I became interested in Einstein's general theory of relativity at the age of ten, when I lived in the small village of Ardsley, one hour north of downtown Manhattan. Although my mother had no college degree, having only a diploma from a vocational high school, she had something that was much more important than a formal education; namely, curiosity and a desire to educate herself. It was this attribute that led her to take art courses at various schools, to learn to play Chopin's Revolutionary Etude on the piano, and to read books on diverse subjects, among which were ones that attempted to make relativity accessible to the layman.
These books that my mother brought into our house, and which she and I discussed quite often, included one by Albert Einstein that had a mathematical appendix that neither of us could understand. This book intrigued me. I felt that if I could understand those funny symbols, I might really comprehend what Einstein was attempting to explain in the book. It was not long before I started borrowing my mother's library card to venture six miles to the Yonkers public library technical book collection, where I might find more books concerning the mathematical symbols I had seen in the appendix to Einstein's book.
I perused books on calculus long enough to learn that one had to understand trigonometry first, and books on trigonometry long enough to learn that one had to understand algebra first. Eventually, I found a book on elementary algebra that I could understand, and I began an odyssey that continues to the present day. During the first two years I digested elementary, intermediate and advanced algebra. This phase culminated in my completing an ancient textbook called College Algebra by Wells, which was published in 1890, and which contained all sorts of goodies that are not to be found in modern textbooks. I then moved on to a study of trigonometry, plane and spherical, and to plane geometry, which I did not enjoy as much as the other mathematical subjects because it involved the elusive concept of "proof" that I did not yet fathom or appreciate. [I was soundly rebuffed by my eighth grade teacher when I sought to ask his opinion concerning a book I had acquired concerning plane geometry. "Get that out of here!"]
About the time I entered high school, I asked clerks at the Barnes and Nobles bookstore in New York City what textbook was used at Columbia University to teach analytic geometry. They obliged, and I started reading the book that they suggested. I found this way of looking at geometry fascinating. At the same time I was desperately looking for a book I could understand on differential and integral calculus. It was extremely frustrating, but I just could not get the basic idea from any of the books I encountered.
It was another of my practices to go to used book stores, where I would ask for permission to rummage through the books nobody wanted. These were often incredibly dusty, but also very cheap. I had picked up the algebra book by Wells for fifty cents. [All prices refer to silver coinage, not the debased currency we employ today.] One day for twenty-five cents I bought a thin book called Early Calculus by Ransom, and I suddenly realized what all those other more formidable books had been talking about. I raced through that book, doing as many of the problems as I could, then proceeded to read a fifty cent copy of Differential and Integral Calculus, by Granville, Smith and Longley, which I found I could now understand, the barrier having been lowered by Ransom.
By my junior year in high school I was reading books on ordinary and partial differential equations, and advanced calculus. After completing them, I felt it was time to go back and see if I could better understand what Einstein and others had been trying to explain in those books to which my mother had introduced me six years earlier. I was delighted when I discovered that I could now understand The Einstein Theory of Relativity by Lieber and Lieber. The only sections that did not make much sense were those that referred back to Newton's theory, about which I knew very little.
I was chagrined when I learned that Eddington's Mathematical Theory of Relativity was out of print. However, in the rare book section of the New York Public Library I discovered a tattered copy of this book, the brittle pages of which were enclosed in a Manila envelope about which was tied a string. This was during my senior year at Ardsley High School, from which, rather remarkably, I was permitted to take off one day a week, specifically so that I could read Eddington. Since my special knot was never disturbed, I surmised that no one else had any interest in reading this book that I found so fascinating. That made my adventure much more exciting.
Although my father had only a sixth grade education, he had held a job with the telephone company all during the years of the Great Depression, no mean accomplishment in those days. While they were comfortable, and were able to buy a small house just before the second world war, my parents had not been able to save much money. Halfway through my senior year, it appeared that only $900 would be available to send me to college. We figured that that would be just enough to finance one year at Columbia University, if I lived at home and commuted. Since I had taught myself most of the things that mattered to me, the thought of not completing college bothered me hardly at all. I continued going to New York City to take advantage of the library, and to feed my other interests, ham radio (W2EVK) and astronomy. Indeed, when I was in high school, TV was quite new, and had an aura not unlike that surrounding computers today. Accordingly, I thought that I would eventually earn my living as a TV technician. I had never met a scientist and did not dream that one could earn a living doing research on gravitation theory. [Nor was science a fit subject for TV in those days.]
One day the science teacher at Ardsley High School called my attention to a yearly science contest sponsored by Westinghouse, the Science Talent Search. Three years earlier, my neighbor, Donald Wahlquist, whose primary interest was astrophysics, had placed among the top three hundred applicants. ["Donny" later assumed his father's name, and is now better known as Hugo Wahlquist, discoverer of the Wahlquist interior solution of Einstein's equations.]
My first "paper" on the subject of relativity was the essay I submitted to the Science Talent Search of 1951. There was also an exam, the likes of which I had never seen before. No one was more surprised than I the morning I received a telephone call from Washington informing me that I was among the forty finalists for that year. Up until that time, I had never been more than one hundred miles from home, and here I was taking the Pullman to Washington, to be interviewed by scientists and to compete with thirty-nine other youngsters for college scholarships. It was pretty heady stuff, especially when I shook hands with President Truman during our visit to the White House, where we heard him talk about problems of what is now called the "third world."
At the final banquet I learned that I had been awarded second prize, a $2,000 scholarship to the college of my choice. [First prize went to a student, Bob Kolenkow, who had built his own computer using electrical relays.] After I returned home, dozens of colleges sent me bulletins and applications. I was amazed by the diversity of things one could study. Finally, after having been invited to the campus, I chose to go to Princeton, where my Science Talent Search scholarship would cover about 25% of my expenses. The Korean war had just provided my mother with a job as a technical illustrator, and the financial aid office at Princeton came up with enough additional money to see me through four years of study.
The Fine Hall mathematics and physics library and the university book store were glorious, unlike anything I had ever seen. Suddenly my world had become much larger. Because I had already mastered the equivalent of several years of college mathematics, I was allowed to enroll in advanced mathematics courses. Thus, my first mathematics course at Princeton was one, nominally about infinite series, in which Prof. D. C. Spencer actually taught us about the then new theory of distributions of Laurent-Schwarz. This was a course I got into because the advanced calculus course of Val Bargmann in which I had enrolled was overflowing with students, and Prof. Bargmann asked any student who did not absolutely have to be in his class to switch to Prof. Spencer's class, which met in an adjoining room at the same time.
There were only ten students in Prof. Spencer's class, and I received the second highest score on the first test, 50%. Prof. Spencer asked Gian-Carlo Rota, [the current chairman of the mathematics department at Harvard,] who got 80%, and me if we would like to sit in on his graduate course, which was concerned with A Complex Tensor Calculus for Kähler Manifolds. [Prof. Spencer later told me that he would not have invited me to sit in on his graduate course if he had known that I was a freshman.] In that course I was exposed for the first time to differential forms, which were destined to play an important role in the future development of relativity theory. Later that year, I took a second undergraduate course from Prof. Spencer that concerned functions of a single complex variable, a subject the elegance of which impressed me greatly.
During my sophomore year I sat in on the graduate relativity course offered by Prof. J. A. Wheeler, and under Prof. Wheeler's inspired supervision I later wrote a thesis concerning geons, gravitational electromagnetic entities. Prof. Wheeler often had famous guests at those relativity classes, and it was there that I heard Niels Bohr speak. The most significant thing, however, about having attended those classes was the opportunity that it afforded in May 1953 of visiting and speaking with Albert Einstein, an unimaginable thrill for a youth of nineteen who had already been fascinated by relativity for nine years.
During the same year I was also able to take a reading course from Sol Bochner, in which I struggled very hard and probably not too effectively with the book Several Complex Variables by S. Bochner and W. T. Martin (Princeton University Press, 1948). At that time I did not realize how important knowing something about functions of several complex variables would become for me. It was just the enthusiasm for complex variables that I had gained from Prof. Spencer's courses that prompted me to enroll in the reading course with Prof. Bochner.
After my graduation from Princeton, I was determined to learn more about "mainstream" physics, especially quantum electrodynamics. I had taken graduate courses from Dicke and from Bargmann on quantum mechanics, but I yearned to learn about things such as Feynman diagrams. I deliberately selected a graduate school, the University of Wisconsin, at which no one professed any interest in the general theory of relativity, and where, supported by fellowships from the National Science Foundation, I immersed myself in the study of (special) relativistic quantum field theory. After completing my Ph. D. in 1958 with Prof. Robert G. Sachs, whose guidance in studying quantum field theory I particularly valued, I continued studying fundamental particle theory as a post-doc at various institutions, for it was not until 1966 that I would return to my first love, the general theory of relativity.
In 1962 I began to teach physics, which I discovered I really loved to do, for I was fortunate enough in those days to have in my classes some very ambitious as well as well-prepared undergraduate students. Indeed, in the sixties it seemed that anything was possible, if one worked hard enough. Those were the days when a president could confidently say that we would place a man on the moon before the end of the decade, and few doubted that it would happen.
I believe that among today's intellectually malnourished youngsters there are many for whom earning a living and being entertained will not be enough. Just as I am sure that language skills [such as recognizing the difference between an adverb and an adjective, the difference between less and fewer, or the difference between the indicative and subjunctive mood, for example,] will be rediscovered when people have more important ideas to communicate, I am confident that mathematical skills will be rediscovered when curiosity about the natural world fosters renewed appreciation of knowledge that was hard won during several millennia. [Will that revival of interest in natural philosophy occur before the earth once again reverts to its flat state, as it did following the decline of Greek civilization, whose scientists had, among other things, determined the earth's radius to within ten percent of its actual value seventeen hundred years before Columbus's speculations?]
Meanwhile, attempts to convince adults to support scientific research by emphasizing the material benefits to be derived therefrom, or to convince youngsters to study science by emphasizing how much fun it is, are, I believe, misguided. Scientific research, like artistic endeavors, should be rewarding for its own sake, and the study of science need not necessarily be "fun." The pursuit of science, like the pursuit of the arts, involves discipline, often accompanied by frustration. These are not lives for everyone, but let us hope that they remain lives for someone!
Rather than material benefits and fun let us emphasize instead our insatiable appetite to understand better our place in the Universe. Let us have no preconceived notion as to where scientific enquiry will lead us, but rather let us have a mind that is receptive to the discovery of unanticipated new ways to look at the Universe that we inhabit. Let the intellectual cathedrals that are our greatest scientific and artistic achievements be testaments to our having been interested in more than just bread and circuses.