should not be tempted to imagine him up there, observing the cosmos and its black holes at close range. As he put it: “There is probably no heaven and no afterlife either. We have this one life to appreciate the grand design of the universe, and for that, I am extremely grateful.” We should be grateful too.

Sunil Mukhi is Professor of Theoretical Physics at the Indian Institute of Science Education and Research, Pune, working in quantum gravity and string theory. He was an organiser of the Strings 2001 conference at the Tata Institute of Fundamental Research, Mumbai, in which Stephen Hawking participated.