# Mass–energy equivalence: Map

### Map showing all locations mentioned on Wikipedia article: In physics, mass–energy equivalence is the concept that the mass of a body is a measure of its energy content. The mass of a body as measured on a scale is always equal to the total energy inside, divided by a constant c2 that changes the units appropriately:

:E = mc^2 \,\!

where E is energy, m is mass, and c is the speed of light in a vacuum, which is .

Mass–energy equivalence was proposed in Albert Einstein's 1905 paper, "Does the inertia of a body depend upon its energy-content?", one of his Annus Mirabilis ("Miraculous Year") Papers. Einstein was not the first to propose a mass–energy relationship, and various similar formulas appeared before Einstein's theory with incorrect numerical coefficients and an incomplete interpretation. Einstein was the first to propose the simple formula and the first to interpret it correctly: as a general principle which follows from the relativistic 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. Using the International System of Units, joules are used to measure energy, kilograms for mass, meters per second for speed. 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. In natural units, the speed of light is set equal to 1, and the formula becomes an identity.

## 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 other forms of energy, like kinetic energy, heat, or light. Kinetic energy or light can also be converted to particles which have mass. The total amount of mass–energy in a closed system remains constant because energy cannot be created or destroyed and, in all of its forms, trapped energy has mass. According to the theory of relativity, mass and energy as commonly understood are two names for the same thing, and one is not changed to the other. Rather, neither one appears without the other. When energy changes type and leaves a system, it takes its mass with it.

### Fast-moving objects and systems of objects

When an object is pushed in the direction of motion, it gains momentum and energy, but when the object is already traveling near the speed of light, it cannot move much faster, no matter how much energy it absorbs. Its momentum and energy continue to increase without bounds, whereas its speed approaches a constant value—the speed of light. This implies that in relativity the momentum of an object cannot be not a constant times the velocity, nor can the kinetic energy be a constant times the square of the velocity.

The relativistic mass is defined as the ratio of the momentum of an object to its velocity, and it depends on the motion of the object. If the object is moving slowly, the relativistic mass is nearly equal to the rest mass and both are nearly equal to the usual Newtonian mass. If the object is moving quickly, the relativistic mass is greater than the rest mass by an amount equal to the mass associated with the kinetic energy of the object. As the object approaches the speed of light, the relativistic mass becomes infinite, because the kinetic energy becomes infinite and this energy is associated with mass.

The relativistic mass is always equal to the total energy (rest energy plus kinetic energy) divided by c2. Because the relativistic mass is exactly proportional to the energy, relativistic mass and relativistic energy are nearly synonyms; the only difference between them is the units. If length and time are measured in natural units, the speed of light is equal to 1, and even this difference disappears. Then mass and energy have the same units and are always equal, so it is redundant to speak about relativistic mass, because it is just another name for the energy. This is why physicists usually reserve the useful short word "mass" to mean rest-mass.

For things made up of many parts, like a nucleus, planet, or star, the relativistic mass is the sum of the relativistic masses (or energies) of the parts, because energies are additive in closed systems. This is not true in systems which are open, however, if energy is subtracted. For example, if a system is bound by attractive forces and the work they do in attraction is removed from the system, mass will be lost. Such work is a form of energy which itself has mass, and thus mass is removed from the system, as it is bound. For example, the mass of an atomic nucleus is less than the total mass of the protons and neutrons that make it up, but this is only true after the energy (work) of binding has been removed in the form of a gamma ray (which in this system, carries away the mass of binding). This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons (in this case, work and mass would need to be supplied). Similarly, the mass of the solar system is slightly less than the masses of sun and planets individually.

The relativistic mass of a moving object is bigger than the relativistic mass of an object that isn't moving, because a moving object has extra kinetic energy. The rest mass of an object is defined as the mass of an object when it is at rest, so that the rest mass is always the same, independent of the motion of the observer: it is the same in all inertial frames.

For a system of particles going off in different directions, the invariant mass of the system is the analog of the rest mass, and is the same for all observers. It is defined as the total energy (divided by c2) in the center of mass frame (where by definition, the system total momentum is zero). A simple example of an object with moving parts but zero total momentum, is a container of gas. In this case, the mass of the container is given by its total energy (including the kinetic energy of the gas molecules), since the system total energy and invariant mass are the same in the reference frame where the momentum is zero, and this reference frame is also the only frame in which the object can be weighed.

## Meanings of the mass–energy equivalence formula Mass–energy equivalence states that any object has a certain energy, even when it isn't moving. In Newtonian mechanics, a motionless body has no kinetic energy, and it may or may not have other amounts of internal stored energy, like 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, all of these energies are much smaller than the mass of the object times the speed of light squared.

In relativity, all of the energy that moves along with an object adds up to the total mass of the body, which measures how much it resists deflection. Each potential and kinetic energy makes a proportional contribution to the mass. Even a single photon traveling in empty space has a relativistic mass, which is its energy divided by c2. If a box of ideal 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.

In relativity, removing energy is removing mass, and the formula m = E/c2 tells you how much mass is lost when energy is removed. In a chemical or nuclear reaction, the mass of the atoms that come out is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light which has the same relativistic mass as the difference (and also the same invariant mass in the center of mass frame of the system). In this case, the E in the formula is the energy released and removed, and the mass m is how much the mass goes down. In the same way, when any kind of energy is added, the increase in the mass is equal to the added energy divided by c2. For example, when water is heated in a microwave oven, the oven adds about of mass for every joule of heat added to the water.

An object moves with different speed in different frames, depending on the motion of the observer, so the kinetic energy in both Newtonian mechanics and relativity is frame dependent. This means that the amount of energy, and therefore the amount of relativistic mass, that an object is measured to have depends on the observer. The rest mass is defined as the mass that an object has when it isn't moving. It also applies to the invariant mass of systems when the system as a whole isn't "moving" (has no net momentum). The rest and invariant masses are the smallest possible value of the mass of the object or system. They are conserved quantities, so long as the system is closed.

The rest mass is almost never additive: the rest mass of an object is not the sum of the rest masses of its parts. The rest mass of an object is the total energy of all the parts, including kinetic energy, as measured by an observer that sees the center of the mass of the object to be standing still. The rest mass adds up only if the parts are standing still and don't attract or repel, so that they don't have any extra kinetic or potential energy. The other possibility is that they have a positive kinetic energy and a negative potential energy that exactly cancels.

The difference between the rest mass of a bound system and of the unbound parts is exactly proportional to the binding energy of the system. A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom; the minuscule mass difference is the energy that is needed to split the molecule into three individual atoms (divided by c2), and which was given off as heat when the molecule formed (this heat had mass). Likewise, a stick of dynamite weighs a little bit more than the fragments after the explosion, so long as the fragments are cooled and the heat removed; the mass difference is the energy/heat that is released when the dynamite explodes (when it escapes, the mass associated with it escapes, but total mass is conserved). The change in mass only happens when the system is open, and the energy escapes. If a stick of dynamite is blown up in a hermetically sealed chamber, the mass of the chamber and fragments, the heat, sound, and light would still be equal to the original mass of the chamber and dynamite. This would in theory also happen, even with a nuclear bomb, if it could be kept in a chamber which did not rupture.

### Massless particles

In relativity, all energy moving along with a body adds up to the total energy, which is exactly proportional to the relativistic mass. Even a single photon, graviton, or neutrino traveling in empty space has a relativistic mass, which is its energy divided by c2. But the rest mass of a photon is slightly subtler to define in terms of physical measurements, because a photon is always moving at the speed of light—it is never at rest.

If you run away from a photon, having it chase you, by moving fast enough in the same direction, when the photon catches up to you the photon would be seen as having less energy, and even less the faster you were traveling when it caught you. As you approach the speed of light, the photon looks redder and redder, by doppler shift (although for a photon the Doppler shift is relativistic), and the energy of a very long-wavelength photon approaches zero. This is why a photon is massless; this means that the rest mass of a photon is zero. A massless particle in relativity is the limit of a particle with very small mass, but which is moving so close to the speed of light, so that it has a non-negligible total energy.

Two photons moving in different directions can't both be made to have arbitrarily small total energy by changing frames, by chasing them. The reason is that in a two-photon system, the energy of one photon is decreased by chasing it, but the energy of the other will increase. Two photons not moving in the same direction have an inertial frame where the combined energy is smallest, but not zero. This is called the center of mass frame or the center of momentum frame; these terms are almost synonyms (the center of mass frame is the special case of a center of momentum frame where the center of mass is put at the origin). If you move at the same direction and speed as the center of mass of the two photons, the total momentum of the photons is zero. Their combined energy E in this frame gives them, as a system, a mass equal to the energy divided by c2. This mass is called the invariant mass of the pair of photons together. It is the smallest mass and energy the system may be seen to have by any observer.

If the photons formed by the collision of a particle and an antiparticle, the invariant mass is the same as the total energy of the particle and antiparticle (their rest energy plus the kinetic energy), in the center of mass frame, where they will automatically be moving in equal and opposite directions (since they have equal momentum in this frame). If the photons are formed by the disintegration of a single particle with a well-defined rest mass, like the neutral pion, the invariant mass of the photons is equal to rest mass of the pion. In this case, the center of mass frame for the pion is just the frame where the pion is at rest, and the center of mass doesn't change. After the two photons are formed, their center of mass is still moving the same way the pion did, and their total energy in this frame adds up to the mass energy of the pion. So the invariant mass of the photons is equal to the pion's rest energy. So by calculating the invariant mass of pairs of photons in a particle detector, pairs can be identified that were probably produced by pion disintegration.

#### Are photons massless?

The photon is currently believed to be strictly massless, but this is an experimental question. If the photon is not a strictly massless particle, it would not move at the exact speed of light. Its speed would be lower and depend on its frequency. Relativity would be unaffected by this; the "speed of light", c, would then not be the actual speed at which light moves, but a constant of nature which is the maximum speed that any object could theoretically attain. It would still be the speed of gravitons, but it would not be the speed of photons.

A massive photon would have other effects as well. Coulomb's law would be modified and the electromagnetic field would have an extra physical degree of freedom. These effects yield more sensitive experimental probes of the photon mass than the frequency dependence of the speed of light. If Coulomb's law is not exactly valid, then that would cause the presence of an electric field inside a hollow conductor when it is subjected to an external electric field. This thus allows one to test Coulomb's law to very high precision. . A null result of such an experiment has set a limit of m\lesssim 10^{-14} eV..

Sharper upper limits have been obtained in experiments designed to detect effects caused by the Galactic vector potential. Although the galactic vector potential is very large because the galactic magnetic field exists on very long length scales, only the magnetic field is observable if the photon is massless. In case of a massive photon, the mass term \frac{1}{2} m^2 A_{\mu}A^{\mu} would affect the galactic plasma. The fact that no such effects are seen implies an upper bound on the photon mass of m 3\times 10^{-27} eV. The galactic vector potential can also be probed directly by measuring the torque exerted on a magnetized ring.. Such methods were used to obtain the sharper upper limit of 10^{-18}eV. given by the Particle Data Group

These sharp limits from the non-observation of the effects caused by the galactic vector potential have been shown to be model dependent. If the photon mass is generated via the Higgs mechanism then the upper limit of m\lesssim 10^{-14} eV from the test of Coulomb's law is valid.

## 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 7Li + p → 2 4He) verified Einstein's formula to an accuracy of ±0.5%.

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 in the total mass of the nuclei that enter and exit the reaction.

In quantum chromodynamics, the modern theory of the nuclear force, most of the mass of the proton and the neutron is explained by special relativity. The mass of the proton is about eighty times greater than the sum of the rest masses of the quarks that make it up, while the gluons have zero rest mass. The extra energy of the quarks and gluons in a region within a proton, as compared to the energy of the quarks and gluons in the QCD vacuum, accounts for over 98% of the mass.

The internal dynamics of the proton are complicated, because they are determined by the quarks exchanging gluons, and interacting with various vacuum condensates. Lattice QCD provides a way of calculating the mass of the proton directly from the theory to any accuracy, in principle. The most recent calculations claim that the mass is determined to better than 4% accuracy, arguably accurate to 1% (see Figure S5 in Dürr et al.). These claims are still controversial, because the calculations cannot yet be done with quarks as light as they are in the real world. This means that the predictions are found by a process of extrapolation, which can introduce systematic errors. It is hard to tell whether these errors are controlled properly, because the quantities that are compared to experiment are the masses of the hadrons, which are known in advance.

These recent calculations are performed by massive supercomputers, and, as noted by Boffi and Pasquini: “a detailed description of the nucleon structure is still missing because ... long-distance behavior requires a nonperturbative and/or numerical treatment..."More conceptual approaches to the structure of the proton are: the topological soliton approach originally due to Tony Skyrme and the more accurate AdS/QCD approach which extends it to include a string theory of gluons, various QCD inspired models like the bag model and the constituent quark model, which were popular in the 1980s, and the SVZ sum rules which allow for rough approximate mass calculations. These methods don't have the same accuracy as the more brute force lattice QCD methods, at least not yet.

But all these methods are consistent with special relativity, and so calculate the mass of the proton from its total energy.

## 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 = 89,875,517,873,681,764 J/kg (≈9.0 × 1016 joules per kilogram)

So one gram of mass 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) Conversions used: 1956 International (Steam) Table (IT) values where one calorie ≡ 4.1868 J and one BTU ≡ 1055.05585262 J. Weapons designers’ conversion value of one gram TNT ≡ 1000 calories used.
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 electromagnetic radiation released in this explosion carried the missing one 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).Assuming a 90/10 alloy of Pt/Ir by weight, a Cp of 25.9 for Pt and 25.1 for Ir, a Pt-dominated average Cp of 25.8, 5.134 moles of metal, and 132 J.K-1 for the prototype. A variation of ±1.5 picograms is of course, much smaller than the actual uncertainty in the mass of the international prototype, which is ±2 micrograms.

A spinning ball will weigh more than a ball that is not spinning.
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.Earth’s gravitational self-energy is 4.6 × 10-10 that of Earth’s total mass, or 2.7 trillion metric tons. Citation: The Apache Point Observatory Lunar Laser-Ranging Operation (APOLLO), T. W. Murphy, Jr. et al. University of Washington, Dept. of Physics ( 132 kB PDF, here.). 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 general theory of 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.

All types of energy contribute to mass, including potential energies. In relativity, interaction potentials are always due to local fields, not to direct nonlocal interactions, because signals can't travel faster than light. The field energy is stored in field gradients or, in some cases (for massive fields), where the field has a nonzero value. The mass associated with the potential energy is the mass–energy of the field energy. The mass associated with field energy can be detected, in principle, by gravitational experiments, by checking how the field attracts other objects gravitationally.

The energy in the gravitational field itself is different. There are several consistent ways to define the location of the energy in a gravitational field, all of which agree on the total energy when space is mostly flat and empty. But because the gravitational field can be made to vanish locally by choosing a free-falling frame, it is hard to avoid making the location dependent on the observer's frame of reference. The gravitational field energy is the familiar Newtonian gravitational potential energy in the Newtonian limit.

## Efficiency

In nuclear reactions, typically only a small fraction of the total mass–energy is converted into heat, light, radiation and motion, into a form which can be used. When an atom fissions, it loses only about 0.1% of its mass, and in a bomb or reactor not all the atoms can fission. In a fission based atomic bomb, the efficiency is only 40%, so only 40% of the fissionable atoms actually fission, and only 0.04% of the total mass appears as energy in the end. In nuclear fusion, more of the mass is released as usable energy, roughly 0.3%. But in a fusion bomb (see nuclear weapon yield), the bomb mass is partly casing and non-reacting components, so that again only about 0.03% of the total mass is released as usable energy.

In theory, it should be possible to convert all the mass in matter into heat and light, but none of the theoretically known methods are practical. One way to convert all rest-mass into usable energy is to annihilate matter with antimatter. But antimatter is rare in our universe, and must be made first. Making the antimatter requires more energy than would be released.

Since most of the mass of ordinary objects is in protons and neutrons, in order to convert all 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. Still, Gerardus '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 proposed by Belavin Polyakov Schwarz and Tyupkin. This process, can in principle convert all the mass of matter into neutrinos and usable energy, but it is normally extraordinarily slow. 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.

Many extensions of the standard model contain magnetic monopoles, and in some 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 anti-monopoles first. The energy required to produce monopoles is believed to be enormous, but magnetic charge is conserved, so that the lightest monopole is stable. All these properties are deduced in theoretical models—magnetic monopoles have never been observed, nor have they been produced in any experiment so far.

The third known method of total mass–energy conversion is using gravity, specifically black holes. Stephen Hawking theorized that black holes radiate thermally with no regard to how they are formed. So it is theoretically possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of Hawking radiation, however, the black hole used will radiate at a higher rate the smaller it is, producing usable powers at only small black hole masses, where usable may for example be something greater than the local background radiation. It is also worth noting that the ambient irradiated power would change with the mass of the black hole, increasing as the mass of the black hole decreases, or decreasing as the mass increases, at a rate where power is proportional to the inverse square of the mass. In a "practical" scenario, mass and energy could be dumped into the black hole to regulate this growth, or keep its size, and thus power output, near constant.

## Background

E=mc2 where m stands for rest mass (invariant mass) m_0, applies most simply to single particles viewed in an inertial frame where they have no momentum. But it also applies to ordinary objects composed of many particles so long as the particles are moving in different directions so the "net" or 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 poker. 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_0 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=mc2 is used with the rest mass or invariant mass of the object, the E given by the equation will be the rest energy of the object, and will change according to the object's internal energy, heat and sound and chemical binding energies (all of which must be added or subtracted from the object), but will not change with the object's overall motion (in the case of systems, the motion of its center of mass). However, if a system is closed, its invariant mass does not vary between different inertial observers (different inertial frames), and is also constant, and conserved.

If the equation E=mc2 is used with the relativistic mass of the object, the energy will be the total energy of the object, which is also conserved so long as no energy is added to or subtracted from the object, However, like the kinetic energy, this total energy will depend on the velocity of the object, and is different in different inertial frames. Thus, this quantity is not invariant between different inertial observers, even though it is constant over time for any single observer. As in the case of rest energy, these relationships for total energy are also true for systems of objects, so long as the system is closed.

Mass–Velocity Relationship

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

:K.E. = \frac{m_0 c^2}\sqrt{1-\frac{v^2}{c^2}} - m_0 c^2,

with v the velocity, and m_0 the rest mass.

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:

:K.E. = \frac{1}{2}m_0 v^2 + ...

Without this second term, there would be an additional contribution in the energy when the particle is not moving.

Einstein found that the total momentum of a moving particle is:

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

and it is this quantity which is conserved in collisions. The ratio of the momentum to the velocity is the relativistic mass, m.

:m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}

And the relativistic mass and the relativistic kinetic energy are related by the formula:

:K.E. = m c^2 - m_0 c^2. \,

Einstein wanted to omit the unnatural second term on the right-hand side, whose only purpose is to make the energy at rest zero, and to declare that the particle has a total energy which obeys:
: E = m c^2 \,

which is a sum of the rest energy m_0 c^2 and the kinetic energy. This total energy is mathematically more elegant, and fits better with the momentum in relativity. But to come to this conclusion, Einstein needed to think carefully about collisions. This expression for the energy implied that matter at rest has a huge amount of energy, and it is not clear whether this energy is physically real, or just a mathematical artifact with no physical meaning.

In a collision process where all the rest-masses are the same at the beginning as at the end, either expression for the energy is conserved. The two expressions only differ by a constant which is the same at the beginning and at the end of the collision. Still, by analyzing the situation where particles are thrown off a heavy central particle, it is easy to see that the inertia of the central particle is reduced by the total energy emitted. This allowed Einstein to conclude that the inertia of a heavy particle is increased or diminished according to the energy it absorbs or emits.

### 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 with an explicit index:

: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 (it is not Lorentz-invariant, in contrast to m_0). The equation holds for moving objects. When the velocity is small, the relativistic mass and the rest mass are almost exactly the same.

• E=mc2 either means E=m0c2 for an object at rest, or E=mrelc2 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=mc2 (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".

Considerable debate has ensued over the use of the concept "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. For example, one view is that only rest mass is a viable concept and is a property of the particle; while relativistic mass is a conglomeration of particle properties and properties of spacetime. A perspective that avoids this debate, due to Kjell Vøyenli, is that the Newtonian concept of mass as a particle property and the relativistic concept of mass have to be viewed as embedded in their own theories and as having no precise connection.

### Low-speed expansion

We can rewrite the expression E = γm0c2 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 m0c2 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 m0c2 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. By considering the emission of energy in different frames, Einstein could show that the rest energy has a real physical meaning.

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 asks:

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), and George Frederick Charles Searle (1897)—to understand how the mass of a charged object depends on the electrostatic field. Because the electromagnetic field carries part of the momentum of a moving charge, it was also suspected that the mass of an electron would vary with velocity near the speed of light. Searle calculated that it is impossible for a charged object to supersede the velocity of light because this would require an infinite amount of energy.

Following Thomson and Searle (1896), Wilhelm Wien (1900), Max Abraham (1902), and Hendrik Lorentz (1904) argued that this relation applies to the complete mass of bodies, because all inertial mass is electromagnetic in origin. The formula of the mass–energy-relation given by them was m=(4/3)E/c^2. Wien went on by stating, that if it is assumed that gravitation is an electromagnetic effect too, then there has to be a strict proportionality between (electromagnetic) inertial mass and (electromagnetic) gravitational mass. This interpretation is in the now discredited electromagnetic worldview, and the formulas that they discovered always included a factor of 4/3 in the proportionality. For example, the formulas given by Lorentz in 1904 for the pre-relativistic longitudinal and transverse masses were (in modern notation):

In July 1905 (published 1906), nearly at the same time when Einstein found the simple relation from relativity, Poincaré was able to explain the reason that the electromagnetic mass calculations always had a factor of 4/3. In order for a particle consisting of positive or negative charge to be stable, there must be some sort of attractive force of non-electrical nature which keeps it together. If the mass–energy of this force field is included in a way which is consistent with relativity theory, the attractive contribution adds an amount -(1/3)E/c^2 to the energy of the bodies, and this explains the discrepancy between the pure electromagnetic theory and relativity.

### 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.

Poincaré
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/c2). 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.

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.

Abraham and Hasenöhrl
Following Poincaré, Max Abraham in 1902 introduced the term "electromagnetic momentum" to maintain the action/reaction principle. Poincaré's result was verified by him, whereby the field density of momentum per cm3 is E/c^2 and E/c per cm2.

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 on the basis of 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, Hasenöhrl 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 the formula in his 1905 Annus Mirabilis paper "Does the Inertia of a Body Depend Upon Its Energy Content?"; rather, the paper states that if a body gives off the energy L in the form of radiation, its mass diminishes by L/c2. (Here, "radiation" means electromagnetic radiation, or light, and mass means the ordinary Newtonian mass of a slow-moving object.) This formulation relates only a change Δm in mass to a change L in energy without requiring the absolute relationship.

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.

First correct derivation (1905)

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 P = Mv.

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

But if the same process is considered 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 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 hasn't 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, discussed above.

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 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 = \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 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

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:

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 perpetuum mobile problem, because on the basis of the mass–energy equivalence he could show that the transport of inertia which 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 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 Samuel Tolver Preston,Bjerknes: S. Tolver Preston's Explosive Idea E = mc2. 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.Bizouard: Poincaré E = mc2 l’équation de Poincaré, Einstein et Planck.

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. However, it raised the question where this energy is coming 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 Ernest Rutherford and 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:

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, rejecting his ideas of 1904, 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 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 based on the rate of molecular diffusion through pores, a now-obsolete process that was then competitive and contributed a fraction of the enriched uranium used in the 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".David Bodanis, E = mc2: A Biography of the World's Most Famous Equation (New York: Walker, 2000).

While Serber's view of the strict lack of need to use mass–energy equivalence in designing the atomic bomb is correct, it does not take into account the pivotal role which this relationship played in making the fundamental leap to the initial hypothesis that large atoms could split into approximately equal halves. In late 1938, while on the winter walk on which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission, Lise Meitner and Otto Robert Frisch made direct use of Einstein's equation to help them understand the quantitative energetics of the reaction which overcame the "surface tension-like" forces holding 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 made use of "packing fraction," or nuclear binding energy values for elements, which Meitner had memorized. These, together with use of E = mc2 allowed them to realize on the spot that the basic fission process was energetically possible:

...We walked up and down in the snow, I on skis and she on foot.
...and gradually the idea took shape... explained by Bohr's idea that the nucleus is like a liquid drop; such a drop might elongate and divide itself...
We knew there were strong forces that would resist, ..just as surface tension.
But nuclei differed from ordinary drops.
At this point we both sat down on a tree trunk and started to calculate on scraps of paper.
...the Uranium nucleus might indeed be an unstable drop, ready to divide itself...
But, ...when the two drops separated they would be driven apart by electrical repulsion, about 200 MeV in all.
Fortunately Lise Meitner remembered how to compute the masses of nuclei... and worked out that the two nuclei formed... would be lighter by about one-fifth the mass of a proton.
Now whenever mass disappears energy is created, according to Einstein's formula E = mc2, and... the mass was just equivalent to 200 MeV; it all fitted!

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