E=mc2

by David Bodanis

  It's just about 3,600 times.

  That was the calculation Newton did, pretty much, in 1666. Imagine a giant clock, where the moon and earth were parts. Newton's rule showed, exactly, how the invisible connecting cogs and rods held the whole swirling contraption together. Anyone reading Newton, and following this argument, could gaze up and understand, for the first time, that the tug of gravity on their body was the same force that reached up, stretching on to the orbit of the moon and forever beyond.

  "My youngest flaunts her mind . . .": Samuel Edwards, The Divine Mistress (London: Cassell, 1971), p. 12.

  . . . memorize cards at the blackjack table: But even this, in her family's opinion, was something she got wrong. "My daughter is mad," her father wrote in exasperation. "Last week she won more than two thousand gold louis at the card tables, and after ordering new gowns . . . she spent the other half on new books She would not understand that no great lord will marry a woman who is seen reading every day." Ibid., p. 11.

  "I was tired of the lazy, quarrelsome life . . .": Voltaire's Memoires; in Edwards, The Divine Mistress, p. 85.

  "is changing staircases into chimneys . . .": Letter from Voltaire to Mme de la Neuville, in Andre Maurel, The Romance of Mme du Châtelet and Voltaire, trans. Walter Mastyn (London: Hatchette, 1930).

  . . . he discovered her with another lover . . . putting him at ease . . . : The various accounts—by servants as well as participants—of this incident are compared in Rene Vaillot's Voltaire en son temps: avec Mme du Châtelet 1/34-1/48, published in French by the Voltaire Foundation, Taylor Institution, Oxford England 1988.

  The occasional visitors from Versailles . . . : The most thorough description is from Mme de Graffigny's Vieprivee de Voltaire et de Mme de Châtelet (Paris, 1820).

  She knew that most people felt energy . . . : The word energy is anachronistic here, for we're describing the period when these concepts were still being formed. But I think it captures the underlying ideas of the time. See, e.g., L. Lau-dan, "The vis visa controversy, a post mortem," Isis, 59 (1968), pp. 131-43.

  Along with various abstract geometric arguments . . .: Galileo had found that freely tumbling objects don't fall at an unchanging rate. Instead of covering a fixed amount of distance each second, they'll cover 1 unit of distance in the first second, 3 units in the second, 5 units in the third, and so on. Add that sequence of odd numbers together, and you get the accumulated distance a falling object travels: In the first second it's 1 unit, in the second it's 4 units (1 + 3), in the third second it's 9 units (1 + 3 + 5), etc. Through a mix of theory and experiment, this was the basis of Galileo's famous result, that accumulated distance is proportional to the square of the amount of time an object's been falling, or d ?t2. Leibniz extended this reasoning.

  "According to [Newton's] doctrine . . .": Richard West-fall, Never at Rest: A Biography of Isaac Newton (Cambridge: Cambridge University Press, 1987), pp. 777-78.

  . . . for du Châtelet it was one of the peak moments of her life . . . : The issue is more complex than either Newton or Leibniz recognized, and it took the impartial du Châtelet to understand what was valid and had to be preserved in both. Newton really did have a good point, despite Leibniz's mocking, for if the stars were spread randomly, why shouldn't gravity simply make them fall towards each other? And Leibniz also had a good point, for he never asserted that there was a perfect interventionist God, but merely that there was an optimal deity, subject to constraints we might not be able to see. This was a very different matter. Voltaire missed the point in his powerful satire Candide, hut it became a fundamental principle in physics. In a variant form, it became central to Einstein's general relativity, where—as we'll see in the epilogue—planets and stars move in optimal paths within the curved spacetime of the universe.

  What effect did it have on Voltaire to see du Châtelet puzzling through these issues? He would constantly be reminded of the contrast between the vast universe and the little "atom of mud" on which vain humans existed—which was one central theme in his work. He would also constantly be reminded of the need to give space for individual genius—a theme that life with the exhausting, exhilarating du Châtelet would no doubt tend to reinforce.

  Willem 'sGravesande: The last name is not a misprint; the symbol '5 means "of the" and is still common in Dutch: The city Den Haag (The Hague) is officially called 's-gravenhage (the hague of the Earls). I'm simplifying a large range of experiments 'sGravesande carried out: He used bullet-shaped ivory cylinders, hollow and solid brass balls, pendulums, scraped clay (of deeply elaborate consistency), supporting frames, and a Laputian-like variety of other contraptions to carry out his contention that "The Properties of Body cannot be known a priori; we must therefore examine Body itself, and nicely consider all its Properties . . ." See his (most beautifully illustrated) Mathematical Elements of Natural Philosophy, Confirm'd by Experiments, trans. J. T. Desaguilliers, especially Book II, ch. 3, 6th edition (London: 1747); the quote is from p. iv.

  "In this, our delightful retreat": Voltaire's Memoires, in Edwards, The Divine Mistress, p. 86.

  "I am pregnant. . .": Letter to Mme de Boufflers, April 3, 1749. In Les lettres de la Marquise du Châtelet, vol. 2, ed. T. Besterman (Geneva: Institut et Musee Voltaire, 1958), p. 247.

  "I have lost the half of myself. . .": Voltaire to d'Argen-tal, in, e.g., Frank Hamel, An Eighteenth-Century Marquise (London: Stanley Paul & Co., 1910), p. 369.

  A car that's racing along at four times another one's speed . . . : A wind of 20 mph is gentle, but a wind of 200 mph is catastrophic, and more like an ignited gas stove exploding. It's a lot more than 10 times as powerful, for it carries 102 or 100 times more energy. That's also why jetliners have to travel so high. Only the thinner air up there lets the plane survive hours in the immense power of a 600-mph storm that the plane's speed produces.

  Athletes perform these complex calculations all the time. Most schoolchildren can toss a ball at 20 mph, but only a few professional athletes can throw a ball at 100 mph. It's "only" five times as fast, but since energy goes up as the square of the speed (E = mv2), the athlete has to generate 25 times as much energy. What's more, she has to do it in only 1/5 the time. (For if the athlete took exactly as long to move her arm as the child did, the ball would come out at only 20 mph.) To pour out 25 times more energy, in 1/5 the time, means she needs to generate 25 X 5 or 125 times more power! Other effects such as air resistance make it even harder. The one factor that does help an adult athlete is having a longer lever arm than a child.

  Only by concentrating on mv2: The point is not that mv2 is "true," while mv1 is not. Newton's concept of momentum— mv1—is quite central to our understanding of the universe. Rather, each definition carves out different domains—different aspects—for us to concentrate on. Shoot a rifle, and the recoil is best understood in terms of mv1; the impact of the bullet in terms of mv2. A rifle and a bullet will have equal amounts of momentum right after the trigger is pulled, but the kickback from a moving gun won't kill you: most of its kinetic energy is carried in its mass, so its velocity will have very little effect on the shooter. The bullet fired out however has so little mass that it carries most of the same momentum in its velocity. It's the square of that velocity—the bullet's kinetic energy— which signals how dangerous its high-speed flight can be to its target.

  This isn't a proof, of course . . . : This is one of the items developed on my Web site.

  That enormous conversion factor . . . of the equation.: If mass were too easily converted fully to energy, pens and pencils around us would start going off with blinding flashes of light, taking much of the Earth's cities with them; most of the physical universe would soon blast out of material existence.

  What saves us is the principle of baryon conservation, which holds, roughly, that the total of protons and neutrons in the universe does not change; they cannot abruptly start disappearing.

  The one time when 100 percent conversion does occur is when ordinary matter bumps against
antimatter. A typical proton in our bodies has a baryon number of +1, but an antimatter antiproton has a baryon number of — i, so if they ever did annihilate, the sum of baryons in the universe wouldn't have changed. We actually experience something related to this every day, for a portion of the radon gas that wafts up from basements or out of walls produces antimatter as part of its decay process. Where that contacts ordinary air molecules or our skin, a (small-scale!) explosion from E=mc2 operating at full power immediately results.

  7. Einstein and the Equation

  . . . rocking his one-year-old . . . : From a slightly later period, see e.g., D. Reichinstein's reminiscences, collected in his Albert Einstein: A Picture of His Life and His Conception of The World, by David Reichinstein (London: Edward Gold-ston, Ltd, 1934).

  . . . as he referred to his notion of God . . . : For a particularly thoughtful analysis, see Max Jammer's Einstein and Religion: Physics and Theology (Princeton, N.J.: Princeton University Press, 1999). A rich compilation of current views by scientists about religion—both pro and con—is in Russell Stannard's Science and Wonders: Conversations about Science and Belief (London: Faber and Faber, 1996), which developed from a BBC radio series.

  "We are in the position . . .": Einstein goes on: "That, it seems to me, is the attitude of the human mind, even the greatest and most cultured, toward God." From a 1929 interview with the then-famous journalist George Sylvester Viereck, reported in Viereck's Glimpses of the Great (London: Duckworth, 1934), p. 372. The wording is probably only approximate, as Viereck at other points admits finding his own shorthand notes hard to decipher.

  In fact she had a passionate nature . . . : The affair is described in Marie Curie: A Life, by Susan Quinn (orig. New York: Simon & Schuster, 1995; London: Mandarin pbk., 1996). Einstein's quote stems from a 1913 hike together, reported on p. 348 of the UK edition.

  He sent off the relativity article . . . : The epoch-making article was rejected for several reasons, not least of which was the sound bureaucratic grounds that it was a printed document and " . . . the regulations insisted upon a hand-written thesis." Carl Seelig, Albert Einstein: A Documentary Biography (London: Staples Press, 1956), p. 88 has the story from Paul Gruner, a supporter of Einstein's, who witnessed this among the Bern faculty.

  ". . . like a man whose skin has been peeled off. . . ": Marianne Weber, Max Weber: A Biography, ed. and trans. Harry Zohn (New York: John Wiley & Sons, 1975), p. 286.

  . . . time advanced smoothly . . . : Einstein wasn't the first to see that Newton's laws were consistent with there being no external "authority" or measurement standard by which our particular activities could be judged—for Newton had seen it as well! But in an intensely theological era, Newton had to be discreet in his thoughts while he was venturing into heresy this way. It was to a great extent to avoid such a God-denying "free float" of time that Newton incorporated absolute time in his Principia. The standard account is in Newton's General Scholium, but do see his more colloquial sequence of explanations in the letters to Richard Bentley (then a young theologian); both are conveniently available in the Norton reader Newton: Texts, Backgrounds, Commentaries, ed. Bernard Cohen and Richard Westfall (New York: Norton, 1995). Would Newton have taken the few simple algebraic steps to develop special relativity if he hadn't been held back by these cautions?

  . . . a world where . . . an easy 30 mph: The image stems from George Gamow's estimable Mr. Tompkins series, on which generations of science lovers were raised. When Gamow wrote it the vision was a great fantasy; I think he would have been pleased that before the twentieth century was over "in February, 1999" a team at Harvard used laser- cooling to produce a substance in such a state "it was within 50 billionths of a degree of absolute zero" that light was measured by outside observers to travel at just under 39 mph in it.

  . . . come down to their original, static weight: The ordinary terms such as weight or bulk up in mass are, again, just to give an indication of what's going on.

  . . . passengers inside would appear to have shrunk . . . : Textbooks usually say that a speeding car's length will be contracted, shrinking to the thickness of tissue paper. But although a direct application of the contraction factor from the note on page 249 would suggest this to be the case, what actually occurs is more subtle, due to such effects as the way light coming from different parts of the car must be emitted at different times. Distortions are similar to the way the three-dimensional Earth gets skewed when it's converted into two-dimensional Mercator projections for maps.

  The Global Positioning System . . . : Along with the corrections applied to GPS satellite signals that do stem from special relativity, substantial effects are also due to general relativistic considerations, as is well surveyed in Clifford M. Will, Was Einstein Right: Putting General Relativity to the Test (Oxford: Oxford University Press, 1993). I love the idea that millions of people, holding GPS receivers at one time or another, have wrapped their hands around devices that contain miniaturized transpositions of the logical sequences that once occurred in Einstein's brain.

  . . . the label relativity . . . : Einstein never used the phrase theory of relativity in his original 1905 paper; this was only suggested by Planck and others a year later. The name he really liked came from Minkowski, in 1908, who referred, accurately, to Einstein's "Invariant Postulates." If that had taken, we'd talk about Albert Einstein and his famous "theory of invariants." But by the time there was a wider move to make such a change, in the 1920s, the original, unwanted label had stuck.

  . . . the wrong impression . . . : "The meaning of relativity has been widely misunderstood," Einstein explained in 1929. "Philosophers play with the word, like a child with a doll. . . . It [relativity] does not mean that everything in life is relative."

  Einstein was misinterpreted, in large part, because many people were ready to misinterpret him. Cezanne had spoken about the need to focus only on what you, personally, see and measure: a patch of red here, a glob of blue there. This was taken to match the way relativity questioned there being an impersonal, "objective" background world, just waiting, like a given interpretation of a Paris boulevard, for everyone to share. More recently, Tom Stop-pard— who likes to undercut conventional perspectives—is happy to have characters in his plays refer to Einsteinian effects that seem to back that view.

  The problem, though, is that these uses have nothing to do with Einstein's work. As mentioned in the main text, the divergence from any usual effects is far too small to be noticed at the speeds where we normally live. Even more central, the fact that the theory actually hinges on a few key invariants being preserved—the speed of light; the consistency and "equality" of any given coordinate frame—is quite the opposite of how the theory is commonly presented. Einstein himself once explained this to an art historian who had been trying to link Cubism with the theory of relativity:

  The essence of the theory of relativity has been incorrectly understood. . . . The theory says only that. . . general laws are such that their form does not depend on the choice of the system of coordinates. This logical demand, however, has nothing to do with how the single, specific case is represented. A multiplicity of systems of coordinates is not [emphasis added] needed for its representation. It is completely sufficient to describe the whole mathematically in relation to one system of coordinates.

  This is quite different in the case of Picasso s painting. . . . This new artistic "language"has nothing in common with the Theory of Relativity.

  The quote is in Paul LaPorte, "Cubism and Relativity, with a Letter of Albert Einstein," Art Journal, 25, no. 3 (1966), p. 246; quoted in Gerald Holton's The Advancement of Science, and Its Burdens (Cambridge, Mass.: Harvard University Press, 1998), p. 109. Holton goes on to make the sensible observation that the notion of a multiplicity of possible observational frames is, in fact, the essence of all modern science, where modern means anything from Galileo in the early 1600s on; multiple and consistent views of a drawing have been a commonplace of architects for a very long t
ime as well.

  . . . both Einstein and Newton . . . impossibly brief periods . . . : Here the laurel has to go to Einstein. Newton is famous for stating that he discovered differential calculus, the composition of light, and universal gravitation, all in the brief period when he was at his mother's farm. But when he recalled this he was already a very old man, talking up the past. The calculations he'd made on the farm had not been very persuasive—instead of the figure of 3,600 we used for the weakness of Earth's gravity at the orbit of the moon, which would "prove" an inverse square law of gravity applied, inaccuracies in measurements of the Earth meant that the best figure he actually got was a far from convincing 4,300 or so. He was also confused about the role of centrifugal force, and whether or not the moon was spinning in a Descartes-style vortex—there was a great deal of work to do when he got back to Cambridge. But then, humility is probably not what's needed for anyone working at these levels.

  The fluently written article "Newton and the Eotvos Experiment" by Curtis Wilson, in his collected works Astronomy from Kepler to Newton: Historical Studies, (London: Variorum Reprints, 1989), is especially good on the subtleties Newton still had to work through. Chapter five of Westfall's Never at Rest: A Biography of Isaac Newton (Cambridge: Cambridge University Press, 1987) further examines Newton's actual accomplishments during the plague years in Lincolnshire; see also The 'Annus Mirabilis" of Sir Isaac Newton, 1666-1966, ed. Robert Palter (Cambridge, Mass.: MIT Press, 1970).

  They would learn what was on offer . . . : What I like about Veblen is that he concentrates on a particular sociointellectual cusp—the intersection of religion with science— which is liable to be especially laden with meaning. We can go deeper into Einstein's own work to see that in operation.

  The first thing that leaps out is Einstein's great belief in unity. One part of traditional physics was built up from conventional Newtonian mechanics, where there were always ways to compare two different observers: to see who was going faster or slower than the other; to determine, objectively, that someone who was speeding in a car and switched a headlight on would make that light beam move along "faster" than would someone in a car that was standing still. But on the other hand, as Einstein realized, the other part of traditional physics built on Maxwell's development of Faraday's work, and that hinged on the speed of light appearing the same from the vantage point of any smoothly moving observer. Both a stationary driver and a moving driver would have to see any headlight beam shoot forward at 670 million mph. To Newton that was impossible. To Maxwell it was indispensable.