E=MC2

In physics, mass–energy equivalence is the concept formulated by Albert Einstein about the relation between mass and energy, which states every mass has an energy equivalent and vice versa. This relationship is expressed using the formula

where E is the energy of a physical system, m is the mass of the system, and c is the speed of light in a vacuum (about 3×108 m/s). In words, energy equals mass multiplied by the speed of light squared. Because the speed of light is a very large number in everyday units, the formula implies that any small amount of matter contains a very large amount of energy. Some of this energy may be released as heat and light by nuclear transformations. This also serves to convert units of mass to units of energy, no matter what system of measurement units used.

Mass–energy equivalence arose originally from special relativity as a paradox described by Henri Poincaré.[1] It was proposed by Einstein in 1905, in the paper "Does the inertia of a body depend upon its energy-content?", one of his Annus Mirabilis ("Miraculous Year") Papers.[2] Einstein was the first to propose that the equivalence of mass and energy is a general principle and a consequence of the symmetries of space and time.

A consequence of the mass–energy equivalence is that 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.Nomenclature The formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.[3][4] 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 L to mean the energy lost by a body in the form of radiation.[2] 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 by L/V2. A remark placed above it informed that the equation was approximated by neglecting "magnitudes of fourth and higher orders" of a series expansion.[5] 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.[6] 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.[7] 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).[8] 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.[9][10] 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.[11] 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.[12] In 1911, Max von Laue gave a more comprehensive proof of M0 = E0/c2 from the stress–energy tensor,[13] which was later (1918) generalized by Felix Klein.[14] Einstein returned to the topic once again after World War II and this time he wrote E = mc2 in the title of his article[15] intended as an explanation for a general reader by analogy.[16] Conservation of mass and energy Main articles: Conservation of energy and Conservation of mass Mass and energy can be seen as two names (and two measurement units) for the same underlying, conserved physical quantity.[17] Thus, the laws of conservation of energy and conservation of (total) mass are equivalent and both hold true.[18] 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."[19] If the conservation of mass law is interpreted as conservation of rest mass, it does not hold true in special relativity. The rest energy (equivalently, rest mass) of a particle can be converted, not "to energy" (it already is energy (mass)), but rather to other forms 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.[20][21] 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 the kinetic 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.[22] 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 the kinetic 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.[23] 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 a photon 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 form adds 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.[24] 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 (Er) 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:[25] 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 Er = pc.