Mass–energy equivalence


 * For other uses, see E=MC2 (disambiguation).

In physics, mass–energy equivalence is the concept that any mass has an associated energy and vice versa. In special relativity this relationship is expressed using the mass–energy equivalence formula


 * $$ E = mc^2 \,$$

where
 * E = energy,
 * m = mass,
 * c = the speed of light in a vacuum (celeritas),
 * and the superscript 2 indicates the squaring of c.

Two definitions of mass in special relativity may be validly used with this formula. If the mass in the formula is the rest mass, the energy in the formula is called the rest energy. If the mass is the relativistic mass, then the energy is the total energy.

The formula was derived by Albert Einstein, who arrived at it in 1905 in the paper "Does the inertia of a body depend upon its energy-content?", one of his Annus Mirabilis ("Wonderful Year") Papers. While Einstein was not the first to propose a mass–energy relationship, and various similar formulas appeared before Einstein's theory, Einstein was the first to propose that the equivalence of mass and energy is a general principle, which is a consequence of the symmetries of space and time.

In the formula, c2 is the conversion factor required to convert from units of mass to units of energy. The formula does not depend on a specific system of units. In the International System of Units, the unit for energy is the joule, for mass the kilogram, and for speed meters per second. Note that 1 joule equals 1 kg·m2/s2. In unit-specific terms, E (in joules) = m (in kilograms) multiplied by (299,792,458 m/s)2.

Conservation of mass and energy
The concept of mass–energy equivalence unites the concepts of conservation of mass and conservation of energy, allowing rest mass to be converted to forms of active energy (such as kinetic energy, heat, or light) while still retaining mass. Conversely, active energy in the form of kinetic energy or radiation can be converted to particles which have rest mass. The total amount of mass/energy in a closed system (as seen by a single observer) remains constant because energy cannot be created or destroyed and, in all of its forms, trapped energy exhibits mass. In relativity, mass and energy are two forms of the same thing, and neither one appears without the other.

Fast-moving object
If a force is applied to an object in the direction of motion, the object gains energy because the force is doing work, but an object cannot be accelerated to the speed of light, regardless of how much energy it absorbs. Its kinetic energy continues to increase without bounds, whereas its speed approaches the (finite) speed of light. This means that in relativity the kinetic energy is not given by 1/2 mv2.

The relativistic mass is the ratio of the momentum of an object to its speed, and it is a quantity that depends on the motion of the observer. If the observer is moving at nearly the same velocity as the object, the relativistic mass is nearly equal to the rest mass, which is also the usual Newtonian mass. If the observer is moving quickly relative to the object, the relativistic mass is bigger than the rest mass.

The relativistic mass is always equal to the total energy divided by c2. The difference between the relativistic mass and the rest mass is the relativistic kinetic energy (divided by c2). Because the relativistic mass is exactly proportional to the energy, relativistic mass and relativistic energy are terms which can be used interchangeably. For this reason, when people talk about the mass of a particle, they are usually talking about its rest mass, which is the same in all inertial frames.

For a system made up of many parts, linked in (nucleus, atom, common object, planet, star . . .), the relativistic mass is the sum of the relativistic masses of the parts, because the energy adds up.

Meanings of the mass–energy equivalence formula


Mass–energy equivalence says that when a body has a mass, it has a certain energy, even when it isn't moving. In Newtonian mechanics, a massive body at rest has no kinetic energy, and it may or may not have other (relatively small) amounts of internal stored energy such as chemical energy or thermal energy, in addition to any potential energy it may have from its position in a field of force. In Newtonian mechanics, none of these energies contributes to the mass.

In relativity, all the energy which moves along with the body adds up to the rest energy of the body, which is proportional to the rest mass of the body. Even a single photon traveling in empty space has a relativistic mass, which is its energy divided by c2. If a box of mirrors contains light, the mass of the box is increased by the energy of the light, since the total energy of the box is its mass.

Although a photon is never "at rest", it still has a rest mass, which is zero. If an observer chases a photon faster and faster, the observed energy of the photon approaches zero as the observer approaches the speed of light. This is why photons are massless. They have zero rest mass even though they have varying amounts of energy and relativistic mass. But, systems of two or more photons moving in different directions (as for example from an electron–positron annihilation) may have zero momentum over all. Their energy E then adds up to an invariant mass m = E/c2, when they are considered as a system.

This formula also gives the amount of mass lost from a body when energy is removed. In a chemical or nuclear reaction, when heat and light are removed, the mass is decreased. So the E in the formula is the energy released or removed, corresponding to a mass m which is lost. In those cases, the energy released and removed is equal in quantity to the mass lost, times c2. Similarly, when energy of any kind is added to a resting body, the increase in the mass is equal to the energy added, divided by c2.

Consequences for nuclear physics
Max Planck pointed out that the mass–energy equivalence formula implied that bound systems would have a mass less than the sum of their constituents, once the binding energy had been allowed to escape. However, Planck was thinking about chemical reactions, where the binding energy is too small to measure. Einstein suggested that radioactive materials such as radium would provide a test of the theory, but even though a large amount of energy is released per atom, only a small fraction of the atoms decay.

Once the nucleus was discovered, experimenters realized that the very high binding energies of the atomic nuclei should allow calculation of their binding energies from mass differences. But it was not until the discovery of the neutron in 1932, and the measurement of its mass, that this calculation could actually be performed (see nuclear binding energy for example calculation). A little while later, the first transmutation reactions (such as $$ \scriptstyle {}^7\mathrm{Li} + \mathrm{p} \rightarrow 2\,{}^4\mathrm{He}$$) verified Einstein's formula to an accuracy of 1%.

The mass–energy equivalence formula was used in the development of the atomic bomb. By measuring the mass of different atomic nuclei and subtracting from that number the total mass of the protons and neutrons as they would weigh separately, one gets the exact binding energy available in an atomic nucleus. This is used to calculate the energy released in any nuclear reaction, as the difference of the binding energies of the nuclei that enter and exit the reaction.

Practical examples
Einstein used the CGS system of units (centimeters, grams, seconds, dynes, and ergs), but the formula is independent of the system of units. In natural units, the speed of light is defined to equal 1, and the formula expresses an identity: E = m. In the SI system (expressing the ratio E / m in joules per kilogram using the value of c in meters per second):


 * E / m = c2 = (299,792,458 m/s)2 = 89,875,517,870,000,000 J/kg (≈9.0 × 1016 joules per kilogram)

So one gram of mass — approximately the mass of a U.S. dollar bill — is equivalent to the following amounts of energy:
 * 89.9 terajoules
 * 24.9 million kilowatt-hours (≈25 GW·h)
 * 21.5 billion kilocalories (≈21 Tcal)
 * 21.5 kilotons of TNT-equivalent energy (≈21 kt)
 * 85.2 billion BTUs

Any time energy is generated, the process can be evaluated from an E = mc2 perspective. For instance, the "Gadget"-style bomb used in the Trinity test and the bombing of Nagasaki had an explosive yield equivalent to 21 kt of TNT. About 1 kg of the approximately 6.15 kg of plutonium in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling (the heat, light and radiation in this case carried the missing gram of mass). This occurs because nuclear binding energy is released whenever elements with more than 62 nucleons fission.

Another example is hydroelectric generation. The electrical energy produced by Grand Coulee Dam’s turbines every 3.7 hours represents one gram of mass. This mass passes to the electrical devices which are powered by the generators (such as lights in cities), where it appears as a gram of heat and light. Turbine designers look at their equations in terms of pressure, torque, and RPM. However, Einstein’s equations show that all energy has mass, and thus the electrical energy produced by a dam's generators, and the heat and light which result from it, all retain their mass, which is equivalent to the energy. The potential energy – and equivalent mass – represented by the waters of the Columbia River as it descends to the Pacific Ocean would be converted to heat due to viscous friction and the turbulence of white water rapids and waterfalls were it not for the dam and its generators. This heat would remain as mass on site at the water, were it not for the equipment which converted some of this potential and kinetic energy into electrical energy, which can be moved from place to place (taking mass with it).

Whenever energy is added to a system, the system gains mass. A spring's mass increases whenever it is put into compression or tension. Its added mass arises from the added potential energy stored within it, which is bound in the stretched chemical (electron) bonds linking the atoms within the spring. Raising the temperature of an object (increasing its heat energy) increases its mass. If the temperature of the platinum/iridium "international prototype" of the kilogram—the world’s primary mass standard—is allowed to change by 1°C, its mass will change by 1.5 picograms (1 pg = 1 × 10–12 g). Note that no net mass or energy is really created or lost in any of these scenarios. Mass/energy simply moves from one place to another. These are some examples of the transfer of energy and mass in accordance with the principle of mass–energy conservation.

Note further that in accordance with Einstein’s Strong Equivalence Principle (SEP), all forms of mass and energy produce a gravitational field in the same way. So all radiated and transmitted energy retains its mass. Not only does the matter comprising Earth create gravity, but the gravitational field itself has mass, and that mass contributes to the field too. This effect is accounted for in ultra-precise laser ranging to the Moon as the Earth orbits the Sun when testing Einstein’s theory of general relativity. According to E=mc2, no closed system (any system treated and observed as a whole) ever loses mass, even when rest mass is converted to energy. This statement is more than an abstraction based on the principle of equivalence, it is a real-world effect.

Potential energy also has mass, but where this mass sits is sometimes difficult to determine. The concept of potential energy is Newtonian, it is defined for the system as a whole. The mass-energy relation together with the law of gravity requires that the potential energy be somewhere, so that its mass can produce a gravitational field. So in relativity, the potential energy always comes from a local field, and it is found wherever the field is varying or has a value which carries energy. Gravitational experiments can locate the field energy, and therefore the potential energy, in principle. The one exception is the gravitational field itself. Because the gravitational field can be made to vanish locally by choosing a free-falling frame, it is difficult to locate gravitational energy in an observer independent way. Still, it is possible to define the location of the gravitational energy consistently in several different ways, all of which agree on the total energy. The field energy in the Newtonian limit is the potential energy of a system.



Although all mass, including that in ordinary objects, is energy, this energy is not always in a form which can be used to generate power. All energy, both usable and unusable, has mass, so when people say that certain reactions "convert" mass into "energy", they mean that the mass is converted into specific types of energy, which can be used to do work, which is sometimes called the "active energy". Practical "conversions" of mass into active energy never make all of the mass into the sort of energy which can be used to do work. For example, in nuclear fission roughly 0.1% of the mass of fissioned atoms is converted to heat energy and radiation. In turn, the mass of fissioned atoms is only part of the mass of the fissionable material: e.g. in a nuclear fission weapon, the efficiency is 40% at most, meaning that 40% of fissionable atoms actually fission. In nuclear fusion roughly 0.3% of the mass of fused atoms is converted to active energy. In thermonuclear weapons (see nuclear weapon yield) some of the bomb mass is casing and non-reacting components, so the efficiency in converting passive energy to active energy, at 6 kilotons TNT equivalent energy output per kilogram of bomb mass (or 6 megatons per metric ton bomb mass), does not exceed 0.03%.

Perfect conversion
One theoretically perfect method of conversion of the rest mass of matter to usable energy is the annihilation of matter with antimatter. In this process, all the mass energy is released as light and heat. However, in our universe, antimatter is rare. To make antimatter requires more energy than would be liberated.

Since most of the mass of ordinary objects is in protons and neutrons, in order to convert all of the mass in ordinary matter to useful energy, the protons and neutrons must be converted to lighter particles. In the standard model of particle physics, the number of protons plus neutrons is nearly exactly conserved in all reactions at moderate energies. Nevertheless, 't Hooft showed that there is a process which will convert protons and neutrons to antielectrons and neutrinos. This is the weak SU(2) instanton discovered by Belavin Polyakov Schwarz and Tyupkin. This process is capable of complete conversion of the mass of matter to usable energy, but it is extraordinarily slow at ordinary energies. Later it became clear that this process will happen at a fast rate at very high temperatures, since then instanton-like configurations will be copiously produced from thermal fluctuations. The temperature required is so high that it would only have been reached shortly after the big bang.

All conservative extensions of the standard model contain magnetic monopoles, and in the usual models of grand unification, these monopoles catalyze proton decay, a process known as the Callan-Rubakov effect. This process would be an efficient mass-energy conversion at ordinary temperatures, but it requires making monopoles and antimonopoles first. The energy required to produce monopoles is enormous, but they are stable so they only need to be produced once.

The third known method of total mass/energy conversion is using gravity, specifically black holes. Stephen Hawking showed that black holes radiate thermally. It is therefore possible to throw matter into a small black hole and use the radiation to power a plant. Unfortunately, this is also impractical for the time being.

Background
E = mc2 where m stands for rest mass (invariant mass), applies most simply to single particles with no net momentum. But it also applies to ordinary objects composed of many particles so long as the particles are moving in different directions so the total momentum is zero. The rest mass of the object includes contributions from heat and sound, chemical binding energies and trapped radiation. Familiar examples are a tank of gas, or a hot bowl of soup. The kinetic energy of their particles, the heat motion and radiation, contribute to their weight on a scale according to E = mc2.

The formula is the special case of the relativistic energy-momentum relationship:



E^2 - (pc)^2 = (m c^2)^2. $$

This equation gives the rest mass of an object which has an arbitrary amount of momentum and energy. The interpretation of this equation is that the rest mass is the relativistic length of the energy-momentum four-vector.

If the equation $$E=mc^2$$ is used with the rest mass of the object, the $$E$$ given by the equation will be the rest energy of the object, and will change with according to the object's internal energy, heat and sound and chemical binding energies, but will not change with the object's overall motion).

If the equation $$E=mc^2$$ is used with the relativistic mass of the object, the energy will be the total energy of the object, which is conserved in collisions with other moving objects.


 * Mass Velocity Relationship

In developing special relativity, Einstein found that the total energy of a moving body is


 * $$E = \frac{m_0 c^2}\sqrt{1-\frac{v^2}{c^2}},$$

with $$ v $$ the velocity. (We are now using $$m_0$$ to denote the rest mass.)

For small velocities, this reduces to


 * $$E = m_0 c^2 + \frac{1}{2}m_0 v^2 + ... $$

Which includes the newtonian kinetic energy, as expected, but also an enormous constant term, which is not zero when the object isn't moving.

The total momentum is:


 * $$P = \frac{m_0 v}\sqrt{1-\frac{v^2}{c^2}}. $$

The ratio of the momentum to the velocity is the relativistic mass, and this ratio is equal to the total energy times c2. The energy and relativistic mass are always related by the famous formula.

While this is suggestive, it does not immediately imply that the energy and mass are equivalent because the energy can always be redefined by adding or subtracting a constant. So it is possible to subtract the $$ m_0 c^2$$ from the expression for $$ E$$ and this is also a valid conserved quantity. Einstein needed to know whether the rest-mass of the object is really an energy, or whether the constant term was just a mathematical convenience with no physical meaning.

In order to see if the $$m_0 c^2$$ is physically significant, Einstein considered processes of emission and absorption. He needed to establish that an object loses mass when it emits energy. He did this by analyzing two photon emission in two different frames.

Relativistic mass
After Einstein first made his proposal, it became clear that the word mass can have two different meanings. The rest mass is what Einstein called m, but others defined the relativistic mass as:


 * $$m_{\mathrm{rel}} = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} . $$

This mass is the ratio of momentum to velocity, and it is also the relativistic energy divided by c2. So the equation $$E=m_{\mathrm{rel}}c^2$$ holds for moving objects. When the velocity is small, the relativistic mass and the rest mass are almost exactly the same.

$$E = mc^2 $$ either means $$E = m_0 c^2 $$ for an object at rest, or $$E = m_{\mathrm{rel}} c^2$$ when the object is moving.

Also Einstein (following Hendrik Lorentz and Max Abraham) used velocity and direction dependent mass concepts (longitudinal and transverse mass) in his 1905 electrodynamics paper and in another paper in 1906. However, in his first paper on $$E=mc^2$$ (1905) he treated m as what would now be called the rest mass. Some claim that (in later years) he did not like the idea of "relativistic mass." When modern physicists say "mass", they are usually talking about rest mass, since if they meant "relativistic mass", they would just say "energy".

Low-speed Expansion
We can rewrite the expression for the energy as a Taylor series:


 * $$E = m_0 c^2 \left[1 + \frac{1}{2} \left(\frac{v}{c}\right)^2 + \frac{3}{8} \left(\frac{v}{c}\right)^4 + \frac{5}{16} \left(\frac{v}{c}\right)^6 + \ldots \right]. $$

For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because $$v/c$$ is small. For low speeds we can ignore all but the first two terms:


 * $$E \approx m_0 c^2 + \frac{1}{2} m_0 v^2 . $$

The total energy is a sum of the rest energy and the Newtonian kinetic energy.

The classical energy equation ignores both the $$m_0 c^2$$ part, and the high-speed corrections. This is appropriate, because all the high order corrections are small. Since only changes in energy affect the behavior of objects, whether we include the $$m_0 c^2$$ part makes no difference, since it is constant. For the same reason, it is possible to subtract the rest energy from the total energy in relativity. In order to see if the rest energy has any physical meaning, it is essential to consider emission and absorption of energy in different frames.

The higher-order terms are extra correction to Newtonian mechanics which become important at higher speeds. The Newtonian equation is only a low speed approximation, but an extraordinarily good one. All of the calculations used in putting astronauts on the moon, for example, could have been done using Newton's equations without any of the higher order corrections.

History
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. But nearly all previous authors thought that the energy which contributes to mass comes only from electromagnetic fields.

Newton: Matter and light
In 1717 Isaac Newton speculated that light particles and matter particles were inter-convertible in "Query 30" of the Opticks, where he states:

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?

Since Newton did not understand light as the motion of a field, he was not speculating about the conversion of motion into matter. Since he did not know about energy, he could not have understood that converting light to matter is turning work into mass.

Electromagnetic rest mass
There were many attempts in the 19th and the beginning of the 20th century - like those of J. J. Thomson (1881), Oliver Heaviside (1888), George Frederick Charles Searle (1896), - to understand how the mass of a charged object varied with the velocity. Because the electromagnetic field carries part of the momentum of a moving charge, it was suspected that the mass of an electron would vary with velocity near the speed of light.

Following Searle (1896), Wilhelm Wien (1900), Max Abraham (1902), and Hendrik Lorentz (1904) concluded that the velocity dependant electromagnetic mass of a body at rest is $$m=(4/3)E/c^2$$. According to them, this relation applies to the complete mass of bodies, because any form of inertial mass was considered to be of electromagnetic origin. Wien went on by stating, that if it is assumed that gravitation is an electromagnetic effect too, than there has to be a strict proportionality between (electromagnetic) inertial mass and (electromagnetic) gravitational mass. To explain the stability of the matter-electron configuration, Poincaré in 1906 introduced some sort of pressure of non-electrical nature, which contributes the amount $$-(1/3)E/c^2$$ to the mass of the bodies, and therefore the 4/3-factor vanishes.

Inertia of energy and radiation
James Clerk Maxwell (1874) and Adolfo Bartoli (1876) found out that the existence of tensions in the ether like the radiation pressure follows from the electromagnetic theory. However, Lorentz (1895) recognized that this led to a conflict between the action/reaction principle and Lorentz's ether theory.
 * Maxwell, Bartoli, Lorentz

In 1900 Henri Poincaré studied this conflict and tried to determine whether the center of gravity still moves with a uniform velocity when electromagnetic fields are included. He noticed that the action/reaction principle does not hold for matter alone, but that the electromagnetic field has its own momentum. The electromagnetic field energy behaves like a fictitious fluid ("fluide fictif") with a mass density of $$E/c^2$$ (in other words $$m=E/c^2$$). If the center of mass frame is defined by both the mass of matter and the mass of the fictitious fluid, and if the fictitious fluid is indestructible - it is neither created or destroyed - then the motion of the center of mass frame remains uniform. But electromagnetic energy can be converted into other forms of energy. So Poincaré assumed that there exists a non-electric energy fluid at each point of space, into which electromagnetic energy can be transformed and which also carries a mass proportional to the energy. In this way, the motion of the center of mass remains uniform. Poincaré said that one should not be too surprised by these assumptions, since they are only mathematical fictions.
 * Poincaré

But Poincaré's resolution led to a paradox when changing frames: if a Hertzian oscillator radiates in a certain direction, it will suffer a recoil from the inertia of the fictitious fluid. In the framework of Lorentz ether theory Poincaré performed a Lorentz boost to the frame of the moving source. He noted that energy conservation holds in both frames, but that the law of conservation of momentum is violated. This would allow a perpetuum mobile, a notion which he abhorred. The laws of nature would have to be different in the frames of reference, and the relativity principle would not hold.

Poincaré's paradox was resolved by Einstein's insight that a body losing energy as radiation or heat was losing a mass of the amount $$m=E/c^2$$. The Hertzian oscillator loses mass in the emission process, and momentum is conserved in any frame. Einstein noted in 1906 that Poincaré's solution to the center of mass problem and his own were mathematically equivalent (see below).

Poincaré came back to this topic in "Science and Hypothesis" (1902) and "The Value of Science" (1905). This time he rejected the possibility that energy carries mass: "... [the recoil] is contrary to the principle of Newton since our projectile here has no mass, it is not matter, it is energy". He also discussed two other unexplained effects: (1) non-conservation of mass implied by Lorentz's variable mass $$\gamma m$$, Abraham's theory of variable mass and Kaufmann's experiments on the mass of fast moving electrons and (2) the non-conservation of energy in the radium experiments of Madame Curie.

Following Poincaré, Max Abraham in 1902-1904 introduced the term "electromagnetic momentum" to maintain the action/reaction principle. Poincaré's result, who according to Abraham gave no proof of his result, was verified by him, whereby the field density of momentum per cm3 is $$E/c^2$$ and $$E/c$$ per cm2.
 * Abraham and Hasenöhrl

In 1904, Friedrich Hasenöhrl specifically associated inertia with radiation in a paper, which was according to his own words very similar to some papers of Abraham. Hasenöhrl suggested that part of the mass of a body (which he called apparent mass) can be thought of as radiation bouncing around a cavity. The apparent mass of radiation depends on the temperature (because every heated body emits radiation) and is proportional to its energy, and he first concluded that $$m=(8/3)E/c^2$$. However, in 1905 Hasenöhrl published a summary of a letter, which was written by Abraham to him. Abraham concluded that Hasenöhrl's formula of the apparent mass of radiation is not correct, and based on his definition of electromagnetic momentum and longitudinal electromagnetic mass Abraham changed it to $$m=(4/3)E/c^2$$, the same value for the electromagnetic mass for a body at rest. Hasenöhrl re-calculated his own derivation and verified Abraham's result. He also noticed the similarity between the apparent mass and the electromagnetic mass. However, Hasenöhrl stated that this energy-apparent-mass relation only holds as long a body radiates, i.e. if the temperature of a body is greater than 0 K.

However, it was suggested that Hasenöhrl had made an error in that he did not include the pressure of the radiation on the cavity shell. If he had included the shell pressure and inertia as it would be included in the theory of relativity, the factor would have been equal to 1 or $$m=E/c^2$$. This calculation assumes that the shell properties are consistent with relativity, otherwise the mechanical properties of the shell including the mass and tension would not have the same transformation laws as those for the radiation. Nobel Prize-winner and Hitler advisor Philipp Lenard claimed that the mass–energy equivalence formula needed to be credited to Hasenöhrl to make it an aryan creation.

Einstein: Mass–energy equivalence
Albert Einstein did not formulate exactly this formula in his 1905 paper "Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?" ("Does the Inertia of a Body Depend Upon Its Energy Content?", published in Annalen der Physik on September 27), one of the articles now known as his Annus Mirabilis Papers. That paper says: ''If a body gives off the energy L in the form of radiation, its mass diminishes by $$L/c^2$$, "radiation" means electromagnetic radiation or light, and mass means the ordinary newtonian mass of a slow moving object. In Einstein's first formulation, it is the difference in the mass '$$\scriptstyle \Delta m\ $$' before and after the ejection of energy that is equal to $$L/c^2$$, not the entire mass '$$ m\ $$' of the object. Objects with zero mass presumably have zero energy, so the extension that all mass is proportional to energy is obvious from this result. In 1905, even the hypothesis that changes in energy are accompanied by changes in mass was untested. Not until the discovery of the first type of antimatter (the positron in 1932) was it found that all of the mass of pairs of resting particles could be converted to radiation.


 * 1905 – First correct derivation

Einstein considered a body at rest with mass M. If the body is examined in a frame moving with nonrelativistic velocity v, it is no longer at rest and in the moving frame it has momentum Mv.

Suppose now that the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy E/2. Since the two pulses are equal, the object remains at rest after the emission since the two beams are equal in strength and carry opposite momentum.

But if we consider the same process in a frame moving with velocity v to the left, the pulse moving to the left will be redshifted while the pulse moving to the right will be blueshifted. 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.

But the object hasn't changed its velocity before or after the emission. Yet in this frame it 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, discussed above.

The velocity is small, so the right moving light is blueshifted by an amount equal to the nonrelativistic Doppler shift factor (1-v/c). The momentum of the light is its energy divided by c, and it is increased by a factor of v/c. So the right moving light is carrying an extra momentum $$\Delta P$$ given by:



\Delta P = {v \over c}{E \over 2c}. \,$$

The left moving light carries a little less momentum, by the same amount $$\Delta P$$. So the total right-momentum in the light is twice $$ \Delta P$$. 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 by this amount:



P' = Mv - 2\Delta P = (M - {E\over c^2})v \,$$

So the change in the object's mass is equal to the total energy lost divided by $$c^2$$. 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. Einstein concludes that all the mass of a body is a measure of its energy content.


 * 1906 – Relativistic center-of-mass theorem

Similar to Poincaré, Einstein concluded in 1906 that the inertia of (electromagnetic) energy is a necessary condition for the center-of-mass theorem to hold in systems, in which electromagnetic fields and matter are acting on each other. On that occasion, Einstein referred to Poincaré's 1900-paper and wrote:

Although the simple formal views, which must be accomplished for the proof of this statement, are already mainly contained in a work by H. Poincaré2, for the sake of clarity I won't rely on that work.

However, Einstein didn't have to introduce fictitious masses and could also avoid the perpetuum mobile problem, because based on the mass–energy equivalence he could show that emission and absorption of em-radiation and therefore the transport of inertia solves the problem. Also Poincaré's rejection of the reaction principle due to the violation of the mass conservation law (as discussed in the preceding section) can be avoided through Einstein's $$E=mc^2$$, because mass conservation appears as a special case of the energy conservation law.

Others
During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various discredited ether theories. In particular, the writings of S. Tolver Preston, and a 1903 paper by Olinto De Pretto,  presented a mass energy relation. De Pretto's paper received recent press coverage, when Umberto Bartocci discovered that there were only three degrees of separation linking De Pretto to Einstein, leading Bartocci to conclude that Einstein was probably aware of De Pretto's work.

Preston and De Pretto, following Le Sage, imagined that the universe was filled with an ether of tiny particles which are always moving at speed c. Each of these particles have a kinetic energy of mc2 up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of 1/2, since Leibniz introduced kinetic energy without it, and the 1/2 is largely conventional in prerelativistic physics. By assuming that every particle has a mass which is the sum of the masses of the ether particles, the authors would conclude that all matter contains an amount of kinetic energy either given by E=mc2 or 2E=mc2 depending on the convention. A particle ether was usually considered unacceptably speculative science at the time, and since these authors didn't formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames.

Independently, Gustave Le Bon in 1905 speculated that atoms could release large amounts of latent energy, reasoning from an all encompassing qualitative philosophy of physics.

Nuclear energy and popular culture
Radioactivity was discovered in 1896, and the source of the energy was initially a mystery. By 1903, Ernest Rutherford and Frederick Soddy had proved that radioactive elements was due to the fact that they decayed into other elements, releasing a great deal of energy in the process. Einstein mentions in his 1905 paper that mass-energy equivalence might perhaps be tested with radioactive decay, which releases enough energy (the quantitative amount known roughly even by 1905) to possibly be "weighed," when missing. But the idea that great amounts of usable energy could be liberated from matter, however, proved initially difficult to substantiate in a practical fashion. Because it had been used as the basis of much speculation, Rutherford himself was once reported in the 1930s to have said that: "Anyone who expects a source of power from the transformation of the atom is talking moonshine."



This changed dramatically after the demonstration of energy released from nuclear fission after the atomic bombings of Hiroshima and Nagasaki in 1945. The equation E=mc2 became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured as early as 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 close-enough linked 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 small role in the Manhattan Project to build the atomic bomb: he had signed a letter to the US President in 1939 urging nuclear research be funded, and this funding over time grew into the bomb development project.

While E=mc2 is useful for understanding the amount of energy released in a fission reaction, it was not strictly necessary to develop the weapon. As the physicist and Manhattan Project participant Robert Serber put it: "Somehow the popular notion took hold long ago that Einstein's theory of relativity, in particular his famous equation E=mc2, plays some essential role in the theory of fission. Albert 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." However the association between E=mc2 and nuclear energy has since stuck, and because of this association, and its simple expression of the ideas of Albert Einstein himself, it has become "the world's most famous equation".