E=mc2 (8 page)
Authors: David Bodanis
For Voltaire that was enough. Newton had spoken, and who was he to argue with Newton, and anyway it seemed such a magnificent vision—and it was backed by such distressingly complicated geometry and calculus— that it was wisest just to nod in confirmation and accept it. But du Châtelet spent a long time in her room with the Watteau paintings, and then at the candle-edged writing table, working through Leibniz's contrary arguments for herself.
Along with various abstract geometric arguments, Leibniz had also focused on the way that Newton's approach left gaps in the world. Diplomats can be sarcastic. He wrote: "According to [Newton's] doctrine, God Almighty wants to wind up his watch from time to time: otherwise it would cease to move. He had not, it seems, sufficient foresight to make it a perpetual motion."
It turned out that concentrating on measurements of energy as being mv
2
avoided this problem. The mv
2
of a cart going due west might be, say, 100 units of energy, and the mv
2
of a second cart going on a collision course due east might be another 100 units. For Newton the two hits canceled each other out, but for Leibniz they added up. When the two carts hit, all the energy they carried remained busily in existence, sending metal parts bouncing and rebounding, heating up the wagon wheels, generally creating an ongoing, reverberating jangle.
In this view of Leibniz's, nothing is lost. The world runs itself; there are no holes or sluicegates where causality and energy rushes away, so that only God would be able to pour them back in. We're alone. God might have been needed at the very beginning, but no longer.
Du Châtelet found some attraction in this analysis, but also recognized why it had languished in the decades since Leibniz had proposed it. This view was too vague; matching Leibniz's personal biases, but without enough objective proof. It was also, as Voltaire got great satisfaction showing in his novel
Candide,
a strangely passive view; suggesting that no fundamental improvements to our worldly condition could be made.
Du Châtelet was known for being burstingly quick in conversation, but at Versailles that had been because she was surrounded by fools, while at Cirey that was the only way to get a word in with Voltaire. When it came to her original work, she was much more methodical, taking her time. After going through the first arguments by Leibniz, and then the standard critiques against them, she—and various specialists she brought in to help—didn't leave it there, but started looking wider, for some practical evidence that would help her make a
choice.
To Voltaire she was clearly "wasting" her time, but for du Châtelet it was one of the peak moments of her life: the research machine she had established at Cirey was finally being used to its full capacity.
She and her colleagues found the decisive evidence in the recent experiments of Willem 'sGravesande, a Dutch researcher who'd been letting weights plummet onto a soft clay floor. If the simple E=mv
1
was true, then a weight going twice as fast as an earlier one would sink in twice as deeply. One going three times as fast would sink three times as deep. But that's not what 'sGravesande found. If a small brass sphere was sent down twice as fast as before, it pushed
four
times as far into the clay. If it was flung down three times as fast, it sank
nine
times as far into the clay.
Which is just what thinking of E=mv
2
would predict. Two squared is four. Three squared is nine. The equation's operation really did seem, in some strange way, fundamental to nature.
'sGravesande had a solid result but wasn't enough of a theoretician to put it all together. Leibniz was a top theoretician but had lacked this detailed experimental finding—his opting for mv
2
had been a bit of a guess. Du Châtelet's work on this topic bridged the gap. She deepened Leibniz's theory, and then embedded the Dutch results within it. Now, finally, there was a strong justification for viewing mv
2
as a fruitful definition of energy.
Her publications had a great effect. Du Châtelet had always been a clear writer, and it helped that Cirey was looked up to as one of the few truly independent research centers. Most English-speaking scientists automatically took Newton's side, while German-speaking ones tended to be just as dogmatically for Leibniz. France had always been the crucial swing vote in the middle, and Du Châtelet's voice was key in finally tilting the debate.
After publishing her work she paused—to take care of her family's finances and to consider what research topic to do next. There were travels with Voltaire, and she was amused that the new generation of courtiers at Versailles had no idea that she was one of the leading interpreters of modern physics in Europe, or that in her spare time she had published original translations of Aristotle and Virgil. Occasionally it would slip, when she did a burst of probability calculations for the gaming table.
Time passed, and they went back to Cirey. The lime trees were growing ("in this, our delightful retreat," as she wrote), and she had even let Voltaire have his vegetable garden. And then, as she hurriedly wrote in a letter to a friend
3 April, 1749
Château de Cirey
I am pregnant and you can imagine . . . how much I fear for my health, even for my life . . . giving birth at the age of forty.
It was the one thing she couldn't control. She'd had children shortly after her marriage, but she had been twenty years younger, and even then it had been dangerous. Being this much older, survival was not very likely. Doctors of the time had no awareness that they should wash their hands or instruments. There were no antibiotics to control the inevitable infection; nothing like oxytocin, which can control uterine bleeding. She didn't rage at the clear incompetence of her era's doctors; she just said to Voltaire that it was sad leaving before she was ready. The length of time before her was very clear: the labor was expected in September. She'd always worked long hours; now she sped up, the candles at the desk where she wrote sometimes burning till dawn.
On September 1, 1749, she wrote to the director of the king's library, stating that he would find in the accompanying package the now complete draft of a major commentary she was doing on Newton. Three days later, the birth began; she survived that, but infection set in, and within a week she died.
Voltaire was beside himself: "I have lost the half of myself—a soul for which mine was made."
In time the focus on energy as being proportional to mv
2
began to seem second nature to physicists. Voltaire's polemical skills, passing on the legacy of his lover, helped give it an even stronger boost. In the following century, Faraday and others used mv
2
—this quantity that might transform but never totally disappeared— as they built up their visions of the conservation of all energy. Du Châtelet's analysis and writing had been an indispensable step, though in time her role came to be forgotten; partly because each new generation of scientists tends to be generally neglectful of their past; partly, perhaps, because it was unsettling to hear that a woman could have directed such a large research effort and helped shape the course of subsequent thought.
The big question, though, is
why.
Why
is
squaring the velocity of what you measure such an accurate way to describe what happens in nature?
One reason is that the very geometry of our world often produces squared numbers. When you move twice as close toward a reading lamp, the light on the page you're reading doesn't simply get twice as strong. Just as with the 'sGravesande experiment, the light's intensity increases
four
times.
When you are at the outer distance, the light from the lamp is spread over a larger area. When you go closer, that same amount of light gets concentrated on a much smaller area.
The interesting thing is that almost
anything
that steadily accumulates will turn out to grow in terms of simple squared numbers. If you accelerate on a road from 20 mph to 80 mph, your speed has gone up by four times. But it won't take merely four times as long to stop if you apply brakes and they lock. Your accumulated energy will have gone up by the square of four, which is sixteen times. That's how much longer your skid will be.
Imagine that skid hooked up to some sort of energy collector. A car that's racing along at four times another one's speed, really will generate—really does carry along—sixteen times as much energy. If someone tried to measure energy as simply equal to mv
1
, they'd miss all this. Only by concentrating on mv
2
do these important aspects come out.
Over time, physicists became used to multiplying an object's mass by the square of its velocity (mv
2
) to come up with such a useful indicator of its energy. If the velocity of a ball or rock was 100 mph, then they knew that the energy it carried would be proportional to its mass times 100 squared. If the velocity is raised as high as it could go, to 670 million mph, it's almost as if the ultimate energy an object will contain should be revealed when you look at its mass times c squared, or its mc
2
. This isn't a proof, of course, but it seemed so natural, so "fitting," that when the expression mc
2
did suddenly appear within Einstein's more detailed calculations, it helped make more plausible his startling conclusion that the seemingly separate domains of energy and mass could be connected, and that the symbol "c"—the speed of light—was the bridge. (For the reader interested in Einstein's actual derivations, the Web site for this book, davidbodanis.com, goes through some of his reasoning.)
The c
2
is crucial in saying how this link operates. If our universe were created differently—if c
2
were a low value—then a small amount of mass would only be transformed into an equally small puff of energy. But in our real universe, and viewed from the small, ponderously rotating planet to which we're consigned, c
2
is a huge number. In units of mph, c is 670 million, and so c
2
is 448,900,000,000,000,000. Visualize the equals sign in the equation as a tunnel or bridge. A very little mass gets enormously magnified whenever it travels through the equation and emerges on the side of energy.
This means that mass is simply the ultimate type of condensed or concentrated energy. Energy is the reverse: it is what billows out as an alternate form of mass under the right circumstances. As an analogy, think of the way that a few wooden twigs going up in flames can produce a great volume of billowing smoke. To someone who'd never seen fire, it would be startling that all that smoke was "waiting" inside the wood. The equation shows that any form of mass can, in theory, be manipulated to expand outward in an analogous way. It also says this will happen far more powerfully than what you would get by simple chemical burning—there is a much greater "expansion." That enormous conversion factor of 448,900,000,000,000,000 is how much any mass gets magnified, if it's ever fully sent across the "=" of the equation.
When Einstein published E=mc
2
in 1905, the equation was at first almost entirely ignored. It simply did not fit in with what most other scientists were doing. The great insights from Faraday and Lavoisier and all the rest were available, but no one else was putting them together this way—hardly anyone even had a hint that one could try.
The world's dominant industries were steel and railways and dyes and agriculture, and that's what ordinary researchers concentrated on. A few universities had specialized labs for more theoretical work, but much of that was in areas that wouldn't have been too surprising to Newton over two centuries before: there were treatises on conventional optics, and sound, and elasticity. There was a little fresh work, on the new and puzzling radio waves, and in areas related to radioactivity, but Einstein was mostly on his own.
We can date to within a month or so the moment when he first saw that E would equal mc
2
. Einstein finished writing his initial paper on relativity by late June 1905, and had the addendum with the equation ready for printing in September, so he probably first realized it some time in July or August. It would likely have been either on one of his walks, or at home after his day job at the patent office. Often his infant son, Hans Albert, was around when he worked, but that wouldn't have been a problem. Visitors recount Einstein contentedly working in the living room of his small apartment, while rocking his one-year-old's bassinet with his free hand, humming or singing to him as needed.
What guided Einstein was that, in his mid-twenties, he found the unknown intriguing. He felt compelled to comprehend what might have been intended for our universe by The Old One (as he referred to his notion of God).
"We are in the position," Einstein explained later, "of a little child entering a huge library, whose walls are covered to the ceiling with books in many different languages. The child knows that someone must have written those books. It does not know who or how. It does not understand the languages in which they are written. The child notes a definite plan in the arrangement of the books, a mysterious order, which it does not comprehend but only dimly suspects."
When the chance came to reach through the gloom, and pluck out The Old One's book that had the shimmering equation E=mc
2
written on its pages, Einstein had been willing to take it.
The reasoning Einstein followed to come up with his extraordinary observation—that mass and energy are one—had begun with the seemingly irrelevant observation that no one could ever catch up with light. But that led, as our space shuttle example suggested, to the insight that energy pouring into a moving object could end up making an outside observer see its mass swell. The argument could also apply in reverse: under the right circumstances an object should be able to pour out energy, generating it from its own mass.
Starting in the 1890s, a few years before Einstein wrote out his equation, a number of investigators had actually seen hints of how this might occur. Several metal-streaked ores that had been brought back from the Congo and Czechoslovakia and other places were found, in laboratories in Paris and Montreal and elsewhere, to be spraying out some sort of mysterious energy beams. If the pebbles were used up as they did this, it wouldn't have been too surprising—one could think that the process was some sort of ordinary burning. But by the best measurements of the time, the energy beams seemed to be pouring out without the pebbles changing in any way.
Marie Curie was one of their first investigators, and indeed in 1898 coined the word
radioactivity for
this active spurting out of radiation. Yet even she, at first, had no understanding that these metals achieved their power by sucking immeasurably tiny portions of their mass out of existence, and switching that mass into the greatly magnified form of sprayed energy. The amounts seemed beyond credibility: a palm-sized chunk of these ores could spray out many trillions of high-speed alpha particles every second, and repeat this for hours and weeks and months, without any loss of weight that anyone could measure.
Later, after Einstein was famous, he met Curie several times, but he never understood her—after one hiking trip he described her as being cold as a herring and constantly complaining. In fact, she had a passionate nature and was deeply in love with an elegant French scientist who was married to someone else. The reason she complained on the hiking trip may have been because she was slowly dying of cancer. Radium was one of these scarcely understood new metals, and Curie had been working with it for years.
The minute traces of radium powder, which she had carried unknowingly on her blouse and hands as she walked across the muddy cobblestones of 1890s Paris and later, had been pouring out energy in accord with the then-unsuspected equation, barely shrinking at all, for thousands of years. They had been spray-releasing part of themselves without getting used up back when they were deep underground in the Belgian mines in the Congo; they continued through her years of experiments, ultimately giving her this killing cancer. More than seventy years later, the dust would still be alive and could squirt out poisonous radiation onto any archivists who were examining her office ledger, or even the cookbooks at her home.
The amount of dust Curie had scattered was measured in millionths of an ounce. But that had been enough, in accord with Einstein's equation, for the radioactive dust to slam into the DNA in her bones, producing the leukemia of which she died; to slam upward, only a fraction more feebly, into the detecting Geiger counters of any such startled archivists so many decades later.
Einstein's equation showed how large the result could be. To work it out for any chunk of mass, take the great speed of light and square that to get an even more immense number. Then, multiply that by the amount of mass you're looking at, and that's how much energy, exactly, the mass will be able to pour out.
It's easy to miss how powerful that idea is. For E=mc
2
says nothing about what sort of mass can fit into the equation! Under the proper circumstances,
any
substance can have its mass exploded outward as energy. This is the power that's around us, encased within the most ordinary rocks and plants and streams. A single page of this book, weighing only a few grams, seems to be just an innocuous, stable mix of cellulose fibers and ink. But if that ink and cellulose could ever be shifted into the form of pure energy, there would be a roaring eruption, greater than that of a large power station exploding. It's easier to access that power in uranium than in ordinary paper—as we'll see later—but that's simply a limitation of our current technology.
The greater the mass being transformed, the more fearsome the power released. Put a single pound of mass into the "m" slot, and after multiplying by the vast 448,900,000,000,000,000 value of c
2
, the equation promises that, in principle, you could get over 10 billion kilowatt hours of energy. This
is
comparable to a huge power station. That's how a small atomic bomb—with a core small enough to fit in your cupped hands—could heave out enough energy to rip open streets and buried fuel lines; to shatter street after street of brick buildings; to tear open the bodies of tens of thousands of soldiers and children and teachers and bus drivers.
A uranium bomb works when less than 1 percent of the mass inside it gets turned into energy. An even larger amount of matter, compressed into a floating star, can warm a planet for billions of years, just by seemingly squeezing part of itself out of existence, and turning those fragments of once-substantial matter into glowing energy.
In 1905, when Einstein first wrote out his equation, he was so isolated that he prepared the main relativity article without footnotes. That's almost unheard of in science. The one acknowledgment Einstein did put in was to his loyal friend Michele Besso, a thirty-something mechanical engineer, working at the patent office, who happened to be the author's friend. Even in 1905 physicists complained of being overburdened. Einstein's articles appeared in a distinguished journal—he'd been keen enough on his career to stay connected by submitting review articles—but one after another, the physicists turning through the journal either skimmed or just ignored this exceptional misfit of an article.
At one point Einstein tried applying for a junior teaching position at the university in Bern, as a way out of the patent office. He sent off the relativity article he was so proud of, along with others he'd written. He was rejected. A little later he applied to a high school, again offering his services as a teacher. The equation was sealed in the envelope with the rest of his application forms. There were twenty-one applicants, and three got called in for interviews. Einstein wasn't one of them.
In time a few scientists did begin to hear of his work, and then jealousy set in. Henri Poincare was one of the glories of Third Republic France, and, along with David Hilbert in Germany, one of the greatest mathematicians in the world. As a young man Poincare had written up the first ideas behind what later became chaos theory; as a student, the story goes, he'd once seen an elderly woman on a street corner knitting, and then, thinking about the geometry of her knitting needles as he walked along the street, he'd hurried back and told her that there was another way she could have done it: he'd independently come up with purling.
By now, though, he was in his fifties, and although he could still get some fresh ideas, he increasingly didn't have the energy to develop them. Or maybe it was more than that. Middle-aged scientists often say that the problem isn't a lack of memory, or the ability to think quickly. It's more a fearfulness at stepping into the unknown. For Poincare had once had the chance of coming close
to
what Einstein was doing.
In 1904 he'd been in the large group of disoriented European intellectuals invited to the World's Fair being held in St. Louis. (Max Weber, the German sociologist, was also there, and he was so startled by the raw energy he saw in America—he described Chicago as being "like a man whose skin has been peeled off "—that it helped jolt him out of a depression he'd been suffering for years.) At the fair, Poincare had actually given a lecture on what he'd labeled a "theory of relativity," but that name is misleading for it only skirted around the edges of what Einstein would soon achieve. Possibly if Poincare had been younger he could have pushed it through to come up with the full results Einstein reached the next year, including the striking equation. But after that lecture, and then the exhausting schedule his St. Louis hosts had for him, the elderly mathematician let it slide. The fact that so many French scientists had turned away from Lavoisier's hands-on approach and instead insisted on a sterile overabstraction only made it harder for Poincare to be immersed in practical physics.
By 1906, realizing that this young man in Switzerland had opened up an immense field, Poincare reacted with the coldest of sulks. Instead of looking closer at this equation, which he could have considered a stepchild, and bringing it in to his Paris colleagues for further development, he kept a severe distance; never speaking of it; seldom mentioning Einstein's name.
Other contemporaries did examine Einstein's work more closely, but tended to miss, at first, such key points as why Einstein selected "c" as being so central. They could understand if relativity and the equation had come from some fresh experimental results; if Einstein had built some new-style apparatus in a laboratory to look more closely at what Marie Curie or others were finding, and so had discoveries which no one else did. But what they could not grasp was that he didn't
have
any labs. The "latest findings" he worked with came from scientists who'd died decades or even centuries before. But that didn't matter. Einstein hadn't come up with his ideas by patiently putting together a range of new results. Instead, as we saw, he just spent a long time "dreamily" thinking about light and speed and what was logically possible in our universe and what wasn't. But it only seemed "dreamy" to outsiders who didn't understand him. What he ended up accomplishing was one of the major intellectual achievements of all time.
For centuries after the birth of mathematically guided science around the seventeenth century, humans thought that they had the main lines of the universe described, and that although there were further details to work out, the "commonsense" properties of the world around us could be taken for granted. We lived in a world where objects kept a constant mass as they moved around; where time advanced smoothly, and everyone could agree about where we were in its flow.
Einstein saw that the universe was different from what everyone had thought. It was, he realized, as if God had restricted us to a small playpen—the surface of the Earth—and had even let us think that what we observed from it was all that really occurred. Yet all the while, stretching further out—around us all the time if we were able to see it—was a further domain, where our intuition no longer applied. Only pure thought would allow us to see what happened there.
The fact of energy and mass being interchangeable, as shown in E=mc
2
, is only one of these fuller consequences. There are others as well, and to recognize them, it helps to imagine a world where instead of the uppermost speed limit being the speed of light at 670 million, it is instead an easy 30 mph. What does Einstein's 1905 relativity paper say we'll see?
The first striking thing we would see
if
we entered that world follows from the space shuttle example. Cars would have their ordinary weight when they were waiting patiently at a red light, but once the light changed to green, they would bulk up in mass as they got faster. It would happen to pedestrians and joggers and bicyclists and indeed everything that moved. A schoolchild, who might weigh 100 pounds on her bicycle when waiting at a corner, would bulk up to 230 pounds once she had pedaled up to 27 mph. If she was fast, or had a downhill slope to help and got up to 29.97 mph, she'd soon have a mass of over 2,000 pounds. Her bicycle would swell up just as much. As soon as she stopped pedaling both she and her bicycle would immediately come down to their original, static weight.
