E = mc2

In physics, mass–energy equivalence is the concept that the mass of an object orsystem is a measure of its energy content. For instance, adding 25 kilowatt-hours(90 megajoules) of any form of energy to any object increases its mass by 1microgram (and, accordingly, its inertia andweight) even though no matter has been added.

A physical system has a property called energy and a corresponding property called mass; the two properties are equivalent in that they are always both present in the same (i.e. constant) proportion to one another. Mass–energy equivalence arose originally from special relativity as a paradox described by Henri Poincaré.^[1] The equivalence of energy E and mass m is reliant on the speed of light c and is described by the famous equation:

Thus, this mass–energy relation states that the universal proportionality factor between equivalent amounts of energy and mass is equal to the speed of light squared. This also serves to convert units of mass to units of energy, no matter what system of measurement units used.

If a body is stationary, it still has some internal or intrinsic energy, called its rest energy. Rest mass and rest energy are equivalent and remain proportional to one another. When the body is in motion (relative to an observer), its total energy is greater than its rest energy. The rest mass (or rest energy) remains an important quantity in this case because it remains the same regardless of this motion, even for the extreme speeds or gravity considered in special and general relativity; thus it is also called the invariant mass.

On the one hand, the equation E = mc² can be applied to rest mass (m or m₀) and rest energy (E₀) to show their proportionality asE₀ = m₀c².^[2]

On the other hand, it can also be applied to the total energy (E_tot or simply E) and total mass of a moving body. The total mass is also called the relativistic mass m_rel. The total energy and total mass are related by E= m_relc².^[3]

Thus, the mass–energy relation E = mc² can be used to relate the rest energy to the rest mass, or to relate the total energy to the total mass. To instead relate the total energy or mass to the rest energy or mass, a generalization of the mass–energy relation is required: the energy–momentum relation.

E = mc² has frequently been invoked as an explanation for the origin of energy in nuclear processes specifically, but such processes can be understood as convertingnuclear potential energy in a manner precisely analogous to the way that chemical processes convert electrical potential energy. The more common association of mass–energy equivalence with nuclear processes derives from the fact that the large amounts of energy released in such reactions may exhibit enough mass that the mass loss (which is called the mass defect) may be measured, when the released energy (and its mass) have been removed from the system; while the energy released in chemical processes is smaller by roughly six orders of magnitude, and so the resultingmass defect is much more difficult to measure. For example, the loss of mass to an atom and a neutron, as a result of thecapture of the neutron and the production of a gamma ray, has been used to test mass–energy equivalence to high precision, as the energy of the gamma ray may be compared with the mass defect after capture. In 2005, these were found to agree to 0.0004%, the most precise test of the equivalence of mass and energy to date. This test was performed in the World Year of Physics 2005, a centennial celebration of Albert Einstein’s achievements in 1905.^[4]

Einstein was not the first to propose a mass–energy relationship (see the Historysection). However, Einstein was the first scientist to propose the E = mc² formula and the first to interpret mass–energy equivalence as a fundamental principle that follows from the relativistic symmetries ofspace and time.

Nomenclature

The formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.^[5][6]

• In “Does the inertia of a body depend upon its energy content?” (1905), Einstein used V to mean the speed of light in a vacuum and Lto mean the energy lost by a body in the form of radiation.^[7] Consequently, the equation E = mc² was not originally written as a formula but as a sentence in German saying that if a body gives off the energy L in the form of radiation, its mass diminishes byL/V². A remark placed above it informed that the equation was approximated by neglecting “magnitudes of fourth and higher orders” of a series expansion.^[8]
• In May 1907, Einstein explained that the expression for energy ε of a moving mass point assumes the simplest form, when its expression for the state of rest is chosen to be ε₀ = μV² (where μ is the mass), which is in agreement with the “principle of the equivalence of mass and energy”. In addition, Einstein used the formula μ =E₀/V², with E₀ being the energy of a system of mass points, in order to describe the energy and mass increase of that system when the velocity of the differently moving mass points is increased.^[9]
• In June 1907, Max Planck rewrote Einstein’s mass–energy relationship as M = (E₀ +pV₀)/c², where p is the pressure and V the volume, in order to express the relation between mass, its “latent energy”, and thermodynamic energy within the body.^[10]Subsequently in October 1907, this was rewritten as M₀ = E₀/c² and given a quantum interpretation by Johannes Stark, who assumed its validity and correctness (Gültigkeit).^[11]
• In December 1907, Einstein expressed the equivalence in the form M = μ + E₀/c² and concluded: A mass μ is equivalent, as regards inertia, to a quantity of energy μc².[…] It appears far more natural to consider every inertial mass as a store of energy.^[12][13]
• In 1909, Gilbert N. Lewis and Richard C. Tolman used two variations of the formula:m = E/c² and m₀ = E₀/c², with E being the energy of a moving body, E₀ its rest energy,m the relativistic mass, and m₀ the invariant mass.^[14] The same relations in different notation were used by Hendrik Lorentz in 1913 (published 1914), though he placed the energy on the left-hand side: ε = Mc² and ε₀= mc², with ε being the total energy (rest energy plus kinetic energy) of a moving material point, ε₀ its rest energy, M the relativistic mass, and m the invariant (or rest) mass.^[15]
• In 1911, Max von Laue gave a more comprehensive proof of M₀ = E₀/c² from thestress–energy tensor,^[16] which was later (1918) generalized by Felix Klein.^[17]
• Einstein returned to the topic once again after World War II and this time he wrote E =mc² in the title of his article^[18] intended as an explanation for a general reader by analogy.^[19]

Conservation of mass and energy

Main articles: Conservation of energy andConservation of mass

Mass and energy can be seen as two names (and two measurement units) for the same underlying, conserved physical quantity.^[20]Thus, the laws of conservation of energy andconservation of (total) mass are equivalent and both hold true.^[21] Einstein elaborated in a 1946 essay that “the principle of the conservation of mass […] proved inadequate in the face of the special theory of relativity. It was therefore merged with the energy [conservation] principle—just as, about 60 years before, the principle of the conservation of mechanical energy had been combined with the principle of the conservation of heat [thermal energy]. We might say that the principle of the conservation of energy, having previously swallowed up that of the conservation of heat, now proceeded to swallow that of the conservation of mass—and holds the field alone.”^[22]

If the conservation of mass law is interpreted as conservation of rest mass, it does not hold true in special relativity. Therest energy (equivalently, rest mass) of a particle can be converted, not “to energy” (it already is energy (mass)), but rather to otherforms of energy (mass) which require motion, such as kinetic energy, thermal energy, or radiant energy; similarly, kinetic or radiant energy can be converted to other kinds of particles which have rest energy (rest mass). In the transformation process, neither the total amount of mass nor the total amount of energy changes, since both are properties which are connected to each other via a simple constant.^[23][24] This view requires that if either energy or (total) mass disappears from a system, it will always be found that both have simply moved off to another place, where they may both be measured as an increase of both energy and mass corresponding to the loss in the first system.

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 a constant times the velocity, nor can thekinetic energy be a constant times the square of the velocity.

A property called the relativistic mass is defined as the ratio of the momentum of an object to its velocity.^[25] Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it. 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 thekinetic energy of the object. As the object approaches the speed of light, the relativistic mass grows infinitely, because the kinetic energy grows infinitely 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.^[3] 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, or invariant mass, and not relativistic mass.

The relativistic mass of a moving object is larger than the relativistic mass of an object that is not 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 things and systems made up of many parts, like an atomic nucleus, planet, or star, the relativistic mass is the sum of the relativistic masses (or energies) of the parts, because energies are additive in isolated 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 energy gained due to the forces of attraction in excess of the work done is removed from the system, then mass will be lost with this removed energy. 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 this energy from binding has been removed in the form of a gamma ray (which in this system, carries away the mass of the energy 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 sum of the individual masses of the sun and planets.

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, even those in relative motion. 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 any reference frame where the momentum is zero, and such a reference frame is also the only frame in which the object can be weighed. In a similar way, the theory of special relativity posits that the thermal energy in all objects (including solids) contributes to their total masses and weights, even though this energy is present as the kinetic and potential energies of the atoms in the object, and it (in a similar way to the gas) is not seen in the rest masses of the atoms that make up the object.

In a similar manner, even photons (light quanta), if trapped in a container space (as aphoton gas or thermal radiation), would contribute a mass associated with their energy to the container. Such an extra mass, in theory, could be weighed in the same way as any other type of rest mass. This is true in special relativity theory, even though individually photons have no rest mass. The property that trapped energy in any formadds weighable mass to systems that have no net momentum is one of the characteristic and notable consequences of relativity. It has no counterpart in classical Newtonian physics, in which radiation, light, heat, and kinetic energy never exhibit weighable mass under any circumstances.

Just as the relativistic mass of an isolated system is conserved through time, so also is its invariant mass. It is this property which allows the conservation of all types of mass in systems, and also conservation of all types of mass in reactions where matter is destroyed (annihilated), leaving behind the energy that was associated with it (which is now in non-material form, rather than material form). Matter may appear and disappear in various reactions, but mass and energy are both unchanged in this process.

Applicability of the strict mass–energy equivalence formula, E = mc²

As is noted above, two different definitions of mass have been used in special relativity, and also two different definitions of energy. The simple equation E = mc² is not generally applicable to all these types of mass and energy, except in the special case that the total additive momentum is zero for the system under consideration. In such a case, which is always guaranteed when observing the system from either its center of mass frame or its center of momentum frame, E =mc² is always true for any type of mass and energy that are chosen. Thus, for example, in the center of mass frame, the total energy of an object or system is equal to its rest mass times c², a useful equality. This is the relationship used for the container of gas in the previous example. It is not true in other reference frames where the center of mass is in motion. In these systems or for such an object, its total energy will depend on both its rest (or invariant) mass, and also its (total) momentum.^[26]

In inertial reference frames other than the rest frame or center of mass frame, the equation E = mc² remains true if the energy is the relativistic energy and the mass is the relativistic mass. It is also correct if the energy is the rest or invariant energy (also the minimum energy), and the mass is the rest mass, or the invariant mass. However, connection of the total or relativistic energy(E_r) with the rest or invariant mass (m₀) requires consideration of the system total momentum, in systems and reference frames where the total momentum has a non-zero value. The formula then required to connect the two different kinds of mass and energy, is the extended version of Einstein’s equation, called the relativistic energy–momentum relation:^[27]

or

Here the (pc)² term represents the square of the Euclidean norm (total vector length) of the various momentum vectors in the system, which reduces to the square of the simple momentum magnitude, if only a single particle is considered. This equation reduces to E = mc² when the momentum term is zero. For photons where m₀ = 0, the equation reduces to E_r = pc.

Meanings of the strict mass–energy equivalence formula, E = mc²

The mass–energy equivalence formula was displayed onTaipei 101during the event of the World Year of Physics 2005.

Mass–energy equivalence states that any object has a certain energy, even when it is stationary. 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 energyor thermal energy, in addition to anypotential 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 (that is, all the energy which is present in the object’s rest frame) contributes to the total mass of the body, which measures how much it resists acceleration. Each potential and kinetic energy makes a proportional contribution to the mass. As noted above, even if a box of ideal mirrors “contains” light, then the individually massless photons still contribute to the total mass of the box, by the amount of their energy divided by c².^[28]

In relativity, removing energy is removing mass, and for an observer in the center of mass frame, the formula m = E/c² indicates how much mass is lost when energy is removed. In a 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 decreases. In the same way, when any sort of energy is added to an isolated system, the increase in the mass is equal to the added energy divided by c². For example, when water is heated it gains about 1.11×10⁻¹⁷ kg 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 relativistic 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 is not moving (or when an inertial frame is chosen such that it is not moving). The term also applies to the invariant mass of systems when the system as a whole is not “moving” (has no net momentum). The rest and invariant masses are the smallest possible value of the mass of the object or system. They also are conserved quantities, so long as the system is isolated. Because of the way they are calculated, the effects of moving observers are subtracted, so these quantities do not change with the motion of the observer.

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 do not attract or repel, so that they do not 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.

Binding energy and the “mass defect”

Main article: binding energy

Whenever any type of energy is removed from a system, the mass associated with the energy is also removed, and the system therefore loses mass. This mass defect in the system may be simply calculated as Δm= ΔE/c², and this was the form of the equation historically first presented by Einstein in 1905. However, use of this formula in such circumstances has led to the false idea that mass has been “converted” to energy. This may be particularly the case when the energy (and mass) removed from the system is associated with the binding energy of the system. In such cases, the binding energy is observed as a “mass defect” or deficit in the new system.

The fact that the released energy is not easily weighed in many such cases, may cause its mass to be neglected as though it no longer existed. This circumstance has encouraged the false idea of conversion ofmass to energy, rather than the correct idea that the binding energy of such systems is relatively large, and exhibits a measurable mass, which is removed when the binding energy is removed. This energy is often released in the form of light and heat, which is too quickly and widely dispersed to be easily weighed, though it does carry mass^{[citation needed]}.

The difference between the rest mass of a bound system and of the unbound parts is the binding energy of the system, if this energy has been removed after binding. For example, 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 in theory weighs a little bit more than the fragments after the explosion, but this is true only so long as the fragments are cooled and the heat removed. In this case the mass difference is the energy/heat that is released when the dynamite explodes, and when this heat escapes, the mass associated with it escapes, only to be deposited in the surroundings which absorb the heat (so that total mass is conserved).

Such a change in mass may only happen when the system is open, and the energy and mass escapes. Thus, 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. If sitting on a scale, the weight and mass would not change. This would in theory also happen even with a nuclear bomb, if it could be kept in an ideal box of infinite strength, which did not rupture or pass radiation.^[24] Thus, a 21.5 kiloton (9×10¹³ joule) nuclear bomb produces about one gram of heat and electromagnetic radiation, but the mass of this energy would not be detectable in an exploded bomb in an ideal box sitting on a scale; instead, the contents of the box would be heated to millions of degrees without changing total mass and weight. If then, however, a transparent window (passing only electromagnetic radiation) were opened in such an ideal box after the explosion, and a beam of X-rays and other lower-energy light allowed to escape the box, it would eventually be found to weigh one gram less than it had before the explosion. This weight loss and mass loss would happen as the box was cooled by this process, to room temperature. However, any surrounding mass which had absorbed the X-rays (and other “heat”) would gain this gram of mass from the resulting heating, so the mass “loss” would represent merely its relocation. Thus, no mass (or, in the case of a nuclear bomb, no matter) would be “converted” to energy in such a process. Mass and energy, as always, would both be separately conserved.

Massless particles

Massless particles have zero rest mass. Their relativistic mass is simply their relativistic energy, divided by c², or m_rel =E/c².^[29][30] The energy for photons is E = hf, where h is Planck’s constant and f is the photon frequency. This frequency and thus the relativistic energy are frame-dependent.

If an observer runs away from a photon in the direction it travels from a source, having it catch up with the observer, then when the photon catches up it will be seen as having less energy than it had at the source. The faster the observer is traveling with regard to the source when the photon catches up, the less energy the photon will have. As an observer approaches the speed of light with regard to the source, the photon looks redder and redder, by relativistic Doppler effect (the Doppler shift is the relativistic formula), 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.

Massless particles contribute rest mass and invariant mass to systems

Two photons moving in different directions cannot both be made to have arbitrarily small total energy by changing frames, or by moving toward or away from them. The reason is that in a two-photon system, the energy of one photon is decreased by chasing after it, but the energy of the other will increase with the same shift in observer motion. Two photons not moving in the same direction will exhibit an inertial framewhere the combined energy is smallest, but not zero. This is called the center of massframe 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). The most that chasing a pair of photons can accomplish to decrease their energy is to put the observer in a frame where the photons have equal energy and are moving directly away from each other. In this frame, the observer is now moving in the same direction and speed as the center of mass of the two photons. The total momentum of the photons is now zero, since their momenta are equal and opposite. In this frame the two photons, as a system, have a mass equal to their total 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. It is only the invariant mass of a two-photon system that can be used to make a single particle with the same rest mass.

If the photons are 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 does not change after it disintegrates into two photons. 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. Thus, by calculating theinvariant mass of pairs of photons in a particle detector, pairs can be identified that were probably produced by pion disintegration.

A similar calculation illustrates that the invariant mass of systems is conserved, even when massive particles (particles with rest mass) within the system are converted to massless particles (such as photons). In such cases, the photons contribute invariant mass to the system, even though they individually have no invariant mass or rest mass. Thus, an electron and positron (each of which has rest mass) may undergoannihilation with each other to produce two photons, each of which is massless (has no rest mass). However, in such circumstances, no system mass is lost. Instead, the system of both photons moving away from each other has an invariant mass, which acts like a rest mass for any system in which the photons are trapped, or that can be weighed. Thus, not only the quantity of relativistic mass, but also the quantity of invariant mass does not change in transformations between “matter” (electrons and positrons) and energy (photons).

Relation to gravity

In physics, there are two distinct concepts ofmass: the gravitational mass and the inertial mass. The gravitational mass is the quantity that determines the strength of thegravitational field generated by an object, as well as the gravitational force acting on the object when it is immersed in a gravitational field produced by other bodies. The inertial mass, on the other hand, quantifies how much an object accelerates if a given force is applied to it. The mass–energy equivalence in special relativity refers to the inertial mass. However, already in the context of Newton gravity, the WeakEquivalence Principle is postulated: the gravitational and the inertial mass of every object are the same. Thus, the mass–energy equivalence, combined with the Weak Equivalence Principle, results in the prediction that all forms of energy contribute to the gravitational field generated by an object. This observation is one of the pillars of the general theory of relativity.

The above prediction, that all forms of energy interact gravitationally, has been subject to experimental tests. The first observation testing this prediction was made in 1919.^[31] During a solar eclipse,Arthur Eddington observed that the light from stars passing close to the Sun was bent. The effect is due to the gravitational attraction of light by the Sun. The observation confirmed that the energy carried by light indeed is equivalent to a gravitational mass. Another seminal experiment, the Pound–Rebka experiment, was performed in 1960.^[32] In this test a beam of light was emitted from the top of a tower and detected at the bottom. The frequency of the light detected was higher than the light emitted. This result confirms that the energy of photons increases when they fall in the gravitational field of the Earth. The energy, and therefore the gravitational mass, of photons is proportional to their frequency as stated by the Planck’s relation.