Mass–Energy Equivalence

Mass–energy equivalence is the principle in relativity that a system’s mass corresponds to a definite amount of energy. Its familiar expression, $E=mc^2$ , states more precisely that the rest energy $E_0$ of a system equals its invariant mass $m$ multiplied by the square of the speed of light in vacuum. Established by Albert Einstein in 1905, the relation explains why changes in stored energy entail changes in mass and provides a quantitative foundation for understanding nuclear and particle reactions. (aps.org)

Meaning and numerical scale

In modern terminology, mass generally means invariant mass: a quantity independent of the observer’s motion. Rest energy is the energy measured in the system’s rest frame, where its total momentum is zero. The relation

E_0=mc^2

therefore describes energy that a system possesses even without motion as a whole. It is not a formula for kinetic energy, nor does it imply that matter must travel at light speed. (atlas-public.web.cern.ch)

In the International System of Units, $c$ is exactly $299{,}792{,}458$ metres per second. Consequently, one kilogram corresponds to approximately $8.99\times10^{16}$ joules of rest energy. Calculating the inverse relation, one joule added to a system at rest corresponds to a mass increase of approximately $1.11\times10^{-17}$ kilograms. These are mass–energy equivalents, not claims that every physical process can release all of an object’s rest energy. (bipm.org)

Older accounts sometimes use “relativistic mass,” defined as total energy divided by $c^2$ . Under that convention, mass increases with speed. The invariant-mass convention instead keeps mass unchanged when an otherwise unchanged object moves faster, while distinguishing its increasing total energy from its rest energy. (einstein-online.info)

Energy, momentum, and motion

For a free particle, rest energy forms part of the broader energy–momentum relation:

E^2=p^2c^2+m^2c^4,

where $E$ is total energy and $p$ is the magnitude of momentum. For a massive particle moving at speed $v$ ,

E=\gamma mc^2,\qquad \gamma=\frac{1}{\sqrt{1-v^2/c^2}}.

Its kinetic energy is therefore $K=(\gamma-1)mc^2$ . At speeds much smaller than $c$ , this approaches the familiar expression $K=\tfrac12mv^2$ of classical mechanics. (sfu.ca)

A photon has zero invariant mass but carries energy and momentum, satisfying $E=pc$ . This does not contradict mass–energy equivalence: a photon has no rest frame, so its energy cannot be interpreted as the rest energy of a massive object. For a collection of particles, the invariant mass is determined by their combined energy and total momentum, rather than simply by adding their individual masses. (sfu.ca)

Composite systems and binding energy

A composite object’s mass includes contributions from its internal motion and interactions. Increasing its internal energy, for example by heating it while leaving its overall momentum zero, increases its rest mass. Conversely, a bound system can have less mass than its separated constituents because energy must be supplied to separate them. (einstein-online.info)

For an atomic nucleus, this difference is called the mass defect. Its nuclear binding energy is

B=\left(\sum_i m_i-M\right)c^2,

where $M$ is the nuclear mass and the $m_i$ are the masses of its separated constituent nucleons. A helium nucleus, for example, has less mass than two free protons and two free neutrons. Chemical bonds produce analogous mass differences, but their associated energies are much smaller than nuclear binding energies. (einstein-online.info)

Nuclear reactions and particle transformations

In an energy-releasing nuclear reaction, the sum of the products’ rest masses is smaller than that of the reactants. The difference appears as kinetic energy, radiation, or other outgoing energy:

Q=(M_{\mathrm{initial}}-M_{\mathrm{final}})c^2.

Both nuclear fission of suitable heavy nuclei and nuclear fusion of suitable light nuclei can release energy in this way. Fusion powers the Sun and other stars; controlled fission supplies heat in nuclear power reactors. (energy.gov)

In particle physics, electron–positron annihilation provides another example. An electron and a positron with negligible initial kinetic energy can annihilate into two oppositely directed photons, each carrying approximately 511 kiloelectronvolts. Their rest energy becomes electromagnetic radiation. The reverse class of process, pair production, creates particles from available energy, subject to conservation of both energy and momentum. (hst-archive.web.cern.ch)

Such transformations do not destroy energy. They change its distribution among rest energy, motion, and radiation. Descriptions of mass being “converted into energy” ordinarily refer to a decrease in the sum of particle rest masses, not to a failure of energy conservation. (einstein-online.info)

Historical development and experimental tests

Einstein’s September 1905 paper, “Does the Inertia of a Body Depend upon Its Energy Content?”, followed his initial presentation of special relativity. He considered a body emitting radiation and concluded that its mass must decrease in proportion to the emitted energy. The argument connected energy content with inertia and suggested radioactive substances as possible experimental tests. (aps.org)

A precision direct test reported in 2005 compared independently measured atomic mass differences with gamma-ray energies associated with neutron capture in silicon and sulfur. Researchers from MIT, NIST, and the Institut Laue–Langevin found agreement with the mass–energy relation at approximately four parts in ten million. The experiment tested the quantitative equality between a measured mass change and its corresponding released energy. (nist.gov)