E=mc2

by David Bodanis

  Faraday, however, still went on about those strange circles and other wending lines from his religious upbringing. The area around an electromagnetic event, Faraday held, was filled with a mysterious "field," and stresses within that field produced what were interpreted as electric currents and the like. He insisted that sometimes you could almost see their essence, as in the curving patterns that iron filings take when they are sprinkled around a magnet. Yet almost no one believed him—except, now, for this young Scot named Maxwell.

  At first glance the two men seemed very different. In his many years of research, Faraday had accumulated over 3,000 paragraphs of dated notebook entries on his experiments, from investigations that began early every morning. Maxwell, however, quite lacked any ability to get a timely start to the day. (When he was told that there was mandatory 6 A.M. chapel at Cambridge University, the story goes that he took a deep breath, and said, "Aye, I suppose I can stay up that late.") Maxwell also had probably the finest mathematical mind of any nineteenth-century theoretical physicist, while Faraday had problems with any conventional math much beyond simple addition or subtraction.

  James Clerk Maxwell

  PHOTO RESEARCHERS, INC.

  But on a deeper level the contact was close. Although Maxwell had grown up in a great baronial estate in rural Scotland, the family name had until recently simply been Clerk, and it was only from an inheritance on the maternal side that they'd acquired the more distinguished Maxwell to tack on. When young James was sent away to boarding school in Edinburgh, the other children—stronger in build, cockily confident with their big-city ways—picked on him: week after week, year after year. James never expressed any anger about it; just once, he quietly remarked: "They never understood me, but I understood them." Faraday also still carried the wounds from his experiences with Sir Humphry Davy in the 1820s, and would relapse into a quiet, watching solitude almost instantly after he'd finished an evening as an apparently ebullient speaker at one of the Royal Institution public lectures.

  When the young Scot and the elderly Londoner corresponded, and then later when they met, they cautiously made contact of a sort they could share with almost no one else. For beyond the personality similarities, Maxwell was such a great mathematician that he was able to see beyond the surface simplicity of Faraday's sketches. It was not the childishness that less gifted researchers mocked it for. ("As I proceeded with the study of Faraday, I perceived that his method . . . was also a mathematical one, though not exhibited in the conventional form of mathematical symbols.") Maxwell took those crude drawings of invisible force lines seriously. They were both deeply religious men; they both appreciated this possibility of God's immanence in the world.

  Back in his 1821 breakthrough, and then in much research after, Faraday had shown ways in which electricity can be turned into magnetism, and vice versa. In the late 1850s, Maxwell extended that idea, into the first full explanation of what Galileo and Roemer had never understood.

  What was happening inside a light beam, Maxwell began to see, was just another variation of this back-and-forth movement. When a light beam starts going forward, one can think of a little bit of electricity being produced, and then as the electricity moves forward it powers up a little bit of magnetism, and as that magnetism moves on, it powers up yet another surge of electricity, and so on like a braided whip snapping forward. The electricity and magnetism keep on leapfrogging over each other in tiny, fast jumps—a "mutual embrace," in Maxwell's words. The light Roemer had seen hurtling across the solar system, and which Maxwell saw bouncing off the stone towers at Cambridge, was merely a sequence of these quick, leapfrogging jumps.

  It was one of the high points of nineteenth-century science; Maxwell's equations summarizing this insight became known as one of the greatest theoretical achievements of all time. But Maxwell was always slightly dissatisfied with what he'd produced. For how exactly did this strangely leapfrogging light wave braid itself along? He didn't know. Faraday didn't know. No one could explain it for sure.

  Einstein's genius was to look closer at what these skittering light waves meant, even though he had to do it largely on his own. He had the confidence to do this: his final high school preparation in Aarau really had been superb, and he'd grown up in a family that encouraged him to always question authority. By the 1890s, when Einstein was a student, Maxwell's formulations were usually taught as a received truth. But Einstein's main professor at the Zurich polytechnic, unimpressed with theory in physics, refused even to teach Maxwell to his undergraduates. (It was Einstein's resentment at being treated like this that led him to address that professor mockingly as Herr Weber rather than the expected Herr Professor Weber—a slight that Weber avenged by refusing to write a proper letter of recommendation for Einstein, leading to his years of isolation at the patent office job.)

  When Einstein cut classes to go to the coffeehouses in Zurich, it was often with accounts of Maxwell's work in hand. He began to explore the leapfrogging of light waves that Maxwell had first uncovered. If light was a wave like any other, Einstein mused, then if you ran after it, could you catch up?

  An example from surfing will show the problem. When you're first out in the water, trying not to let everyone on shore see how scared you are, the waves slosh past you. But once you force yourself to stand up on your surfboard, you can glide shoreward as the wave of water pushing you seems to be standing still around you. Be bold enough—or foolhardy enough—to do this in the extreme surf off Hawaii, and an entire curving tube of water might seem to be at rest around, above, and beside you.

  Only in 1905 did the full insight hit Einstein. Light waves are different from everything else. A surfer's water wave can appear to hold still, because all the parts of the wave take up a steady position in relation to one another. That's why you can glance out from your surfboard, and see a hovering sheet of water. Light is not like that, however. Light waves keep themselves going only by virtue of one part moving forward and so powering up the next part. (The electricity part of the light wave shimmers forward, and that "squeezes" out a magnetic part; then that magnetic part, as it powers up, creates a further "surge" of electricity so the rushing cycle starts repeating.) Whenever you think you're racing forward fast enough to have pulled up next to a light beam, look harder and you'll see that whatever part you thought you were close to is powering up a further part of the light beam that is still hurtling away from you.

  To catch up with a streak of light and see it standing still would be like saying, "I want to see the blurred arcs of a thrilling juggling act, but only if the balls are not moving." You can't do it. The only way you'll see a blur from the juggling balls is if they're moving fast.

  Einstein concluded that light can exist only when a light wave is actively moving forward. It was an insight that had been lurking in Maxwell's work for over forty years, but no one had recognized it.

  This new realization about light changed everything; for the speed of light becomes the fundamental speed limit in our universe: nothing can go faster.

  It's easy to misunderstand this. If you were traveling at 669,999,999 mph, couldn't you pump in more fuel, and go the few mph faster—to 670,000,000, and then to 670,000,0001—to take you past the speed of light? But the answer is that you can't, and it's not a quirk about the present state of earthly technology.

  A good way to recognize this is to remember that light isn't just a number, it is a physical process. There's a big difference. If I say that -273 (negative 273) is the lowest number that there is, you could rightly answer that I'm wrong: that -274 is lower, and -275 is lower yet, and that you can keep on going forever. But suppose we were dealing with temperatures. The temperature of a substance is a readout of how much its inner parts are moving, and at some point they're going to stop vibrating entirely. That happens at about -273 degrees on the centigrade scale, and that's why -273 degrees is said to be "absolute zero" when you're talking about temperature. Pure numbers might be able to go lower, but physical things
can't: a coin or a snowmobile or a mountain can't vibrate any less than not vibrating at all.

  So it is with light. The 670,000,000 mph figure that Roemer measured for the light speeding down from Jupiter is a statement about what that light is like. It's a physical "thing." Light will always be a quick leapfrogging of electricity out from magnetism, and then of magnetism leaping out from electricity, all swiftly shooting away from anything trying to catch up to it. That's why its speed can be an upper limit.

  . . .

  It's an interesting enough observation, but a cynic might say, so what if there is an ultimate speed limit? How should that affect all the solid objects that move around within the universe? You can clamp a label saying "Warning: No Speed Over 670,000,000 mph Can Be Achieved," on the signs by a busy road, but the traffic whirring past will be unaffected.

  Or will it? This is where Einstein's whole argument finally circles back: where he showed that light's curious properties—the fact that it inherently squiggles away from you, and is therefore the ultimate speed limit-really enters into the nature of energy and mass. An example modified from one that Einstein himself used can suggest how it might happen.

  Suppose a super space shuttle is blasting along very close to the speed of light. Under normal circumstances, when that space shuttle is going slowly, the fuel energy that's pumped into the engines would just raise its speed. But things are different when the shuttle is right at the very edge of the speed of light. It can't go much faster.

  The pilot of the shuttle doesn't want to accept this, and starts frantically leaping up and down on the thruster control to get the vessel to go faster. But of course the pilot sees any light beam that's ahead still squirting out of reach at the full speed of "c." That's what light does for any observer. Despite the pilot's best efforts, the shuttle is not gaining on it. So what happens?

  Think of frat boys jammed into a phone booth, their faces squashed hard against the glass walls. Think of a parade balloon, with an air hose pumping into it that can't be turned off. The whole balloon starts swelling, far beyond any size for which it was intended. The same thing would happen to the shuttle. The engines are roaring with energy, but that can't raise the shuttle's speed, for nothing goes faster than light. But the energy can't just disappear, either.

  As a result, the energy being pumped in gets "squeezed" into becoming mass. Viewed from outside, the solid mass of the shuttle starts to grow. There's only a bit of swelling at first, but as you keep on pouring in energy, the mass will keep on increasing. The shuttle will keep on swelling.

  It sounds preposterous, but there's evidence to prove it. If you start to speed up small protons, which have one "unit" of mass when they're standing still, at first they simply move faster and faster, as you'd expect. But then, when they get close to the speed of light, an observer really will see the protons begin to change. It's a regular event at the accelerators outside of Chicago, and at CERN (the European center for nuclear research) near Geneva, and everywhere else physicists work. The protons first "swell" to become two units of mass—twice as much as they were at the start—then three units, then on and on, as the power continues to be pumped in. At speeds of 99.9997 percent of "c," the protons end up 430 times bigger than their original size. (So much power is drained from nearby electricity stations that the main experiments are often scheduled to run late at night, so that nearby residents won't complain about their lights dimming.)

  What's happening is that energy that's pumped into the protons or into our imagined shuttle has to turn into extra mass. Just as the equation states: that "E" can become "m," and "m" can become "E."

  That's what explains the "c" in the equation. In this example, as you push up against the speed of light, that's where the linkages between energy and mass become especially clear. The figure "c" is merely a conversion factor telling you how that linkage operates.

  Whenever you link two systems that developed separately, there will need to be some conversion factor. To go from centigrade to Fahrenheit, you multiply the centigrade figure by 9/5, and then add 32. To go from centimeters to inches there's another rule: you multiply the centimeters by 0.3937.

  The conversion factors seem arbitrary, but that's because they link measurement systems that evolved separately. Inches, for example, began in medieval England, and were based on the size of the human thumb. Thumbs are excellent portable measuring tools, since even the poorest individuals could count on regularly carrying them along to market. Centimeters, however, were popularized centuries later, during the French Revolution, and are defined as one billionth of the distance from the equator to the North Pole, passing by Paris. It's no wonder the two systems don't fit together smoothly.

  For centuries, energy and mass had also seemed to be entirely separate things. They evolved without contact. Energy was thought of as horsepower or kilowatt hours; mass was measured in pounds or kilos or tons. No one thought of connecting the units. No one glimpsed what Einstein did, that there could be a "natural" transfer between energy and mass, as we saw with the shuttle example, and that "c" is the conversion factor linking the two.

  The reader might wonder when we'll get to the theory of relativity. The answer is that we've already been using it! All these points about a speeding shuttle and its expanding mass are central to what Einstein published in 1905.

  Einstein's work changed the two separate visions scientists had taken from the nineteenth-century work on conservation laws. Energy isn't conserved, and mass isn't conserved—but that doesn't mean there is chaos. Instead, there's actually a deeper unity, for there's a link between what happens in the energy domain and what happens in the seemingly distinct mass domain. The amount of mass that's gained is always going to be balanced by an equivalent amount of energy that's lost.

  Lavoisier and Faraday had seen only part of the truth. Energy does not stand alone, and neither does mass. But the sum of mass plus energy will always remain constant.

  This, finally, is the ultimate extension of the separate conservation laws the eighteenth- and nineteenth-century scientists had once thought complete. The reason this effect had remained hidden, unsuspected, all the time before Einstein, is that the speed of light is so much higher than the ordinary motions we're used to. The effect is weak at walking speed, or even at the speed of locomotives or jets, but it's still there. And as we'll see, the linkage is omnipresent in our ordinary world: all the energy is held quiveringly ready inside even the most ordinary substances.

  Linking energy and mass via the speed of light was a tremendous insight, but there's still one more detail to get clear. A famous cartoon shows Einstein at a board, trying out one possibility after another: E=mc1, E=mc2, E=mc3,. . . But he didn't really do it that way, arriving at the squaring of "c" by mere chance.

  So why did the conversion factor turn out to be c2?

  2 6

  Enlarging a number by "squaring" it is an ancient procedure. A garden that has four paving slabs on one edge, and four on the other, doesn't have eight stones in it. It has 16.

  The convenient shorthand that summarizes this action of building up a "square"—of multiplying a number by itself—went through almost the same range of permutations as did the Western typography for the equals sign. But why should it appear in physics equations? The story of how an equation with a "squared" in it came to be plucked from all other possibilities for representing the energy of a moving object takes us back to France once more—to the early 1700s—and the generation halfway between Roemer and Lavoisier.

  In February 1726, the thirty-one-year-old playwright Francois-Marie Arouet was convinced he'd successfully gate-crashed the establishment in France. He'd risen from the provinces to receive grants from the king, acceptance at the homes of noblemen, and one evening was even being dined at the gated home of the Due de Sully. A servant interrupted the meal: there was a gentleman outside to see Arouet.

  He went out and probably had a moment to recognize the carriage of the Chevalier de Rohan, an unpl
easant, yet staggeringly rich man whom he'd mocked in public when they'd recently attended a play at the Comedie Francaise. Then de Rohan's bodyguards got to work, beating Arouet while de Rohan watched, delighted, from inside his carriage, "supervising the workers," as he later described it. Somehow, Arouet managed to get back inside the gates, and into de Sully's home. But instead of sympathy or even outrage, Arouet encountered only laughter. De Sully and his friends were amused: a preposterous wordsmith had been put in his place by someone who really mattered. Arouet vowed to avenge himself; he would challenge de Rohan to a duel, and kill him.

  That was getting too serious. De Rohan's family had a word with the authorities; there was a police hunt; Arouet was arrested, then put in the Bastille.

  When he finally got out he crossed the Channel, falling in love with England, and especially—estate agents take note—with the bucolic wonderland of Wandsworth, far from the grime of the busy city. He was exhilarated to find that there was a new concept in the air, the works of Newton, which represented what could be the opposite of the ancient, locked-in aristocratic system he'd known in France.

  Newton had created a system of laws that seemed to detail, with superb accuracy, how every part of our universe moved about. The planets swung through space at a rate and in directions that Newton's laws described; a cannonball fired in the air would land exactly where Newton's calculations of its trajectory showed that it would land.

  It really was as if we were living inside a vast windup clock, and all the laws Newton had seen were simply the gears and cogs that made it work. But if we could demand a rational explanation of the grand universe beyond our planet, Arouet wondered, shouldn't we be able to demand the same down here on earth? France had a king who demanded obedience, on the grounds that he was God's regent on earth. Aristocrats got authority from the king, and it was impious to question this. But what if the same analysis used in science by Newton could be used to reveal the role of money or vanity or other hidden forces in the political world as well?