Chandra: A biographical portrait The complexities of three countries—India, England, and the US—helped produce a scientist of rare stature and greatness. Kameshwar C. Wali December 2010, page 38 The simple is the seal of the true. And beauty is the splendor of truth. With the words above, Subrahmanyan Chandrasekhar, popularly known as Chandra, concluded his Nobel Prize lecture on 8 December 1983. Toward the end of his talk, he was describing black holes in the astronomical universe, explaining the simplicity in the underlying physics and the beauty of their mathematical description within the framework of Einstein’s theory of relativity. “They are,” he said, “the most perfect macroscopic objects there are in the universe.” The Nobel Prize in Physics brought an extremely private and somewhat shy individual into the limelight. For newspaper journalists and broadcast interviewers, neither the simplicity of the physics of the black holes nor the mathematical beauty of their description was of major concern; the pronunciation of Chandra’s full name seemed to present them with an astronomical difficulty in and of itself. The announcement of the prize he shared with William Fowler was greeted with joy and appreciation throughout the scientific world, and he was soon inundated with telephone calls, telegrams, and letters of congratulations and good wishes from his former students, associates, heads of scientific institutions, and governments. Most considered the prize belated and long overdue. But for Chandra, who had been critical of the atmosphere it creates—and of the ways in which some people seemed to go after it—it was to a large extent distorting to science and its true Figure 1
Transcript
Chandra: A biographical portrait
The complexities of three countries—India, England, and the US—helped produce a scientist of rare stature and greatness.Kameshwar C. Wali December 2010, page 38
The simple is the seal of the true. And beauty is the splendor of truth.
With the words above, Subrahmanyan Chandrasekhar, popularly known as Chandra, concluded his Nobel Prize lecture on 8 December 1983. Toward the end of his talk, he was describing black holes in the astronomical universe, explaining the simplicity in the underlying physics and the beauty of their mathematical description within the framework of Einstein’s theory of relativity. “They are,” he said, “the most perfect macroscopic objects there are in the universe.”
The Nobel Prize in Physics brought an extremely private and somewhat shy individual into the limelight. For newspaper journalists and broadcast interviewers, neither the simplicity of the physics of the black holes nor the mathematical beauty of their description was of major concern; the pronunciation of Chandra’s full name seemed to present them with an astronomical difficulty in and of itself.
The announcement of the prize he shared with William Fowler was greeted with joy and appreciation throughout the scientific world, and he was soon inundated with telephone calls, telegrams, and letters of congratulations and good wishes from his former students, associates, heads of scientific institutions, and governments. Most considered the prize belated and long overdue. But for Chandra, who had been critical of the atmosphere it creates—and of the ways in which some people seemed to go after it—it was to a large extent distorting to science and its true pursuit. He had never considered himself as a possible candidate, since his areas of research, pursued in a single-minded quest for a personal perspective, had not led him into areas that were in the science spotlight.
Lahore and Madras, 1910–25
Chandra was born on 19 October 1910 (“19-10-1910,” as Chandra was fond of saying with a rare chuckle) in Lahore, Pakistan (then a part of colonial British India). His father, Chandrasekhara
Subrahmanyan Ayyar, was in the government service, the deputy auditor-general of the North Western Railways. Chandra was the first son and the third child in a family of four sons and six daughters. His mother, Sitalakshmi, was a woman of great talent and intellectual attainment. She married young and received only a few years of elementary schooling, yet she managed to continue her education while bearing 10 children and learned English well enough to adapt Henrik Ibsen’s A Doll’s House and translate a long story by Tolstoy into Tamil. Intensely ambitious for her children, she was to play a pivotal role in Chandra’s career.
Being the first son, Chandra inherited the name of his grandfather, Ramanathan Chandrasekhar (referred to as R. C. hereafter). R. C. had been the first in the family to depart from traditional village life and pursue an English education. If, after graduating from high school in 1881, he had continued his college education and completed his BA degree as expected, he probably would have ended up in a high British government post. But he took his Western education seriously: He read English literature and philosophy extensively, studied mathematics and physics, and in general pursued what interested him most rather than what was required of him. This remarkable person, 1 who transformed the lives of his future generations, built a fine home library, which proved to be a very valuable resource to them. Chandra inherited not only his grandfather’s name but also his independent streak in the pursuit of knowledge.
Chandra’s early education was at home under the tutelage of his parents and private tutors. When he was 11 and the family had permanently settled in Madras (now Chennai), he began his regular schooling at the Hindu High School in the city’s Triplicane neighborhood.
Chandra found formal school neither easy nor pleasant. His education at home under private tutors had allowed him the freedom to study what he liked (mainly English and arithmetic). Now he was suddenly required to study history, geography, and general science and was subjected to periodic examinations. It was a disappointing first year, but the promise of the following year’s curriculum, which included algebra and geometry, was enough to get him excited. Without waiting for classes to begin, he got the books and studied on his own during the summer vacation. By the time he started his second year, he knew all the geometry and all the algebra the school was going to teach, and in fact more. He kept up his studying during the following three vacations and did extremely well in high school; he became a freshman at Presidency College in Madras when he was only 15 years old.
Those early years of learning were happy years for Chandra. Though the family was growing—a new child every two years or so—his father’s income provided a comfortable life. In 1924,
Chandra’s father built his own house, named Chandra Vilas, in Mylapore, a prestigious suburb of Madras. And because he was in the railway services, he and the family received free railway travel or reduced fares, so they got together more frequently than they could otherwise. The children traveled to all parts of India—a privilege few Indians could afford. Grandfather R. C.’s efforts had paved the way for a new urban life for his children and grandchildren.
Education and urban life could not completely change centuries of tradition, however. Chandra’s father, a highly cultivated individual, widely read and traveled, was a traditional father. He was authoritarian and demanded unquestioned obedience. Reserved and undemonstrative, he remained aloof from his children; they in turn could not share their innermost thoughts or feelings with him. Deeper connections were left to Chandra’s mother. Sitalakshmi was the vital force of the family, keeping it together, helping the children with their studies, and meeting their needs. Without imposing strict religious discipline, she infused them with the cultural heritage and ideals of Hinduism. Chandra, the eldest son, held a special place in her heart.
Presidency College, 1925–30
Chandra’s freshman and sophomore years (1925–27) proceeded smoothly. After he completed his second year with distinction in physics, chemistry, and mathematics, Chandra’s next step was to work toward a bachelor’s degree.
He wanted to take honors mathematics; he had not only excelled in his mathematics studies, he had long been under the spell of the legendary Srinivasa Ramanujan. Chandra was not quite 10 years old when his mother
told him of the death of a famous Indian mathematician named Ramanujan who had gone to England some years earlier, collaborated with some famous English mathematicians, and returned to India only recently with international fame as a great mathematician. Ever since, Ramanujan was a source of inspiration.
Unfortunately, Chandra’s father had different ideas. He wanted Chandra to aim for the Indian Civil Service examination to become an ICS officer in His Majesty’s government. That was certainly the practical thing for such a brilliant young man to do. From his high-school days, however, Chandra had determined to pursue a career in pure science. He had as an example his uncle Chandrasekhara Venkata Raman (popularly known as C. V. Raman), who had resigned a high-level government post to pursue an academic and research career in physics. Although
Chandra wanted to study pure mathematics, as a compromise to his father he opted to study physics and enrolled himself for a BSc honors degree.
The year 1928 proved an extraordinary time for Chandra. First of all, in February and March of that year, Raman, along with Kariamanickam Srinivasa Krishnan, made a fundamental discovery in the molecular scattering of light, later to become known as the Raman effect. Chandra spent the summer months in Calcutta, staying with Raman and working in the laboratory where the discovery was made. He knew enough theoretical physics to participate in the excitement and even explain to the experimentalists the significance of the discovery. He came to know Krishnan very well. Although 12 years apart in age, the two struck up a friendship that lasted through Krishnan’s lifetime.
Soon after his return to Madras, Chandra learned from Krishnan that Arnold Sommerfeld was to visit India on a lecture tour in the fall, and Madras and Presidency College were on his itinerary. For Chandra that was most exciting news—a rare opportunity to hear the famous man, especially since he had read Sommerfeld’s book Atomic Structure and Spectral Lines and worked through it on his own. Chandra dreamed of meeting him, impressing him with his knowledge of atomic physics, and discussing plans for his research. Indeed, after the lecture in the science college, Chandra made arrangements to see him in his hotel room the following day. Chandra approached him with the brash confidence of a young undergraduate, but Sommerfeld shocked him by telling him that the quantum theory in the book was outdated. He told Chandra about recent discoveries—Erwin Schrödinger’s wave mechanics and the new quantum mechanics of Werner Heisenberg, Paul Dirac, Wolfgang Pauli, and others. Chandra had also studied classical Maxwell–Boltzmann statistics. Sommerfeld told him that too had undergone a fundamental change in the light of the new quantum mechanics. Seeing a crestfallen young student before him, Sommerfeld offered Chandra the galley proofs of his as yet unpublished paper that contained an account of the new Fermi–Dirac quantum statistics and its application to the electron theory of metals.
Chandra would later characterize that encounter as the “single most important event” in his life. He immediately launched on a serious study of the new developments in atomic theory. Sommerfeld’s paper was sufficient for him to learn about Fermi–Dirac statistics and prepare, within a few months, a paper entitled “The Compton Scattering and the New Statistics.” Chandra thought it significant enough to merit publication in the Proceedings of the Royal Society. But the society required the papers to be communicated by a fellow, a member of the society. As he was browsing through the newly arrived journals in the university library, he had come across Ralph Fowler, a fellow, who had just published his pioneering paper on the theory of white
dwarfs based on the new Fermi–Dirac statistics. So Chandra sent his paper to Fowler, who agreed to communicate it and got it published. That chance circumstance was to have a profound influence on Chandra’s future scientific career.
Along with his studies, Chandra continued his research, and by the end of his second undergraduate year he had a formidable list of papers to his name. His final year in college was equally eventful. First Heisenberg came through Madras on a lecture tour in October 1929. Krishnan had put Chandra in charge of showing Heisenberg around Madras. A day alone with the famous Heisenberg was an exhilarating experience for young Chandra. In addition, his activities and his prominence in his studies and research had attracted the attention of Lalitha Doraiswamy, a fellow undergraduate who would become his wife. A few months later, in January 1930, he attended the Indian Science Congress Association meeting in Allahabad. He met the celebrated astrophysicist Meghnad Saha and his students and was pleasantly surprised to know his work had become well known. Chandra had the honor of being a dinner guest in the company of some distinguished senior Indian scientists. To top it all, on his return, he was called into the college principal’s office. Principal Philip Fyson told him, in strict confidence, that he was going to be offered a Government of India scholarship to continue his studies in England. The scholarship was special, more or less created for him. On 22 May he received official notification that he had been awarded the scholarship and that he could proceed to make the necessary travel arrangements. Chandra decided to go to Cambridge University and study under the guidance of Fowler.
The opportunity to go abroad for advanced studies, ordinarily so difficult a matter both financially and logistically, came to Chandra so unexpectedly and so easily. Nonetheless, he had to face a difficult personal conflict. His mother had been ill since the summer of 1928, just before his encounter with Sommerfeld, and her illness had taken a serious turn. Although she had ups and downs, after two years of every kind of treatment it had become clear she was not going to get well again. If he went to England, he might never see her again.
Tradition and pressure from friends and relatives mounted against leaving his mother in such a condition. But Sitalakshmi herself intervened. Her insistence and persuasion and her solemn desire not to stand in the way of his future prevailed. With her promise of getting well, she persuaded the reluctant Chandra to proceed. Chandra left India on 31 July 1930, leaving behind a loving and caring family, and Lalitha.
Cambridge and Copenhagen, 1930–33
Before leaving India, Chandra had studied Fowler’s paper more carefully and further developed the theory of white dwarfs to obtain a more detailed picture of them. On his long voyage from India, as a result of musings and calculations, he found Fowler’s theory needed modifications to include special relativistic effects that led to a startling conclusion: There was an upper limit on the mass of a star that could become a white dwarf in its terminal stage (see the article by Freeman Dyson, page 44), and that limit could be expressed in terms of fundamental atomic constants.
When Chandra met Fowler for the first time, he handed over two papers, one he had completed in India and the other about the startling discovery he had made on his journey. Fowler was extremely impressed with this young, new student who exhibited so much independence and initiative. After some discussion, Fowler was quite pleased with the first paper, which had extended his own work. However, he was not so sure of the second paper. He offered to send it to Edward Arthur Milne, who Fowler thought was more familiar with the subject. After getting no response from Fowler or Milne for months and seeing no possibility of its publication in Monthly Notices of the Royal Astronomical Society, Chandra sent it to the Astrophysical Journal on 12 November 1930; it was published the following July.2
Among the lectures Chandra attended during his first year were Dirac’s lectures on quantum mechanics. He had studied Dirac’s book on his own, but he nevertheless attended the lectures faithfully, even though Dirac essentially copied onto the blackboard from his book. Dirac became his official adviser during the second term when Fowler left Cambridge on sabbatical, and Chandra came to know Dirac quite well. “He was very human, extremely cordial to me in a personal way,” Chandra recalled. “Even though he was not very much interested in what I was doing, he used to have me for tea in his rooms in St. John’s about once a month. He also came to my rooms for tea and, on some Sundays, used to drive me out to fields outside Cambridge where we used to go for long walks.”
Chandra continued to do research on relativistic ionization and on stellar atmospheres and began a correspondence with Milne, who was quite receptive of his work. Milne’s encouragement as well as his critical comments were of great help to Chandra during those early days. Within six months they had established a strong rapport, and Milne suggested collaboration and joint publications. Chandra’s research efforts were recognized—he was elected to Trinity College’s Sheep Shanks Exhibition, a special honor bestowed every year to one candidate for proficiency in astrophysics, with an award of £40. He received a congratulatory note from Arthur Eddington with an invitation to meet him on 23 May 1931.
However, on 21 May 1931, Chandra received a devastating telegram:
Mother passed away Thursday 2PM Bear patiently.
Chandra used to write home twice a week to his father and at least once a month to his younger brother Balakrishnan, and also to his mother in Tamil; they were probably the only diversion from his routine. The letters to his father reveal in depth Chandra’s life: his work, study, and leisure routines; his worries; his excursions and walks; the scientists around him; financial details (how he spent and saved); and his diet and his health (as indicated by his weight). His mother’s health was constantly on his mind, and in every letter he inquired about it. As her health went through rapid ups and downs, he always hoped she would get better. But that was not to be.
Alone, with no one to share his grief, he went to the riverbank, sat, and wept. Bear patiently, he told himself. He kept his appointment with Eddington two days after the news, received congratulations, and discussed his work, all the while feeling empty inside.
Work was the only panacea for loneliness and grief. He was working on stellar coefficients of absorption with Milne and had plans to spend the summer in Oxford. But after the news of his mother’s death, he felt the need for a change from the drudgery of Cambridge and the past 11 months of ceaseless study and research. He thought a few months of diversion on the continent would provide the necessary relief. He spent the summer at the Institute for Theoretical Physics in Göttingen, Germany, where Max Born was working. Although the summer was supposed to be a vacation for Chandra, it became mostly a change of place and a change of study topics. But it helped to broaden his circle of friends on the continent. He returned to Cambridge in early September to begin his second academic year.
As he continued his research and piled up publications, some in collaboration with Milne, a conflict slowly brewed in Chandra’s mind. He was in astrophysics by sheer chance—on his own, he had found a problem to work on. He began to have nagging doubts about the value of the work he was doing in astrophysics, as he was not receiving any encouragement in Cambridge. What about his true love—pure mathematics? He had come into physics due to the insistence of his father. Failing to pursue pure mathematics, he felt he would be happier if he could devote himself to pure physics, which he saw as the frontier field in which fundamental discoveries were taking place. The star-studded Cavendish Laboratory was the center of activity.
Dirac was his mentor, but because of a sense of loyalty to Fowler, he hesitated to tell Dirac he wanted to switch fields from astrophysics to theoretical physics. Finally, toward the end of his
second year, he revealed to Dirac how unhappy he was in Cambridge and with what he was doing. Dirac, not being in astrophysics, was in no position to convince him otherwise. But he was very nice and understanding, and sympathized with Chandra’s situation. He strongly urged Chandra to go to Niels Bohr’s Institute for Theoretical Physics in Copenhagen, where he would find a better climate with friendly men who, though younger, were “big men” in physics.
Chandra took Dirac’s suggestion and spent his final year in Copenhagen. The atmosphere at the institute was indeed, as Dirac indicated, quite unlike Cambridge. It was extremely friendly and truly international. Chandra found himself in a group of enthusiastic young people, including Max Delbrück, George Placzek, Victor Weisskopf, E. J. Williams, and Léon Rosenfeld. Chandra was particularly drawn to Rosenfeld, from Belgium, whose fiancée was studying astrophysics. There were also frequent visitors, including Oskar Klein and Heisenberg. With new friendships, tea every Sunday at Bohr’s house, and walks and bicycle rides in the country, Chandra’s life took on a new communal dimension.
He was also happy to be working on a problem in physics that Dirac had suggested: generalizing Fermi–Dirac statistics to more than two particles. Unfortunately, that did not work out. Chandra believed he had solved the problem and wrote a paper titled “On the Statistics of Similar Particles.” Bohr and Rosenfeld read the paper and Bohr communicated it for publication to the Proceedings of the Royal Society. But Dirac found an error and convinced Chandra that he had not solved the problem Dirac had suggested. The paper had to be withdrawn.
Chandra had hoped to change fields from astrophysics to pure theoretical physics, but his lack of success with Dirac’s problem put an end, at least for then, to that idea. Physics was the fundamental science, and while Chandra socialized with physicists like Weisskopf, Delbrück, Hans Kopfermann, and others who appeared to be at the hub of important discoveries, he was not part of their science.
As December came along, it became clear to Chandra that he had to get his thesis ready to get his degree before the end of his scholarship in August 1933. Back to astrophysics, he set himself to prepare for his thesis a series of papers on distorted polytropes. (A polytrope is gaseous material in equilibrium under its own gravity and in which the pressure and density have a power-law relationship.)
A fellowship and an encounter, 1933–36
The degree became just a formality. Fowler did not find it necessary even to read Chandra’s thesis. Chandra felt his future to be bleak, however. His scholarship would end in August and he would be required to return to India as soon as he completed his PhD degree. He was also under pressure from his father to return, but there was no promise of a suitable position that would allow him to continue his research. He was determined to extend his stay in Europe. He would seek support from Cambridge and Copenhagen; if nothing materialized, he had sufficient savings to stay at least six months anywhere in Europe. With little hope, he applied for a fellowship at Trinity College, a wild dream. If it came true, he would have four more years in Cambridge with free rooms in the college, dining privileges at the high table, and an allowance of £300 per year. Fowler was not very optimistic, though—the fellowship was open to candidates from all fields and the competition was formidable.
The dream did come true. The only other Indian who had been elected a Trinity fellow was Ramanujan some 16 years before. Chandra’s Cambridge life became more enjoyable. He was no longer as lonely. He felt assured that his work would be appreciated. Astrophysics was going to be his predominant area of research, at least for the next four years. As a Trinity fellow, he could become a fellow of the Royal Astronomical Society on his own merit and did so without much ado. A trip to London to attend RAS meetings every second Friday of the month became a routine in his life and allowed him to make a mark on the tradition-ridden, hierarchical scientific surroundings. The Trinity fellowship also brought an opportunity to visit Russia during the summer of 1934.
The Russian visit renewed Chandra’s interest in his own earlier work on the theory of white dwarfs. Neither Fowler nor Milne appreciated the startling discovery he had made. During the intervening years, he had occupied himself with other problems. In Russia, he gave talks about his white dwarf work, and Viktor Ambartsumian, in particular, was quite enthusiastic about his discovery. Ambartsumian suggested Chandra should work out the exact, complete theory devoid of some simplifying assumptions he had made.
During the fall months of 1934, Chandra involved himself in detailed, tedious numerical calculations in order to obtain as exact a theory of the white dwarf as one could construct within the framework of relativistic quantum statistics and the known features of stellar interiors. He accomplished the task by the end of 1934 and submitted two papers to the RAS. At the society’s invitation, he presented a brief account of his results at the January 1935 meeting. His findings raised challenging and fundamental questions: What happens to the more massive stars as they continue to collapse? Are there other terminal stages different from white dwarfs? Instead of
getting appreciation and recognition for a fundamental discovery, Chandra unexpectedly faced what amounted to a public humiliation. No sooner had he presented his paper than Eddington, who had been his mentor and who had followed his work closely, ridiculed the basic idea of relativistic degeneracy on which Chandra’s work was based. Eddington characterized the theory as amounting to reductio ad absurdum behavior of the star, tantamount to stellar buffoonery.3 Chandra sought the support of eminent physicists, who without exception agreed that his derivations were flawless, but Eddington’s authority prevailed among the astronomers as he continued to attack the theory.
Eddington’s denunciation was a traumatic experience for Chandra. In the face of such opposition, he decided to gracefully withdraw from the controversy instead of engaging in a dogged fight. He stopped further work on the theory of white dwarfs and went on to research in other areas. As he said,4
I foresaw for myself some thirty to forty years of scientific work, and I simply did not think it was productive to constantly harp on something which was done. It was much better for me to change the field of interest and go into something else. If I was right, then it would be known as right. For myself, I was positive that a fact of such clear significance for evolution of the stars would in time be established or disproved. I didn’t see a need to stay there, so I just left it.
More than two decades passed before the Chandrasekhar limit became an established fact. It has been hailed as one of the most important discoveries of the last century, since it paved the way to the discovery of the other two presently known terminal stages of stars: neutron stars and black holes.
A voyage to the New World, 1936
During the fall of 1935, Chandra received an offer of a lectureship at Harvard University from Harlow Shapley, director of the Harvard College Observatory. The appointment could begin in December or January and required at least three months’ stay. Chandra accepted the offer and had a highly successful and productive first visit to America from 30 December 1935 to 25 March 1936. He attended the American Astronomical Society meeting at Princeton University, gave 10 lectures at the observatory, and at an invitation from its director, Otto Struve, visited the Yerkes Observatory in Williams Bay, Wisconsin. A future for Chandra in America seemed to chart itself without any effort on his part. He received two offers, one from Harvard and the other
from Yerkes. At Harvard, he would join the Society of Fellows, and at Yerkes he would have a research associateship. He received the latter offer aboard ship during his return voyage to Cambridge, and it came directly from Robert Hutchins, the president of the University of Chicago, with a prepaid cable for his answer.
Chandra chose to accept the offer from Yerkes, persuaded by Struve’s arguments in favor of close cooperation between a theorist like him and observational astronomers. With Gerard Kuiper and Bengt Strömgren (a good friend from Copenhagen) also coming to Yerkes, a formidable group of young theorists and observational astronomers was in the making. Thus Chandra felt that as far as his scientific career was concerned, his immediate future had been virtually settled for him. It was time to think of other matters before setting forth to America. He had been away from home for nearly six years and it was time to return. He planned a short trip of three months. There was also the matter of marriage. Chandra had met Lalitha when both were undergraduate students at Presidency College, and they had developed an “understanding” of a lifelong commitment. Though the intervening six years had raised concerns of their future, once they met again in Madras all the doubts and uncertainties vanished. They were married on 11 September 1936, and after one short month in Cambridge they set forth to America.
Williams Bay and Chicago, 1937–95
Chandra and Lalitha, newly married, arrived in the US in 1937, and Chandra joined the faculty of the University of Chicago at Yerkes Observatory. He immediately took on the task of developing a graduate program in astronomy and astrophysics. It wasn’t too long before his reputation as a teacher, his youth, and his enthusiasm for research began to attract students from all parts of the world. As a teacher and a lecturer, Chandra was a grand master who brought elegance and scholarship that literally charmed his listeners and kept them spellbound. He was also the sole editor of the Astrophysical Journal during the years 1952–71. He played a decisive role in transforming the journal, which had been essentially the private property of the University of Chicago, into the national journal of the American Astronomical Society and one of the foremost astrophysics journals in the world.
Chandra and Lalitha lived in Williams Bay for the next 27 years. In 1964 they moved to the Hyde Park neighborhood of Chicago, near the university. Elected a fellow of the Royal Society of London in 1944, Chandra was named the Morton D. Hull Distinguished Service Professor at the University of Chicago in 1946 and remained at the university until his death in 1995.
The judgment of posterity
Chandra often told his life story as follows:
I left India and went to England in 1930. I returned to India in 1936 and married a girl who had been waiting for six years, came to Chicago, and lived happily thereafter.
It may be so. But the Chandra one knows is the product of the complexities of three widely different countries: India, the land of his birth with its ancient culture and traditions, which undoubtedly influenced his early
childhood and youth; England, the land of colonial masters, where his scientific research mushroomed and matured; and finally America, his adopted homeland, where he continued his research and became one of the foremost scientists of the 20th century.
Although there have been many scientists whose discoveries had perhaps greater impact and whose names have become more illustrious, in my opinion Chandra stands alone for his single-minded pursuit of his science and his devotion to the life of the mind. His extraordinary success in his scientific work was marked by an extraordinary effort, an intensity, a fervor for completeness, elegance, and above all else a personal, aesthetic perspective that extended beyond his well-known scientific papers and monographs. For example, when he was chosen for the University of Chicago’s 1975 Ryerson Lecture, Chandra said that his preparations for his talk, “Shakespeare, Newton, and Beethoven, or Patterns of Creativity,”
consisted in reading several biographies of Shakespeare, his sonnets (in A. L. Rowse’s editions) very carefully, and listening with the text (together with Ruth and Norman Lebovitz) to all the great tragedies (in their Marlowe editions); reading several biographies of Beethoven (particularly Turner’s and Sullivan’s); and similarly reading several biographies of Newton; besides, the lives of Rutherford, Faraday, Michelson, Moseley, Maxwell, Einstein, Rayleigh, Abel; and books and essays by Hadamard, Poincaré, and Hardy; and the works of Keats and Shelley and most particularly Shelley’s A Defense of Poetry and King-Hele’s biography of Shelley.5
Chandra often quoted a letter from Milne:
Posterity, in time, will give us all our true measure and assign to each of us our due and humble place. He really succeeds who perseveres according to his lights, unaffected by fortune, good or bad. And it is well to remember there is no correlation between the judgment of posterity and the judgment of contemporaries.
This first centennial celebration of Chandra’s birth may or may not be the moment to determine the true measure of posterity. However, the Chandrasekhar Centennial Symposium, held at the University of Chicago in October, and this special issue of Physics Today mark the beginning of that posterity’s judgment to bestow on him his due place as a scientist of rare stature and greatness.
Kamesh Wali is the distinguished research professor emeritus in the department of physics at Syracuse University in Syracuse, New York.
References
1. For more details, see C. S. Ayyar, “Family History” (1946), Subrahmanyan Chandrasekhar Papers, box 6, folder 4, Special Collections Research Center, University of Chicago Library.
2. S. Chandrasekhar, Astrophys. J. 74, 81 (1931). 3. For details, see, for instance, K. C. Wali, Chandra: A Biography of S. Chandrasekhar, U.
Chicago Press, Chicago (1991), p. 124. 4. Ref. 3, p. 146. 5. K.C. Wali, ed., A Scientific Autobiography: S. Chandrasekhar, World Scientific,
Hackensack, NJ (in press).
Chandrasekhar’s role in 20th-century science
Once the astrophysics community had come to grips with a calculation performed by a 19-year-old student sailing off to graduate school, the heavens could never again be seen as a perfect and tranquil dominion.Freeman DysonDecember 2010, page 44
In 1946 Subrahmanyan Chandrasekhar gave a talk at the University of Chicago entitled “The Scientist.” 1 He was then 35 years old, less than halfway through his life and less than a third of the way through his career as a scientist, but already he wa reflecting deeply on the meaning and purpose of his work. His talk was one of a series of public lectures organized by Robert Hutchins, then the chancellor of the university. The list of speakers is impressive, and included Frank Lloyd Wright, Arnold Schoenberg, and Marc Chagall. That list proves two things. It shows that Hutchins was an impresario with remarkable powers of persuasion, and that he already recognized Chandra as a world-class artist whose medium happened to be theories of the universe rather than music or paint. I say “Chandra” because that is the name his friends used for him when he was alive.
Basic science and derived science
Chandra began his talk with a description of two kinds of scientific inquiry. “I want to draw your attention to one broad division of the physical sciences which has to be kept in mind, the division into a basic science and a derived science. Basic science seeks to analyze the ultimate constitution of matter and the basic concepts of space and time. Derived science, on the other hand, is concerned with the rational ordering of the multifarious aspects of natural phenomena in terms of the basic concepts.”
As examples of basic science, Chandra mentioned the discovery of the atomic nucleus by Ernest Rutherford and the discovery of the neutron by James Chadwick. Each of those discoveries was made by a simple experiment that revealed the existence of a basic building block of the universe. Rutherford discovered the nucleus by shooting alpha particles at a thin gold foil and observing that some of the particles bounced back. Chadwick discovered the neutron by shooting alpha particles at a beryllium target and observing that the resulting radiation collided with other
nuclei in the way expected for a massive neutral twin of the proton. As an example of derived science, Chandra mentioned the discovery by Edmond Halley in 1705 that the comet now bearing his name had appeared periodically in the sky at least four times in recorded history and that its elliptical orbit was described by Newton’s law of gravitation. He also noted the discovery by William Herschel in 1803 that the orbits of binary stars are governed by the same law of gravitation operating beyond our solar system. The observations of Halley and Herschel did not reveal new building blocks, but they vastly extended the range of phenomena that the basic science of Newton could explain.
Chandra also described the particular examples of basic and derived science that played the decisive role in his own intellectual development. In 1926, when Chandra was 15 years old but already a physics student at Presidency College in Madras (now Chennai), India, Enrico Fermi and Paul Dirac independently discovered the basic concepts of Fermi–Dirac statistics: If a bunch of electrons is distributed over a number of quantum states, each quantum state can be occupied by at most one electron, and the probability that a state is occupied is a simple function of the temperature. Those basic properties of electrons were a cornerstone of the newborn science of quantum mechanics. They paved the way to the solution of one of the famous unsolved problems of condensed-matter physics, explaining why the specific heats of solid materials decrease with temperature and go rapidly to zero as the temperature goes to zero.
Two years later, in 1928, the famous German professor Arnold Sommerfeld, one of the chief architects of quantum mechanics, visited Presidency College. Chandra was well prepared. He had read and understood Sommerfeld’s classic textbook, Atomic Structure and Spectral Lines. He boldly introduced himself to Sommerfeld, who took the time to tell him about the latest work of Fermi and Dirac. Sommerfeld gave the young Chandra the galley proofs of his paper on the electron theory of metals, a yet-to-be-published article that gave the decisive confirmation of Fermi–Dirac statistics. Sommerfeld’s paper was a masterpiece of derived science, showing how the basic concepts of Fermi and Dirac could explain in detail why metals exist and how they behave. The Indian undergraduate was one of the first people in the world to read it.
Two years after his meeting with Sommerfeld, at the ripe old age of 19, Chandra sailed on the steamship Pilsna to enroll as a graduate student at Cambridge University. He was to work there with Ralph Fowler, who had used Fermi–Dirac statistics to explain the properties of white dwarf stars—stars that have exhausted their supply of nuclear energy by burning hydrogen to make helium or carbon and oxygen. White dwarfs collapse gravitationally to a density many thousands of times greater than normal matter, and then slowly cool down by radiating away their residual
heat. Fowler’s triumph of derived science included a calculation of the relation between the density and mass of a white dwarf, and his result agreed well with the scanty observations available at that time. With the examples of Sommerfeld and Fowler to encourage him, Chandra was sailing to England with the intention of making his own contribution to derived science.
A sea change
Aboard the Pilsna, Chandra quickly found a way to move forward. The calculations of Sommerfeld and Fowler had assumed that the electrons were nonrelativistic particles obeying the laws of Newtonian mechanics. That assumption was certainly valid for Sommerfeld. Electrons in metals at normal densities have speeds that are very small compared with the speed of light. But for Fowler, the assumption of Newtonian mechanics was not so safe. Electrons in the central regions of white dwarf stars might be moving fast enough to make relativistic effects important. So Chandra spent his free time on the ship repeating Fowler’s calculation of the behavior of a white dwarf star, but with the electrons obeying the laws of Einstein’s special relativity instead of the laws of Newton. Fowler had calculated that for a given chemical composition, the density of a white
dwarf would be proportional to the square of its mass. That made sense from an intuitive point of view. The more massive the star, the stronger the force of gravity and the more tightly the star would be squeezed together. The more massive stars would be smaller and fainter, which explained the fact that no white dwarfs much more massive than the Sun had been seen.
To his amazement, Chandra found that the change from Newton to Einstein has a drastic effect on the behavior of white dwarf stars. It makes the matter in the stars more compressible, so that the density becomes greater for a star of given mass. The density does not merely increase faster as the mass increases, it tends to infinity as the mass reaches a finite value, the Chandrasekhar limit. Provided its mass is below the limit, physicists can model a white dwarf star with relativistic electrons and obtain a unique mass–density relation; there are no models for white dwarfs with mass greater than the Chandrasekhar limit. The limiting mass depends on the chemical composition of the star. For stars that have burned up all their hydrogen, it is about 1.5 times the mass of the Sun.
Chandra finished his calculation before he reached England and never had any doubt that his conclusion was correct. When he arrived in Cambridge and showed his results to Fowler, Fowler
was friendly but unconvinced and unwilling to sponsor Chandra’s paper for publication by the Royal Society in London. Chandra did not wait for Fowler’s approval but sent a brief version of the paper to the Astrophysical Journal in the US.2 The journal sent it for refereeing to Carl Eckart, a famous geophysicist who did not know much about astronomy. Eckart recommended that it be accepted, and it was published a year later. Chandra had a cool head. He had no wish to engage in public polemics with the British dignitaries who failed to understand his argument. He published his work quietly in a reputable astronomical journal and then waited patiently for the next generation of astronomers to recognize its importance. Meanwhile, he would remain on friendly terms with Fowler and the rest of the British academic establishment, and he would find other problems of derived science that his mastery of mathematics and physics would allow him to solve.
The decline and fall of Aristotle
Astronomers had good reason in 1930 to react with skepticism to Chandra’s statements. The implications of his discovery of a limiting mass were totally baffling. All over the sky, we see an abundance of stars cheerfully shining with masses greater than the limit. Chandra’s calculation says that when those stars burn up their nuclear fuel, there will exist no equilibrium states into which they can cool down. What, then, can a massive star do when it runs out of fuel? Chandra had no answer to that question, and neither did anyone else when he raised it in 1930.
The answer was discovered in 1939 by J. Robert Oppenheimer and his student Hartland Snyder. They published their solution in a paper, “On Continued Gravitational Contraction.”3 In my opinion, it was Oppenheimer’s most important contribution to science. Like Chandra’s contribution nine years earlier, it was a masterpiece of derived science, taking some of Einstein’s basic equations and showing that they give rise to startling and unexpected consequences in the real world of astronomy. The difference between Chandra and Oppenheimer was that Chandra started with the 1905 theory of special relativity, whereas Oppenheimer started with Einstein’s 1915 theory of general relativity. In 1939 Oppenheimer was one of the few physicists who took general relativity seriously. At that time it was an unfashionable subject, of interest mainly to philosophers and mathematicians. Oppenheimer knew how to use it as a working tool, to answer questions about real objects in the sky.
Oppenheimer and Snyder accepted Chandra’s conclusion that there exists no static equilibrium state for a cold star with mass larger than the Chandrasekhar limit. Therefore, the fate of a massive star at the end of its life must be dynamic. They worked out the solution to the equations of general relativity for a massive star collapsing under its own weight and discovered that the star is in a state of permanent free fall—that is, the star continues forever to fall inward toward its center. General relativity allows that paradoxical behavior because the time measured by an observer outside the star runs faster than the time measured by an observer inside the star. The time measured on the outside goes all the way from now to the end of the universe, while the time measured on the inside runs only for a few days. During the gravitational collapse, the inside observer sees the star falling freely at high speed, while the outside observer sees it quickly slowing down. The state of permanent free fall is, so far as we know, the actual state of every massive object that has run out of fuel. We know that such objects are abundant in the universe. We call them black holes.
With several decades of hindsight, we can see that Chandra’s discovery of a limiting mass and the Oppenheimer–Snyder discovery of permanent free fall were major turning points in the history of science. Those discoveries marked the end of the Aristotelian vision that had dominated astronomy for 2000 years: the heavens as the realm of peace and perfection, contrasted with Earth as the realm of strife and change. Chandra and Oppenheimer demonstrated that Aristotle was wrong. In a universe dominated by gravitation, no peaceful equilibrium is possible. During the 1930s, between the theoretical insights of Chandra and Oppenheimer, Fritz Zwicky’s systematic observations of supernova explosions confirmed that we live in a violent universe.4 In the same decade, Zwicky discovered the dark matter whose gravitation dominates the dynamics of large-scale structures. After 1939, astronomers slowly and reluctantly abandoned the Aristotelian universe as more evidence accumulated of violent events in the heavens. Radio and x-ray telescopes revealed a universe full of shock waves and high-temperature plasmas, with outbursts of extreme violence associated in one way or another with black holes.
Every child learning science in school and every viewer watching popular scientific documentary programs on television now knows that we live in a violent universe. The “violent universe” has become a part of the prevailing culture. We know that an asteroid collided with Earth 65 million years ago and caused the extinction of the dinosaurs. We know that every heavy atom of silver or gold was cooked in the core of a massive star before being thrown out into space by a supernova explosion. We know that life survived on our planet for billions of years because we are living in a quiet corner of a quiet galaxy, far removed from the explosive violence that we see all around
us in more turbulent parts of the universe. Astronomy has changed its character totally during the past 100 years. A century ago the main theme of astronomy was to explore a quiet and unchanging landscape. Today the main theme is to observe and explain the celestial fireworks that are the evidence of violent change. That radical transformation in our picture of the universe began on the good ship Pilsna when the 19-year-old Chandra discovered that there can be no stable equilibrium state for a massive star.
New ideas confront the old order
It has always seemed strange to me that the work of the three main pioneers of the violent universe—Chandra, Oppenheimer, and Zwicky—received so little recognition and acclaim at the time when it was done. Those discoveries were neglected, in part, because all three pioneers came from outside the astronomical profession. The professional astronomers of the 1930s were conservative in their view of the universe and in their social organization. They saw the universe as a peaceful domain that they knew how to explore with the standard tools of their trade. They were not inclined to take seriously the claims of interlopers with new ideas and new tools. It was easy for the astronomers to ignore the outsiders because the new discoveries did not fit into the accepted ways of thinking and the discoverers did not fit into the established astronomical community.
In addition to those general considerations, which applied to all three of the scientists, individual circumstances contributed to the neglect of their work. For Chandra, the special circumstances were the personalities of Arthur Eddington and Edward Arthur Milne, who were the leading astronomers in England when Chandra arrived from India. Eddington and Milne had their own theories of stellar structure in which they firmly believed; both of those were inconsistent with Chandra’s calculation of a limiting mass. The two astronomers promptly decided that Chandra’s calculation was wrong and never accepted the physical facts on which it was based.
Zwicky confronted an even worse situation at Caltech, where the astronomy department was dominated by Edwin Hubble and Walter Baade. Zwicky belonged to the physics department and had no official credentials as an astronomer. Hubble and Baade believed that Zwicky was crazy, and he believed that they were stupid. Both beliefs had some basis in fact. Zwicky had beaten the astronomers at their own game of observing the heavens, using a wide-field camera that could cover the sky 100 times faster than could other telescope cameras existing at that time. Zwicky then made an enemy of Baade by accusing him of being a Nazi. As a result of that and other incidents, Zwicky’s discoveries were largely ignored for the next 20 years.
The neglect of Oppenheimer’s greatest contribution to science was mostly due to an accident of history. His paper with Snyder, establishing in four pages the physical reality of black holes, was published in the Physical Review on 1 September 1939, the same day Adolf Hitler sent his armies into Poland and began World War II. In addition to the distraction created by Hitler, the same issue of the Physical Review contained the monumental paper by Niels Bohr and John Wheeler on the theory of nuclear fission—a work that spelled out, for all who could read between the lines, the possibilities of nuclear power and nuclear weapons. 5 It is not surprising that the understanding of black holes was pushed aside by the more urgent excitements of war and nuclear energy.
Each of the three pioneers, after a brief period of revolutionary discovery and a short publication, lost interest in fighting for the revolution. Chandra enjoyed seven peaceful years in Europe before moving to America, mostly working, without revolutionary implications, on the theory of normal stars. Zwicky, after finishing the sky survey that revealed dark matter and several types of supernovae, became involved in military problems as World War II was beginning; ultimately, he became an expert in rocketry. Oppenheimer, after discovering the most important astronomical consequence of general relativity, turned his attention to mundane nuclear explosions and became the director of the Los Alamos laboratory.
When I tried in later years to start a conversation with Oppenheimer about the importance of black holes in the evolution of the universe, he was as unwilling to talk about them as he was to talk about his work at Los Alamos. Oppenheimer suffered from an extreme form of the prejudice prevalent among theoretical physicists, overvaluing pure science and undervaluing derived science. For Oppenheimer, the only activity worthy of the talents of a first-rate scientist was the search for new laws of nature. The study of the consequences of old laws was an activity for graduate students or third-rate hacks. He had no desire in later years to return to the study of black holes, the area in which he had made his most important contribution to science. Indeed, Oppenheimer might have continued to make important contributions in the 1950s, when black holes were an unfashionable subject, but he preferred to follow the latest fashion. Oppenheimer and Zwicky did not, like Chandra, live long enough to see their revolutionary ideas adopted by a younger generation and absorbed into the mainstream of astronomy.
From stellar structure to Shakespeare
Chandra would spend 5–10 years on each field that he wished to study in depth. He would take a year to master the subject, a few more years to
publish a series of journal articles demolishing the problems that he could solve, and then a few more years writing a definitive book that surveyed the subject as he left it for his successors. Once the book was finished, he left that field alone and looked for the next topic to study.
That pattern was repeated eight times and recorded in the dates and titles of Chandra’s books. An Introduction to the Study of Stellar Structure (University of Chicago Press, 1939) summarizes his work on the internal structure of white dwarfs and other types of stars. Principles of Stellar Dynamics (University of Chicago Press, 1942) describes his highly original work on the statistical theory of stellar motions in clusters and in galaxies. Radiative Transfer (Clarendon Press, 1950) gives the first accurate theory of radiation transport in stellar atmospheres. Hydrodynamic and Hydromagnetic Stability (Clarendon Press, 1961) provides a foundation for the theory of all kinds of astronomical objects—including stars, accretion disks, and galaxies—that may become unstable as a result of differential rotation. Ellipsoidal Figures of Equilibrium (Yale University Press, 1969) solves an old problem by finding all the possible equilibrium configurations of an incompressible liquid mass rotating in its own gravitational field. The problem had been studied by the great mathematicians of the 19th century—Carl Jacobi, Richard Dedekind, Peter Lejeune Dirichlet, and Bernhard Riemann—who were unable to determine which of the various configurations were stable. In the introduction to his book, Chandra remarks,
These questions were to remain unanswered for more than a hundred years. The reason for this total neglect must in part be attributed to a spectacular discovery by Poincaré, which channeled all subsequent investigations along directions which appeared rich with possibilities; but the long quest it entailed turned out in the end to be after a chimera.
After the ellipsoidal figures opus came a gap of 15 years before the appearance of the next book, The Mathematical Theory of Black Holes (Clarendon Press, 1983). Those 15 years were the time during which Chandra worked hardest and most intensively on the subject closest to his heart: the precise mathematical description of black holes and their interactions with surrounding fields and particles. His book on black holes was his farewell to technical research, just as The Tempest was William Shakespeare’s farewell to writing plays. After the book was published, Chandra lectured and wrote about nontechnical themes, about the works of Shakespeare and Beethoven and Shelley, and about the relationship between art and science. A collection of his lectures for the general public was published in 1987 with the title Truth and Beauty.1
During the years of his retirement, he spent much of his time working his way through Newton’s Principia. Chandra reconstructed every proposition and every demonstration, translating the geometrical arguments of Newton into the algebraic language familiar to modern scientists. The results of his historical research were published shortly before his death in his last book, Newton’s “Principia” for the Common Reader (Clarendon Press, 1995). To explain why he wrote the book, he said, “I am convinced that one’s knowledge of the Physical Sciences is incomplete without a study of the Principia in the same way that one’s knowledge of Literature is incomplete without a knowledge of Shakespeare.”6
Chandra’s work on black holes was the most dramatic example of his commitment to derived science as a tool for understanding nature. Our basic understanding of the nature of space and time rests on two foundations: first, the equations of general relativity discovered by Einstein, and second, the black hole solutions of those equations discovered by Karl Schwarzschild and Roy Kerr and explored in depth by Chandra. To write down the basic equations is a big step toward understanding, but it is not enough. To reach a real understanding of space and time, it is necessary to construct solutions of the equations and to explore all their unexpected consequences. Chandra never said that he understood more about space and time than Einstein, but he did. So long as Einstein did not accept the existence of black holes, his understanding of space and time was far from complete.
When I was a student at Cambridge, I studied with Chandra’s friend Godfrey Hardy, a pure mathematician who shared Chandra’s views about British imperialism and Indian politics. When I came, Hardy was old and he spent most of his time writing books. With the arrogance of youth, I asked Hardy why he wasted his time writing books instead of doing research. Hardy replied, “Young men should prove theorems. Old men should write books.” That was good advice that I have never forgotten. Chandra followed it too. I do not know whether he learned it from Hardy.
This article is based on a talk I gave for the Chandrasekhar Centennial Symposium at the University of Chicago on 16 October 2010.
Freeman Dyson is a retired professor at the Institute for Advanced Study in Princeton, New Jersey.
References
1. S. Chandrasekhar, Truth and Beauty: Aesthetics and Motivations in Science, U. Chicago Press, Chicago (1987).
2. S. Chandrasekhar, Astrophys. J. 74, 81 (1931). 3. J. R. Oppenheimer, H. Snyder, Phys. Rev. 56, 455 (1939). 4. See, for example, F. Zwicky, Morphological Astronomy, Springer, Berlin (1957), sec. 8
and 9. 5. N. Bohr, J. A. Wheeler, Phys. Rev. 56, 426 (1939). 6. S. Chandrasekhar, Curr. Sci. 67, 495 (1994). 7. Ref. 2, reprinted in K. C. Wali, A Quest for Perspectives: Selected Works of S.
Chandrasekhar, with Commentary, vol. 1, Imperial College Press, London (2001), p. 13.
Some memories of Chandra
Five noted scientists, all close colleagues and friends of Subrahmanyan Chandrasekhar, share thoughts and memories of the man whose centennial we celebrate. Robert M. WaldDecember 2010, page 49
I first met Subrahmanyan Chandrasekhar in December 1972, but did not get to know him well until early 1976, more than a year after I arrived at the University of Chicago as a postdoc in the relativity group. For nearly 20 years after that, until his death in 1995, we interacted on an almost daily basis. My memories of those conversations and interactions have faded considerably over the past 15 years—I simply do not have Chandra’s remarkable ability to recall all details of events that occurred long ago. However, the overall impression that Chandra left on me and many other scientists is something that will never fade away.
To many who met him but did not get to know him well, Chandra must have seemed an exceptionally austere and formidable figure—an impression with a great deal of validity. Of all the scientists I have met, Chandra had the highest standards for both intellectual rigor and personal integrity. He applied those standards most uncompromisingly to himself, but he also did not tolerate failings by others in such matters. He was particularly intolerant of scientists motivated primarily by the hope of receiving recognition from others rather than by a deep, inner conviction that their work was of importance and interest, whatever anyone else might think. He was equally intolerant of scientists who rested on their laurels or were otherwise lazy or sloppy, rather than applying their full intellectual efforts toward their work. It was not unusual for Chandra to ask questions of a seminar speaker that were aimed at discerning the speaker’s convictions or at probing how carefully the speaker had thought through the relevant issues. Often those were uncomfortable moments for the speaker.
To get to know Chandra well, a barrier first had to be crossed, a barrier undoubtedly enhanced by the man’s impeccable dress—a suit and tie on all occasions—and by his impeccable speech and manners. It is unfortunate that this barrier had the effect of isolating him from a portion of the scientific community. I believe all that was needed to cross the barrier was some expression to him of the depth of one’s passion for research or other intellectual endeavors. With the barrier
crossed, the very sensitive, caring, and above all loyal nature of Chandra’s personality would become readily apparent. The combination of those very human qualities with Chandra’s almost superhuman discipline, self-sacrifice, and dedication to science had a profound and lasting effect on all who knew him.
In his scientific career of more than 65 years, Chandra’s enthusiasm for the pursuit of science never declined, nor did his fortitude in carrying out major projects. I do not recall a single instance in which he appeared to be motivated by personal gain, nor a single occasion when he made an excuse for not doing something he felt should be done. If he thought a visit to a collaborator or other scientist would help advance his research, he would make the visit without seeking reimbursement for his travel expenses. Similarly, he never requested summer salary from his NSF grant. It appears that the free pursuit of his own scientific research was so important to Chandra that he did not want it tainted or encumbered with issues involving personal gain or accountability.
Chandra will be remembered for the next hundred years and beyond primarily for his truly major contributions to a remarkably broad range of areas in physics and astronomy. He ensured that his scientific legacy will pass on to future generations in unadulterated form by writing a definitive monograph on each of the topics on which he worked. It is highly appropriate that Chandra be remembered primarily for his scientific work. But it also is important that he be remembered for his personal qualities.
To convey a more complete picture of what Chandra was like as a person, I present four reminiscences from scientists who knew him well. John Friedman, professor of physics at the University of Wisconsin–Milwaukee, was one of Chandra’s last students and closely collaborated with him in the early 1970s. Abhay Ashtekar, Eberly Professor of Physics at the Pennsylvania State University, was a student in the Chicago relativity group in the early 1970s, a postdoc in the group in the late 1970s, and a close friend of Chandra’s thereafter. Valeria Ferrari, professor of physics at the University of Rome I (“La Sapienza”), was Chandra’s closest collaborator during the last 10 years of his life. Roger Penrose, emeritus Rouse Ball Professor of Mathematics at Oxford University, was someone whose research Chandra particularly admired and whose scientific advice Chandra sought when he encountered particularly challenging problems. The excerpts below were written about a year after Chandra’s death and are taken from S. Chandrasekhar: The Man Behind the Legend, edited by Kameshwar C. Wali (Imperial College Press, 1997). They are reprinted here with the permission of the publisher and the authors.
John Friedman
Despite the fact that he was still sole editor of The Astrophysical Journal, Chandra spent as much time on research as did his most dedicated students. Beginning his work by 5 am, he finished each 13-hour workday late in the evening. As part of his moral instruction to us, Chandra did not hesitate to point out that by the time his colleagues arrived in the morning, he had already put in half as many hours as they would work in a day. He described a visit to Caltech mainly by noting that the physicists had spent several evenings during the week at cocktail parties. How, he asked, could they get anything done if this was the way they lived? If a few supremely talented physicists could afford
such lapses, Chandra placed himself (and, of course, us) among that vast majority for whom success in science was a matter of character.
In my last year of graduate work, Chandra and [his wife] Lalitha were scheduled to spend six months at Oxford, and Chandra asked me to come with him to finish up my thesis work, a collaboration with him on the stability of rapidly rotating “configurations,” none of which had, at that time, been observed. [My wife] Paula, Mack (our six-month old son), and I traveled to Oxford in time for the great blackout of ’72, one of the miners’ strikes.
In the darkness of that winter, when Chandra went home to his apartment with Lalitha and I to the row house we rented from the Rev. Gauntlett of Maid Marion Way in Nottingham, we worked by candlelight. . . . It was dim and as damp as England’s winters have always been. I might have been feeling a little down myself, tired from our son’s cries and straining to check equations in the dark. But when I came in to work, Chandra’s meticulous script was as elegant as ever, lengthy error-free art, ink on bond. He smiled with mischievous pleasure that I had also been working by candle. Amid 13th century stone walls, built to sequester from the town a secular clergy that once comprised Oxford, he was obviously proud that we each had again spent a day and an evening showing our devotion. It was, he said, as if we were medieval scribes.
The beautiful hand in which his equations were written mirrored Chandra’s understanding of the equations themselves. For most physicists on the mathematical side, equations are viewed abstractly in a way that highlights the properties their expressions share as operators on a Hilbert space, while astrophysicists usually take from mathematics only what is needed for the problem at hand. Chandra, however, fell in neither camp. For his time, Chandra was, to my knowledge,
unique in the way he treated the equations of relativistic astrophysics seriously as objects in themselves, their structure clear in the manner he displayed them, their meaning to be found in this structure. That mathematics was the language of nature he never doubted, and he served nature all his life.
Chandra was also unique in the way he combined a deep understanding of classical mathematics, of astrophysics and of the history of science, particularly the history of classical physics and astronomy. [Andrzej] Trautman and Roger Penrose were then the physicists to whom Chandra seemed closest in temperament and perspective, while his interests were closest to those of the astrophysical relativists, Kip Thorne and James Bardeen. The understanding that grew from Chandra’s history distinguished the problems he worked on, and the unmatched artistry with which he handled his language of equations distinguished their solutions. He was as devoted to science as anyone I have ever met.
Abhay Ashtekar
I first met Chandra when I arrived at the University of Chicago as a green graduate student in ’71. He had just turned sixty. I had done my undergraduate work in India and to me—as to most other Indian students in science—Chandra’s stature was god-like. We had heard of the innumerable discoveries he had made whose meaning and scope we understood only in the vaguest terms. But there was a feeling of awe and admiration and a conviction that for a single person to accomplish all this, he had to be superhuman. And so, I was very surprised when I first met him. Yes, he did have that pristine air about him, and yes, everything he did—the way he dressed, the way he sat in seminars, even the hard-backed chairs he chose to sit on—everything had an aura about it that set him apart. One immediately sensed a refined, dignified and austere personality, just of the type one would expect of a legendary figure like him. Yet, when it came to science, there was unexpected openness. He treated us, students in the newly formed relativity group at Chicago, as if we were his colleagues, his equals. He would come to all seminars, including the ones given by students. He would ask us technical questions with genuine interest. When discussions began, he seemed to become genuinely young, almost one of us. I still remember the smile that would light up his face in the middle of a talk when he heard a beautiful result. Sometimes, when he had cracked a hard problem, something that he found truly satisfying, he would tell us about it. The joy he experienced was so manifest and so contagious!
Chandra was a master storyteller; I have yet to encounter his equal. He had such a fantastic memory for dates and details that, in the anecdotes he recounted, everything became alive. And his anecdotes ranged from incidents that took place in the lofty halls of the Trinity College in Cambridge to his small cabin in the ship he took across the North Sea when he went to Russia. He would recount the events as if they had happened yesterday. We would later shake our heads in astonishment. For, here was Chandra telling about a storm he encountered during the North Sea passage in 1934, or his interesting meetings with the then President of the University of Chicago in 1946, with such clarity and in such detail that we could not have matched in describing events that took place in our own lives just a year before!
I still vividly recall the first time that I heard him tell a story. The students and post-docs in the relativity group had organized a potluck dinner. Chandra and his wife Lalitha came with a delicious vegetarian casserole. When it came to coffee time, there was some unease about how the event was going to end. Do we just say good-bye and leave? Students had planned the menu well but hadn’t thought of anything specific as an after-dinner activity. So, there was some unease. Chandra got up spontaneously and told us some wonderful ghost stories—one told to him by Dirac! They were short, dry and crisp and we all gasped when the punch line came and then laughed. Then other people got up to tell other stories and the evening ended in a relaxed and friendly mood.
Valeria Ferrari
My collaboration with S. Chandrasekhar started in October of 1983. We had met in Rome after the X International Conference on General Relativity, held in Padova in the summer of 1983, and he had invited me to work with him on some relations existing between the mathematical theory of black holes and exact solutions of Einstein’s equations possessing two spacelike Killing vectors.
I arrived in Chicago a few days after he had been awarded the Nobel Prize. I was afraid that the commitments associated with such an important event would prevent Chandra from working with me. But my
fears were unwarranted, because he was more interested in the work we were doing than in giving interviews to the press. Our first paper was completed in two weeks.
For me, this first interaction with Chandra was surprising in many respects. Knowing the breadth and wide range of his scientific accomplishments and having listened to his lectures at
conferences, I had nurtured the idea that he was very strict and rigorous, a man totally and exclusively dedicated to science, and so overwhelming that it would be difficult for me even to talk to him. But Chandra turned out to be entirely different from my preconceptions. In our work, for example, he never used his authority to impose his view on a subject; we always discussed and confronted our ideas as if we were on the same footing. At the same time he was an extraordinary teacher, and shared with me his knowledge and the secrets of his technical ability.
I had to change my views also about Chandra’s personality. In spite of his strict appearance, he was a very warm person, to whom friendship was of great importance. Although I came to know him only during the last twelve years of his life, from many episodes that he narrated to me I think that this had always been the case. For example, in remembering [Arthur] Eddington, with whom he had had the famous scientific dispute that strongly affected his life and his career, he never expressed feelings of resentment or disrespect. I was surprised to learn that while Eddington attacked Chandra’s work in international conferences (he characterized the theory of the limiting mass for the white dwarfs [as] “a stellar buffoonery”), in private they remained on good terms, joining for tea or for a bicycle ride. Chandra was convinced that Eddington’s opposition to his theory was motivated by honest scientific disagreement, and his enormous respect, admiration and affection for him were unharmed by these events. At that time Chandra was in his mid-twenties. Chandra told me that when he used to see Eddington walking the streets of Cambridge with an umbrella under his arm, he thought that this was the picture of a man who had dedicated his life to the pursuit of science and finally had reached a sense of harmony and contentment. Thinking of his own future, he would think that he would also experience a similar sense of harmony, peace and contentment in his old age. “But,” he would add, “it hasn’t turned out that way.” He had a feeling of disappointment because the hope for contentment and a peaceful outlook on life as a result of single-minded pursuit of science had remained unfulfilled. I used to wonder, how could a man like Chandra have this feeling of discontentment about his life? Chandra did not exactly know the reason himself. However, I used to feel a sense of relief in seeing that the excitation for a new result, or the occurrence of a problem difficult to solve, was always able to divert his mind from these sad thoughts.
Roger Penrose
This world has seen some scientists of extraordinary ability—some who are quick and often arrogant, others cautious and possessing genuine humility. Among that small proportion who are of real and rare distinction are the very few who are truly great. It has been my
considerable good fortune to have made the acquaintance of some four or five of those that fall into this final category, but only one of them could I claim to have known at all well—Subrahmanyan Chandrasekhar. .
My acquaintance with Chandra dates back to 1962, when I first encountered him at the Warsaw International Conference on General Relativity and Gravitation. That occasion had a particular significance for Chandra with regard to general relativity, as it marked his entry into the world of general relativists. In fact, he attended that meeting as a “student,” as his way to acquaint himself best with the current activity in that subject.
Why did Chandra have such determination, at the age of 51, to break entirely into a new field, demanding the learning of many new concepts and techniques, where much of the vast expertise that he had built up over many decades would have little direct relevance? It would be natural to suppose, and as I would strongly suspect myself, that it was his desire finally to address the profound conundrum that his early work had thrown up, dating back to his calculations in 1930 on the boat from India to England—that white dwarf stars of more than about one and one-half solar masses cannot sustain themselves against gravitational collapse. It seems clear that even at that time, Chandra was basically aware of the awesome implications of this conclusion, namely that the collapse of the star must eventually take it out of the realm of known physics and into an area shrouded in puzzlement and mystery. But he was by nature an extremely cautious individual, as is made manifest in the modest way he stated his conclusion:
The life-history of a star of small mass must be essentially different from the life-history of a star with large mass. For a star of small mass the natural white-dwarf stage is an initial step towards complete extinction. A star of large mass cannot pass into the white-dwarf stage and one is left speculating on other possibilities.
He was not the sort who would attempt, without due preparation, to make “authoritative” assessments of the likely fate of the material of a body indulging in gravitational collapse. There are, indeed, still many possible loopholes in the arguments which lead to the final conclusion that has now become an accepted implication of present-day theory—that, at least in some cases, the fate of a body in gravitational collapse must be to encounter a space-time singularity, representing, for the constituents of that body, an end to time!
The issue had been at the root of his difficulties with Eddington, when Eddington had so unfairly attacked his work at a meeting of the Royal Astronomical Society in 1935. Eddington, also, was aware of the implications of Chandra’s findings, but regarded this as a reductio ad absurdum and
preferred to move along his own highly speculative route towards a fundamental theory, thereby rejecting the sound reasoning within the accepted tenets of procedure that had characterized what Chandra had achieved. Chandra appears to have been deeply hurt by Eddington’s reaction—the reaction of a man whom Chandra had previously so admired and looked up to. In response, Chandra turned his back on Cambridge and on the immediate problems thrown up by the structure of white dwarfs, apparently devoting his attention entirely to other problems. Yet the question of the ultimate fate of a gravitationally collapsing body must have continued to nag at his physical understandings for many intervening years—even while he was engaging in thorough studies of matters pertaining to quite other astrophysical questions. It is almost as though he had made a tactical retreat, circling around and exploring the details of the surrounding terrain—stellar dynamics, radiative transfer, and the stability of various types of astrophysical structures—before he felt ready for an assault on the profound issue that his early work had uncovered.
His assault was carefully prepared, and required many years of study of the intricacies of Einstein’s general relativity. Not only did he familiarize himself with the standard mathematical techniques and conceptual notions that had been developed for that subject over the years, but he engaged the assistance of certain relativists, such as Andrzej Trautman (and even myself), who had specialist knowledge of some of the less familiar modern mathematical procedures, to give a series of lectures in Chicago to him, his co-workers, and students.
Chandra’s first contributions in which he was able to bring general relativity to bear on astrophysical questions showed that there were additional instabilities, beyond those of Newtonian theory, making their mark earlier than had been expected, and leading even more surely to the ultimate situation of a black-hole fate for a collapsing star. He then moved to the study of black holes themselves, and became fascinated by the beauty of these structures—particularly the Kerr geometry that pertains to a stationary rotating black hole, the ultimate configuration of gravitational collapse. He eventually referred to black holes, in the prologue to his epic book on the subject, The Mathematical Theory of Black Holes, as “the most perfect macroscopic objects that there are in the universe.” .
His fascination with black holes gained as much from aesthetics as from a desire to push forward the frontiers of scientific knowledge. In his later years Chandra became quite explicit as to the importance of aesthetic qualities in science and in his own work in particular.
This brings out what must surely be one of Chandra’s very special qualities: his profound appreciation of the beauty of mathematical formulae. This appreciation extended into pure mathematics as well as applied, and he had an especial admiration for the work of Srinivasa Ramanujan. (He often expressed to me his delight in the fact that the only known photograph of Ramanujan was one that he had himself retrieved. Ramanujan had served as an important inspiration for Chandra in his early aspirations to become a scientist.) Chandra’s wonderful way with mathematical formulae must have been a quality that benefited him also through his earlier work—and provided a thread of continuity throughout his scientific researches in various disparate fields of endeavour. However, this quality is particularly apparent in his work in relativity theory. No doubt he was struck by the fact that the closer his researches took him to fundamental issues in physics—in the analysis of the very nature of space-time—the greater was the mathematical elegance that he encountered in the equations.
What are the qualities that stand out in my memories of Chandra? That he was a great and prolific scientist, there is no doubt, and a deeply individual original thinker. He was enormously systematic and well organized, and he worked incredibly hard. He was a rigorous and somewhat autocratic taskmaster, but he had a genuine appreciation of quality in others. He was a loyal friend, reliable, and totally honest. He was deeply sensitive, but proud. He was a difficult man to criticize, and on occasion his pride might get the better of him—but he would be scrupulously generous with his critics if he could be found to be in error. He was polite and enormously dignified: a greatly cultured individual with a feeling for what is valuable in humanity wherever it might be found. He respected life in all forms (he was a strict vegetarian) and had deep appreciation of the works of Nature. He particularly valued the arts and took great pleasure in them, perceiving profound links between artistic and scientific values. [See the article on page 57.]
How did he view the status of his own scientific contributions in relation to his initial aspirations? One recalls Chandra’s distinctive way of working—reminiscent of the great mathematician David Hilbert—whereby (in essence) Chandra would devote different decades of his life to different topics, culminating each with a definitive book, and leaving each topic behind when he embarked on the next. What does one conclude from this? It might seem that these decades must have represented, to him, completed work that would be neatly wrapped up in the final book. Perhaps so; yet I detected a restlessness in him indicative of a dissatisfaction with what he had ever been able to achieve.
I suspect that his work in relativity theory was what brought him closest to the ultimate goals that he was striving for. He must have derived great satisfaction from his study of black holes, but there were always profound questions left open—and the more that were resolved, the more new ones would appear. Moreover, in his black-hole work, it was the vicinity of the horizon that was being studied, and this lay far outside the central region where the matter of the collapsing star would meet its fate. To gain insights into the nature of this region one must study the space-time singularities—where space and time themselves reach their final termination. Chandra’s work on colliding plane waves must surely have been directed towards gaining an understanding of these singularities, for they provide specific models where one can examine the generation of singularities explicitly.
It is inevitable that the results of this work must remain inconclusive, despite the power and insights that Chandra and his associates were able to provide. If the problem of the ultimate fate of a collapsing star—or a collapsing universe—remains unresolved, it is no discredit to him. He opened our eyes to this profound and deeply important problem and he made great strides towards resolving it. Quite apart from all his other achievements, that in itself might be thought to be enough for any man.
Robert Wald is the Charles H. Swift Distinguished Service Professor in the department of physics at the University of Chicago.
Beauty and the quest for beauty in science
Science, like the arts, admits aesthetic criteria; we seek theories that display “a proper conformity of the parts to one another and to the whole” while still showing “some strangeness in their proportion.”
S. ChandrasekharDecember 2010, page 57(Originally published July 1979, page 25)
The topic to which I have been asked to address myself is a difficult one, if one is to avoid the trivial and the banal. Besides, my knowledge and my experience, such as they are, compel me to limit myself, entirely, to the theoretical aspects of the physical sciences—limitations, most serious. I must, therefore, begin by asking for your patience and your forbearance.
All of us are sensitive to Nature’s beauty. It is not unreasonable that some aspects of this beauty are shared by the natural sciences. But one may ask the question as to the extent to which the quest for beauty is an aim in the pursuit of science. On this question, Poincaré is unequivocal. In one of his essays he has written:
The Scientist does not study nature because it is useful to do so. He studies it because he takes pleasure in it; and he takes pleasure in it because it is beautiful. If nature were not beautiful, it would not be worth knowing and life would not be worth living…. I mean the intimate beauty which comes from the harmonious order of its parts and which a pure intelligence can grasp.
And Poincaré goes on to say,
It is because simplicity and vastness are both beautiful that we seek by preference simple facts and vast facts; that we take delight, now in following the giant courses of the stars, now in scrutinizing with a microscope that prodigious smallness which is also a vastness, and, now in seeking in geological ages the traces of the past that attracts us because of its remoteness.
Commenting on these observations of Poincaré, J. W. N. Sullivan, the author of perceptive biographies of both Newton and Beethoven, wrote (in the Athenaeum for May 1919):
Since the primary object of the scientific theory is to express the harmonies which are found to exist in nature, we see at once that these theories must have an aesthetic value. The measure of
the success of a scientific theory is, in fact, a measure of its aesthetic value, since it is a measure of the extent to which it has introduced harmony in what was before chaos.
It is in its aesthetic value that the justification of the scientific theory is to be found, and with it the justification of the scientific method. Since facts without laws would be of no interest, and laws without theories would have, at most, only a practical utility, we see that the motives which guide the scientific man are, from the beginning, manifestations of the aesthetic impulse…. The measure in which science falls short of art is the measure in which it is incomplete as science….
In a perceptive essay on “Art and Science,” the distinguished art critic, Roger Fry (who may be known to some of you through Virginia Woolf’s biography of him), begins by quoting Sullivan and continues:
Sullivan boldly says: “It is in its aesthetic value that the justification of the scientific theory is to be found and with it the justification of the scientific method.” I should like to pose to S. [Sullivan] at this point the question whether a theory that disregarded facts would have equal value for science with one which agreed with facts. I suppose he would say No; and yet so far as I can see there would be no purely aesthetic reason why it should not.
I shall return to this question which Roger Fry raises and suggest an answer different from what Fry presumes that Sullivan would have given. But I shall pass on now to Fry’s observations comparing the impulses of an artist and of a scientist.
From the merest rudiments of pure sensation up to the highest efforts of design, each point in the process of art is inevitably accompanied by pleasure: it cannot proceed without it…. It is also true that the recognition of inevitability in thought is normally accompanied by pleasurable emotion; and that the desire for this mental pleasure is the motive force which impels to the making of scientific theory. In science the inevitability of the relations remains equally definite and demonstrable, whether the emotion accompanies it or not, whereas, in art, an aesthetic harmony simply does not exist without the emotional state. The harmony in art is not true unless it is felt with emotion…. In art the recognition of relations is immediate and sensational—perhaps we ought to consider it curiously akin to those cases of mathematical geniuses who have immediate intuition of mathematical relations which it is beyond their powers to prove….
Let me pass on from these generalities to particular examples of what scientists have responded to as beautiful.
My first example is related to Fry’s observation with respect to what mathematical geniuses perceive as true with no apparent cause. The Indian mathematician, Srinivasa Ramanujan (whose dramatic emergence into mathematical fame in 1915 may be known to some of you) left a large number of notebooks (one of which was discovered only a few years ago). In these notebooks Ramanujan has recorded several hundred formulae and identities. Many of these have been
proved only recently by methods which Ramanujan could not have known. G. N. Watson, who spent several years of his life proving many of Ramanujan’s identities, has written:
The study of Ramanujan’s work and the problem to which it gives rise, inevitably recalls to mind Lamé’s remark that, when reading Hermite’s papers on modular functions, “on a la chair de poule [one has goose bumps].” I would express my own attitude with more prolixity by saying that such a formula as,
gives me a thrill which is indistinguishable from the thrill which I feel when I enter the Sagrestia Nuova of Capelle Medicee and see before me the austere beauty of “Day,” “Night,” “Evening,” and “Dawn” which Michelangelo has set over the tombs of Giuliano de’ Medici and Lorenzo de’ Medici.
An example of a very different kind is provided by Boltzmann’s reaction to one of Maxwell’s papers on the dynamical theory of gases in which Maxwell shows how one can solve exactly for the transport coefficients in a gas in which the intermolecular force varies as the inverse fifth power of the intermolecular distance. Here is Boltzmann:
Even as a musician can recognize his Mozart, Beethoven, or Schubert after hearing the first few bars, so can a mathematician recognize his Cauchy, Gauss, Jacobi, Helmholtz, or Kirchhoff after the first few pages. The French writers reveal themselves by their extreme formal elegance, while the English, especially Maxwell, by their dramatic sense. Who, for example, is not familiar
with Maxwell’s memoirs on his dynamical theory of gases? … The variations of the velocities are, at first, developed majestically; then from one side enter the equations of state; and from the other side, the equations of motion in a central field. Ever higher soars the chaos of formulae. Suddenly, we hear, as from kettle drums, the four beats “put n = 5.” The evil spirit V (the relative velocity of the two molecules) vanishes; and, even as in music, a hitherto dominating figure in the bass is suddenly silenced, that which had seemed insuperable has been overcome as if by a stroke of magic…. This is not the time to ask why this or that substitution. If you are not swept along with the development, lay aside the paper. Maxwell does not write programme music with explanatory notes…. One result after another follows in quick succession till at last, as the unexpected climax, we arrive at the conditions for thermal equilibrium together with the expressions for the transport coefficients. The curtain then falls!
I have started with these two simple examples to emphasize that one does not have to go to the largest canvasses to find beauty in science. But the largest canvasses do provide the best examples. I shall consider two of them.
Einstein’s discovery of the general theory of relativity has been described by Hermann Weyl as a supreme example of the power of speculative thought, while Landau and Lifshitz consider the theory as probably the most beautiful of all existing physical theories. And Einstein himself wrote at the end of his first paper announcing his field equations: “Scarcely anyone who fully understands this theory can escape from its magic.” I shall return later to consider wherein the source of this magic lies. Meantime, I want to contrast, in parallel with Einstein’s expressed reaction to his theory, the feelings of Heisenberg at the moment of his discovery of quantum mechanics. We are fortunate in having Heisenberg’s own account. He writes:
It had become clear to me what precisely had to take the place of the Bohr–Sommerfeld quantum conditions in an atomic physics working with none but observable magnitudes. It also became obvious that with this additional assumption, I had introduced a crucial restriction into the theory. Then I noticed that there was no guarantee that . . . the principle of the conservation of energy would apply…. Hence I concentrated on demonstrating that the conservation law held; and one evening I reached the point where I was ready to determine the individual terms in the energy table [Energy Matrix]…. When the first terms seemed to accord with the energy principle, I became rather excited, and I began to make countless arithmetical errors. As a result, it was almost three o’clock in the morning before the final result of my computations lay before me. The energy principle had held for all the terms, and I could no longer doubt the mathematical consistency and coherence of the kind of quantum mechanics to which my
calculations pointed. At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structure nature had so generously spread out before me.
In the context of these statements by Einstein and by Heisenberg on their discoveries, it is of interest to recall the following conversation between Heisenberg and Einstein which Heisenberg has recorded. Here is an extract:
If nature leads us to mathematical forms of great simplicity and beauty—by forms, I am referring to coherent systems of hypotheses, axioms, etc.—to forms that no one has previously encountered, we cannot help thinking that they are “true,” that they reveal a genuine feature of nature…. You must have felt this too: the almost frightening simplicity and wholeness of the relationships which nature suddenly spreads out before us and for which none of us was in the least prepared.
These remarks of Heisenberg find an echo in the following lines of Keats:
Beauty is truth, truth beauty—that is allYe know on earth, and all ye need to know.
At this point, I should like to return to Roger Fry’s question I quoted earlier, namely, what one should make of a theory which is aesthetically satisfying but which one believes is not true.
Freeman Dyson has quoted Weyl as having told him: “My work always tried to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.” I inquired of Dyson whether Weyl had given an example of his having sacrificed truth for beauty. I learned that the example which Weyl gave was his gauge theory of gravitation, which he had worked out in his Raum–Zeit–Materie. Apparently, Weyl became convinced that this theory was not true as a theory of gravitation; but still it was so beautiful that he did not wish to abandon it and so he kept it alive for the sake of its beauty. But much later, it did turn out that Weyl’s instinct was right after all, when the formalism of gauge invariance was incorporated into quantum electrodynamics.
Another example which Weyl did not mention, but to which Dyson drew attention, is Weyl’s two-component relativistic wave equation of the neutrino. Weyl discovered this equation and the physicists ignored it for some thirty years because it violated parity invariance. And again, it turned out that Weyl’s instincts were right.
We have evidence, then, that a theory developed by a scientist, with an exceptionally well-developed aesthetic sensibility, can turn out to be true even if, at the time of its formulation, it appeared not to be so. As Keats wrote a long time ago, “What the imagination seizes as beauty must be truth—whether it existed before or not.”
It is, indeed, an incredible fact that what the human mind, at its deepest and most profound, perceives as beautiful finds its realization in external nature.
What is intelligible is also beautiful.
We may well ask: how does it happen that beauty in the exact sciences becomes recognizable even before it is understood in detail and before it can be rationally demonstrated? In what does this power of illumination consist?
These questions have puzzled many thinkers from the earliest times. Thus, Heisenberg has drawn attention, precisely in this connection, to the following thought expressed by Plato in the Phaedrus:
The soul is awestricken and shudders at the sight of the beautiful, for it feels that something is evoked in it that was not imparted to it from without by the senses, but has always been already laid down there in the deeply unconscious region.
The same thought is expressed in the following aphorism of David Hume: “Beauty in things exists in the mind which contemplates them.”
Kepler was so struck by the harmony of nature as revealed to him by his discovery of the laws of planetary motion that in his Harmony of the World, he wrote:
Now, it might be asked how this faculty of the soul, which does not engage in conceptual thinking and can therefore have no prior knowledge of harmonic relations, should be capable of recognizing what is given in the outward world…. To this, I answer that all pure Ideas, or archetypal patterns of harmony, such as we are speaking of, are inherently present in those who
are capable of apprehending them. But they are not first received into the mind by a conceptual process, being the product, rather, of a sort of instinctive intuition and innate in those individuals.
More recently, Pauli, elaborating on these ideas of Kepler, has written:
The bridge, leading from the initially unordered data of experience to the Ideas, consists in certain primeval images pre-existing in the soul—the archetypes of Kepler. These primeval images should not be located in consciousness or related to specific rationally formulizable ideas. It is a question, rather, of forms belonging to the unconscious region of the human soul, images of powerful emotional content, which are not thought, but beheld, as it were, pictorially. The delight one feels, on becoming aware of a new piece of knowledge, arises from the way such pre-existing images fall into congruence with the behavior of the external objects….
Pauli concludes with
One should never declare that theses laid down by rational formulation are the only possible presuppositions of human reason.
This congruence between pre-existing images and external reality, to which Pauli refers, once intensely experienced appears to have the consequence that it develops over-confidence in judgment and values in the person who has had such an experience. For otherwise, how can one understand statements, such as these, made by some of the great scientists:
‣ “It is thermodynamics gone mad,” by Lord Kelvin, one of the founders of thermodynamics, commenting on Boltzmann’s derivation of Stefan’s law.‣ “You look at it from the point of view of the star; I look at it from the point of view of Nature,” by Eddington in a controversial discussion with me.‣ “I disagree with most physicists at the present time just at this point,” by Dirac in the context of his views on the extant methods of renormalization in quantum electrodynamics.‣ “It really looked as if, for the first time, we had a framework wide enough to include the entire spectrum of elementary particles and their interactions fulfilling my dream of 1933,” by Heisenberg in 1957 in the context of his ill-fated collaboration with Pauli on a unified field theory.‣ “God does not throw dice,” by Einstein; or, even more provokingly, ‣ “When judging a physical theory, I ask myself, whether I would have made the Universe in that way, had I been God,” also by Einstein.
In the context of these last statements by Einstein, it may be well to remember Bohr’s remonstrance “Nor is it our business to prescribe to God how he should run the world!”
Perhaps it is in terms of this over-confidence that one must try to understand the comparative sterility of once great minds. For as Claude Bernard has said, “Those who have an excessive faith in their ideas are not fitted to make discoveries.” I am clearly treading on dangerous ground. But it does provide me the opportunity to draw attention to a fact which has been a source of considerable puzzlement to me: it concerns the very different ways—at least, so they seem to me—in which great writers, poets, and musicians on the one hand and great scientists on the other, appear to grow and to mature.
It is not uncommon that in considering the works of a great writer or a great composer one distinguishes an early, a middle, and a late period. And it is almost always the case that the progression from the early, to the middle, and to the late periods is one of growing depth and excellence. In some cases, as in the cases of Shakespeare and Beethoven, the latest works are the greatest. This fact is forcibly described by J. Dover Wilson in his delineation of the growth of Shakespeare’s art in his great tragedies.
From 1601 to 1608 he is absorbed in tragedy; and the path he treads during these eight years may be likened to a mountain track which, rising gently from the plain, grows ever narrower, until at the climax of the ascent it dwindles to the thinnest razor-edge, a glacial arete, with the abyss on either hand, and then once again grows secure for foothold as it broadens out and gradually descends into the valley beyond.
Eight plays compose this tragic course. The first, Julius Caesar, written a little before the tragic period proper, is a tragedy of weakness not of evil. In Hamlet the forces of evil are active and sinister, though still the prevailing note is weakness of character. Othello gives us Shakespeare’s earliest creation of a character wholly evil, and at the same time Iago’s victim is blameless—human weakness is no longer allowed to share the responsibility with heaven. King Lear carries us right to the edge of the abyss, for here horror is piled upon horror and pity on pity, to make the greatest monument of human misery and despair in the literature of the world…. Shakespeare came very near to madness in Lear.
Yet he pushed forward: Macbeth, Antony and Cleopatra (one of the very greatest of Shakespeare’s plays), and Coriolanus followed in succession. And Dover Wilson asks: “How did Shakespeare save his soul alive in this, one of the most perilous and arduous adventures ever
undertaken by the spirit of man?” Shakespeare survived; and he survived only to follow his great tragedies by those wonderful plays, Winter’s Tale and Tempest.
I am afraid that I have, perhaps, digressed a little too long in detailing to you the growth of Shakespeare’s art. But I did want to emphasize to you the magnitude of that growth. And I am sure that one can say very similar things about Beethoven’s late compositions which include the Hammerklavier Sonata, the Missa Solemnis, and above all, his last quartets.
While Shakespeare and Beethoven are probably unique in treading the razor-edge at the very end of their lives and surviving, there are others who illustrate, at a somewhat more modest level, the same consistent ascent to higher peaks of accomplishment. But I am not aware of a single instance of a scientist of whom the same can be said. His early successes are often his last successes. (I am here excluding the cases of those who, like Coates, Galois, Abel, Ramanujan, and Majorana, died in their youth. In these cases, we do not know how they may have fared had they lived past their prime.) In any event, he seems unable to sustain a constant and a continuous ascent. Why is this the case? I shall not, however, attempt to answer this question but pass on to some more concrete considerations.
The question to which I now wish to address myself is how one may evaluate scientific theories as works of art in the manner of literary or art criticisms. The case of general relativity provides a good example, since almost everyone is agreed that it is a beautiful theory. I think it is useful to inquire wherein the source of this beauty lies. It will not do, I think, to dismiss such an inquiry with an assertion such as Dirac’s (made in a different context):
[Mathematical beauty] cannot be defined any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating.
Nor do I think that one should be satisfied with a remark such as Born’s
It [the general theory of relativity] appeared to me like a great work of art, to be enjoyed and admired from a distance.
(Parenthetically, may I say, quite frankly, that I do not know what to make of Born’s remark. Has the general theory of relativity to be admired only from a distance? Does it not require study and development like any other branch of the physical sciences?)
In spite of the inherent difficulties which beset such discussions, I shall attempt to clarify why the general theory of relativity appeals to our aesthetic sense and why we consider it as beautiful. For this purpose, it is necessary to adopt some criteria for beauty. I shall adopt two.
The first is the criterion of Francis Bacon:
There is no excellent beauty that hath not some strangeness in the proportion!
(Strangeness, in this context, has the meaning “exceptional to a degree that excites wonderment and surprise.”)
The second criterion, as formulated by Heisenberg, is complementary to Bacon’s:
Beauty is the proper conformity of the parts to one another and to the whole.
That the general theory of relativity has some strangeness in the proportion, in the Baconian sense, is manifest. It consists primarily in relating, in juxtaposition, two fundamental concepts which had, till then, been considered as entirely independent: the concepts of space and time, on the one hand, and the concepts of matter and motion on the other. Indeed, as Pauli wrote in 1919, “The geometry of space–time is not given; it is determined by matter and its motion.” In the fusion of gravity and metric that followed, Einstein accomplished in 1915 what Riemann had prophesied in 1854, namely, that the metric field must be causally connected with matter and its motion.
Perhaps the greatest strangeness in the proportion consists in our altered view of spacetime. As Eddington wrote: “Space is not a lot of points close together; it is a lot of distances interlocked.”
There is another aspect of Einstein’s founding of his general theory of relativity that continues to be a marvel. It is this.
We can readily concede that Newton’s laws of gravitation require to be modified to allow for the finiteness of the velocity of light and to disallow instantaneous action at a distance. With this concession, it follows that the deviations of the planetary orbits from the Newtonian predictions must be quadratic in v/c where v is a measure of the velocity of the planet in its orbit and c is the velocity of light. In planetary systems, these deviations, even in the most favorable cases, can amount to no more than a few parts in a million. Accordingly, it would have been entirely sufficient if Einstein had sought a theory that would allow for such small deviations from the predictions of the Newtonian theory by a perturbative treatment. That would have been the
normal way. But that was not Einstein’s way: he sought, instead, for an exact theory. And he arrived at his field equations by qualitative arguments of a physical nature combined with an unerring sense for mathematical elegance and simplicity. The fact that Einstein was able to arrive at a complete physical theory by such speculative thought is the reason why, when we follow his thoughts, we feel as “though a wall obscuring truth has collapsed” (Weyl).
The foregoing remarks apply only to the foundations of the theory leading to the field equations. We must now ask whether, on further examination, the theory satisfies the second criterion for beauty, namely, “the conformity of the parts to one another, and to the whole.” The theory most abundantly satisfies this criterion while revealing at every stage a “strangeness in the proportion.” Let me give a few illustrations.
Consider, first, the solutions which the general theory of relativity allow for black holes. As is known, black holes partition the three-dimensional space into two regions, an inner region, bounded by a smooth two-dimensional null-surface, which (the inner region) is incommunicable to the space outside which is, in turn, asymptotically flat. It is a startling fact that with these very simple and necessary restrictions, the general theory allows for stationary black holes a single unique two-parameter family of solutions. This is the Kerr family in which the two parameters are the mass and the angular momentum of the black hole. What is even more remarkable, the metric for this family of solutions is explicitly known. The Kerr metric is axisymmetric and represents a black hole rotating about the axis of symmetry.
The axisymmetric character of the Kerr geometry clearly guarantees that the energy of a test particle describing a geodesic, as well as its component of the angular momentum about the axis of symmetry, will be conserved. In addition to these two conserved quantities, the Kerr geometry unexpectedly allows for the test particle a third conserved quantity (discovered by Brandon Carter). In consequence, the Hamilton–Jacobi equation, governing the motion of a test particle, is separable in its variables; and the solution of the geodesic equations can be reduced to quadratures. This was surprising enough. But what is even more surprising is that all the equations of mathematical physics—the scalar wave equation, Maxwell’s equations, Dirac’s equation, and the equations governing the propagation of gravitational waves—all, are separable in Kerr geometry (even as they are in Minkowskian geometry) and can, therefore, be solved explicitly.
One experiences similar astonishment when we realize that the singularity theorems of Penrose and Hawking require that our universe must necessarily have originated in a singularity and that,
in consequence, we are compelled to contemplate the nature of the physical processes that will occur at densities of the order of 1093 g/cm3, in volumes with linear dimensions of the order of 10−33 cm, and in intervals of time of the order of 10−44 seconds—dimensions which must stagger even this audience.
Or again, Hawking’s theorem that the surface area of a black hole must always increase suggests the identification of the surface area with the thermodynamic entropy of the black hole; and this leads to an intimate connection between thermodynamics, geometry, and gravity.
There is clearly no lack of strangeness in the proportion in all these!
Everything I have said so far is in conformity with the two criteria of beauty with which I started. But there is yet another aspect of the matter which remains to be considered.
When Henry Moore visited the University of Chicago some ten years ago, I had the occasion to ask him how one should view sculptures: from afar or from near by. Moore’s response was that the greatest sculptures can be viewed— indeed, should be viewed—from all distances since new aspects of beauty will be revealed in every scale. Moore cited the sculptures of Michelangelo as examples. In the same way, the general theory of relativity reveals strangeness in the proportion at any level in which one may explore its consequences. One illustration must suffice.
If one enlarges Einstein’s equations to the Einstein–Maxwell equations, that is, the field equations appropriate for space pervaded by an electromagnetic field, and seeks spherically symmetric solutions, one obtains a solution describing a black hole with a mass and an electric charge. This solution was discovered by Reissner and Nordström as a generalization of the well known one of Schwarzschild. Because of the charge of the black hole, it is clear that if an electromagnetic wave is incident on the black hole, a certain fraction of the incident electromagnetic energy will be reflected back in the form of gravitational waves. Conversely, if a gravitational wave is incident on the black hole, a certain fraction of the incident gravitational energy will be reflected back in the form of electromagnetic waves. The remarkable fact is that the two fractions are identically the same, that is, for all frequencies. This result was not expected and the underlying cause for it is still not known. This example illustrates how strangeness in the proportion is revealed by the general theory of relativity at all levels of exploration. And it is this fact, more than any other, that contributes to the unparalleled beauty of the general theory of relativity.
So far, my remarks have been confined to what we may all concede as great ideas conceived by great minds. It does not, however, follow that beauty is experienced only in the context of great ideas and by great minds. This is no more true than that the joys of creativity are restricted to a fortunate few. They are, indeed, accessible to each one of us provided we are attuned to the perception of strangeness in the proportion and the conformity of the parts to one another and to the whole. And there is satisfaction also to be gained from harmoniously organizing a domain of science with order, pattern, and coherence. Examples of such organizations are Jacobi’s Vorlesungen über Dynamik, Boltzmann’s Vorlesungen über Gas Theorie, Sommerfeld’s Atombau und Spektrallinen, Dirac’s Principles of Quantum Mechanics, and the various gems of exposition which Schrödinger wrote in his later years. The translucence of the eternal splendor through material phenomena (of which Plotinus spoke) are made iridescent in these books.
May I conclude then by suggesting that each of us, in our own modest ways, can achieve satisfaction in our quest for beauty in science like the players in Virginia Woolf’s The Waves:
There is a square; there is an oblong. The players take the square and place it upon the oblong. They place it very accurately; they make a perfect dwelling place. Very little is left outside. The structure is now visible; what was inchoate is here stated; we are not so various or so mean; we have made oblongs and stood them upon squares. This is our triumph; this is our consolation.
This article was based on a lecture given at the International Symposium in Honor of Robert R. Wilson on 27 April 1979, at the Fermi National Accelerator Laboratory. The Proceedings of the Symposium were published by Fermilab.
At the time of this article, Subrahmanyan Chandrasekhar was the Morton D. Hull Distinguished Service Professor in the Departments of Astronomy and Physics and in the Enrico Fermi Institute of the University of Chicago.
Supplemental Material
Interview of S. Chandrasekhar by S. Weart on May 17, 1977, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, www.aip.org/history/ohilist/4551_1.html
Interview of Dr. S. Chandrasekhar by Kevin Krisciunas on October 6, 1987, Niels Bohr Library & Archives, American Institute of Physics, College Park, MD USA, www.aip.org/history/ohilist/4552.html
