Six Not-So-Easy Pieces

Einstein’s Relativity, Symmetry, and Space-Time


By Richard P. Feynman

By Robert B. Leighton

By Matthew Sands

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Learn about Einstein's theory of relativity from a physics Nobel laureate and "one of the greatest minds of the twentieth century" (New York Review of Books) in six memorable lessons

It was Richard Feynman's outrageous and scintillating method of teaching that earned him legendary status among students and professors of physics. From 1961 to 1963, Feynman delivered a series of lectures at the California Institute of Technology that revolutionized the teaching of physics. In Six Not-So-Easy Pieces, taken from these famous Lectures on Physics, Feynman delves into one of the most revolutionary discoveries in twentieth-century physics: Einstein's theory of relativity. The idea that the flow of time is not a constant, that the mass of an object depends on its velocity, and that the speed of light is a constant no matter what the motion of the observer, at first seemed shocking to scientists and laymen alike. But as Feynman shows, these tricky ideas are not merely dry principles of physics, but things of beauty and elegance.
No one — not even Einstein himself — explained these difficult, anti-intuitive concepts more clearly, or with more verve and gusto, than Feynman. Filled with wonderful examples and clever illustrations, Six Not-So-Easy Pieces is the ideal introduction to the fundamentals of physics by one of the most admired and accessible physicists of all time.
“There is no better explanation for the scientifically literate layman.” –Washington Post Book World


Also by Richard P. Feynman
The Character of Physical Law
Elementary Particles and the Laws of Physics:
The 1986 Dirac Memorial Lectures (with Steven Weinberg)
Feynman Lectures on Computation
(edited by Anthony J. G. Hey and Robin Allen)
Feynman Lectures on Gravitation (with Fernando B. Morinigo and
William G. Wagner; edited by Brian Hatfield)
The Feynman Lectures on Physics
(with Robert B. Leighton and Matthew Sands)
The Meaning of It All: Thoughts of a Citizen-Scientist
Photon-Hadron Interactions
Perfectly Reasonable Deviations from the Beaten Track:
The Letters of Richard P. Feynman
The Pleasure of Finding Things Out:
The Best Short Works of Richard P. Feynman
QED: The Strange Theory of Light and Matter
Quantum Mechanics and Path Integrals (with A. R. Hibbs)
Six Easy Pieces:
Essentials of Physics Explained by Its Most Brilliant Teacher
Statistical Mechanics: A Set of Lectures
Surely You're Joking, Mr. Feynman!
Adventures of a Curious Character (with Ralph Leighton)
The Theory of Fundamental Processes
What Do You Care What Other People Think?
Further Adventures of a Curious Character
(with Ralph Leighton)

The unqualified success and popularity of Six Easy Pieces sparked a clamor, from the general public, students, and professional scientists alike, for more Feynman in book and audio. So we went back to the original The Feynman Lectures on Physics and to the Archives at Caltech to see if there were more "easy" pieces. There were not. But there were many not-so-easy lectures that, although they contain some mathematics, are not too difficult for beginning science students; and for the student and the layperson, these six lectures are every bit as thrilling, as absorbing, and as much fun as the first six.
Another difference between these not-so-easy pieces and the first six is that the topics of the first six spanned several fields of physics, from mechanics to thermodynamics to atomic physics. These new six pieces you hold in your hand, however, are focused around a subject which has evoked many of the most revolutionary discoveries and amazing theories of modern physics, from black holes to worm holes, from atomic energy to time warps; we are talking, of course, about Relativity. But even the great Einstein himself, the father of Relativity, could not explain the wonders, workings, and fundamental concepts of his own theory as well as could that guy from Noo Yawk, Richard P. Feynman, as reading the chapters or listening to the CDs will prove to you.
We wish to thank Roger Penrose for his penetrating Introduction to this collection; Brian Hatfield and David Pines for their invaluable advice in the selection of the six lectures; and the California Institute of Technology's Physics Department and Institute Archives, in particular Judith Goodstein, for helping to make this book/CD project happen.

To understand why Richard Feynman was such a great teacher, it is important to appreciate his remarkable stature as a scientist. He was indeed one of the outstanding figures of twentieth-century theoretical physics. His contributions to that subject are central to the whole development of the particular way in which quantum theory is used in current cutting-edge research and thus to our presentday pictures of the world. The Feynman path integrals, Feynman diagrams, and Feynman rules are among the very basic tools of the modern theoretical physicist—tools that are necessary for the application of the rules of quantum theory to physical fields (e.g., the quantum theory of electrons, protons, and photons), and which form an essential part of the procedures whereby one makes these rules consistent with the requirements of Einstein's Special Relativity theory. Although none of these ideas is easy to appreciate, Feynman's particular approach always had a deep clarity about it, sweeping away unnecessary complications in what had gone before. There was a close link between his special ability to make progress in research and his particular qualities as a teacher. He had a unique talent that enabled him to cut through the complications that often obscure the essentials of a physical issue and to see clearly into the deep underlying physical principles.
Yet, in the popular conception of Feynman, he is known more for his antics and buffoonery, for his practical jokes, his irreverence towards authority, his bongo-drum performing, his relationships with women, both deep and shallow, his attendance at strip clubs, his attempts, late in life, to reach the obscure country of Tuva in central Asia, and many other schemes. Undoubtedly, he must have been extraordinarily clever, as his lightning quickness at calculation, his exploits involving safe-cracking, outwitting security services, deciphering ancient Mayan texts—not to mention his eventual Nobel Prize—clearly demonstrate. Yet none of this quite conveys the status that he unquestionably has amongst physicists and other scientists, as one of the deepest and most original thinkers of this century.
The distinguished physicist and writer Freeman Dyson, an early collaborator of Feynman's at a time when he was developing his most important ideas, wrote in a letter to his parents in England in the spring of 1948, when Dyson was a graduate student at Cornell University, "Feynman is the young American professor, half genius and half buffoon, who keeps all physicists and their children amused with his effervescent vitality. He has, however, as I have recently learned, a great deal more to him than that. ..." Much later, in 1988, he would write: "A truer description would have said that Feynman was all genius and all buffoon. The deep thinking and the joyful clowning were not separate parts of a split personality.... He was thinking and clowning simultaneously."1 Indeed, in his lectures, his wit was spontaneous, and often outrageous. Through it he held his audiences' attention, but never in a way that would distract from the purpose of the lecture, which was the conveying of genuine and deep physical understanding. Through laughter, his audiences could relax and be at ease, rather than feel daunted by what might otherwise be somewhat intimidating mathematical expressions and physical concepts that are tantalizingly difficult to grasp. Yet, although he enjoyed being center stage and was undoubtedly a showman, this was not the purpose of his expositions. That purpose was to convey some basic understanding of underlying physical ideas and of the essential mathematical tools that are needed in order to express these ideas properly.
Whereas laughter played a key part of his success in holding an audience's attention, more important to the conveying of understanding was the immediacy of his approach. Indeed, he had an extraordinarily direct no-nonsense style. He scorned airy-fairy philosophizing where it had little physical content. Even his attitude to mathematics was somewhat similar. He had little use for pedantic mathematical niceties, but he had a distinctive mastery of the mathematics that he needed, and could present it in a powerfully transparent way. He was beholden to no one, and would never take on trust what others might maintain to be true without himself coming to an independent judgment. Accordingly, his approach was often strikingly original whether in his research or teaching. And when Feynman's way differed significantly from what had gone before, it would be a reasonably sure bet that Feynman's approach would be the more fruitful one to follow.
Feynman's preferred method of communication was verbal. He did not easily, or often, commit himself to the printed word. In his scientific papers, the special "Feynman" qualities would certainly come through, though in a somewhat muted form. It was in his lectures that his talents were given full reign. His exceedingly popular "Feynman Lectures" were basically edited transcripts (by Robert B. Leighton and Matthew Sands) of lectures that Feynman gave, and the compelling nature of the text is evident to anyone who reads it. The Six Not-So-Easy Pieces that are presented here are taken from those accounts. Yet, even here, the printed words alone leave something significantly missing. To sense the full excitement that Feynman's lectures exude, I believe that it is important to hear his actual voice. The directness of Feynman's approach, the irreverence, and the humor then become things that we can immediately share in. Fortunately, there are recordings of all the lectures presented in this book, which give us this opportunity—and I strongly recommend, if the opportunity is there, that at least some of these audio versions are listened to first. Once we have heard Feynman's forceful, enthralling, and witty commentary, in the tones of this streetwise New Yorker, we do not forget how he sounds, and it gives us an image to latch on to when we read his words. But whether we actually read the chapters or not, we can share something of the evident thrill that he himself feels as he explores—and continually re-explores—the extraordinary laws that govern the workings of our universe.
The present series of six lectures was carefully chosen to be of a level a little above the six that formed the earlier set of Feynman lectures entitled Six Easy Pieces (published by Addison Wesley Longman in 1995). Moreover, they go well together and constitute a superb and compelling account of one of the most important general areas of modern theoretical physics.
This area is relativity, which first burst forth into human awareness in the early years of this century. The name of Einstein figures preeminently in the public conception of this field. It was, indeed, Albert Einstein who, in 1905, first clearly enunciated the profound principles which underlie this new realm of physical endeavor. But there were others before him, most notably Hendrik Antoon Lorentz and Henri Poincaré, who had already appreciated most of the basics of the (then) new physics. Moreover, the great scientists Galileo Galilei and Isaac Newton, centuries before Einstein, had already pointed out that in the dynamical theories that they themselves were developing, the physics as perceived by an observer in uniform motion would be identical with that perceived by an observer at rest. The key problem with this had arisen only later, with James Clerk Maxwell's discovery, as published in 1865, of the equations that govern the electric and magnetic fields, and which also control the propagation of light. The implication seemed to be that the relativity principle of Galileo and Newton could no longer hold true; for the speed of light must, by Maxwell's equations, have a definite speed of propagation. Accordingly, an observer at rest is distinguished from those in motion by the fact that only to an observer at rest does the light speed appear to be the same in all directions. The relativity principle of Lorentz, Poincaré, and Einstein differs from that of Galileo and Newton, but it has this same implication: the physics as perceived by an observer in uniform motion is indeed identical with that perceived by an observer at rest.
Yet, in the new relativity, Maxwell's equations are consistent with this principle, and the speed of light is measured to have a definite fixed value in every direction, no matter in what direction or with what speed the observer might be moving. How is this magic achieved so that these apparently hopelessly incompatible requirements are reconciled? I shall leave it to Feynman to explain—in his own inimitable fashion.
Relativity is perhaps the first place where the physical power of the mathematical idea of symmetry begins to be felt. Symmetry is a familiar idea, but it is less familiar to people how such an idea can be applied in accordance with a set of mathematical expressions. But it is just such a thing that is needed in order to implement the principles of special relativity in a system of equations. In order to be consistent with the relativity principle, whereby physics "looks the same" to an observer in uniform motion as to an observer at rest, there must be a "symmetry transformation" which translates one observer's measured quantities into those of the other. It is a symmetry because the physical laws appear the same to each observer, and "symmetry," after all, asserts that something has the same appearance from two distinct points of view. Feynman's approach to abstract matters of this nature is very down to earth, and he is able to convey the ideas in a way that is accessible to people with no particular mathematical experience or aptitude for abstract thinking.
Whereas relativity pointed the way to additional symmetries that had not been perceived before, some of the more modern developments in physics have shown that certain symmetries, previously thought to be universal, are in fact subtly violated. It came as one of the most profound shocks to the physical community in 1957, as the work of Lee, Yang, and Wu showed, that in certain basic physical processes, the laws satisfied by a physical system are not the same as those satisfied by the mirror reflection of that system. In fact, Feynman had a hand in the development of the physical theory which is able to accommodate this asymmetry. His account here is, accordingly, a dramatic one, as deeper and deeper mysteries of nature gradually unfold.
As physics develops, there are mathematical formalisms that develop with it, and which are needed in order to express the new physical laws. When the mathematical tools are skillfully tuned to their appropriate tasks, they can make the physics seem much simpler than otherwise. The ideas of vector calculus are a case in point. The vector calculus of three dimensions was originally developed to handle the physics of ordinary space, and it provides an invaluable piece of machinery for the expression of physical laws, such as those of Newton, where there is no physically preferred direction in space. To put this another way, the physical laws have a symmetry under ordinary rotations in space. Feynman brings home the power of the vector notation and the underlying ideas for expressing such laws.
Relativity theory, however, tells us that time should also be brought under the compass of these symmetry transformations, so a four-dimensional vector calculus is needed. This calculus is also introduced to us here by Feynman, as it provides the way of understanding how not only time and space must be considered as different aspects of the same four-dimensional structure, but the same is true of energy and momentum in the relativistic scheme.
The idea that the history of the universe should be viewed, physically, as a four-dimensional space-time, rather than as a three-dimensional space evolving with time is indeed fundamental to modern physics. It is an idea whose significance is not easy to grasp. Indeed, Einstein himself was not sympathetic to this idea when he first encountered it. The idea of space-time was not, in fact, Einstein's, although, in the popular imagination it is frequently attributed to him. It was the Russian/German geometer Hermann Minkowski, who had been a teacher of Einstein's at the Zurich Polytechnic, who first put forward the idea of four-dimensional space-time in 1908, a few years after Poincaré and Einstein had formulated special relativity theory. In a famous lecture, Minkowski asserted: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of unity between the two will preserve an independent reality."2
Feynman's most influential scientific discoveries, the ones that I have referred to above, stemmed from his own space-time approach to quantum mechanics. There is thus no question about the importance of space-time to Feynman's work and to modern physics generally. It is not surprising, therefore, that Feynman is forceful in his promotion of space-time ideas, stressing their physical significance. Relativity is not airy-fairy philosophy, nor is space-time mere mathematical formalism. It is a foundational ingredient of the very universe in which we live.
When Einstein became accustomed to the idea of space-time, he took it completely into his way of thinking. It became an essential part of his extension of special relativity—the relativity theory I have been referring to above that Lorentz, Poincaré, and Einstein introduced—to what is known as general relativity. In Einstein's general relativity, the space-time becomes curved, and it is able to incorporate the phenomenon of gravity into this curvature. Clearly, this is a difficult idea to grasp, and in Feynman's final lecture in this collection, he makes no attempt to describe the full mathematical machinery that is needed for the complete formulation of Einstein's theory. Yet he gives a powerfully dramatic description, with insightful use of intriguing analogies, in order to get the essential ideas across.
In all his lectures, Feynman made particular efforts to preserve accuracy in his descriptions, almost always qualifying what he says when there was any danger that his simplifications or analogies might be misleading or lead to erroneous conclusions. I felt, however, that his simplified account of the Einstein field equation of general relativity did need a qualification that he did not quite give. For in Einstein's theory, the "active" mass which is the source of gravity is not simply the same as the energy (according to Einstein's E=mc2); instead, this source is the energy density plus the sum of the pressures, and it is this that is the source of gravity's inward accelerations. With this additional qualification, Feynman's account is superb, and provides an excellent introduction to this most beautiful and self-contained of physical theories.
While Feynman's lectures are unashamedly aimed at those who have aspirations to become physicists—whether professionally or in spirit only—they are undoubtedly accessible also to those with no such aspirations. Feynman strongly believed (and I agree with him) in the importance of conveying an understanding of our universe—according to the perceived basic principles of modern physics—far more widely than can be achieved merely by the teaching provided in physics courses. Even late in his life, when taking part in the investigations of the Challenger disaster, he took great pains to show, on national television, that the source of the disaster was something that could be appreciated at an ordinary level, and he performed a simple but convincing experiment on camera showing the brittleness of the shuttle's O-rings in cold conditions.
He was a showman, certainly, sometimes even a clown; but his overriding purpose was always serious. And what more serious purpose can there be than the understanding of the nature of our universe at its deepest levels? At conveying this understanding, Richard Feynman was supreme.
December 1996

(from The Feynman Lectures on Physics)
Toward the end of his life, Richard Feynman's fame had transcended the confines of the scientific community. His exploits as a member of the commission investigating the space shuttle Challenger disaster gave him widespread exposure; similarly, a best-selling book about his picaresque adventures made him a folk hero almost of the proportions of Albert Einstein. But back in 1961, even before his Nobel Prize increased his visibility to the general public, Feynman was more than merely famous among members of the scientific community—he was legendary. Undoubtedly, the extraordinary power of his teaching helped spread and enrich the legend of Richard Feynman.
He was a truly great teacher, perhaps the greatest of his era and ours. For Feynman, the lecture hall was a theater, and the lecturer a performer, responsible for providing drama and fireworks as well as facts and figures. He would prowl about the front of a classroom, arms waving, "the impossible combination of theoretical physicist and circus barker, all body motion and sound effects," wrote The New York Times. Whether he addressed an audience of students, colleagues, or the general public, for those lucky enough to see Feynman lecture in person, the experience was usually unconventional and always unforgettable, like the man himself.
He was the master of high drama, adept at riveting the attention of every lecture-hall audience. Many years ago, he taught a course in Advanced Quantum Mechanics, a large class comprised of a few registered graduate students and most of the Caltech physics faculty. During one lecture, Feynman started explaining how to represent certain complicated integrals diagrammatically: time on this axis, space on that axis, wiggly line for this straight line, etc. Having described what is known to the world of physics as a Feynman diagram, he turned around to face the class, grinning wickedly. "And this is called THE diagram!" Feynman had reached the denouement, and the lecture hall erupted with spontaneous applause.
For many years after the lectures that make up this book were given, Feynman was an occasional guest lecturer for Caltech's freshman physics course. Naturally, his appearances had to be kept secret so there would be room left in the hall for the registered students. At one such lecture the subject was curved space-time, and Feynman was characteristically brilliant. But the unforgettable moment came at the beginning of the lecture. The supernova of 1987 had just been discovered, and Feynman was very excited about it. He said, "Tycho Brahe had his supernova, and Kepler had his. Then there weren't any for 400 years. But now I have mine." The class fell silent, and Feynman continued on. "There are 1011 stars in the galaxy. That used to be a huge number. But it's only a hundred billion. It's less than the national deficit! We used to call them astronomical numbers. Now we should call them economical numbers." The class dissolved in laughter, and Feynman, having captured his audience, went on with his lecture.
Showmanship aside, Feynman's pedagogical technique was simple. A summation of his teaching philosophy was found among his papers in the Caltech Archives, in a note he had scribbled to himself while in Brazil in 1952:
First figure out why you want the students to learn the subject and what you want them to know, and the method will result more or less by common sense.
What came to Feynman by "common sense" were often brilliant twists that perfectly captured the essence of his point. Once, during a public lecture, he was trying to explain why one must not verify an idea using the same data that suggested the idea in the first place. Seeming to wander off the subject, Feynman began talking about license plates. "You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW 357. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!" A point that even many scientists fail to grasp was made clear through Feynman's remarkable "common sense."
In 35 years at Caltech (from 1952 to 1987), Feynman was listed as teacher of record for 34 courses. Twenty-five of them were advanced graduate courses, strictly limited to graduate students, unless undergraduates asked permission to take them (they often did, and permission was nearly always granted). The rest were mainly introductory graduate courses. Only once did Feynman teach courses purely for undergraduates, and that was the celebrated occasion in the academic years 1961 to 1962 and 1962 to 1963, with a brief reprise in 1964, when he gave the lectures that were to become The Feynman Lectures on Physics.
At the time there was a consensus at Caltech that freshman and sophomore students were getting turned off rather than spurred on by their two years of compulsory physics. To remedy the situation, Feynman was asked to design a series of lectures to be given to the students over the course of two years, first to freshmen, and then to the same class as sophomores. When he agreed, it was immediately decided that the lectures should be transcribed for publication. That job turned out to be far more difficult than anyone had imagined. Turning out publishable books required a tremendous amount of work on the part of his colleagues, as well as Feynman himself, who did the final editing of every chapter.
And the nuts and bolts of running a course had to be addressed. This task was greatly complicated by the fact that Feynman had only a vague outline of what he wanted to cover. This meant that no one knew what Feynman would say until he stood in front of a lecture hall filled with students and said it. The Caltech professors who assisted him would then scramble as best they could to handle mundane details, such as making up homework problems.
Why did Feynman devote more than two years to revolutionizing the way beginning physics was taught? One can only speculate, but there were probably three basic reasons. One is that he loved to have an audience, and this gave him a bigger theater than he usually had in graduate courses. The second was that he genuinely cared about students, and he simply thought that teaching freshmen was an important thing to do. The third and perhaps most important reason was the sheer challenge of reformulating physics, as he understood it, so that it could be presented to young students. This was his specialty, and was the standard by which he measured whether something was really understood. Feynman was once asked by a Caltech faculty member to explain why spin 1/2 particles obey Fermi-Dirac statistics. He gauged his audience perfectly and said, "I'll prepare a freshman lecture on it." But a few days later he returned and said, "You know, I couldn't do it. I couldn't reduce it to the freshman level. That means we really don't understand it."
This specialty of reducing deep ideas to simple, understandable terms is evident throughout The Feynman Lectures on Physics


  • “There is no better explanation for the scientifically literate layman.”—Washington Post Book World

On Sale
Mar 22, 2011
Page Count
184 pages
Basic Books

Richard P. Feynman

About the Author

Richard P. Feynman was Richard Chace Tolman Professor of Theoretical Physics at the California Institute of Technology. He was awarded the 1965 Nobel Prize for his work on the development of quantum field theory. He was also one of the most famous and beloved figures of the twentieth century, both in physics and as a public intellectual.

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