A Brief Guide to the Great Equations (25 page)

Here the famous equation – energy is equal to the mass times the speed of light squared – appears with the corrective factor ß, which takes into account the effect when the body is in motion. That is, let’s take a piece of matter such as an electron. At rest, every electron has the same mass; it’s as if nature stamped that mass on each electron when it was created. Whenever that electron is weighed in its own reference frame, it always has that mass. Now suppose we look at it from another reference frame, in which the electron is moving. If
E
=
mc
²
, and
c
is a constant, then
m
and
e
have to vary in exactly the same way as the energy increases. The electron’s inertial mass – its mass from its own rest frame – does not change. But its mass as you measure it in the laboratory, where you see it as moving, changes. And ß, the compensation factor, is the transformation that tells you what to multiply by to get the rest mass. Leaving out that compensating factor gives you what he calls in a footnote the ‘simplifying stipulation μ
V
²
= ε
₀
.’
²²

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 μ
c
²
.’
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:
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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.’
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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
=
mc
²
. ‘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.’
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