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Excerpt: 'E = mc2'
By David Bodanis
It may be the most famous equation on earth, but few people understand the theory and complexity behind the simple set of letters and numbers that make up "E = mc2." David Bodanis explains this cornerstone of science in easy-to-understand language, delving into the biography of the man who arrived at the equation -- Albert Einstein -- the equation's impact, and what each part of it means.
Einstein and the Equation
When Einstein published E=mc2 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 mc2. 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=mc2 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=mc2 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 c2, 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 twentyone 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 Poincar 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 Poincar 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 Poincar 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, Poincar 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 Poincar 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 Poincar to be immersed in practical physics.
By 1906, realizing that this young man in Switzerland had opened up an immense field, Poincar 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=mc2, 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.
At the same time, cars, bicycles, and even pedestrians would undergo another change. Depending on the position from which it was being watched a 12-foot car would undergo distortions such as parts of it appearing smaller (and its position shifted) as it roared toward us. At 29.9 mph parts of it would be tiny. The driver and passengers inside would appear to have shrunk just as much, and, again, as soon as they stopped, would settle back and return to their normal appearance.
As the cars hurried by, we'd not only see them as getting more massive and changing size, we'd also notice that time seemed to be slowing down inside of them. If the driver reached to turn on the CD player, we'd see his hand move in extreme slow motion. Once the player was on and the sound was coming out, we'd hear the sound waves transmitted with painful slowness, transforming even early Michael Jackson warbles into heavy, dirge like chants.
In this view of the universe there is no "true" perspective some sort of traffic helicopter hovering above this odd city from which one can assert that, yes, the cars are undergoing these strange changes, but the bystanders who aren't moving are unchanged and clearly "normal." For why should the bystanders have some favored status, while it's only the moving cars that are changing? In fact, the drivers of the cars, or the schoolgirl on her bicycle, will have no sensation that they're changing. The bicyclist will look around her, and see that her handlebars and her body and her backpack haven't become heavier. Rather, to her it's the people left behind who will seem weird. They'll be the ones whose mass has swelled.
The passengers in the car will agree. Their CD player is fine, they'll say, and the young Michael Jackson is warbling along as quickly as ever. It's the people outside the car who seem slow, with hotel doormen seeming to lift their arms in laborious heaviness, then puffing out their cheeks like stately deep-sea fish whenever they blow a whistle to hail a taxi.
These effects are summarized in relativity by saying that when someone watches an object recede away from them, that object will be seen to undergo mass dilation, length changes, and time dilation. The bystanders will see it in the car; the driver of the car, looking back, will see it in the bystanders.
The first time one reads of this, it seems like nonsense. Even Einstein found it hard to accept as with the inexplicable tension he felt in his long talk with Michele Besso, on the summer day when he was still trying to work out these relations. But it's only hard to accept because we never actually interact with each other at speeds close to the 670 million mph of light (and the effects are too slight to notice at our ordinary speeds). Think, for example, of a portable music player at a picnic. To someone standing next to it, it's loud. To someone who walks a few hundred yards away, the music is soft. We accept that there's no answer about how loud it "really" is. But that's simply because we're capable of walking quickly enough to cover that few hundred yards in a brief time. To an ant or some smaller creature, one that took many generations to migrate far enough from the music player to detect any change in its volume, our view that music can appear to be at different volumes to different observers would seem crazy.
The Web site gives the details of how physicists show that all this must follow from such simple observations as light's constancy. But there are a number of ordinary objects around us, which always do work at the high speeds where these effects become apparent. The electrons that shoot from the back of traditional TV sets to the screen at the front, for example, travel so fast that they really will respond to us as if they've grown in mass as they travel. Engineers have to take that into account when they design the magnets that focus the screen's image. If they didn't we would see a blur. The Global Positioning System (GPS) of navigational satellites, which fly overhead and beam down location signals for cars and jets and hikers, are also traveling so fast that from our perspective, time on board them seems to be slow. The circuits in the handheld GPS location devices we use to locate our positions, or in the larger GPS devices that banks use to synchronize payments, are programmed to correct for this in exact accord with the equations Einstein worked out in 1905.
Einstein never especially liked the label relativity for what he'd created. He thought it gave the wrong impression, suggesting that anything goes: that no exact results any longer occur. That's not so. The predictions are precise. The label is also misleading because all Einstein's equations are cohesive, and exactly linked up. Although each of us might view things in the universe differently, there will be enough synchronization where these different views join to ensure that it all fits. The old notions that mass never changes and that time flows at the same rate for everyone made sense when people only noticed the ordinary, slow-moving objects around them. In the true wider universe, however, they're not correct but there are exact laws to explain how they change.
This is an achievement that has occurred very few times in history. Imagine being able to make a shimmering crystalline model, small enough to hold in your closed fist. Now open your hand and see the entire universe soar out; glowing into full existence. Newton was the first person to have done that, back in the 1600s: conceiving a complete system of the world, that could be described in but a handful of equations, yet also contained the rules for how to move out from the summary and go on to creating the full world. Einstein was the next.
Just to make the bond more impressive, both Einstein and Newton achieved much of their work in impossibly brief periods in their mid-twenties. For Newton, back at his mother's Lincolnshire farm after his university had been closed because of the Plague, there were about eighteen months in which he did fundamental work on developing calculus, conceiving the law of universal gravitation, as well as working on key concepts for a mechanics that would apply throughout the universe. For Einstein, in a period of under eight months in 1905 and while still putting in full days at the patent office, Monday through Saturday there was his first theory of relativity, and E=mc2, as well as his work that helped lay the path for lasers, computer chips, key aspects of the modern pharmaceutical and bio-engineering industry, and all Internet switching devices. He really was, as Newton described himself when similarly in his mid-twenties, "in the prime of my age for invention." In each area, Einstein pushed beyond what was known; he unified fields that had remained separate, questioning assumptions that everyone until then had simply accepted.
The few researchers around 1905 who had uncovered a small part of what he later deduced had no chance of matching him. Poincar got closer than almost anyone else, but when it came to breaking our usual assumptions about time's flow or the nature of simultaneity, he backed off, unable to consider the consequences of such a new view.
Why was Einstein so much more successful? It's tempting to say it was just a matter of being brighter than everyone else. But several of Einstein's Bern friends were highly intelligent, while someone like Poincar would have been off the scale on any IQ test. Thorstein Veblen once wrote a curious little essay that I think gets at a deeper reason. Suppose, Veblen began, a young boy learns that everything in the Bible is true. He then goes to a secular high school, or university, and is told that's wrong. "What you learned at your mother's knee is entirely false. What we teach you here, however, will be entirely true." Some students would say, Oh, fine, I'll accept that. But others will be more suspicious. They'd been fooled once before, taking on faith an entire traditional world. They're not going to be fooled again. They would learn what was on offer, but always hold it critically, as just one possibility among others. Einstein was Jewish, and even though his immediate family wasn't observant, this meant he was immersed in a culture with different views about personal responsibility, justice, and belief in authority than the standard German and Swiss consensus.
There's more, though. When Einstein was a little boy, he was fascinated with how magnets worked. But instead of being teased about it by his parents, they accepted his interest. How did magnets work? There had to be a reason, and that reason had to be based on another reason, and maybe if you traced it all the way, you'd reach . . . what would you reach?
At one time, in the Einstein household, there had been a very clear answer to what would ultimately be reached. When his grandparents had been growing up, most Jews in Germany were still close to traditional Orthodoxy. It was a world suffused by the Bible, as well as by the crisply rational accumulated analysis of the Talmud. What counted was to push through to the very edge of what was knowable, and comprehend the deepest patterns God had decreed for our world. Einstein had gone through an intense religious period when he was approaching his teens, though by the time he was at the Aarau high school that literal belief was gone. Yet the desire to see the deepest underpinnings was still there, as was the trust that you would find something magnificent waiting if you made it that far. There was a waiting "slot": things could be clarified, and in a comprehensible, rational way. At one time the slot had been filled in by religion. It could easily enough be extended now to science. Einstein had great confidence that the answers were waiting to be found.
It also helped that Einstein had the space to explore his ideas. The patent job meant that he didn't have to churn out academic papers ("a temptation to superficiality," Einstein wrote, "which only strong characters can resist"), but rather he could work on his ideas for as long as it took. Most of all, his family trusted him, which is a great boost to confidence, and they also encouraged a playful, distancing tone. It's just what's needed for "stepping back" from ordinary assumptions, and imagining such oddities as a space shuttle pushed up against a barrier at the speed of light, or someone chasing toward a skedaddling beam of light.
His sister, Maja, later gave a hint of this gently self-teasing tone. When Einstein got in a temper as a little child, she recounted, he sometimes threw things at her. Once it was a large bowling ball; another time he used a child's hoe "to try to knock a hole" in her head. "This should suffice," she commented, "to show that it takes a sound skull to be the sister of an intellectual." When she described the high school Greek teacher who complained that nothing would ever become of her brother, she added: "And in fact Albert Einstein never did attain a professorship of Greek grammar."
To crank it all forward, there need to be driving tensions, and these Einstein had aplenty. There was the failure of being in his mid-twenties, isolated from other serious scientists, when university friends were already making careers for themselves. There was also thunderous guilt from seeing the difficulties his father was having in his own business career. Einstein had grown up with his father fairly prosperous in the electrical contracting business in Munich, but when Einstein was a teenager, possibly because key contracts stopped being given to Jewish firms, his father had moved the family to Italy to set up again. In the move, and in a series of near-successes that never quite made it, his father was exhausted in paying back loans to a brother-in-law, the constantly nagging Uncle Rudolf "The Rich" (as Einstein mockingly called him). It wrecked his father's health; yet through it all the family had insisted on finding the money to pay for Einstein to study. ("He is oppressed by the thought that he is a burden on us, people of modest means," as his father had remarked in the 1901 letter.) There was a huge obligation for Einstein to show he had been worth it after that.
Eventually a few other physicists did begin to pay attention to Einstein, sometimes visiting Bern to talk over the equation and other results. It was just what Einstein and Besso had hoped for, but it also meant that they started being pulled apart. For Einstein was gradually going beyond the ideas his best friend could follow. Although Besso was bright, he'd chosen a life in industry. ("I prodded him very much to become a [university teacher], but I doubt . . . he'll do it. He simply doesn't want to.") Besso couldn't follow the next level.
Besso adored his younger friend, and had gone out of his way to help him back when Einstein was still a student. He even tried, hard, in their evenings sharing Gruyre and sausages and tea, to keep up with the further ideas Einstein was seeing now. Einstein himself was kind about the growing distance from his friends. He never declared to Besso that he was no longer interested in him. They continued country walks, stops for a drink, musical evenings, and practical jokes with the others. But it's a bit like two old school friends breaking off once both have started moving separate ways at university, or in their first jobs afterward. Each one would really like things not to be like that, but everything they care about now is pulling them apart. They can talk about the old days when they're together, but the enthusiasm is forced, even though neither of them wants to admit it. A similar distancing happened with Einstein's wife, Mileva. She'd been a physics student with him, and very bright. Men in the sciences rarely marry fellow specialists and Einstein was almost smug to his college friends about how lucky he'd been. His first letters to her had started neutrally:
Zurich, Wednesday [16 February 1898] I have to tell you what material we covered. . . . Hurwitz lectured on differential equations (exclusive of partial ones), also on Fourier series. . . .
But the relationship developed, as extracts from a series of letters written in August and September 1900 show:
Once again a few lazy and dull days flitted past my sleepy eyes, you know, such days on which one gets up late because one cannot think of anything proper to do, then goes out until the room has been made up. . . . Then one hangs around and looks halfheartedly forward to the meal. . . .However things turn out, we are getting the most delightful life in the world. Beautiful work, and together. . . . Be cheerful, dear sweetheart. Kissing you tenderly, your Albert.
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