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In physics, mass–energy equivalence defines the relationship between mass and energy in a system’s rest frame, where the two values differ only by a constant and the units of measurement. The principle is described by Albert Einstein's famous formula: E = m c 2 {\displaystyle E=m\,c^{2}} The formula defines the energy E of a particle in its rest frame as the product of mass m with the speed of light squared ( c 2 ). Equivalently, the mass of a particle at rest is equal to its energy E divided by the speed of light squared ( c 2 ). Because the speed of light is a large number in everyday units (approximately 3 × 108 meters per second), the formula implies that a small amount of rest mass corresponds to an enormous amount of energy, which is independent of the composition of the matter. Rest mass, also called invariant mass, is the mass that is measured when the system is at rest. The rest mass is a fundamental physical property that remains independent of mome...

Mass–energy equivalence states that all objects having mass, called massive objects, also have corresponding intrinsic energy, even when they are stationary. In the rest frame of an object, where by definition it is motionless and so has no momentum, the mass and energy are equivalent and they differ only by a constant, the speed of light squared. 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. These energies tend to be much smaller than the mass of the object multiplied by the speed of light squared, which is on the order of 1019 Joules for a mass of one kilogram. Due to this principle, the mass of the atoms that come out of a nuclear reaction is less than the mass of the atoms that go in, and the difference in mass shows up as heat and light with the same equivalent energy as...

An object moves with different speeds in different frames of reference, depending on the motion of the observer. This implies the kinetic energy, in both Newtonian mechanics and relativity, is frame dependent , so that the amount of relativistic energy that an object is measured to have depends on the observer. The relativistic mass of an object is given by the relativistic energy divided by c2 . Because the relativistic mass is exactly proportional to the relativistic energy, relativistic mass and relativistic energy are nearly synonyms; the only difference between them is the units. The rest mass or invariant mass of an object is defined as the mass an object has when its rest frame, when it is not moving. The rest mass is typically denoted as just mass by physicists, though experiments have shown that the gravitational mass of an object depends on its total energy and not just its rest mass. The rest mass is the same for all inertial frames, as it is independent of the motion of ...

In some reactions matter particles can be destroyed and their associated energy released to the environment as other forms of energy, such as light and heat. One example of such a conversion takes place in elementary particle interactions, where the rest energy is transformed into kinetic energy. Such conversions between types of energy happen in nuclear weapons, in which the protons and neutrons in atomic nuclei lose a small fraction of their original mass, though the mass lost is not due to the destruction of any smaller constituents. Nuclear fission allows a tiny fraction of the energy associated with the mass to be converted into usable energy such as radiation, in the decay of the uranium, for instance, about 0.1% of the mass of the original atom is lost. In theory, it should be possible to destroy matter and convert all of the rest-energy associated with matter into heat and light, but none of the theoretically known methods are practical. One way to harness all the energy assoc...

Unlike a system's energy in an inertial frame, the relativistic energy ( E r {\displaystyle E_{r}} ) of a system depends on both the rest mass ( m 0 {\displaystyle m_{0}} ) and the total momentum of the system. The extension of Einstein's equation to these systems is given by:note E r 2 − | p → | 2 c 2 = m 0 2 c 4 E r 2 − ( p c ) 2 = ( m 0 c 2 ) 2 {\displaystyle {\begin{aligned}E_{r}^{2}-|{\vec {p}}\,|^{2}c^{2}&=m_{0}^{2}c^{4}\\E_{r}^{2}-(pc)^{2}&=(m_{0}c^{2})^{2}\end{aligned}}} or where the ( p c ) 2 {\displaystyle (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 is called the energy–momentum relation and reduces to ...

Using the Lorentz factor, γ , the energy–momentum can be rewritten as E = γmc 2 and expanded as a power series: E = m 0 c 2 1 + 1 2 ( v c ) 2 + 3 8 ( v c ) 4 + 5 16 ( v c ) 6 + … . {\displaystyle E=m_{0}c^{2}\left1+{\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, all but the first two terms can be ignored: E ≈ m 0 c 2 + 1 2 m 0 v 2 . {\displaystyle E\approx m_{0}c^{2}+{\frac {1}{2}}m_{0}v^{2}.} In classical mechanics, both the m 0 c 2 term and the high-speed corrections are ignored. The initial value of the energy is arbitrary, as only the change in energy can be measured, so the m 0 c 2 term ...

Application to nuclear physics edit The nuclear binding energy is the minimum energy that is required to disassemble the nucleus of an atom into its component parts. The mass of an atom is less than the sum of the masses of its constituents due to the attraction of the strong nuclear force. The difference between the two masses is called the mass defect and is related to the binding energy through Einstein's formula. The principle is used in modeling nuclear fission reactions and it implies a great amount of energy can be released by the nuclear fission chain reactions used in both nuclear weapons and nuclear power. A water molecule weighs a little less than two free hydrogen atoms and an oxygen atom. The minuscule mass difference is the energy needed to split the molecule into three individual atoms (divided by c 2 ), which was given off as heat when the molecule formed (this heat had mass). Similarly, a stick of dynamite in theory weighs a little bit more than the fragments afte...

While Einstein was the first to have correctly deduced the mass–energy equivalence formula, he was not the first to have related energy with mass, though nearly all previous authors thought that the energy that contributes to mass comes only from electromagnetic fields. Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps. Developments prior to Einstein edit Eighteenth century theories on the correlation of mass and energy included Isaac Newton in 1717, who speculated that light particles and matter particles were interconvertible in "Query 30" of the Opticks , where he asks: "Are not the gross bodies and light convertible into one another, and may not bodies receive much of their activity from the particles of light which enter their composition?" Swedish scientist and theologian Emanuel Swedenborg, in his Principia of 1734 theorized that all matte...