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Ancestors of E = mc2

  • By David Bodanis
  • Posted 10.11.05
  • NOVA

Historian David Bodanis subtitled his best seller E = mc2 "A Biography of the World's Most Famous Equation." Like most biographies, it includes stories of its subject's ancestors—in this case, innovative thinkers whose groundbreaking experiments shaped Einstein's understanding of mass, energy, and light. Their pioneering efforts helped make Einstein's great epiphany possible. With Bodanis's permission, we've adapted excerpts of E = mc2 to assemble snapshots of these ancestors.

"E" is for energy

Introduction equals m c squared

The word energy is surprisingly new and can only be traced in its modern sense to the mid-1800s. It wasn't that people before then had not recognized that there were different powers around—the crackling of static electricity and the billowing gust of wind that snaps out a sail, for example. It's just that they were thought of as unrelated things. There was no overarching notion of "Energy" within which all these diverse events could fit.

One of the men who took a central role in changing this was Michael Faraday. Faraday's work showed a profound link between electricity and magnetism, and helped lead the scientific community to see that every other form of energy was connected. Scientists of the Victorian era came to believe that energy could change its form, but the total amount of energy would always remain precisely the same. The principle was called the law of conservation of energy.

A painted portrait of Michael Faraday.

Michael Faraday didn't attend Oxford, Cambridge, or even what we call secondary school, yet he became one of England's most prominent scientists. Enlarge Photo credit: public domain/Wikimedia

Michael Faraday (1791-1867)

Faraday, the son of a blacksmith, was once a very good apprentice bookbinder. He had no interest, however, in spending his life binding books. When he was 20, a shop visitor offered him tickets to a series of lectures at the Royal Institution. Sir Humphry Davy was speaking on electricity and on the hidden powers that must exist behind the surface of our visible universe.

Davy's lectures captivated Faraday. He crafted an impressive book from his lecture notes, complete with drawings of the demonstration apparatus, and presented it to Davy. Not long after, Davy hired Faraday as a lab assistant.

For years, Faraday's new position was less than ideal. Sometimes Davy behaved as a warm mentor, but at other times he would seem angry and push him away. Faraday's prospects of becoming a great scientist in his own right changed, though, when Davy asked him to investigate an extraordinary finding out of Denmark.

Until then, everyone knew that electricity and magnetism were as unrelated as any two forces could be. Electricity was the crackling and hissing stuff that came from batteries. Magnetism was different, an invisible force that made navigators' needles tug forward. Yet a lecturer in Copenhagen had now found that if you switched on the current in an electric wire, any compass needle put on top of the wire would turn slightly to the side. Davy asked Faraday to work on why this might occur.

The discovery that sparked a revolution

In the late summer of 1821, Faraday designed a landmark experiment. He imagined that a whirling tornado of invisible circular lines swirled around a magnet. If he were right, then a loosely dangling wire could be tugged along, caught in those mystical circles like a small boat getting caught up in a whirlpool. He propped up a magnet next to a dangling copper wire. When he connected a battery to the wire, he had the discovery of the century.

A copper wire hanging over a magnet that rests on a base.

In Faraday's apparatus, a copper wire hangs with one end near a magnet in a dish of mercury. When charged with electricity, the wire revolves around the magnet. The simple experiment unites electricity, magnetism, and motion. Enlarge Photo credit: © DK Limited/Corbis

What Faraday invented, in his basement laboratory, was the basis of the electric engine. Ultimately one could attach heavy objects to a similar wire, and they would be tugged along as well. Apart from the countless practical applications, Faraday's work gave science a new concept: electromagnetic rotation.

With Faraday's experiment, the crackling of electricity and the silent force fields of a magnet—and now even the speeding motion of a fast twirling copper wire—were seen as linked. As the amount of electricity went up, the available magnetism would go down. Faraday's invisible whirling lines were the tunnel—the conduit—through which magnetism could pour into electricity and vice versa. The full concept of "Energy" had still not been formed, but Faraday's discovery brought it closer.

A triumph turned sour

It was the high point of Faraday's life—and then Sir Humphry Davy accused him of stealing the whole idea. After a few months Davy backed off, but he never apologized, and he left the charges to dangle. Faraday never spoke out against Davy. But for years after the charges of plagiarism, he stayed warily away from front-line research. Only when Davy died, in 1829, did Faraday resume work on electromagnetism.

Faraday went on to make other important discoveries, including the principle behind the electric transformer and generator, innovations that fueled the Industrial Revolution.

In his day, Faraday was celebrated as a great experimenter, but many elite scientists spurned Faraday's more theoretical notions, particularly his vision that the area around an electromagnetic event is filled with a mysterious "field," and his idea that light itself might be an electromagnetic phenomenon.

A magnet surrounded by iron shavings.

Faraday thought the essence of electromagnetic fields was apparent in the curving patterns that iron filings take when they are sprinkled around a magnet. Enlarge Photo credit: NASA

"=" is for equals

(but it's not as simple as you think)

A good equation is not simply a formula for computation. Nor is it a balance scale confirming that two items you suspected were nearly equal really are the same. Instead, scientists started using the "=" symbol as something of a telescope for new ideas—a device for directing attention to fresh, unsuspected realms. This is how Einstein used the "=" in his 1905 equation. Einstein made a giant intellectual leap when he realized that mass and energy are interchangeable—that with the right "conversion factor" of c2, they can straddle an equal sign.

But where did the seemingly mundane symbol at the heart of Einstein's profound equation come from? It can be traced to an enterprising academic of the 1500s named Robert Recorde.

Birth of the equal sign

Bibles of the 14th century often had text that looked much like telegrams:
IN THE BEGINNING GOD CREATED THE HEAVEN AND THE EARTH
AND THE EARTH WAS WITHOUT FORM AND VOID AND DARKNESS
WAS UPON THE FACE OF THE DEEP

Major typographic symbols were locked in rather quickly once printing began at the end of the 1400s. Texts began to be filled in with the old "?" symbols and the newer "!" marks. Minor symbols took longer.

Through the mid-1500s there was still space for entrepreneurs to set their own mark by establishing minor symbols. In 1543, Robert Recorde, a pioneering mathematics textbook writer in Great Britain, tried to promote the new-style "+" sign, which had achieved some popularity on the Continent. The book he wrote didn't make his fortune, so in the next decade he tried again, this time with a symbol, which probably had roots in old logic texts, that he was sure would take off.

In the best style of advertising hype everywhere, he even tried to give it a unique selling point: "...And to avoide the tediouse repetition of these woordes: is equalle to: I will sette ... a pair of parallels, or ... lines of one lengthe, thus: ====== bicause noe .2. thynges, can be moare equalle..."

It doesn't seem that Recorde gained from his innovation, for it remained in bitter competition with the equally plausible "//" and even with the bizarre "[;" symbol, which the powerful German printing houses were trying to promote. But by Shakespeare's time a generation later Recorde's victory was finally certain.

E=mc2, with various alternatives for a

Just imagine how Einstein's famous equation might have looked if one of these other proposed symbols for "equals" had triumphed. Enlarge Photo credit: © WGBH Educational Foundation

"m" is for mass

For a long time the concept of "mass" had been like the concept of energy before the 19th century. There were a lot of different material substances around, but it was not clear how they related to each other, if at all.

Antoine-Laurent Lavoisier, as much as anyone else, first showed that all the seemingly diverse bits of tree and rock and iron on Earth—all the "mass" there is—really were parts of a single connected whole. Through decades of meticulous experiments, aided by his able wife Marie Anne, Lavoisier proved that the substances that fill our universe can be burned, squeezed, shredded, or hammered to bits, but they won't disappear. The different sorts floating around just combine or recombine. The total amount of mass, however, remains the same. This principle—the law of conservation of mass—was one of the great scientific achievements of the 1700s.

Antoine-Laurent Lavoisier (1743-1794)

Lavoisier was born into an aristocratic Parisian family. As a teenager, he studied botany, astronomy, and mathematics, but it was chemistry, above all, that became his passion. It was an experimental science suited to his character, for Lavoisier had the meticulous nature of an accountant with a soul that could soar.

Painted portrait of Antoine-Laurent Lavoisier.

To discover the law of conservation of mass, it would take a person with a great sense of finicky precision, someone willing to spend time measuring even tiny shifts in weight or size—someone like Antoine-Laurent Lavoisier. Enlarge Photo credit: © Archivo Iconografico, S.A./Corbis

Lavoisier demonstrated his romanticism in 1771 by rescuing the innocent 13-year-old daughter of his boss Jacques Paulze from a forced marriage to an uncouth ogre of a man. His own marriage to Marie Anne turned out to be a good one, despite the difference in age, and despite the fact that the handsome 28-year-old Lavoisier soon shifted back to the stupendously boring accountancy work he did for Paulze, collecting taxes for Louis XVI's government.

Lavoisier kept a vast tax-churning organization in operation, working long hours, six days a week on average, for the next 20 years. Only in his spare time—an hour or two in the morning, and then one full day each week—did he focus on his science. But he called that single day his "jour de bonheur"—his "day of happiness."

Proving that nature is "a closed system"

Lavoisier's devotion to detail allowed him to investigate a fundamental scientific question: Can matter ever be completely destroyed or created? Or, when it comes to the substances around us, is nature "a closed system"? Lavoisier explored, for example, what happens when objects rust. While intuition might suggest that a rusted piece of metal weighs less than a pristine one, Lavoisier took nothing on trust. He built an entirely closed apparatus and set it up in a drawing room of his house.

Aided by Marie Anne, Lavoisier put various substances in the apparatus, sealed it tight, and applied heat or started an actual burn to speed up the rusting. Once everything had cooled down, the pair took out the mangled or rusty or otherwise burned-up metal and weighed it, and also carefully measured how much air was lost. Each time they got the same, unexpected result. What they found, in modern terms, was that a rusted sample does not weigh less. It doesn't even weigh the same. It weighs more.

What was happening? There was the same amount of stuff overall, yet now the oxygen that had been in the gases floating above was no longer in the air. But it had not disappeared. It had simply stuck onto the metal. With his state-of-the-art weighing machine, Lavoisier showed that matter can move around from one form to another, yet it will not burst in and out of existence.

Rusted gears.

Lavoisier not only proved that metal weighs more when it rusts, he also first identified and named the gas involved in the process—oxygen. Enlarge Photo credit: © iStockphoto

A life cut short, but a long legacy

With all of Lavoisier's accurate weighing and chemical analysis, other researchers were able to start tracing how that conservation happened in practice—as with his working out how oxygen molecules cascaded from the air to stick to iron. Lavoisier became known as the father of modern chemistry but only decades after his violent demise.

It was primarily his day job as a tax collector that led Lavoisier to the guillotine at the age of 51. It didn't help, however, that in the years before the French Revolution, Lavoisier had made an enemy of Jean-Paul Marat, a frustrated scientist who became a captain of the Reign of Terror.

Lavoisier was tried, convicted, and guillotined in one day. Legend has it that an appeal to spare his life was rejected by the judge with a curt "The Republic has no need of geniuses." Yet one and a half years after his death, the French government exonerated Lavoisier and sent Marie Anne his confiscated belongings along with a note: "To the widow of Lavoisier, who was falsely convicted."

A painting of the execution of Louis XVI.

While Lavoisier's execution wasn't captured in a period painting, like the death of Louis XVI, his tax collecting for the king led to a similar end. Enlarge Photo credit: © Stefano Bianchetti/Corbis

"C" is for celeritas

E is the vast domain of energies, and m is the material stuff of the universe. But c is simply the speed of light. (Celeritas is Latin for "swiftness.") How can this particular speed—what might seem an arbitrary number—control the link between all the mass and all the energy in the universe?

Before Einstein could have possibly thought of using c, someone had to confirm that light travels at a finite speed. Galileo was the first person to clearly conceive of measuring the speed of light. It took a brash young Danish astronomer named Ole Roemer, however, to accomplish the feat. He did it through one of the tensest scientific showdowns of the 17th century.

Ole Roemer's great challenge

In 1671, when Roemer was just 21 years old, he was recruited from Denmark to work at the new Paris Observatory headed by Jean-Dominique Cassini. Others might have been humbled to meet the great Cassini, a world authority on the planet Jupiter and especially on the orbits of its satellites. But Roemer was cockily proud, enough to challenge Cassini and try making his own name.

Painted portrait of Cassini.

Giovanni Cassini became known as Jean-Dominique Cassini when he reigned at the Paris Observatory. Enlarge Photo credit: MacTutor History of Mathematics Archive, University of St. Andrews, Scotland

Cassini had a problem with the innermost moon of Jupiter, the one called Io. It was supposed to orbit its planet every 42 and a half hours. But it never stuck honestly to schedule. Everyone—even Cassini—assumed that the problem was in how Io traveled. Possibly it was ungainly and wobbled during its orbit. Young Roemer, though, reversed the problem. The question wasn't how Io was moving. It was how Earth was moving in relationship to Jupiter.

Cassini and almost everyone of the day assumed that light traveled as an instantaneous flash, but Roemer supposed that light took some time to travel the great distance from Jupiter. In the summer, if Earth was closer to Jupiter, the light's journey would be shorter, and Io's image would arrive sooner. In the winter, though, if Earth had swung around to the other side of the solar system, it would take a lot longer for Io's signal to reach us.

The day of reckoning

By the late summer of 1676, Roemer had an exact figure for how many extra minutes light took to fly that extra distance when Earth was far from Jupiter. At the public forum of a journal all serious astronomers read, he proclaimed a challenge: Io would appear from behind Jupiter the following November 9 not at 5:27 p.m., as Cassini calculated, but ten minutes later, at 5:37.

A painted portrait of an older Ole Roemer.

While shown here as a proud middle-aged man, Roemer was in his mid-20s when he pitted himself against one of the most powerful astronomers of the era. Enlarge Photo credit: Rundetaarn

On November 9, observatories in France and across Europe had their telescopes ready. 5:27 p.m. arrived. No Io. 5:30 arrrived. Still no Io. 5:35 p.m. And then it appeared, at 5:37 and 49 seconds exactly. And yet Cassini declared he had not been proven wrong! It was so far away, so hard to see exactly, that perhaps those clouds from Jupiter's upper atmosphere were producing a distorting haze.

Roemer had performed an impeccable experiment, with a clear prediction, yet Europe's astronomers still did not accept that light traveled at a finite speed. Cassini's supporters won: the official line remained that the speed of light was just a mystical, unmeasureable figure.

Roemer gave up and went back to Denmark. Only 50 years later did further experiments convince astronomers that he had been right. The value Roemer had estimated for light's speed was close to the actual speed of light, which is about 670,000,000 mph.

It's fast, but what is it?

While the speed of light was pinpointed in the 1600s, the exact nature of light was poorly understood until centuries later. The story picks up in the late 1850s, when an elderly Michael Faraday began to correspond with James Clerk Maxwell, a slender Scot still in his 20s.

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. Maxwell, who probably had the finest mathematical mind of any 19th-century theoretical physicist, extended the idea.

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.

Faraday's much maligned hunch that light was an electromagnetic phenomenon had been correct after all.

"2" is for squared

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, of course. So why did the conversion factor turn out to be c2?

The story of how an equation with a "squared" in it came to be plucked from all other possibilities takes us to France in the early 1700s—to a woman who, in her outspoken brilliance, was out of step with her time.

A portrait of Emilie du Châtelet.

As a teenager, Emilie du Châtelet pored through Descartes' analytic geometry. As a grown woman, she was one of Newton's greatest interpreters. Enlarge Photo credit: public domain/Wikimedia

Emilie du Châtelet (1706-1749)

As a girl, Emilie de Breteuil lived with her family overlooking the Tuileries gardens in Paris, in an apartment with 30 rooms and 17 servants. But although her brothers and sisters turned out as might be expected, Emilie was different, as her father wrote: "My youngest flaunts her mind, and frightens away the suitors."

Despite her father's fears, Emilie had many suitors. At the age of 19, she chose one of the least objectionable courtiers as a husband. He was a wealthy soldier named du Châtelet who would conveniently be on distant campaigns much of the time. It was a pro forma arrangement, and in the custom of the time, her husband accepted her having affairs while he was away.

When she was a 27-year-old mother of three, du Châtelet began perhaps the most passionate affair of her life—a true partnership of heart and mind. Her lover, the writer Voltaire, recounted later, "In the year 1733 I met a young lady who happened to think nearly as I did." She and Voltaire shared deep interests: in political reform, in the fun of fast conversations, and, above all, in advancing science as much as they could.

A portrait of Voltaire.

Voltaire, perhaps the most renowned intellectual of the Enlightenment movement, wrote that du Châtelet had "a soul for which mine was made." Enlarge Photo credit: public domain/Wikimedia

Correcting Voltaire and improving on Newton

Together, du Châtelet and Voltaire turned her husband's château at Cirey, in northeastern France, into a base for scientific research with a library comparable to that of the Academy of Sciences in Paris, as well as the latest laboratory equipment from London.

When they engaged in their teasing, mock battling, it wasn't the case of a widely read man deciding when to let his young lover win. Du Châtelet was the real investigator of the physical world, and the one who decided that there was one key question that had to be turned to now: what is energy?

Most people felt energy was already sufficiently understood. Voltaire had covered the seemingly ordained truths in his own popularizations of Newton: an object's energy is simply the product of its mass times its velocity, or mv1. If a five-pound ball is going 10 mph, it has 50 units of energy. But du Châtelet knew there was a competing, albeit highly theoretical view proposed by Gottfried Leibniz, the great German natural philosopher and mathematician. For Leibniz, the important factor was mv2.

A weighty test

Du Châtelet 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 = mv1 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. It if was flung down three times as fast, it sank nine times as far into the clay.

Du Châtelet deepened Leibniz's theory and then embedded the Dutch results within it. Now, finally, there was a strong justification for viewing mv2 as a fruitful definition of energy.

du Châtelet's geometric drawings from <i>Institutions physiques</i>.

These drawings are from du Châtelet's Institutions physiques, her elaboration on the ideas of Leibniz. She finished a major commentary on Newton just before her death. Enlarge Photo credit: University of Hamburg

End of a partnership

Du Châtelet was one of the leading interpreters of modern physics in Europe as well as a master of mathematics, linguistics, and the art of courtship. But there was one thing she couldn't control. In April of 1749, she wrote to Voltaire, "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." 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.

She survived the birth the next fall, 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."

Einstein's new take on an old formula

Over time, physicists became used to multiplying an object's mass by the square of its velocity (mv2) to come up with 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 mc2.

This isn't a proof, of course, but it seemed so natural, so "fitting," that when the expression mc2 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.

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