Mass–energy equivalence

• 1. Introduction

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  Mass-energy equivalence is one of the most consequential ideas in the history of physics, and it can be written in five characters: E equals mc squared. Albert Einstein proposed it on the 21st of November 1905, in a paper with a deceptively plain title: "Does the inertia of a body depend upon its energy content?" The answer, he argued, was yes. Mass and energy are not two separate things. They are two expressions of the same underlying quantity, related by the square of the speed of light. Because light travels at roughly 300 million metres per second, squaring that number produces an almost incomprehensible multiplier. A tiny scrap of matter holds an enormous reservoir of energy. What does that actually mean for atoms, for stars, for nuclear weapons, and for our understanding of gravity itself? And how did an equation written in a modest 1905 paper end up on the cover of Time magazine next to a mushroom cloud?

• 2. Before Einstein

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  Isaac Newton, in "Query 30" of his Opticks in 1717, asked whether light and matter might be convertible into one another. It was a speculation, not a proof. Emanuel Swedenborg, the Swedish scientist and theologian, theorised in his Principia of 1734 that all matter is ultimately composed of dimensionless points of pure and total motion. In 1873, Russian physicist Nikolay Umov pointed out a proportional relationship between mass and energy for ether. English engineer Samuel Tolver Preston in 1875 and Italian industrialist Olinto De Pretto in 1903 each imagined a universe filled with ether particles moving at the speed of light, each carrying a kinetic energy that suggested a mass-energy relation. None of them had the physical framework to make the idea rigorous. The French polymath Henri Poincaré came closer. In 1900, he associated electromagnetic radiation energy with a fictitious fluid possessing momentum and mass, an attempt to save the center-of-mass theorem in Lorentz's theory. His treatment, however, led to what he himself described as radiation paradoxes. In 1905, also independently of Einstein, the French polymath Gustave Le Bon speculated that atoms could release large amounts of latent energy. The pieces were scattered across half a century of physics. What was missing was a general principle grounded in the symmetries of space and time, and Einstein was the first to supply it.

• 3. The 1905 Derivation

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  Einstein did not write out the familiar formula directly in his 1905 paper. Instead, he stated in German that if a body gives off energy in the form of radiation, its mass diminishes by a corresponding amount. He reached this conclusion through a thought experiment involving a body emitting two pulses of light in opposite directions. Examining the same emission from a moving reference frame, he found that the body must lose mass equal to the total energy emitted divided by the square of the speed of light. He concluded that the mass of a body is a measure of its energy content. The derivation was not immediately accepted without challenge. German theoretical physicist Max Planck argued in 1907 that Einstein's proof was only valid to a first approximation. American physicist Herbert Ives in 1952 and Israeli physicist Max Jammer in 1961 asserted that the derivation was circular. Philosophers John Stachel and Roberto Torretti argued Ives was wrong and that Einstein's derivation was sound. American physics writer Hans Ohanian, writing in 2008, sided with Stachel and Torretti against Ives, but argued Einstein's derivation was flawed for entirely different reasons. Despite this scholarly debate, Einstein's conclusion was not in dispute. In 1906 he himself acknowledged the earlier work of Poincaré on related questions, and in the same year he showed that the transport of inertia accompanying radiation emission and absorption resolves the paradox Poincaré had identified. Einstein elaborated the meaning in a 1946 essay, writing that the principle of conservation of mass proved inadequate in the face of special relativity, and was merged with the energy conservation principle, which, as he put it, now held the field alone.

• 4. What Rest Mass Really Means

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  Rest mass, also called invariant mass, is a fundamental physical property of matter that does not change with velocity. An object at rest possesses an intrinsic energy equal to its mass multiplied by the square of the speed of light, which for one kilogram is on the order of 100 quadrillion joules. That energy is present even when nothing is moving. A water molecule weighs slightly less than the sum of two free hydrogen atoms and one oxygen atom. The difference is the energy released as heat when the molecule formed. A stick of dynamite, by the same logic, weighs slightly more than its fragments after the explosion; the difference accounts for the energy and heat released. If that same stick of dynamite were detonated inside a perfectly sealed, infinitely strong chamber, the total mass of the chamber, its contents, the heat, the light, and the sound would remain unchanged. The mass has not disappeared; it has changed form. This principle applies to atomic nuclei as well. The mass of an atomic nucleus is less than the combined mass of the protons and neutrons that make it up. That gap is called the mass defect, and it equals the binding energy required to pull the nucleus apart. A spinning ball is more massive than a stationary one, by exactly the energy equivalent of its rotation. The Earth itself is more massive because it rotates, and the rotational energy of the Earth exceeds 10 to the power of 24 joules, which corresponds to more than 10 million kilograms of extra mass.

• 5. Energy, Gravity, and Light

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  Physics recognises two kinds of mass: gravitational mass, which determines how strongly an object generates and responds to gravitational fields, and inertial mass, which measures resistance to acceleration. The weak equivalence principle, already assumed within Newtonian gravity, holds that these are identical. Combined with mass-energy equivalence, this leads to a striking prediction: all forms of energy, not just matter, generate gravity. The test came on the 29th of May 1919, during a solar eclipse. The English astronomer and physicist Arthur Eddington observed that light from stars passing close to the Sun was bent. The Sun's gravity was deflecting the path of light itself, confirming that the energy carried by light behaves as gravitational mass. A second key experiment, the Pound-Rebka experiment, was performed in 1960. A beam of light emitted from the top of a tower was detected at the bottom at a higher frequency than it was emitted. Photons falling through Earth's gravitational field gain energy. Since the energy of a photon is proportional to its frequency, as stated by the Planck-Einstein relation, this result is a direct consequence of mass-energy equivalence applied to light. Photons have no rest mass, but they carry energy and momentum, and in a confined system they contribute to total mass. A sealed container of light would weigh more than an empty one, by an amount equal to the total energy of the trapped light divided by the square of the speed of light.

• 6. Nuclear Energy and the Bomb

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  After the discovery of radioactivity in 1897, physicists noticed that the energy released in radioactive processes is roughly one million times greater than that involved in any known chemical reaction. New Zealand physicist Ernest Rutherford and British radiochemist Frederick Soddy proposed in 1903 that a huge reserve of latent energy is stored within matter. Rutherford went further in 1904, speculating that if the rate of disintegration of radioactive elements could ever be controlled, an enormous amount of energy could be extracted from a small quantity of matter. Yet as late as 1933, Rutherford was reported to have declared that anyone who expected a source of power from transforming the atom was talking moonshine. The picture changed in 1932 with the discovery of the neutron, which made it possible to calculate mass differences for individual atomic nuclei directly. By 1933, the reaction of lithium-7 with protons yielding two alpha particles allowed Einstein's equation to be verified to within plus or minus 0.5 percent. In late 1938, Austrian-Swedish physicist Lise Meitner and British physicist Otto Robert Frisch used Einstein's equation during a winter walk to work out the energetics of what they were calling atomic fission. They used nuclear binding energy values to confirm that splitting a uranium nucleus was energetically possible. After the atomic bombings of Hiroshima and Nagasaki in 1945, the equation became fixed in the public mind as the symbol of nuclear power. It appeared on page 2 of the Smyth Report, the official US government account of the Manhattan Project. By 1946, Time magazine placed Einstein's portrait next to a mushroom cloud bearing the equation on its cover. Einstein himself had cosigned a letter to the US president in 1939 urging research into atomic energy, warning that a bomb was theoretically possible. That letter persuaded President Roosevelt to devote a significant portion of the wartime budget to atomic research. Einstein had no security clearance and his only scientific contribution to the project was an inconsequential analysis of an isotope separation method. Manhattan Project physicist Robert Serber noted that Einstein's theory of relativity is not actually required to describe the fission process, which is a non-relativistic phenomenon at its core.

• 7. Converting All of Matter

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  Nuclear fission converts only a small fraction of mass into energy. In the decay of uranium, about 0.1 percent of the original atom's mass is released. Full conversion of matter into energy is theoretically possible but practically out of reach by all known methods. The most complete conversion would involve annihilating matter with antimatter, releasing all the rest-energy as radiation. Antimatter, however, is vanishingly rare in the universe. CERN estimated in 2011 that producing and storing antimatter requires more than a billion times the energy that would be released when it annihilates. A different pathway, predicted by the Standard Model, involves a process shown by Gerard 't Hooft in which protons and neutrons convert into antielectrons and neutrinos through what is called the weak SU(2) instanton. This process was described by the physicists Alexander Belavin, Alexander Markovich Polyakov, Albert Schwarz, and Yu. S. Tyupkin. Under ordinary conditions it proceeds at an extraordinarily slow rate, but it was later shown to occur rapidly at the extreme temperatures that existed shortly after the Big Bang. Some extensions of the Standard Model also predict that magnetic monopoles could catalyse proton decay through what is called the Callan-Rubakov effect, offering in principle another efficient mass-energy conversion path. Even black holes enter the calculation. Stephen Hawking theorised that matter thrown into a black hole could generate usable heat through what is now called Hawking radiation, though larger black holes radiate less than smaller ones, making only small black holes viable sources of usable power by this method.

• 8. The Equation as an Object

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  According to the Einstein Papers Project at the California Institute of Technology and Hebrew University of Jerusalem, only four known copies of the equation written in Einstein's own hand survive. One is a letter in German addressed to Ludwik Silberstein. That letter, held in Silberstein's archives, was sold at auction on the 21st of May 2021 for 1.2 million dollars by RR Auction of Boston, Massachusetts. Einstein first published the equation using a lowercase c rather than the uppercase V in 1907. In later papers he adopted the notation that is now standard. Original offprints of the 1907 paper from Jahrbuch der Radioaktivite und Elektronik are rare; one was listed for sale at 60,000 dollars. The equation began as a sentence in German, became a formula in a physics paper, and eventually appeared on the cover of the world's most widely read news magazine. Few ideas in science have made that journey.