While
Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass, though nearly all previous authors thought that the energy that contributes to mass comes only from electromagnetic fields. Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.
Developments prior to Einstein 's
Opticks, published in 1717, Newton speculated on the equivalence of mass and light. Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist
Isaac Newton in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the
Opticks, where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?" Swedish scientist and theologian
Emanuel Swedenborg, in his
Principia of 1734 theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it. During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various
ether theories. In 1873 the Russian physicist and mathematician
Nikolay Umov pointed out a relation between mass and energy for ether in the form of , where . English engineer
Samuel Tolver Preston in 1875 and the Italian industrialist and
geologist Olinto De Pretto in 1903, following physicist
Georges-Louis Le Sage, imagined that the universe was filled with an
ether of tiny particles that always move at speed . Each of these particles has a kinetic energy of up to a small numerical factor, giving a mass–energy relation. In 1905, independently of Einstein, French polymath
Gustave Le Bon speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative
philosophy of physics.
Electromagnetic mass There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists
J. J. Thomson in 1881 and
Oliver Heaviside in 1889, and
George Frederick Charles Searle in 1897, German physicists
Wilhelm Wien in 1900 and
Max Abraham in 1902, and the Dutch physicist
Hendrik Antoon Lorentz in 1904—to understand how the mass of a charged object depends on the
electrostatic field. This concept was called
electromagnetic mass, and was considered as being dependent on velocity and direction as well. Lorentz in 1904 gave the following expressions for longitudinal and transverse electromagnetic mass: m_{L}=\frac{m_{0}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=\frac{m_{0}}{\sqrt{1-\frac{v^2}{c^2}}} , where m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}} Another way of deriving a type of electromagnetic mass was based on the concept of
radiation pressure. In 1900, French polymath
Henri Poincaré associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass
Einstein: mass–energy equivalence in 1921 Einstein did not write the exact formula in his 1905
Annus Mirabilis paper "Does the Inertia of an object Depend Upon Its Energy Content?"; He has stated that the laws of conservation of energy and conservation of mass are "one and the same". Einstein elaborated in a 1946 essay that "the principle of the conservation of mass… proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy
conservation principle—just as, about 60 years before, the principle of the
conservation of mechanical energy had been combined with the principle of the conservation of heat [thermal energy]. We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone."
Mass–velocity relationship 's own handwriting from 1912 In developing
special relativity, Einstein found that the
kinetic energy of a moving body is E_k = m_0 c^2( \gamma -1 ) = m_0 c^2\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1\right), with the
velocity, the rest mass, and the Lorentz factor. He included the second term on the right to make sure that for small velocities the energy would be the same as in classical mechanics, thus satisfying the
correspondence principle: E_k = \frac{1}{2}m_0 v^2 + \cdots Without this second term, there would be an additional contribution in the energy when the particle is not moving.
Einstein's view on mass Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906. In Einstein's first 1905 paper on , he treated as what would now be called the
rest mass, In modern physics terminology, relativistic energy is used in lieu of relativistic mass and the term "mass" is reserved for the rest mass.
Einstein's 1905 derivation Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles:E_{k}=mc^{2}\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right). Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?", one of his
Annus Mirabilis papers. Here, Einstein used to represent the speed of light in vacuum and to represent the
energy lost by a body in the form of radiation. Einstein used a body emitting two light pulses in opposite directions, having energies of before and after the emission as seen in its rest frame. As seen from a moving frame, becomes and becomes . Einstein obtained, in modern notation:\left(H_{0}-E_{0}\right)-\left(H_{1}-E_{1}\right)=E\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right). He then argued that can only differ from the kinetic energy by an additive constant, which givesK_{0}-K_{1}=E\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right). Neglecting effects higher than third order in after a
Taylor series expansion of the right side of this yields:K_{0}-K_{1}=\frac{E}{c^2}\frac{v^2}{2}. Einstein concluded that the emission reduces the body's mass by , and that the mass of a body is a measure of its energy content. The correctness of Einstein's 1905 derivation of was criticized by German theoretical physicist
Max Planck in 1907, who argued that it is only valid to first approximation. Another criticism was formulated by American physicist
Herbert Ives in 1952 and the Israeli physicist
Max Jammer in 1961, asserting that Einstein's derivation is based on
begging the question. Other scholars, such as American and Chilean
philosophers
John Stachel and
Roberto Torretti, have argued that Ives' criticism was wrong, and that Einstein's derivation was correct. American physics writer
Hans Ohanian, in 2008, agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons.
Relativistic center-of-mass theorem of 1906 Like Poincaré, Einstein concluded in 1906 that the inertia of electromagnetic energy is a necessary condition for the center-of-mass theorem to hold. On this occasion, Einstein referred to Poincaré's 1900 paper and wrote: "Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré2, for the sake of clarity I will not rely on that work." In Einstein's more physical, as opposed to formal or mathematical, point of view, there was no need for fictitious masses. He could avoid the
perpetual motion problem because, on the basis of the mass–energy equivalence, he could show that the transport of inertia that accompanies the emission and absorption of radiation solves the problem. Poincaré's rejection of the principle of action–reaction can be avoided through Einstein's , because mass conservation appears as a special case of the
energy conservation law.
Further developments There were several further developments in the first decade of the twentieth century. In May 1907, Einstein explained that the expression for energy of a moving mass point assumes the simplest form when its expression for the state of rest is chosen to be (where is the mass), which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein used the formula , with being the energy of a system of mass points, to describe the energy and mass increase of that system when the velocity of the differently moving mass points is increased. Max Planck rewrote Einstein's mass–energy relationship as in June 1907, where is the pressure and the volume to express the relation between mass, its latent energy, and thermodynamic energy within the body. Subsequently, in October 1907, this was rewritten as and given a quantum interpretation by German physicist
Johannes Stark, who assumed its validity and correctness. In December 1907, Einstein expressed the equivalence in the form and concluded: "A mass is equivalent, as regards inertia, to a quantity of energy . […] It appears far more natural to consider every inertial mass as a store of energy." American
physical chemists
Gilbert N. Lewis and
Richard C. Tolman used two variations of the formula in 1909: and , with being the relativistic energy (the energy of an object when the object is moving), is the rest energy (the energy when not moving), is the relativistic mass (the rest mass and the extra mass gained when moving), and is the rest mass. The same relations in different notation were used by Lorentz in 1913 and 1914, though he placed the energy on the left-hand side: and , with being the total energy (rest energy plus kinetic energy) of a moving material point, its rest energy, the relativistic mass, and the invariant mass. In 1911, German physicist
Max von Laue gave a more comprehensive proof of from the
stress–energy tensor, which was later generalized by German mathematician
Felix Klein in 1918. Einstein returned to the topic once again after
World War II and this time he wrote in the title of his article intended as an explanation for a general reader by analogy.
Alternative version An alternative version of Einstein's
thought experiment was proposed by American theoretical physicist
Fritz Rohrlich in 1990, who based his reasoning on the
Doppler effect. Like Einstein, he considered a body at rest with mass . If the body is examined in a frame moving with nonrelativistic velocity , it is no longer at rest and in the moving frame it has momentum . Then he supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy . In its rest frame, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum. However, if the same process is considered in a frame that moves with velocity to the left, the pulse moving to the left is
redshifted, while the pulse moving to the right is
blue shifted. The blue light carries more momentum than the red light, so that the momentum of the light in the moving frame is not balanced: the light is carrying some net momentum to the right. The object has not changed its velocity before or after the emission. Yet in this frame it has lost some right-momentum to the light. The only way it could have lost momentum is by losing mass. This also solves Poincaré's radiation paradox. The velocity is small, so the right-moving light is blueshifted by an amount equal to the nonrelativistic
Doppler shift factor . The momentum of the light is its energy divided by , and it is increased by a factor of . So the right-moving light is carrying an extra momentum given by: \Delta P = {v \over c}{E \over 2c} . The left-moving light carries a little less momentum, by the same amount . So the total right-momentum in both light pulses is twice . This is the right-momentum that the object lost. 2\Delta P = v {E\over c^2} . The momentum of the object in the moving frame after the emission is reduced to this amount: P' = Mv - 2\Delta P = \left(M - {E\over c^2}\right)v . So the change in the object's mass is equal to the total energy lost divided by . Since any emission of energy can be carried out by a two-step process, where first the energy is emitted as light and then the light is converted to some other form of energy, any emission of energy is accompanied by a loss of mass. Similarly, by considering absorption, a gain in energy is accompanied by a gain in mass.
Radioactivity and nuclear energy was prominently indicated on the cover of
Time magazine in July 1946. It was quickly noted after the discovery of
radioactivity in 1897 that the total energy due to radioactive processes is about one million times greater than that involved in any known molecular change, raising the question of where the energy comes from. After eliminating the idea of absorption and emission of some sort of Lesagian ether particles, the existence of a huge amount of latent energy, stored within matter, was proposed by New Zealand physicist
Ernest Rutherford and British radiochemist
Frederick Soddy in 1903. Rutherford also suggested that this internal energy is stored within normal matter as well. He went on to speculate in 1904: "If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous amount of energy could be obtained from a small quantity of matter." Einstein's equation does not explain the large energies released in radioactive decay, but can be used to quantify them. The theoretical explanation for radioactive decay is given by the
nuclear forces responsible for holding atoms together, though these forces were still unknown in 1905. The enormous energy released from radioactive decay had previously been measured by Rutherford and was much more easily measured than the small change in the gross mass of materials as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a
calorimeter was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which was known by then to release enough energy to possibly be "weighed," when missing from the system. However, radioactivity seemed to proceed at its own unalterable pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking
moonshine." This outlook changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single
nuclides and their reactions to be calculated directly, and compared with the sum of masses for the particles that made up their composition. In 1933, the energy released from the reaction of
lithium-7 plus protons giving rise to two
alpha particles, allowed Einstein's equation to be tested to an error of ±0.5%. However, scientists still did not see such reactions as a practical source of power, due to the energy cost of accelerating reaction particles. After the very public demonstration of huge energies released from nuclear fission after the
atomic bombings of Hiroshima and Nagasaki in 1945, the equation became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured on page 2 of the
Smyth Report, the official 1945 release by the US government on the development of the atomic bomb, and by 1946 the equation was linked closely enough with Einstein's work that the cover of
Time magazine prominently featured a picture of Einstein next to an image of a
mushroom cloud emblazoned with the equation. Einstein himself had only a minor role in the
Manhattan Project: he had
cosigned a letter to the U.S. president in 1939 urging funding for research into atomic energy, warning that an atomic bomb was theoretically possible. The letter persuaded Roosevelt to devote a significant portion of the wartime budget to atomic research. Without a security clearance, Einstein's only scientific contribution was an analysis of an
isotope separation method in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information to fully work on the problem. While is useful for understanding the amount of energy potentially released in a fission reaction, it was not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at 200
MeV (which was directly possible, using a quantitative
Geiger counter, at that time). The physicist and Manhattan Project participant
Robert Serber noted that somehow "the popular notion took hold long ago that Einstein's theory of relativity, in particular his equation , plays some essential role in the theory of fission. Einstein had a part in alerting the United States government to the possibility of building an atomic bomb, but his theory of relativity is not required in discussing fission. The theory of fission is what physicists call a non-relativistic theory, meaning that relativistic effects are too small to affect the dynamics of the fission process significantly." There are other views on the equation's importance to nuclear reactions. In late 1938, the Austrian-Swedish and British physicists
Lise Meitner and
Otto Robert Frisch—while on a winter walk during which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission—directly used Einstein's equation to help them understand the quantitative energetics of the reaction that overcame the "surface tension-like" forces that hold the nucleus together, and allowed the fission fragments to separate to a configuration from which their charges could force them into an energetic
fission. To do this, they used
packing fraction, or nuclear
binding energy values for elements. These, together with use of allowed them to realize on the spot that the basic fission process was energetically possible.
Einstein's equation written According to the Einstein Papers Project at the
California Institute of Technology and
Hebrew University of Jerusalem, there remain only four known copies of this equation as written by Einstein. One of these is a letter written in
German to
Ludwik Silberstein, which was in Silberstein's archives, and sold at
auction for $1.2 million,
RR Auction of
Boston, Massachusetts said on May 21, 2021. Einstein first published the equation using lowercase 'c' instead of upper case 'V' in 1907. In later papers he adopted the now
common notation. {{Cite journal |last=Mendelson |first=K. S. |year=2006 |title=The story of
c |journal=
American Journal of Physics |volume=74 |issue=11 |pages=995–997 |doi=10.1119/1.2238887 ==See also==