A Brief Guide to the Great Equations

by Robert Crease

  Later that year, Einstein changed his symbols to use c rather than V for the speed of light. The theory of relativity contains a result of ‘extraordinary theoretical importance’, he says; that ‘the inertial mass and the energy of a physical system appear in it as things of the same kind. With respect to inertia, a mass μ is equivalent to an energy content of magnitude μc2.’23 Over the next few years, Einstein worked out more fully the mass-energy principle and its implications. And in a manuscript on relativity theory of 1912, at the beginning of a discussion of the subject, he writes the above formula using m in place of μ, and a script L (as in the first version) in place of ε then crosses it out and writes E. From then on, he sticks with E and c, and we now have the familiar equation plus the corrective factor where q (sometimes written as v) stands for the velocity:24

  Enter the Atomic Nucleus

  After any great scientific discovery, the question inevitably arises as to why the phenomenon or principle had not been discovered before. The answer is usually complicated, and several factors enter into play. One is that scientists often had bumped into it before, but had either ignored it, or misunderstood it, or incompletely described it. Such was indeed the case with the conversion of mass and energy. Another factor is that the existing scientific knowledge may be structured in a way to discourage people from seeing the phenomenon or principle as possible. That, too, was an element here, for mass and energy were viewed as entirely separate categories of nature, obeying separate laws. Finally, situations where the phenomenon or principle may show up in ways that scientists can easily explore may be rare and the effect tiny. That was also true, for conversions of mass into energy, or vice versa, are rarely observed in ordinary life. As Einstein wrote, ‘It is as though a man who is fabulously rich should never spend or give away a cent; no one could tell how rich he was.’25

  Was the wealthy man spending any money? And if so, where? In his ‘Energy Content’ paper, Einstein expressed himself about the prospect far more cautiously than in his enthusiastic letter to Habicht. ‘Perhaps’, Einstein wrote, ‘it will prove possible to test this theory’ using substances such as radium that emit energy in the form of radiation. But the size of the effect, he remarked in a paper written shortly thereafter, would be ‘immeasurably small’, and cited a calculation by Max Planck that radium’s mass loss would be ‘outside the experimentally accessible range for the time being.’26 Still, Einstein added, ‘it is possible that radioactive processes will be detected in which a significantly higher percentage of the mass of the original atom will be converted into the energy of a variety of radiations than in the case of radium.’

  The discovery of the nucleus in 1911 did little for the moment to open any doors to testing the mass-energy concept, and matters remained unchanged for over two decades. But in 1932, two key developments made the mass-energy concept not only useful but indispensable for explaining aspects of the universe from its smallest to its largest dimensions – from atomic structure to stellar explosions. The first was the discovery of the neutron by British physicist James Chadwick. Physicists now had a good picture of the basic structural elements of the nucleus: protons and neutrons. Where did the energy come from that held them together? The clue was provided by a second key discovery of 1932, when British physicists John Cockroft and Ernest Walton used a new instrument in physics, a particle accelerator, to bombard lithium nuclei with protons, producing a nuclear transformation: the lithium nucleus plus the proton turned into two helium nuclei. Cockroft and Walton were able to measure the masses and energies of the initial states (lithium nuclei and protons) and of the final states (helium nuclei). They discovered a net mass loss, and energy gain – and established that, within experimental error, the mass loss was accounted for by Einstein’s formula. The total inertial mass afterward, that is, was less by an amount equal to the increase in kinetic energy in the reaction divided by the speed of light squared. This was the first confirmation of Einstein’s mass-energy equation, and it quickly became indispensable in atomic physics. The difference between the mass of particles inside and outside the nucleus was known as the ‘packing fraction’, and the total mass difference between all such particles inside and outside is called the binding energy. Meanwhile, physicists were also learning that the energy of starlight came from mass-energy changing transformations in stellar interiors. In the 1930s, the concepts of packing fraction and binding energy made Einstein’s equation a well-used tool of science, from atomic physics to astrophysics.

  Physicists knew that even a small fractional conversion of mass to energy generated a lot more energy than any other kind of process they knew about. Still, the energy generated by any single nucleus – even if all of its binding energy were released – was far too minuscule for any practical purpose. For this reason, for the rest of the decade, nuclear energy seemed a distant, even ridiculous, prospect, the stuff of dreamers and fanatics. Almost to the end of the 1930s, nearly all physicists thought that the prospect of being able to release and control nuclear energy was far-fetched, even crazy. In 1921, Einstein was cornered by a young man proposing to produce a weapon based on E = mc2. ‘Its foolishness is evident at first glance’, Einstein replied.27 In a 1933 interview, physicist Ernest Rutherford called the idea ‘moonshine.’ Einstein compared it to shooting in the dark at scarce birds. And in 1936, Danish physicist Niels Bohr, discussing instances when collisions between particles and nuclei that are so energetic that the nuclei explode, remarked that this would not ‘bring us any nearer to the much discussed problem of releasing nuclear energy for practical purposes.’ Bohr added, ‘Indeed, the more our knowledge of nuclear reactions advances the remoter this goal seems to be.’28

  By then, however, a series of events had already begun to unfold that would transform the world’s appreciation for mass-energy conversions. The scientific and political events of this now-familiar story, with an international cast of characters, moved forward with a swiftness and drama that, even in outline, is still breathtaking well over half a century later.

  Immediately after Chadwick’s discovery of the neutron in 1932, physicists realized that the particle was an excellent tool for studying atomic nuclei. In the mid-1930s, as fascism grew in Europe, Italian physicist Enrico Fermi began bombarding elements of the periodic table with neutrons, going systematically from beginning to end, producing heavier, radioactive versions of each. When he reached the heaviest known element, uranium, he got strange results, and he thought he was creating brand-new, ‘transuranic’ elements.

  German scientists Otto Hahn and Fritz Strassman discovered that Fermi was wrong; adding neutrons to uranium actually produced lighter, already familiar elements. In December 1938 they mailed the news to Lise Meitner, a former co-worker who had fled Nazi Germany for Sweden. With her physicist nephew, Otto Frisch, Meitner realized that the bombardment was in effect splitting the nuclei, which Frisch named ‘fission’ after consulting with a biologist. Frisch and Meitner sent a landmark paper on nuclear fission to Nature, which published it in February 1939 – but by then Frisch had told Niels Bohr, who was about to embark on a boat for the U.S. Bohr and his companion broke the news to U.S. physicists the day they arrived, in mid-January, at the Princeton physics department journal club. The following week Columbia University physicists conducted the first fission experiment on U.S. soil, while word spread around the country like wildfire. Scientists first read about it not in journals but in newspapers. Most realized that fission – a process in which one uranium nucleus, in splitting and releasing neutrons able to split more nuclei – raised the possibility of a chain reaction, with massive numbers of nuclei splitting and releasing energy all at once – and thus of the possibility of a new, particularly terrifying type of bomb. This, just as Europe was on the brink of war.

  In March 1939, Fermi (who meanwhile had fled Fascist Italy for the U.S., first to Columbia University in New York City and then to the University of Chicago) and other physicists began formally speaking to U.S. government
officials about possible military applications. In July, two scientists visited Einstein at his summer home in Peconic, Long Island, to seek his help. ‘I never thought of that!’ Einstein exclaimed after learning of the possibility of a chain reaction. Two weeks later, he signed an urgent letter to President Roosevelt informing him of ‘some recent work’ that ‘leads me to expect that the element uranium may be turned into a new and important source of energy in the immediate future.’ In the last 4 months, the letter continued, the possibility has arisen of using uranium to set up a chain reaction. ‘This new phenomenon would also lead to the construction of bombs, and it is conceivable – though much less certain – that extremely powerful bombs of a new type may thus be constructed.’

  In September 1939, Nazi Germany invaded Poland. In October, Einstein’s letter was presented to Roosevelt. In February 1940, the federal government awarded a $6,000 grant to study the phenomenon, dubbed the Manhattan Project. Several nations that were participants in the growing hostilities, including Germany, the Soviet Union, Japan, and Great Britain, began atomic bomb research. But events moved forward swiftly only in the U.S.

  On December 2, 1942, less than a year after the project began serious work, the world’s first controlled chain reaction took place at the Metallurgical Laboratory in a squash court in the west stands of the University of Chicago’s football field, clinching the project’s possibility (the news was communicated by the improvised code that ‘the Italian navigator [that is, Fermi] has just landed in the new world’). President Roosevelt then approved $400,000 for the project, leading to the construction of a huge isotope separation plant at Oak Ridge, Tennessee, and a plutonium production plant in Hanford, Washington. J. Robert Oppenheimer, the scientific head of the project, found a safe, remote site for the actual construction of the bomb atop a mesa in Los Alamos, New Mexico, and scientists began moving there in March 1943.

  The Manhattan Project culminated in a test explosion at Alamo-gordo, New Mexico, on July 16, 1945. Scientists are used to witnessing new phenomena only in clinical lab conditions, but the Trinity test was different. That cold morning in the desert, Los Alamos scientists crouched down clutching pieces of welder’s glass to protect their eyes. Suddenly a fireball erupted that was brighter than the sun, giving off heat that warmed their faces from 20 miles away. Slowly a white cloud rose tens of thousands of feet high, making some fear that they had unleashed force beyond their control, and reminding Oppenheimer (he said later) of scriptural passages about the apocalypse. It was, Abraham Pais wrote, ‘one of the most spectacular events in the history of the world.’29

  Three weeks later, on August 6, 1945, the first atomic bomb incinerated the Japanese city of Hiroshima. The next day, headlines all over the world revealed the existence of a new and particularly horrible type of bomb that, as The New York Times put it, ‘was the first time that Prof. Albert Einstein’s theory of relativity has been put to practical use outside the laboratory.’30 On August 9, 3 days after the bombing of Hiroshima, another atomic bomb destroyed the city of Nagasaki.

  The equation E = mc2 had not played any direct role in the events leading up to the Manhattan Project, except as a key ingredient of the nuclear physics theory by which fission was understood. The atomic bomb, involving as it did the conversion of matter into energy via fission, was only an example of the equation E = mc2 – and a rare one in the comings and goings of life on earth – not an outgrowth of it. But the equation almost immediately became associated with it, partly thanks to Atomic Energy for Military Purposes, a report written by Princeton physicist Henry D. Smyth, an official of the Manhattan Project.

  The Smyth report, as it was called, was released to the public on August 11, 2 days after the bombing of Nagasaki. ‘A weapon has been developed that is potentially destructive beyond the wildest nightmares of the imagination’, Smyth wrote. It was created ‘not by the devilish inspiration of some warped genius but by the arduous labour of thousands of normal men and women working for the safety of their country.’ The report’s intended audience was ‘engineers and scientific men’ who might be able to ‘explain the potentialities of atomic bombs to their fellow citizens.’ But the book was a crossover hit, becoming a huge popular success and selling over a hundred thousand copies in its first 5 months.31

  Right at the beginning, the Smyth report used E = mc2 as a cornerstone for explaining the weapon. An early implication of relativity theory, it said, was the equivalence between mass and energy.

  To most practical physicists and engineers this appeared a mathematical fiction of no practical importance. Even Einstein would hardly have foreseen the present applications, but as early as 1905 he did clearly state that mass and energy were equivalent and suggested that proof of this equivalence might be found by the study of radioactive substances. He concluded that the amount of energy, E, equivalent to a mass, m, was given by the equation

  E = mc2

  where c is the velocity of light.

  The Smyth report was the mediating document through which nonscientists learned about the Manhattan Project. It more than any other single document made E = mc2 an emblem of atomic energy and weaponry.

  Celebrity Status

  Ethnographers say that when two cultures interact, they do not meet all of a piece but through ‘congeners’ through which certain members of one culture look at, try to understand, and respond to the other. Congeners can include artifacts, rituals, practices, and art; fear, fascination, and exoticism usually play a role. A congener is like a little lens that allows the members of the one culture to approach the other culture in a focused way, to get an introductory grip. A congener is thus more than a symbol or logo of the other culture, but guides and disciplines curiosity and fascination into a first interaction with it.

  The equation E = mc2 served as a congener in this sense, between a public anxious for information about atomic energy and the scientific developments that made it possible. In the process, it grew into even more – a symbol of physics, of science, even of knowledge – to the point where it acquired a mythic status.

  The French intellectual Roland Barthes once wrote an essay on Einstein in which he noted that, while photographs of Einstein often show him next to a blackboard covered with impenetrable symbols and equations, cartoons of him often portray him, chalk in hand, next to a clean blackboard on which he has just written down this particular formula as if it just came to him out of the blue. Barthes observed of the symbolic character of this equation that it restores the image of ‘knowledge reduced to a formula…science entirely contained in a few letters.’ It has become a Gnostic image: ‘the unity of nature, the ideal possibility of a fundamental reduction of the world, the unfastening power of the word, the age-old struggle between a secret and an utterance, the idea that total knowledge can only be discovered all at once, like a lock which suddenly opens after a thousand unsuccessful attempts.’ Barthes’s essay helps explain the transformation of this equation from a scientific tool into a congener.

  Einstein himself began to use the equation in its now-famous, simplified form. In April 1946, the first issue of a new popular magazine, Science Illustrated, appeared with an article written by Einstein entitled ‘E = mc2.’ ‘It is customary’, he wrote, ‘to express the equivalence of mass and energy (though somewhat inexactly) by the formula E = mc2.’32

  Then, on July 1, 1946, less than a year after the explosions over Hiroshima and Nagasaki, and just shy of 41 years after it first appeared in its initial form, E = mc2 made the cover of Time magazine. The issue coincided with an atomic test in the South Pacific. The Time cover juxtaposed a portrait of the now white-haired, 66-year-old Einstein, whom it called a ‘shy, almost saintly, childlike little man’, next to a fiery mushroom cloud rising above the hulks of warships. Red flames at the base gave way to orange and purple on the column, topped by a gray mushroom cap. On it was inscribed a now-in-famous equation: E = mc2. Now it was a celebrity.

  Interlude

  CRAZY IDEAS
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br />   In order to understand this we need some crazy idea. Has anyone a crazy idea?

  – Niels Bohr

  The scientist and science writer Jeremy Bernstein writes that he occasionally fantasizes the following:

  It is the year 1905 and I am a professor of physics at the University of Bern. The phone rings and a person I have never heard of identifies himself as a patent examiner in the Swiss National Patent Office. He says that he has heard I give lectures on electromagnetic theory and that he has developed some ideas which might interest me. ‘What sort of ideas?’ I ask a bit superciliously. He begins discussing some crazy sounding notions about space and time. Rulers contract when they are set in motion; a clock on the equator goes at a slower rate than the identical clock when it is placed at the North Pole; the mass of an electron increases with its velocity; whether or not two events are simultaneous depends on the frame of reference of the observer; and so on. How would I have reacted?1

  Bernstein’s thought experiment highlights an occupational side-effect of writing about science: letters from strangers bearing crazy ideas. Once upon a time these letters arrived in brown envelopes in crabbed handwritten script; today they are emailed with links to flashy Web pages. The usual subjects are astrophysics, cosmology, unification theories, and the overthrow of Western science. Einstein is often invoked, either as emblematic of mainstream science (as the author’s archenemy) or of the misunderstood loner outside it (as the author’s precursor). One or more of the following are generally mentioned: gravity, electromagnetism, and planetary orbits; the most primitive and readily recognizable versions mention also psychic phenomena, astrological signs, herbal remedies, the stock market, baseball scores, and rock lyrics. Many crackpot letters explode into italics, boldface, and CAPITAL LETTERS in a way reminiscent of far-right newsletters and software licensing agreements. Some authors warn of a government or scientific conspiracy to suppress their ideas; others are generously allowing you into the Fold.2