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Feynman is, as ever, utterly ingenious in his ease of explication here, especially given that the concepts covered in this volume are more advanced than those covered in the previous one, “Six Easy Pieces” (as cheekily indicated by the title, “Six Not-So-Easy Pieces”)
In this collection of transcribed lectures, once again taken from the fantastically popular series of undergraduate lectures he gave at Caltech in the early ‘60s (just a few years before he won the 1965 Nobel Prize for his work on quantum electrodynamics and path integrals), Feynman focuses almost exclusively on relativity, beginning with Galilean and Newtonian relativity before moving on to Einsteinian relativity—first the special and the then the general theory.
He covers Special Relativity with brilliance and lucidity, enumerating the history of scientific thought underpinning Einstein’s revolutionary leap—from Galileo and Huygens, to Faraday, Maxwell, Lorentz, and Mach—and then moving on to the contemporaneous experimental results (such as the famous Michelson-Morley experiment) which by the end of the 20th century had revealed serious fissures in the foundations of physics. The most notable of these fissures (for the development of relativity, at least) was the incompatibility between Newtonian mechanics and Maxwell’s electromagnetic field equations, which—unlike Newton’s ideas, enshrined in his “Principia Mathematica,” which had taken on an aura of almost divine infallibility in the roughly two centuries since its publication—had only been completed about fifty years previous. As anyone who reads popular science books know, all of this is extremely well-tread ground in the ever-expanding world of mass-market physics writing—from the bestselling books of Stephen Hawking to the irresistibly fun and witty works of Neil deGrasse Tyson, to the brilliantly evocative, almost poetically imagistic writing of Carlo Rovelli. Where I found Feynman to be refreshingly different, though, is in his willingness to dig into the actual mathematics behind the science, demonstrating, for example, how the formula for Galilean transformations led directly to the far more accurate (both theoretically and experimentally) equation for the Lorentz transformation; or how the formula for Newton’s Second Law (F = d(mv)/dt) was brilliantly tweaked by Einstein, who corrected for Newton’s inaccurate assumption that mass is a constant when he came to the (stunning) conclusion that the mass of a body increases with velocity. Feynman gives us Einstein’s corrected formula, in which m has the value
m = m[0] ➗ [the square root of] (1 - v^2/c^2),
“where the ‘rest mass’ m[0] represents the mass of a body that is not moving.”
In my experience, most contemporary popular science writers are utterly allergic to putting equations in their books—unless they’re buried in the footnotes way in the back (I remember one such writer saying something to the effect that for every equation he put in one of his books, its sales were cut in half). Because Feynman makes a point of using equations, not as off-putting esoterica to be avoided at all costs, but as an essential part of his teaching toolkit, he’s able to show much more clearly the evolution of Einstein’s thinking, presenting it for what is actually was: a meticulously thought-out scientific and mathematical conclusion—one which he drew from centuries of thought about the nature of relativistic motion, as well as the more recent discovery of the electromagnetic field equations and the finite velocity of light—not some sort of divine revelation, as is all-too-frequently implied (and which, I must admit, has an alluring quality to it, as it makes for an easier, neater story). Presenting the reader with the actual mathematics allows Feynman to dispel this myth and to show, step by step, the crucial thought processes that led to the incredible intellectual breakthrough that was Special Relativity. No matter how many clever analogies one is presented with—or, for that matter, illustrations of train cars and light clocks and so forth—one can’t fully grasp the many steps that lead to the real scientific theory until one can understand the equations which underpin it. Which isn’t to say that analogies aren’t useful, necessary tools—they are. Not only when trying to gain an understanding of a concept whose mathematics are utterly beyond one’s ken (the general theory, for example, requires a much higher level of mathematical understanding—or quantum mechanics, which, for most mere mortals, is an area of almost breathtaking abstruseness), but also, crucially, for the many modern scientific theories—including the Special Relativity—that go completely against the grain of our intuition. In terms of the basic formulae, though, with Special Relativity all one needs is an understanding of high school-level math to apprehend the steps that Einstein took to arrive at the conclusions he did. And by giving the actual equations—at least in the case of SR, where the underlying mathematics, if not the fairly mind-boggling conclusions drawn from it, are at least relatively (ha!) simple—Feynman is able to peel away the ornamentation of analogy to reveal the substructure beneath.
I have to say that as I read this book, I found myself wishing that more of today’s science writers would take Feynman’s approach—forget whatever their publishers might be telling them about their book sales and respect the intellect of their readership. The publishers might just be in for a surprise.
As brilliant as Feynman’s chapters on Special Relativity are—and they really are quite brilliant—the chapters on General Relativity are truly inspired. As the mathematics are far, far more advanced for the general theory (differential geometry, Riemann curvature, etc.) than for the special theory, and because these lectures were designed for an undergraduate audience, Feynman has to rely much more on analogies than equations here. Regardless of your level of mathematical proficiency, though, the concepts of General Relativity are not ones that human brains are evolved to understand intuitively. Just look at Einstein himself. After discovering the special theory in 1905, it took Einstein a full decade of wrestling with the extreme subtlety of the mathematics and all of its bizarre implications before he was finally able to complete the general theory (and this is Einstein we’re talking about, the guy whose name is a synonym for genius!). And to this day, many still view General Relativity as the single greatest achievement of human creativity and intellect. Needless to say the conclusions of the general theory, far more than those of SR, fly directly in the face of common-sense intuition and everyday experience. And here is where Feynman’s brilliance as a teacher really shines through. His analogies are concise, his explanations sparkling. He reminds me of no one so much as Carlo Rovelli, the Italian physicist whose books (“Seven Brief Lesson on Physics,” “Reality Is Not What it Seems,” “The Order of Time”) are more like Feynman’s than any other contemporary author I’ve come across, including such luminaries as Hawking, Roger Penrose (who, incidentally, wrote the introduction for “Six Not-So-Easy Pieces”), Leonard Susskind, Brian Greene, and Sean Carroll. (I would put Janna Levin in this class, too, though the only book of hers that I’ve read so far is “Black Hole Blues,” which is partly, if not mostly, also a narrative history of LIGO and the search for and ultimate discovery of gravitational waves.) Every science reader, of course, has their own favorite science writers. To me, what writers like Feynman and Rovelli (as well as Einstein himself, for that matter) seem to share that sets them apart is an intense artistic sensibility (for example, Rovelli begins each chapter in “The Order of Time” with a verse from Horace’s “Odes,” and Feynman, well—just read his memoir or one of the many biographies of the guy!), and stylistically a kind of poetic pithiness that makes reading their work such a unique experience. They’re simultaneously brilliantly lucid and poetically succinct; concise, compact, and perfectly cogent, while not avoiding or sacrificing any of the more difficult material or underestimating the intelligence of their readers.
Feynman was not just a one-of-a-kind physicist, but also a one-of-a-kind person, and I highly recommend his memoir “Surely You’re Joking, Mr. Feynman,” which catalogues his many picaresque adventures, as well as his profound creativity in all areas of life.
As quoted in the preface, Feynman wonders aloud whether, if he can’t explain a concept to an undergraduate student, he even understands the idea himself. Going by this standard for comprehension—and if his explanations here are any indication—he understood the concepts of modern physics better than almost anyone, before or since.
In this collection of transcribed lectures, once again taken from the fantastically popular series of undergraduate lectures he gave at Caltech in the early ‘60s (just a few years before he won the 1965 Nobel Prize for his work on quantum electrodynamics and path integrals), Feynman focuses almost exclusively on relativity, beginning with Galilean and Newtonian relativity before moving on to Einsteinian relativity—first the special and the then the general theory.
He covers Special Relativity with brilliance and lucidity, enumerating the history of scientific thought underpinning Einstein’s revolutionary leap—from Galileo and Huygens, to Faraday, Maxwell, Lorentz, and Mach—and then moving on to the contemporaneous experimental results (such as the famous Michelson-Morley experiment) which by the end of the 20th century had revealed serious fissures in the foundations of physics. The most notable of these fissures (for the development of relativity, at least) was the incompatibility between Newtonian mechanics and Maxwell’s electromagnetic field equations, which—unlike Newton’s ideas, enshrined in his “Principia Mathematica,” which had taken on an aura of almost divine infallibility in the roughly two centuries since its publication—had only been completed about fifty years previous. As anyone who reads popular science books know, all of this is extremely well-tread ground in the ever-expanding world of mass-market physics writing—from the bestselling books of Stephen Hawking to the irresistibly fun and witty works of Neil deGrasse Tyson, to the brilliantly evocative, almost poetically imagistic writing of Carlo Rovelli. Where I found Feynman to be refreshingly different, though, is in his willingness to dig into the actual mathematics behind the science, demonstrating, for example, how the formula for Galilean transformations led directly to the far more accurate (both theoretically and experimentally) equation for the Lorentz transformation; or how the formula for Newton’s Second Law (F = d(mv)/dt) was brilliantly tweaked by Einstein, who corrected for Newton’s inaccurate assumption that mass is a constant when he came to the (stunning) conclusion that the mass of a body increases with velocity. Feynman gives us Einstein’s corrected formula, in which m has the value
m = m[0] ➗ [the square root of] (1 - v^2/c^2),
“where the ‘rest mass’ m[0] represents the mass of a body that is not moving.”
In my experience, most contemporary popular science writers are utterly allergic to putting equations in their books—unless they’re buried in the footnotes way in the back (I remember one such writer saying something to the effect that for every equation he put in one of his books, its sales were cut in half). Because Feynman makes a point of using equations, not as off-putting esoterica to be avoided at all costs, but as an essential part of his teaching toolkit, he’s able to show much more clearly the evolution of Einstein’s thinking, presenting it for what is actually was: a meticulously thought-out scientific and mathematical conclusion—one which he drew from centuries of thought about the nature of relativistic motion, as well as the more recent discovery of the electromagnetic field equations and the finite velocity of light—not some sort of divine revelation, as is all-too-frequently implied (and which, I must admit, has an alluring quality to it, as it makes for an easier, neater story). Presenting the reader with the actual mathematics allows Feynman to dispel this myth and to show, step by step, the crucial thought processes that led to the incredible intellectual breakthrough that was Special Relativity. No matter how many clever analogies one is presented with—or, for that matter, illustrations of train cars and light clocks and so forth—one can’t fully grasp the many steps that lead to the real scientific theory until one can understand the equations which underpin it. Which isn’t to say that analogies aren’t useful, necessary tools—they are. Not only when trying to gain an understanding of a concept whose mathematics are utterly beyond one’s ken (the general theory, for example, requires a much higher level of mathematical understanding—or quantum mechanics, which, for most mere mortals, is an area of almost breathtaking abstruseness), but also, crucially, for the many modern scientific theories—including the Special Relativity—that go completely against the grain of our intuition. In terms of the basic formulae, though, with Special Relativity all one needs is an understanding of high school-level math to apprehend the steps that Einstein took to arrive at the conclusions he did. And by giving the actual equations—at least in the case of SR, where the underlying mathematics, if not the fairly mind-boggling conclusions drawn from it, are at least relatively (ha!) simple—Feynman is able to peel away the ornamentation of analogy to reveal the substructure beneath.
I have to say that as I read this book, I found myself wishing that more of today’s science writers would take Feynman’s approach—forget whatever their publishers might be telling them about their book sales and respect the intellect of their readership. The publishers might just be in for a surprise.
As brilliant as Feynman’s chapters on Special Relativity are—and they really are quite brilliant—the chapters on General Relativity are truly inspired. As the mathematics are far, far more advanced for the general theory (differential geometry, Riemann curvature, etc.) than for the special theory, and because these lectures were designed for an undergraduate audience, Feynman has to rely much more on analogies than equations here. Regardless of your level of mathematical proficiency, though, the concepts of General Relativity are not ones that human brains are evolved to understand intuitively. Just look at Einstein himself. After discovering the special theory in 1905, it took Einstein a full decade of wrestling with the extreme subtlety of the mathematics and all of its bizarre implications before he was finally able to complete the general theory (and this is Einstein we’re talking about, the guy whose name is a synonym for genius!). And to this day, many still view General Relativity as the single greatest achievement of human creativity and intellect. Needless to say the conclusions of the general theory, far more than those of SR, fly directly in the face of common-sense intuition and everyday experience. And here is where Feynman’s brilliance as a teacher really shines through. His analogies are concise, his explanations sparkling. He reminds me of no one so much as Carlo Rovelli, the Italian physicist whose books (“Seven Brief Lesson on Physics,” “Reality Is Not What it Seems,” “The Order of Time”) are more like Feynman’s than any other contemporary author I’ve come across, including such luminaries as Hawking, Roger Penrose (who, incidentally, wrote the introduction for “Six Not-So-Easy Pieces”), Leonard Susskind, Brian Greene, and Sean Carroll. (I would put Janna Levin in this class, too, though the only book of hers that I’ve read so far is “Black Hole Blues,” which is partly, if not mostly, also a narrative history of LIGO and the search for and ultimate discovery of gravitational waves.) Every science reader, of course, has their own favorite science writers. To me, what writers like Feynman and Rovelli (as well as Einstein himself, for that matter) seem to share that sets them apart is an intense artistic sensibility (for example, Rovelli begins each chapter in “The Order of Time” with a verse from Horace’s “Odes,” and Feynman, well—just read his memoir or one of the many biographies of the guy!), and stylistically a kind of poetic pithiness that makes reading their work such a unique experience. They’re simultaneously brilliantly lucid and poetically succinct; concise, compact, and perfectly cogent, while not avoiding or sacrificing any of the more difficult material or underestimating the intelligence of their readers.
Feynman was not just a one-of-a-kind physicist, but also a one-of-a-kind person, and I highly recommend his memoir “Surely You’re Joking, Mr. Feynman,” which catalogues his many picaresque adventures, as well as his profound creativity in all areas of life.
As quoted in the preface, Feynman wonders aloud whether, if he can’t explain a concept to an undergraduate student, he even understands the idea himself. Going by this standard for comprehension—and if his explanations here are any indication—he understood the concepts of modern physics better than almost anyone, before or since.
