Mass–energy equivalence (Wikitext)

{{Short description|Relativity concept expressed as E {{=}} mc²}}
{{redirect2|E{{=}}MC²|E{{=}}mc²}}
[[File:M87 jet.jpg|thumb|Mass near the [[M87*]] black hole is converted into a very energetic [[astrophysical jet]], stretching five thousand [[Light-year|light years]]]]
In [[physics]], '''mass–energy equivalence''' is the relationship between [[mass]] and [[energy]] in a system's [[rest frame]], where the two quantities differ only by a multiplicative constant and the units of measurement.<ref name=Serway1217>{{Cite book|last1=Serway |first1=Raymond A.|title=Physics for scientists and engineers with modern physics|last2=Jewett |first2=John W. |last3=Peroomian |first3=Vahé|date=5 March 2013|isbn=978-1-133-95405-7|edition=9th|location=Boston, MA|oclc=802321453|pages=1217–1218}}</ref><ref name=Günther>{{Citation|last1=Günther|first1=Helmut|title=Einstein's Energy–Mass Equivalence|date=2019|url=https://doi.org/10.1007/978f=The Special Theory of Relativity: Einstein’s World in New Axiomatics|pages=97–105|editor-last=Günther|editor-first=Helmut|place=Singapore|publisher=Springer|language=en|doi=10.1007/978-981-13-7783-9_7|isbn=978-981-13-7783-9|access-date=2020-10-14|last2=Müller|first2=Volker|s2cid=209978258|editor2-last=Müller|editor2-first=Volker|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080229/https://link.springer.com/chapter/10.1007%2F978-981-13-7783-9_7|url-status=live}}</ref> The principle is described by the physicist [[Albert Einstein]]'s formula:&nbsp;<math qid=Q35875>E = mc^2</math>.<ref name="famous">{{cite book |title=E=mc<sup>12!</sup>: A Biography of the World's Most Famous Equation |edition=illustrated |first1=David |last1=Bodanis |publisher=Bloomsbury Publishing |year=2009 |isbn=978-0-8027-1821-1|at=preface|url=https://books.google.com/books?id=8TX2tFLZ7gYC }}</ref> In a [[reference frame]] where the system is moving, its [[relativistic energy]] and [[Mass in special relativity|relativistic mass]] (instead of [[Rest Mass|rest mass]]) obey the same formula.

The formula defines the energy {{math|''E''}} of a particle in its rest frame as the product of mass ({{math|''m''}}) with the [[speed of light]] squared ({{math|''c''<sup>2</sup>}}). Because the speed of light is a large number in everyday units (approximately {{cvt|300000|km/s|mi/s|comma=gaps|sigfig=3|disp=x| or }}), the formula implies that a small amount of "rest mass", measured when the system is at rest, corresponds to an enormous amount of energy, which is independent of the composition of the [[matter]].

Rest mass, also called [[invariant mass]], is a fundamental [[physical property]] that is independent of [[momentum]], even at extreme speeds approaching the speed of light. Its value is the same in all [[inertial frame of reference|inertial frames of reference]]. [[Massless particle]]s such as [[photon]]s have zero invariant mass, but massless [[free particle]]s have both momentum and energy.

The equivalence principle implies that when energy is lost in [[chemical reaction]]s, [[nuclear reaction]]s, and other [[energy transformation]]s, the [[Physical system|system]] will also lose a corresponding amount of mass. The energy, and mass, can be released to the environment as [[radiant energy]], such as [[light]], or as [[thermal energy]]. The principle is fundamental to many fields of physics, including [[nuclear physics|nuclear]] and [[particle physics]].

Mass–energy equivalence arose from [[special relativity]] as a [[paradox]] described by the French [[polymath]] [[Henri Poincaré]] (1854–1912).<ref name=action>{{Cite journal| author=Poincaré, H. | year=1900 | title=La théorie de Lorentz et le principe de réaction | journal=Archives Néerlandaises des Sciences Exactes et Naturelles | volume =5 | pages =252–278| title-link=s:fr:La théorie de Lorentz et le principe de réaction|language=fr|trans-title=[http://physicsinsights.org/poincare-1900.pdf The Theory of Lorentz and The Principle of Reaction]}}</ref> Einstein was the first to propose the equivalence of mass and energy as a general principle and a consequence of the [[Spacetime symmetries|symmetries of space and time]]. The principle first appeared in "Does the inertia of a body depend upon its energy-content?", one of his [[annus mirabilis papers|''annus mirabilis'' papers]], published on 21 November 1905.<ref name="inertia">{{Cite journal|last=Einstein|first=A.|date=1905|title=Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?|journal=Annalen der Physik|language=de|volume=323|issue=13|pages=639–641|doi=10.1002/andp.19053231314|bibcode=1905AnP...323..639E|trans-title=[http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ Does the Inertia of a Body Depend Upon its Energy-Content?]|issn=1521-3889|doi-access=free}}</ref> The formula and its relationship to momentum, as described by the [[energy–momentum relation]], were later developed by other physicists.

==Description==
{{Special relativity sidebar}}

Mass–energy equivalence states that all objects having [[mass]], or ''massive objects'', have a 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 equal or they differ only by a constant factor, the [[speed of light]] squared ({{math|''c''<sup>2</sup>}}).<ref name=Serway1217 /><ref name=Günther /> 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 (physics)|field of force]]. These energies tend to be much smaller than the mass of the object multiplied by {{math|''c''<sup>2</sup>}}, which is on the order of 10<sup>17</sup>&nbsp;[[joule]]s 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 the difference. In analyzing these explosions, Einstein's formula can be used with {{mvar|E}} as the energy released (removed), and {{mvar|m}} as the change in mass.

In [[Theory of relativity|relativity]], all the energy that moves with an object (i.e., the energy as measured in the object's rest frame) contributes to the total mass of the body, which measures how much it resists [[acceleration]]. If an isolated box of ideal mirrors could contain light, the individually massless photons would contribute to the total mass of the box by the amount equal to their energy divided by {{math|''c''<sup>2</sup>}}.<ref>{{Cite book|last1=Puri|first1=H. S.|last2=Hans|first2=S. P.|url=https://books.google.com/books?id=hrBe52GPHrYC|title=Mechanics, 2E|date=2003-07-01|publisher=Tata McGraw-Hill Education|isbn=978-0-07-047360-7|language=en|page=[https://books.google.com/books?id=hrBe52GPHrYC&pg=PA433 433]}}</ref> For an observer in the rest frame, removing energy is the same as removing mass and the formula {{math|1=''m'' = ''E''/''c''<sup>2</sup>}} indicates how much mass is lost when energy is removed.<ref>{{Cite book|last=Serway, Raymond A.|url=https://www.worldcat.org/oclc/802321453|title=Physics for scientists and engineers with modern physics.|others=Jewett, John W., Peroomian, Vahé.|date=5 March 2013|isbn=978-1-133-95405-7|edition=Ninth|location=Boston, MA|oclc=802321453|page=1386}}</ref> In the same way, when any energy is added to an isolated system, the increase in the mass is equal to the added energy divided by {{math|''c''<sup>2</sup>}}.<ref name=griffithsElectro512>{{Cite book|last=Griffiths, David J.|url=https://www.worldcat.org/oclc/40251748|title=Introduction to electrodynamics|date=1999|publisher=Prentice Hall|isbn=978-0-13-805326-0|edition=3rd|location=Upper Saddle River, N.J.|oclc=40251748|page=512|access-date=2020-10-15|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080229/https://www.worldcat.org/title/introduction-to-electrodynamics/oclc/40251748|url-status=live}}</ref>

==Mass in special relativity==
{{main|Mass in special relativity}}[[File:E=mc²-explication.svg|thumb|{{math|1=''E'' = ''mc''{{smallsup|2}}}}—In [[SI units]], the energy {{math|''E''}} is measured in [[Joules]], the mass {{math|''m''}} is measured in [[kilograms]], and the [[speed of light]] is measured in [[meters]] per [[second]].]]An object moves at different speeds in different [[Frame of reference|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 {{math|''c''<sup>2</sup>}}.<ref name="Tipler">{{Cite book|last1=Tipler|first1=Paul Allen|last2=Llewellyn|first2=Ralph A.|url= https://www.worldcat.org/oclc/49894577|title=Modern physics.|date=2003|publisher=W.H. Freeman|isbn=978-0-7167-4345-3|edition=4th|location=New York|oclc=49894577|pages=87–88}}</ref> Because the relativistic mass is exactly proportional to the relativistic energy, relativistic mass and relativistic energy are nearly [[synonym]]ous; the only difference between them is the [[unit of measurement|units]]. The ''rest mass'' or [[invariant mass]] of an object is defined as the mass an object has in its rest frame, when it is not moving with respect to the observer. Physicists typically use the term ''mass'', though experiments have shown an object's gravitational mass depends on its total energy and not just its rest mass.{{Citation needed|date=February 2021|reason=experiments which specifically address this, unless what was meant was that the gravitational mass of a system is the invariant mass of the system as opposed to the sum of the component invariant masses}} The rest mass is the same for all [[inertial frame]]s, as it is independent of the motion of the observer, it is the smallest possible value of the relativistic mass of the object. Because of the attraction between components of a system, which results in potential energy, the rest mass is almost never [[Additive function|additive]]; in general, the mass of an object is not the sum of the masses of its parts.<ref name="griffithsElectro512" /> The rest mass of an object is the total energy of all the parts, including kinetic energy, as observed from the center of momentum frame, and potential energy. The masses add up only if the constituents are at rest (as observed from the center of momentum frame) and do not attract or repel, so that they do not have any extra kinetic or potential energy.<ref group="note">They can also have a positive kinetic energy and a negative potential energy that exactly cancels.</ref> Massless particles are particles with no rest mass, and therefore have no intrinsic energy; their energy is due only to their momentum.

===Relativistic mass===
Relativistic mass depends on the motion of the object, so that different observers in relative motion see different values for it. The relativistic mass of a moving object is larger than the relativistic mass of an object at rest, because a moving object has kinetic energy. If the object moves slowly, the relativistic mass is nearly equal to the [[rest mass]] and both are nearly equal to the classical inertial mass (as it appears in [[Newton's laws of motion]]). If the object moves 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. Massless particles also have relativistic mass derived from their kinetic energy, equal to their relativistic energy divided by {{math|''c''<sup>2</sup>}}, or {{math|1=''m''{{ssub|rel}} = ''E''/''c''<sup>2</sup>}}.<ref>{{Cite book|last=Mould|first=Richard A.|url=https://books.google.com/books?id=lfGE-wyJYIUC|title=Basic Relativity|date=2001-11-01|publisher=Springer Science & Business Media|isbn=978-0-387-95210-9|language=en|page=[https://books.google.com/books?id=lfGE-wyJYIUC&pg=PA126 126]}}</ref><ref>{{Cite book|last=Chow|first=Tai L.|url=https://books.google.com/books?id=dpnpMhw1zo8C|title=Introduction to Electromagnetic Theory: A Modern Perspective|date=2006|publisher=Jones & Bartlett Learning|isbn=978-0-7637-3827-3|language=en|page=[https://books.google.com/books?id=dpnpMhw1zo8C&pg=PA392 392]|access-date=2016-02-22|archive-date=2016-12-02|archive-url=https://web.archive.org/web/20161202172249/https://books.google.com/books?id=dpnpMhw1zo8C|url-status=live}}</ref> The speed of light is one in a system where length and time are measured in [[natural units]] and the relativistic mass and energy would be equal in value and dimension. As it is just another name for the energy, the use of the term ''relativistic mass'' is redundant and physicists generally reserve ''mass'' to refer to rest mass, or invariant mass, as opposed to relativistic mass.<ref name=elementaryParticles>{{Cite book|last=Griffiths, David J.|title=Introduction to elementary particles|date=2008|publisher=Wiley-VCH|isbn=978-3-527-40601-2|edition=2nd, rev.|location=Weinheim [Germany]|oclc=248969635|page=101}}</ref><ref name=serway>{{Cite book|last=Serway, Raymond A.|title=Physics for scientists and engineers with modern physics.|others=Jewett, John W., Peroomian, Vahé.|date=5 March 2013|isbn=978-1-133-95405-7|edition=Ninth|location=Boston, MA|oclc=802321453|page=1219}}</ref> A consequence of this terminology is that the [[conservation of mass|mass is not conserved]] in special relativity, whereas [[Momentum#Conservation|the conservation of momentum]] and [[conservation of energy]] are both fundamental laws.<ref name=elementaryParticles />

===Conservation of mass and energy===
{{Main|Conservation of energy|Conservation of mass}}

The conservation of energy is a universal principle in physics and holds for any interaction, along with the conservation of momentum.<ref name=elementaryParticles /> The classical conservation of mass, in contrast, is violated in certain relativistic settings.<ref name=serway /><ref name=elementaryParticles /> This concept has been experimentally proven in a number of ways, including the conversion of mass into kinetic energy in nuclear reactions and other interactions between [[elementary particle]]s.<ref name=serway /> While modern physics has discarded the expression 'conservation of mass', in older terminology a [[relativistic mass]] can also be defined to be equivalent to the energy of a moving system, allowing for a ''conservation of relativistic mass''.<ref name=elementaryParticles /> Mass conservation breaks down when the energy associated with the mass of a particle is converted into other forms of energy, such as kinetic energy, thermal energy, or [[radiant energy]]. Similarly, kinetic or radiant energy can be used to create particles that have mass, always conserving the total energy and momentum.<ref name=elementaryParticles />

===Massless particles===
Massless particles have zero rest mass. The [[Planck–Einstein relation]] for the energy for [[photon]]s is given by the equation {{math|1=''E'' = ''hf''}}, where {{mvar|h}} is the [[Planck constant]] and {{mvar|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 the photon travels from a source, and it catches up with the observer, the observer sees it 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 would be seen to have. As an observer approaches the speed of light with regard to the source, the [[redshift]] of the photon increases, according to the [[relativistic Doppler effect]]. The energy of the photon is reduced and as the wavelength becomes arbitrarily large, the photon's energy approaches zero, due to the massless nature of photons, which does not permit any intrinsic energy.

===Composite systems===
{{see also|Mass in special relativity#The mass of composite systems}}

For closed systems made up of many parts, like an [[atomic nucleus]], planet, or star, the relativistic energy is given by the sum of the relativistic energies of each of the parts, because energies are additive in these systems. If a system is [[Binding energy#Mass-energy relation|''bound'']] by attractive forces, and the energy gained in excess of the work done is removed from the system, then mass is lost with this removed energy. The mass of an atomic nucleus is less than the total mass of the [[proton]]s and [[neutron]]s that make it up.<ref name=Serway1386>{{Cite book|last=Serway, Raymond A.|url=https://www.worldcat.org/oclc/802321453|title=Physics for scientists and engineers with modern physics.|others=Jewett, John W., Peroomian, Vahé.|date=5 March 2013|isbn=978-1-133-95405-7|edition=Ninth|location=Boston, MA|oclc=802321453|page=1386|access-date=15 October 2020|archive-date=21 February 2021|archive-url=https://web.archive.org/web/20210221080236/https://www.worldcat.org/title/physics-for-scientists-and-engineers-with-modern-physics/oclc/802321453|url-status=live}}</ref> This mass decrease is also equivalent to the energy required to break up the nucleus into individual protons and neutrons. This effect can be understood by looking at the potential energy of the individual components. The individual particles have a force attracting them together, and forcing them apart increases the potential energy of the particles in the same way that lifting an object up on earth does. This energy is equal to the work required to split the particles apart. The mass of the [[Solar System]] is slightly less than the sum of its individual masses.

For an isolated system of particles moving 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 {{math|''c''<sup>2</sup>}}) in the [[center of momentum frame]]. The ''center of momentum frame'' is defined so that the system has zero total momentum; the term [[center of mass]] frame is also sometimes used, where the ''center of mass frame'' is a special case of the center of momentum frame where the center of mass is put at the origin. 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's 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, 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.<ref name=griffithsElectro512 /> Similarly, even photons, if trapped in an isolated container, would contribute their energy to the mass of the container. Such extra mass, in theory, could be weighed in the same way as any other type of rest mass, 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 consequences of relativity. It has no counterpart in classical Newtonian physics, where energy never exhibits weighable mass.<ref name=griffithsElectro512 />

===Relation to gravity===
Physics has two concepts of mass, the gravitational mass and the inertial mass. The gravitational mass is the quantity that determines the strength of the [[gravitational 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 weak [[equivalence 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 prediction that all forms of energy interact gravitationally has been subject to experimental tests. One of the first observations testing this prediction, called the [[Eddington experiment]], was made during the [[Solar eclipse of May 29, 1919]].<ref>{{Cite journal|last1=Dyson|first1=F.W.|author2=Eddington, A.S.|author3=Davidson, C.R.|name-list-style=amp|date=January 1920|title=IX. A determination of the deflection of light by the sun's gravitational field, from observations made at the total eclipse of May 29, 1919|journal=Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character|language=en|volume=220|issue=571–581|pages=291–333|doi=10.1098/rsta.1920.0009|bibcode=1920RSPTA.220..291D|issn=0264-3952|doi-access=free}}</ref><ref>{{Cite journal|last=Stanley|first=Matthew|date=2003-03-01|title='An Expedition to Heal the Wounds of War' The 1919 Eclipse and Eddington as Quaker Adventurer|url=https://www.journals.uchicago.edu/doi/10.1086/376099|journal=Isis|volume=94|issue=1|pages=57–89|doi=10.1086/376099|pmid=12725104|bibcode=2003Isis...94...57S|s2cid=25615643|issn=0021-1753|access-date=2020-10-22|archive-date=2020-08-05|archive-url=https://web.archive.org/web/20200805053416/https://www.journals.uchicago.edu/doi/10.1086/376099|url-status=live}}</ref> During the [[solar eclipse]], the English [[astronomer]] and physicist [[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.<ref>{{Cite journal|last1=Pound|first1=R. V.|last2=Rebka|first2=G. A.|date=1960-04-01|title=Apparent Weight of Photons|journal=Physical Review Letters|language=en|volume=4|issue=7|pages=337–341|doi=10.1103/PhysRevLett.4.337|bibcode=1960PhRvL...4..337P|issn=0031-9007|doi-access=free}}</ref> 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.

==Efficiency==

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.<ref name=Serway1217 /> One example of such a conversion takes place in elementary particle interactions, where the rest energy is transformed into kinetic energy.<ref name=Serway1217 /> 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.<ref name="bulletin1950">{{Cite journal|last=Bethe|first=Hans A.|date=1950-04-01|title=The Hydrogen Bomb|url=https://doi.org/10.1080/00963402.1950.11461231|journal=Bulletin of the Atomic Scientists|volume=6|issue=4|pages=99–104|doi=10.1080/00963402.1950.11461231|bibcode=1950BuAtS...6d..99B|issn=0096-3402}}</ref> 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 associated with mass is to annihilate matter with [[antimatter]]. [[baryon asymmetry|Antimatter is rare in our universe]], however, and the known mechanisms of production require more usable energy than would be released in annihilation. [[CERN]] estimated in 2011 that over a billion times more energy is required to make and store antimatter than could be released in its annihilation.<ref>{{Cite web|title=Making antimatter {{!}} Angels & Demons - The science behind the story|url=https://angelsanddemons.web.cern.ch/antimatter/making-antimatter.html|access-date=2020-10-15|website=angelsanddemons.web.cern.ch|archive-date=2020-11-01|archive-url=https://web.archive.org/web/20201101023510/https://angelsanddemons.web.cern.ch/antimatter/making-antimatter.html|url-status=live}}</ref>

As most of the mass which comprises ordinary objects resides in protons and neutrons, converting all the energy of ordinary matter into more useful forms requires that the protons and neutrons be converted to lighter particles, or particles with no mass at all. In the [[Standard Model of particle physics]], the [[baryon number|number of protons plus neutrons]] is nearly exactly conserved. Despite this, [[Gerard 't Hooft]] showed that there is a process that converts protons and neutrons to [[antielectron]]s and [[neutrino]]s.<ref>{{Cite journal|last='t Hooft|first=G.|date=1976-12-15|title=Computation of the quantum effects due to a four-dimensional pseudoparticle|url=http://dx.doi.org/10.1103/physrevd.14.3432|journal=Physical Review D|volume=14|issue=12|pages=3432–3450|doi=10.1103/physrevd.14.3432|bibcode=1976PhRvD..14.3432T|issn=0556-2821|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080229/https://journals.aps.org/prd/abstract/10.1103/PhysRevD.14.3432|url-status=live}}</ref> This is the weak [[SU(2)]] [[instanton]] proposed by the physicists [[Alexander Belavin]], [[Alexander Markovich Polyakov]], [[Albert Schwarz]], and Yu. S. Tyupkin.<ref>{{Cite journal|last1=Belavin|first1=A.A.|last2=Polyakov|first2=A.M.|last3=Schwartz|first3=A.S.|last4=Tyupkin|first4=Yu.S.|date=October 1975|title=Pseudoparticle solutions of the Yang-Mills equations|url=http://dx.doi.org/10.1016/0370-2693(75)90163-x|journal=Physics Letters B|volume=59|issue=1|pages=85–87|doi=10.1016/0370-2693(75)90163-x|bibcode=1975PhLB...59...85B|issn=0370-2693|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080231/https://www.sciencedirect.com/science/article/abs/pii/037026937590163X?via%3Dihub|url-status=live}}</ref> This process, can in principle destroy matter and convert all the energy of matter into neutrinos and usable energy, but it is normally extraordinarily slow. It was later shown that the process occurs rapidly at extremely high temperatures that would only have been reached shortly after the [[Big Bang]].<ref>{{cite journal | last1 = Klinkhammer | first1 = F. | author-link2 = Nicholas Manton | last2 = Manton | first2 = N. | year = 1984| title = A Saddle Point Solution in the Weinberg Salam Theory | journal = Physical Review D | volume = 30 | issue = 10| page = 2212 | doi = 10.1103/PhysRevD.30.2212 | bibcode = 1984PhRvD..30.2212K }}</ref>

Many extensions of the standard model contain [[magnetic monopole]]s, and in some models of [[grand unification theory|grand unification]], these monopoles catalyze [[proton decay]], a process known as the [[Callan–Rubakov effect]].<ref>{{cite journal | last1 = Rubakov | first1 = V. A. | year = 1988 | title = Monopole Catalysis of Proton Decay | journal = Reports on Progress in Physics | volume = 51 | issue = 2| pages = 189–241 | doi = 10.1088/0034-4885/51/2/002 | s2cid = 250904729 }}</ref> This process would be an efficient mass–energy conversion at ordinary temperatures, but it requires making monopoles and anti-monopoles, whose production is expected to be inefficient. Another method of completely annihilating matter uses the gravitational field of black holes. The British [[theoretical physicist]] [[Stephen Hawking]] theorized<ref>{{cite journal | last1 = Hawking | first1 = S.W. | year = 1974 | title = Black Holes Explosions? | journal = Nature | volume = 248 | issue = 5443| page = 30 | doi = 10.1038/248030a0 | bibcode = 1974Natur.248...30H | s2cid = 4290107 }}</ref> it is possible to throw matter into a black hole and use the emitted heat to generate power. According to the theory of [[Hawking radiation]], however, larger black holes radiate less than smaller ones, so that usable power can only be produced by small black holes.

==Extension for systems in motion==
{{main|Energy–momentum relation}}

Unlike a system's energy in an inertial frame, the relativistic energy (<math>E_{\rm rel}</math>) of a system depends on both the rest mass (<math>m_0</math>) and the total momentum of the system. The extension of Einstein's equation to these systems is given by:<ref>{{Cite book|last=Forshaw|first=Jeffrey Robert|url=https://www.worldcat.org/oclc/291193458|title=Dynamics and relativity|date=2009|publisher=John Wiley & Sons|others=Smith, A. Gavin.|isbn=978-0-470-01459-2|location=Chichester, UK|oclc=291193458|page=259|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080230/https://www.worldcat.org/title/dynamics-and-relativity/oclc/291193458|url-status=live}}</ref><ref>{{Cite book|last=McMahon|first=David|url=https://www.worldcat.org/oclc/61684277|title=Relativity demystified|date=2006|publisher=McGraw-Hill|isbn=978-0-07-145545-9|location=New York|oclc=61684277|chapter=1: Special relativity}}</ref><ref group="note">Some authors state the expression equivalently as <math>E = \gamma m_0 c^2</math> where <math>\gamma</math> is the [[Lorentz factor]].</ref>

:<math>\begin{align}
E_{\rm rel}^2 - |\mathbf{p} |^2 c^2 &= m_0^2 c^4 \\
E_{\rm rel}^2 - (pc)^2 &= (m_0 c^2)^2
\end{align}</math>

or

<math>\begin{align}
E_{\rm rel} = \sqrt{ (m_0 c^2)^2 + (pc)^2 } \,\!
\end{align}</math>

where the <math>(pc)^2</math> 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 <math>E_{\rm rel} = mc^2</math> when the momentum term is zero. For photons where <math>m_0 = 0</math>, the equation reduces to <math>E_{\rm rel} = pc</math>.

==Low-speed expansion==
Using the [[Lorentz factor]], {{math|''γ''}}, the energy–momentum can be rewritten as {{math|''E'' {{=}} ''γmc''<sup>2</sup>}} and expanded as a [[power series]]:
:<math>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]. </math>

For speeds much smaller than the speed of light, higher-order terms in this expression get smaller and smaller because {{math|{{sfrac|''v''|''c''}}}} is small. For low speeds, all but the first two terms can be ignored:
:<math>E \approx m_0 c^2 + \frac{1}{2} m_0 v^2. </math>

In [[classical mechanics]], both the {{math|''m''<sub>0</sub>''c''<sup>2</sup>}} 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 {{math|''m''<sub>0</sub>''c''<sup>2</sup>}} term is ignored in classical physics. While the higher-order terms become important at higher speeds, the Newtonian equation is a highly accurate low-speed approximation; adding in the third term yields:
:<math>E \approx m_0 c^2 + \frac{1}{2}m_0 v^2 \left(1 + \frac{3v^2}{4c^2}\right)</math>.
The difference between the two approximations is given by <math>\tfrac{3v^2}{4c^2}</math>, a number very small for everyday objects. In 2018 NASA announced the [[Parker Solar Probe]] was the fastest ever, with a speed of {{convert|153,454|mph|m/s}}.<ref>{{Cite web|title=Parker Solar Probe Becomes Fastest-Ever Spacecraft – Parker Solar Probe|url=https://blogs.nasa.gov/parkersolarprobe/2018/10/29/parker-solar-probe-becomes-fastest-ever-spacecraft/|access-date=2020-08-25|website=blogs.nasa.gov|language=en-US|archive-date=2020-08-17|archive-url=https://web.archive.org/web/20200817022956/https://blogs.nasa.gov/parkersolarprobe/2018/10/29/parker-solar-probe-becomes-fastest-ever-spacecraft/|url-status=live}}</ref> The difference between the approximations for the Parker Solar Probe in 2018 is <math>\tfrac{3v^2}{4c^2} \approx 3.9 \times 10^{-8}</math>, which accounts for an energy correction of four parts per hundred million. The [[gravitational constant]], in contrast, has a standard [[relative uncertainty]] of about <math>2.2 \times 10^{-5}</math>.<ref>{{Cite web|title=CODATA Value: Newtonian constant of gravitation|url=https://physics.nist.gov/cgi-bin/cuu/Value?bg|access-date=2020-08-25|website=physics.nist.gov|archive-date=2011-08-27|archive-url=https://web.archive.org/web/20110827153649/http://www.physics.nist.gov/cgi-bin/cuu/Value?bg|url-status=live}}</ref>

==Applications==
===Application to nuclear physics===
{{main|Nuclear binding energy|Mass defect}}
[[File:USS Enterprise (CVAN-65), USS Long Beach (CGN-9) and USS Bainbridge (DLGN-25) underway in the Mediterranean Sea during Operation Sea Orbit, in 1964.jpg|thumb|right|Task Force One, the world's first nuclear-powered task force. {{USS|Enterprise|CVN-65|2}}, {{USS|Long Beach|CGN-9|2}} and {{USS|Bainbridge|CGN-25|2}} in formation in the Mediterranean, 18 June 1964. ''Enterprise'' crew members are spelling out Einstein's mass–energy equivalence formula {{math|1=''E'' = ''mc''<sup>2</sup>}} on the flight deck.]]

The [[nuclear binding energy]] is the minimum energy that is required to disassemble the nucleus of an atom into its component parts.<ref>{{Cite book|last=Rohlf|first=James William.|url=https://www.worldcat.org/oclc/29563946|title=Modern physics from [alpha] to Z⁰|date=1994|publisher=John Wiley|isbn=978-0-471-57270-1|edition=1st|location=New York|oclc=29563946|page=20}}</ref> 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]].<ref name=radiopharm>{{Citation|last=Rösch|first=Frank|title=The Basics of Nuclear Chemistry and Radiochemistry: An Introduction to Nuclear Transformations and Radioactive Emissions|date=2019|url=https://doi.org/10.1007/978-3-319-98947-1_3|work=Radiopharmaceutical Chemistry|pages=27–61|editor-last=Lewis|editor-first=Jason S.|place=Cham|publisher=Springer International Publishing|language=en|doi=10.1007/978-3-319-98947-1_3|isbn=978-3-319-98947-1|s2cid=134052082|access-date=2020-10-14|editor2-last=Windhorst|editor2-first=Albert D.|editor3-last=Zeglis|editor3-first=Brian M.}}</ref> The difference between the two masses is called the ''mass defect'' and is related to the binding energy through Einstein's formula.<ref name=radiopharm /><ref>{{Cite book|last=Serway, Raymond A.|title=Physics for scientists and engineers with modern physics.|others=Jewett, John W., Peroomian, Vahé.|date=5 March 2013|isbn=978-1-133-95405-7|edition=9th|location=Boston, MA|oclc=802321453|page=1419}}</ref><ref>{{Cite book|last1=Frisch|first1=David H|url=https://www.worldcat.org/oclc/222569|title=Elementary particles|last2=Thorndike|first2=Alan M|date=1964|publisher=D. Van Nostrand|location=Princeton, N.J.|language=en|oclc=222569|pages=11–12}}</ref> 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 reaction]]s used in both [[nuclear weapon]]s 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 {{math|''c''<sup>2</sup>}}), 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 after the explosion; in this case the mass difference is the energy and heat that is released when the dynamite explodes. Such a change in mass may only happen when the system is open, and the energy and mass are allowed to escape. 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]].{{refn|group=note|name="A. Wheeler, 1992. pp. 248"|See Taylor and Wheeler<ref>{{Cite book|last=Taylor, Edwin F.|url=https://www.worldcat.org/oclc/25165077|title=Spacetime physics: introduction to special relativity |date=1992|publisher=W.H. Freeman|others=Wheeler, John Archibald, 1911-2008.|isbn=978-0-7167-2327-1|edition=2nd|location=New York|oclc=25165077|pages=248–249}}</ref> for a discussion of mass remaining constant after detonation of nuclear bombs, until heat is allowed to escape.}} Thus, a 21.5&nbsp;[[TNT equivalent|kiloton]] ({{val|9|e=13|u=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 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 that absorbed the X-rays (and other "heat") would ''gain'' this gram of mass from the resulting heating, thus, in this case, the mass "loss" would represent merely its relocation.

===Practical examples===
Einstein used the [[centimetre–gram–second system of units]] (cgs), but the formula is independent of the system of units. In natural units, the numerical value of the speed of light is set to equal 1, and the formula expresses an equality of numerical values: {{math|1=''E'' = ''m''}}. In the [[International System of Units|SI]] system (expressing the ratio {{math|{{sfrac|''E''|''m''}}}} in [[joules]] per kilogram using the value of {{math|''c''}} in [[metre per second|meters per second]]):<ref>{{cite book |title=Megawatts and Megatons: The Future of Nuclear Power and Nuclear Weapons |edition=illustrated |first1=Richard L. |last1=Garwin |first2=Georges |last2=Charpak |publisher=University of Chicago Press |year=2002 |isbn=978-0-226-28427-9 |page=[https://books.google.com/books?id=1YgBR6shTckC&pg=PA17 17] |url=https://books.google.com/books?id=1YgBR6shTckC}}</ref>

:{{math|1={{Sfrac|''E''|''m''}} =}} {{math|1=''c''<sup>2</sup> = ({{val|299792458|u=m/s}})<sup>2</sup> =}} {{math|{{val|89875517873681764|u=J/kg}}}} (≈&nbsp;9.0 × 10<sup>16</sup> joules per kilogram).

So the energy equivalent of one kilogram of mass is
*89.9&nbsp;[[joules|petajoules]]
*25.0&nbsp;billion [[kilowatt-hour]]s (≈&nbsp;25,000&nbsp;[[GW·h]])
*21.5&nbsp;trillion [[calorie|kilocalories]] (≈&nbsp;21&nbsp;Pcal)<ref group="note" name="Conversion">Conversions used: 1956 International (Steam) Table (IT) values where one calorie ≡&nbsp;4.1868&nbsp;J and one BTU ≡&nbsp;1055.05585262&nbsp;J. Weapons designers' conversion value of one&nbsp;gram TNT ≡ 1000&nbsp;calories used.</ref>
*85.2&nbsp;trillion [[British thermal unit|BTUs]]<ref group="note" name="Conversion"/>
*0.0852 [[Quad (unit)|quads]]

or the energy released by combustion of the following:
*21 500&nbsp;[[kiloton]]s of [[TNT equivalent|TNT-equivalent]] energy (≈&nbsp;21&nbsp;Mt)<ref group="note" name="Conversion"/>
*{{val|2630000000}} [[litre]]s or {{val|695000000}} US [[gallon]]s of automotive [[Gasoline#Combustion energy content|gasoline]]

Any time energy is released, the process can be evaluated from an {{math|1=''E'' = ''mc''<sup>2</sup>}} perspective. For instance, the "[[Fat Man|Gadget]]"-style bomb used in the [[Trinity test]] and the [[bombing of Nagasaki]] had an explosive yield equivalent to 21&nbsp;kt of TNT.<ref>{{Cite web|last=John|first=Malik|date=September 1985|title=The Yields of the Hiroshima and Nagasaki Nuclear Explosions|url=https://permalink.lanl.gov/object/tr?what=info:lanl-repo/lareport/LA-08819|access-date=1 October 2020|website=Los Alamos National Laboratories|archive-date=13 October 2020|archive-url=https://web.archive.org/web/20201013123611/https://permalink.lanl.gov/object/tr?what=info%3Alanl-repo%2Flareport%2FLA-08819|url-status=live}}</ref> About 1&nbsp;kg of the approximately 6.15&nbsp;kg of [[plutonium]] in each of these bombs fissioned into lighter elements totaling almost exactly one gram less, after cooling. The electromagnetic radiation and kinetic energy (thermal and blast energy) released in this explosion carried the missing gram of mass.

Whenever energy is added to a system, the system gains mass, as shown when the equation is rearranged:
* A [[spring (device)|spring's]] mass increases whenever it is put into compression or tension. Its mass increase arises from the increased 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 [[thermal energy]]) increases its mass. For example, consider the world's primary mass standard for the kilogram, made of [[platinum]] and [[iridium]]. If its temperature is allowed to change by 1&nbsp;°C, its mass changes by 1.5 picograms (1&nbsp;pg = {{val|1|e=-12|u=g}}).<ref group=note>Assuming a 90/10 alloy of Pt/Ir by weight, a {{math|''C<sub>p</sub>''}} of 25.9 for Pt and 25.1 for Ir, a Pt-dominated average {{math|''C<sub>p</sub>''}} of 25.8, 5.134&nbsp;moles of metal, and 132&nbsp;J⋅K<sup>−1</sup> for the prototype. A variation of ±1.5&nbsp;picograms is much smaller than the uncertainty in the mass of the international prototype, which is ±2&nbsp;micrograms.</ref>
* A spinning ball has greater mass than when it is not spinning. Its increase of mass is exactly the equivalent of the mass of [[Rotational energy|energy of rotation]], which is itself the sum of the kinetic energies of all the moving parts of the ball. For example, [[the Earth]] itself is more massive due to its rotation, than it would be with no rotation. The rotational energy of the Earth is greater than 10<sup>24</sup> Joules, which is over 10<sup>7</sup> kg.<ref>{{Cite news|last=Allain|first=Rhett|date=2009-06-22|title=Rotational Energy of the Earth as an energy source|magazine=Wired|url=https://www.wired.com/2009/06/rotational-energy-of-the-earth-as-an-energy-source/|access-date=2020-10-14|issn=1059-1028|archive-date=2020-10-16|archive-url=https://web.archive.org/web/20201016032556/https://www.wired.com/2009/06/rotational-energy-of-the-earth-as-an-energy-source/|url-status=live}}</ref>

==History==
{{further|History of special relativity}}
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.<ref>{{Cite book|last=Whittaker, E. T.|author-link=E. T. Whittaker|title=[[A History of the Theories of Aether and Electricity]]|date=1989|publisher=Dover Publications|isbn=978-0-486-26126-3|location=New York|oclc=20357018|volume=2|chapter=The relativity theory of Poincaré and Lorentz}}</ref><ref name="mill">{{Cite book|last=Miller, Arthur I.|url=https://www.worldcat.org/oclc/5894058|title=Albert Einstein's special theory of relativity: emergence (1905) and early interpretation, 1905-1911|date=1981|publisher=Addison-Wesley Pub. Co., Advanced Book Program|isbn=978-0-201-04680-9|location=Reading, Mass.|section=Some Others Who Discussed an Association Between Energy and Mass|oclc=5894058|pages=339–340}}</ref><ref name="darr">{{Citation|author=Darrigol, O.|chapter-url=https://www.worldcat.org/oclc/317084635|title=Einstein, 1905-2005: Poincaré Seminar 2005|chapter=The Genesis of the theory of relativity|date=2006|publisher=Birkhäuser Verlag|others=Damour, Thibault.|isbn=978-3-7643-7436-5|location=Basel|oclc=317084635|pages=1–22|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080237/https://www.worldcat.org/title/einstein-1905-2005-poincare-seminar-2005/oclc/317084635|url-status=live}}</ref> Once discovered, Einstein's formula was initially written in many different notations, and its interpretation and justification was further developed in several steps.<ref name=jammer2 /><ref name=hecht>{{Cite journal|last=Hecht|first=Eugene|date=June 2011|title=How Einstein confirmed E0=mc2|url=http://aapt.scitation.org/doi/10.1119/1.3549223|journal=American Journal of Physics|language=en|volume=79|issue=6|pages=591–600|doi=10.1119/1.3549223|bibcode=2011AmJPh..79..591H|issn=0002-9505|access-date=2020-10-14|archive-date=2019-04-05|archive-url=https://web.archive.org/web/20190405015020/https://aapt.scitation.org/doi/10.1119/1.3549223|url-status=live}}</ref>

===Developments prior to Einstein===
[[File:Portrait of Sir Isaac Newton, 1689.jpg|thumb|In the revised English edition of [[Isaac Newton]]'s ''[[Opticks]]'', published in 1717, Newton speculated on the equivalence of mass and light.]]

Eighteenth century theories on the correlation of mass and energy included that devised by the English scientist [[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?"<ref>{{Cite web|title=Selected Queries from Isaac Newton's Opticks {{!}} Inters.org|url=http://inters.org/newton-opticks-queries|access-date=2020-10-14|website=inters.org|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080253/http://inters.org/newton-opticks-queries|url-status=live}}</ref> Swedish scientist and theologian [[Emanuel Swedenborg]], in his ''[[The Principia (book)|Principia]]'' of 1734 theorized that all matter is ultimately composed of dimensionless points of "pure and total motion". He described this motion as being without force, direction or speed, but having the potential for force, direction and speed everywhere within it.<ref>{{Cite book|last=Swedenborg|first=Emanuel|url=https://books.google.com/books?id=1keP6ZXitBYC|title=Principia rerum naturalium sive Novorum tentaminum phaenomena mundi elementaris philosophice explicandi...|date=1734|publisher=sumptibus Friderici Hekelii|language=la|page=32|chapter=De Simplici Mundi vel Puncto naturali}}</ref><ref>{{Cite book|last=Swedenborg|first=Emanuel|url=https://www.worldcat.org/oclc/863755|title=The principia: or, The first principles of natural things, being new attempts toward a philosophical explanation of the elementary world.|translator-last=Clissold|translator-first=Augustus|translator-link=Augustus Clissold|date=1845|pages=55–57|publisher=W. Newbery; O. Clapp|location=London; Boston|language=en|oclc=863755|access-date=2020-10-14}}</ref>

During the nineteenth century there were several speculative attempts to show that mass and energy were proportional in various [[Aether theories|ether theories]].<ref>{{Cite book|last=Kragh|first=Helge|url=https://www.jstor.org/stable/j.ctv10crfmk|title=Quantum generations: a history of physics in the twentieth century|pages=3–12|date=1999|isbn=978-0-691-21419-1|language=en|oclc=1159003206|chapter=Fin-de-Siècle Physics: A World Picture in Flux|publisher=Princeton University Press|doi=10.2307/j.ctv10crfmk |jstor=j.ctv10crfmk|s2cid=243126061 }}</ref> In 1873 the Russian physicist and mathematician [[Nikolay Umov]] pointed out a relation between mass and energy for ether in the form of {{math|1=''Е'' = ''kmc''<sup>2</sup>}}, where {{math|0.5 ≤ ''k'' ≤ 1}}.<ref>''Умов Н. А.'' Избранные сочинения [N.A. Umov. Selected Works].(1950) М. — Л.. (in Russian)</ref> The writings of the English engineer [[Samuel Tolver Preston]],<ref>{{Cite book|last=Preston|first=S. Tolver|url=https://www.worldcat.org/oclc/5834362|title=Physics of the ether|date=1875|publisher=E. & F.N. Spon|location=London; New York|language=en|oclc=5834362|access-date=23 October 2020}}</ref> and a 1903 paper by the Italian industrialist and [[geologist]] [[Olinto De Pretto]],<ref>{{Cite book|last1=Bartocci|first1=U|url=https://www.worldcat.org/oclc/44897464|title=Albert Einstein e Olinto De Pretto: la vera storia della formula più famosa del mondo|last2=Bonicelli|first2=Bianca Maria|date=1999|publisher=Andromeda|location=Bologna|language=it|oclc=44897464|access-date=2020-10-14}}</ref><ref>{{Cite news|last=Carroll|first=Rory|date=1999-11-11|title=Einstein's E=mc2 'was Italian's idea'|language=en-GB|work=The Guardian|url=https://www.theguardian.com/world/1999/nov/11/rorycarroll|access-date=2020-10-23|issn=0261-3077|archive-date=2020-10-23|archive-url=https://web.archive.org/web/20201023034128/https://www.theguardian.com/world/1999/nov/11/rorycarroll|url-status=live}}</ref> presented a mass–energy relation. Italian mathematician and math historian [[Umberto Bartocci]] observed that there were only [[six degrees of separation|three degrees of separation]] linking De Pretto to Einstein, concluding that Einstein was probably aware of De Pretto's work.<ref>{{Cite book|last1=Bartocci|first1=U|url=https://www.worldcat.org/oclc/44897464|title=Albert Einstein e Olinto De Pretto: la vera storia della formula più famosa del mondo|last2=Bonicelli|first2=Bianca Maria|date=1999|publisher=Andromeda|location=Bologna|language=it|oclc=44897464|section=Pretto, O. ''Reale Instituto Veneto Di Scienze, Lettere Ed Arti'', LXIII, II, 439–500|access-date=2020-10-14}}</ref><ref>{{Cite web|url=http://www.cartesio-episteme.net/fis/depret-bombay.htm|title=Information about the "De Pretto-Einstein case"|website=www.cartesio-episteme.net}}</ref>Preston and De Pretto, following physicist [[Georges-Louis Le Sage]], imagined that the universe was filled with an [[Aether (classical element)|ether]] of tiny particles that always move at speed {{mvar|c}}. Each of these particles has a kinetic energy of {{math|''mc''<sup>2</sup>}} up to a small numerical factor. The nonrelativistic kinetic energy formula did not always include the traditional factor of {{sfrac|2}}, since German polymath [[Gottfried Leibniz]] introduced kinetic energy without it, and the {{sfrac|2}} is largely conventional in prerelativistic physics.<ref>{{Cite journal|last=Prentis|first=Jeffrey J.|date=August 2005|title=Why is the energy of motion proportional to the square of the velocity?|url=http://aapt.scitation.org/doi/10.1119/1.1927550|journal=American Journal of Physics|language=en|volume=73|issue=8|pages=701–707|doi=10.1119/1.1927550|bibcode=2005AmJPh..73..701P|issn=0002-9505}}</ref> By assuming that every particle has a mass that is the sum of the masses of the ether particles, the authors concluded that all matter contains an amount of kinetic energy either given by {{math|1=''E'' = ''mc''<sup>2</sup>}} or {{math|1=2''E'' = ''mc''<sup>2</sup>}} depending on the convention. A particle ether was usually considered unacceptably speculative science at the time,<ref>{{Cite journal|last=Worrall|first=John|date=1985-03-01|title=Reviews|url=https://academic.oup.com/bjps/article/36/1/81/1477386|journal=The British Journal for the Philosophy of Science|language=en|volume=36|issue=1|pages=81–85|doi=10.1093/bjps/36.1.81|issn=0007-0882|access-date=2020-10-14}}</ref> and since these authors did not formulate relativity, their reasoning is completely different from that of Einstein, who used relativity to change frames.

In 1905, and independent of Einstein, French polymath [[Gustave Le Bon]] speculated that atoms could release large amounts of latent energy, reasoning from an all-encompassing qualitative philosophy of physics.<ref>{{Cite book|last=Le Bon|first=Gustave|url=https://www.worldcat.org/oclc/875679536|title=The evolution of forces|at=[http://www.rexresearch.com/lebonfor/evforp1.htm#p1b3ch2 The Energetical Explanation of Phenomena]|date=2014|isbn=978-1-4942-9965-1|language=en|oclc=875679536|access-date=2020-10-14}}</ref><ref>{{Cite journal|last=Bizouard|first=Christian|date=2004|title=E = mc2 l'équation de Poincaré, Einstein et Planck: Henri Poincare et la physique|url=https://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=16384743|journal=E = mc2 l'équation de Poincaré, Einstein et Planck: Henri Poincare et la physique|issue=4|pages=35–37|issn=0151-0304}}</ref>

====Electromagnetic mass====
{{Main|Electromagnetic mass}}
There were many attempts in the 19th and the beginning of the 20th century—like those of British physicists [[J. J. Thomson]] in 1881 and [[Oliver Heaviside]] in 1889, and [[George Frederick Charles Searle]] in 1897, German physicists [[Wilhelm Wien]] in 1900 and [[Max Abraham]] in 1902, and the Dutch physicist [[Hendrik Antoon Lorentz]] in 1904—to understand how the mass of a charged object depends on the [[electrostatic field]].<ref>{{Cite book|last=Whittaker, E. T.|author-link=E. T. Whittaker|title=[[A History of the Theories of Aether and Electricity]]|date=1989|publisher=Dover Publications|isbn=978-0-486-26126-3|location=New York|oclc=20357018|volume=1|chapter=The followeres of Maxwell}}</ref> This concept was called [[electromagnetic mass]], and was considered as being dependent on velocity and direction as well. Lorentz in 1904 gave the following expressions for longitudinal and transverse electromagnetic mass:
:<math>m_{L}=\frac{m_{0}}{\left(\sqrt{1-\frac{v^{2}}{c^{2}}}\right)^{3}},\quad m_{T}=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}} </math>,

where
:<math>m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}</math>

Another way of deriving a type of electromagnetic mass was based on the concept of [[radiation pressure]]. In 1900, French polymath [[Henri Poincaré]] associated electromagnetic radiation energy with a "fictitious fluid" having momentum and mass<ref name=action />
:<math>m_{em}=\frac{E_{em}}{c^2}\,.</math>

By that, Poincaré tried to save the center of mass theorem in Lorentz's theory, though his treatment led to radiation paradoxes.<ref name="darr" />

Austrian physicist [[Friedrich Hasenöhrl]] showed in 1904 that electromagnetic [[cavity radiation]] contributes the "apparent mass"
:<math>m_{0}=\frac{4}{3}\frac{E_{em}}{c^{2}}</math>

to the cavity's mass. He argued that this implies mass dependence on temperature as well.<ref>{{Cite web|date=2011-08-23|title=Did Einstein discover E = mc2?|url=https://physicsworld.com/a/did-einstein-discover-e-mc2/|access-date=2020-10-14|website=[[Physics World]]|language=en-GB|archive-date=2020-10-16|archive-url=https://web.archive.org/web/20201016144605/https://physicsworld.com/a/did-einstein-discover-e-mc2/|url-status=live}}</ref>

===Einstein: mass–energy equivalence===
[[File:Einstein 1921 by F Schmutzer - restoration.jpg|thumb|Photo of [[Albert Einstein]] in 1921]]

Einstein did not write the exact formula {{math|1=''E'' = ''mc''<sup>2</sup>}} in his 1905 [[Annus Mirabilis Papers|''Annus Mirabilis'' paper]] "Does the Inertia of an object Depend Upon Its Energy Content?";<ref name="inertia" /> rather, the paper states that if a body gives off the energy {{mvar|L}} in the form of radiation, its mass diminishes by {{math|{{sfrac|''L''|''c''<sup>2</sup>}}}}.<ref group=note>Here, "radiation" means [[electromagnetic radiation]], or light, and mass means the ordinary Newtonian mass of a slow-moving object.</ref> This formulation relates only a change {{math|Δ''m''}} in mass to a change {{mvar|L}} in energy without requiring the absolute relationship. The relationship convinced him that mass and energy can be seen as two names for the same underlying, conserved physical quantity.<ref>{{Cite journal|last=Hecht|first=Eugene|date=September 2009|title=Einstein on mass and energy|url=http://aapt.scitation.org/doi/10.1119/1.3160671|journal=American Journal of Physics|language=en|volume=77|issue=9|pages=799–806|doi=10.1119/1.3160671|bibcode=2009AmJPh..77..799H|issn=0002-9505|quote=Einstein was unequivocally against the traditional idea of conservation of mass. He had concluded that mass and energy were essentially one and the same; 'inert mass is simply latent energy.' He made his position known publicly time and again…|access-date=2020-10-14|archive-date=2019-05-28|archive-url=https://web.archive.org/web/20190528171923/https://aapt.scitation.org/doi/10.1119/1.3160671|url-status=live}}</ref> He has stated that the laws of conservation of energy and conservation of mass are "one and the same".<ref>{{Cite journal|last=Einstein|first=Albert|date=1940-05-24|title=Considerations Concerning the Fundaments of Theoretical Physics|url=https://www.science.org/doi/10.1126/science.91.2369.487|journal=Science|language=en|volume=91|issue=2369|pages=487–492|doi=10.1126/science.91.2369.487|issn=0036-8075|pmid=17847438|bibcode=1940Sci....91..487E|quote=There followed also the principle of the equivalence of mass and energy, with the laws of conservation of mass and energy becoming one and the same.|access-date=2020-10-14|archive-date=2020-07-11|archive-url=https://web.archive.org/web/20200711070127/https://science.sciencemag.org/content/91/2369/487|url-status=live}}</ref> 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 law|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."<ref>{{cite book|url= https://books.google.com/books?id=SYPbH6xCbUMC&pg=PA14 |first=Albert |last=Einstein |title=The Theory of Relativity (And Other Essays) |publisher=Citadel Press |year=1950 |page=14|isbn=978-0-8065-1765-0 }}</ref>

====Mass–velocity relationship====
[[File:E mc 2 IMG 0859.jpg|thumb|The equation in [[Albert Einstein]]'s own handwriting from 1912]]

In developing [[special relativity]], Einstein found that the [[kinetic energy#Relativistic kinetic energy of rigid bodies|kinetic energy]] of a moving body is
:<math>E_k = m_0 c^2( \gamma -1 ) = m_0 c^2\left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}} - 1\right),</math>

with {{math|''v''}} the [[velocity]], {{math|''m''{{sub|0}}}} the rest mass, and {{math|''γ''}} the Lorentz factor.

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, thus satisfying the [[correspondence principle]]:
:<math id=kineticEnergy>E_k = \frac{1}{2}m_0 v^2 + \cdots </math>

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

====Einsteins's view on mass====
{{see also|Mass in special relativity#Early developments: transverse and longitudinal mass}}

Einstein, following Lorentz and Abraham, used velocity- and direction-dependent mass concepts in his 1905 electrodynamics paper and in another paper in 1906.<ref>{{Cite journal|last=Einstein|first=A.|date=1905|title=Zur Elektrodynamik bewegter Körper|journal=Annalen der Physik|language=de|volume=322|issue=10|pages=891–921|doi=10.1002/andp.19053221004|bibcode=1905AnP...322..891E|trans-title=[http://www.fourmilab.ch/etexts/einstein/specrel/www/ On the Electrodynamics of Moving Bodies]|doi-access=free}}</ref><ref>{{Cite journal|last=Einstein|first=A.|date=1906|title=Über eine Methode zur Bestimmung des Verhältnisses der transversalen und longitudinalen Masse des Elektrons|url=http://doi.wiley.com/10.1002/andp.19063261310|journal=Annalen der Physik|language=de|volume=326|issue=13|pages=583–586|doi=10.1002/andp.19063261310|bibcode=1906AnP...326..583E|trans-title=[https://einsteinpapers.press.princeton.edu/vol2-trans/221 On a method for the determination of the ratio of the transverse and the longitudinal mass of the electron]|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080247/https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19063261310|url-status=live}}</ref> In Einstein's first 1905 paper on {{math|1=''E'' = ''mc''<sup>2</sup>}}, he treated {{mvar|m}} as what would now be called the ''rest mass'',<ref name="inertia" /> and it has been noted that in his later years he did not like the idea of "relativistic mass".<ref name=Okun>{{Cite journal|last=Okun|first=Lev B.|date=June 1989|title=The Concept of Mass|url=http://dx.doi.org/10.1063/1.881171|journal=Physics Today|volume=42|issue=6|pages=31–36|doi=10.1063/1.881171|bibcode=1989PhT....42f..31O|issn=0031-9228}}</ref>

In older physics terminology, relativistic energy is used in lieu of relativistic mass and the term "mass" is reserved for the rest mass.<ref name=elementaryParticles /> Historically, there has been considerable debate over the use of the concept of "relativistic mass" and the connection of "mass" in relativity to "mass" in Newtonian dynamics. 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. Another view, attributed to Norwegian physicist 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.<ref name="Jammer">{{Cite book|last=Jammer|first=Max|url= https://www.worldcat.org/oclc/614715841|title=Concepts of mass in contemporary physics and philosophy|date=2000|publisher=Princeton University Press|isbn=978-1-4008-1219-6 |page=51|location=Princeton, N.J.|oclc=614715841}}</ref><ref name="Vøyenli">{{Cite journal|last1=Eriksen|first1=Erik|last2=Vøyenli|first2=Kjell|date=February 1976|title=The classical and relativistic concepts of mass|url=http://link.springer.com/10.1007/BF00708670|journal=Foundations of Physics|language=en|volume=6|issue=1|pages=115–124|doi=10.1007/BF00708670|bibcode=1976FoPh....6..115E|s2cid=120139174|issn=0015-9018}}</ref>

====Einstein's 1905 derivation====
Already in his relativity paper "On the electrodynamics of moving bodies", Einstein derived the correct expression for the kinetic energy of particles:
:<math>E_{k}=mc^{2}\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)</math>.

Now the question remained open as to which formulation applies to bodies at rest. This was tackled by Einstein in his paper "Does the inertia of a body depend upon its energy content?", one of his [[Annus Mirabilis papers]]. Here, Einstein used {{math|''V''}} to represent the speed of light in vacuum and {{math|''L''}} to represent the [[energy]] lost by a body in the form of radiation.<ref name="inertia"/> Consequently, the equation {{math|1=''E'' = ''mc''<sup>2</sup>}} was not originally written as a formula but as a sentence in German saying that "if a body gives off the energy {{math|''L''}} in the form of radiation, its mass diminishes by {{math|{{sfrac|''L''|''V''<sup>2</sup>}}}}." A remark placed above it informed that the equation was approximated by neglecting "magnitudes of fourth and higher orders" of a [[Series (mathematics)|series expansion]].<ref group=note>See the sentence on the last page 641 of the original German edition, above the equation
{{math|1=''K''<sub>0</sub> − ''K''<sub>1</sub> = {{sfrac|''L''|''V''<sup>2</sup>}} {{sfrac|''v''<sup>2</sup>|2}}}}. See also the sentence above the last equation in the English translation, {{math|1=''K''<sub>0</sub> − ''K''<sub>1</sub> = {{sfrac|1|2}}({{sfrac|''L''|''c''<sup>2</sup>}})''v''<sup>2</sup>}}, and the comment on the symbols used in ''About this edition'' that follows the translation.</ref> Einstein used a body emitting two light pulses in opposite directions, having energies of {{math|''E''<sub>0</sub>}} before and {{math|''E''<sub>1</sub>}} after the emission as seen in its rest frame. As seen from a moving frame, this becomes {{math|''H''<sub>0</sub>}} and {{math|''H''<sub>1</sub>}}. Einstein obtained, in modern notation:
:<math>\left(H_{0}-E_{0}\right)-\left(H_{1}-E_{1}\right)=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)</math>.

He then argued that {{math|''H'' − ''E''}} can only differ from the kinetic energy {{math|''K''}} by an additive constant, which gives
:<math>K_{0}-K_{1}=E\left(\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}-1\right)</math>.

Neglecting effects higher than third order in {{math|{{sfrac|''v''|''c''}}}} after a [[Taylor series]] expansion of the right side of this yields:
:<math>K_{0}-K_{1}=\frac{E}{c^{2}}\frac{v^{2}}{2}.</math>

Einstein concluded that the emission reduces the body's mass by {{math|{{sfrac|''E''|''c''<sup>2</sup>}}}}, and that the mass of a body is a measure of its energy content.

The correctness of Einstein's 1905 derivation of {{math|1=''E'' = ''mc''<sup>2</sup>}} was criticized by German theoretical physicist [[Max Planck]] in 1907, who argued that it is only valid to first approximation. Another criticism was formulated by American physicist [[Herbert Ives]] in 1952 and the Israeli physicist [[Max Jammer]] in 1961, asserting that Einstein's derivation is based on [[begging the question]].<ref name=jammer2>{{Cite book|last=Jammer|first=Max|url=https://www.worldcat.org/oclc/37546758|title=Concepts of mass: in classical and modern physics|orig-year=1961|date=1997|publisher=Dover Publications|isbn=978-0-486-29998-3|location=Mineola, N.Y.|oclc=37546758|page=51}}</ref><ref>{{Cite journal|last=Ives|first=Herbert E.|date=1952-08-01|title= Derivation of the Mass-Energy Relation|url=https://www.osapublishing.org/abstract.cfm?URI=josa-42-8-540|journal=Journal of the Optical Society of America|language=en|volume=42|issue=8 |page=540|doi=10.1364/JOSA.42.000540 |issn=0030-3941}}</ref> Other scholars, such as American and Chilean [[philosopher]]s [[John Stachel]] and [[Roberto Torretti]], have argued that Ives' criticism was wrong, and that Einstein's derivation was correct.<ref>{{Cite journal|last1=Stachel|first1=John|last2=Torretti|first2=Roberto|date=August 1982|title=Einstein's first derivation of mass–energy equivalence|url=http://aapt.scitation.org/doi/10.1119/1.12764|journal=American Journal of Physics|language=en|volume=50|issue=8|pages=760–763|doi=10.1119/1.12764|bibcode=1982AmJPh..50..760S|issn=0002-9505|access-date=2020-10-14|archive-date=2019-05-28|archive-url=https://web.archive.org/web/20190528172007/https://aapt.scitation.org/doi/10.1119/1.12764|url-status=live}}</ref> American physics writer [[Hans Ohanian]], in 2008, agreed with Stachel/Torretti's criticism of Ives, though he argued that Einstein's derivation was wrong for other reasons.<ref>{{Cite journal|last=Ohanian|first=Hans C.|date=May 2009|title=Did Einstein prove E=mc2?|url=https://linkinghub.elsevier.com/retrieve/pii/S1355219809000112|journal=Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics|language=en|volume=40|issue=2|pages=167–173|doi=10.1016/j.shpsb.2009.03.002|bibcode=2009SHPMP..40..167O|access-date=2020-10-14}}</ref>

====Relativistic center-of-mass theorem of 1906====
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: "Although the merely formal considerations, which we will need for the proof, are already mostly contained in a work by H. Poincaré<sup>2</sup>, for the sake of clarity I will not rely on that work."<ref>{{Cite journal|last=Einstein|first=A.|date=1906|title=Das Prinzip von der Erhaltung der Schwerpunktsbewegung und die Trägheit der Energie|trans-title=[https://einsteinpapers.press.princeton.edu/vol2-trans/214 The Principle of Conservation of Motion of the Center of Gravity and the Inertia of Energy]|url=http://doi.wiley.com/10.1002/andp.19063250814|journal=Annalen der Physik|language=de|volume=325|issue=8|pages=627–633|doi=10.1002/andp.19063250814|bibcode=1906AnP...325..627E|s2cid=120361282 |quote=Trotzdem die einfachen formalen Betrachtungen, die zum Nachweis dieser Behauptung durchgeführt werden müssen, in der Hauptsache bereits in einer Arbeit von H. Poincaré enthalten sind<sup>2</sup>, werde ich mich doch der Übersichtlichkeit halber nicht auf jene Arbeit stützen.|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080259/https://onlinelibrary.wiley.com/doi/abs/10.1002/andp.19063250814|url-status=live}}</ref> 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 ''[[perpetual motion]]'' problem because, on the basis of the mass–energy equivalence, he could show that the transport of inertia that 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 {{math|1=''E'' = ''mc''<sup>2</sup>}}, because mass conservation appears as a special case of the [[energy conservation law]].

====Further developments====
There were several further developments in the first decade of the twentieth century. In May 1907, Einstein explained that the expression for energy {{math|''ε''}} of a moving mass point assumes the simplest form when its expression for the state of rest is chosen to be {{math|1=''ε''<sub>0</sub> = ''μV''<sup>2</sup>}} (where {{math|''μ''}} is the mass), which is in agreement with the "principle of the equivalence of mass and energy". In addition, Einstein used the formula {{math|1=''μ'' = {{sfrac|''E''<sub>0</sub>|''V''<sup>2</sup>}}}}, with {{math|''E''<sub>0</sub>}} being the energy of a system of mass points, to describe the energy and mass increase of that system when the velocity of the differently moving mass points is increased.<ref>{{Cite journal|last=Einstein|first=A.|date=1907|title=Über die vom Relativitätsprinzip geforderte Trägheit der Energie|trans-title=[https://einsteinpapers.press.princeton.edu/vol2-trans/252 On the Inertial of Energy Required by the Relativity Principle]|url=http://doi.wiley.com/10.1002/andp.19073280713|journal=Annalen der Physik|language=de|volume=328|issue=7|pages=371–384|doi=10.1002/andp.19073280713|bibcode=1907AnP...328..371E}}</ref> Max Planck rewrote Einstein's mass–energy relationship as {{math|1=''M'' = {{sfrac|''E''<sub>0</sub> + ''pV''<sub>0</sub>|''c''<sup>2</sup>}}}} in June 1907, where {{math|''p''}} is the pressure and {{math|''V''<sub>0</sub>}} the volume to express the relation between mass, its latent energy, and thermodynamic energy within the body.<ref>{{Cite journal|last=Planck|first=M.|date=1908|title=Zur Dynamik bewegter Systeme|trans-title=[[s:Translation:On the Dynamics of Moving Systems|On the Dynamics of Moving Systems]]|url=http://doi.wiley.com/10.1002/andp.19083310602|journal=Annalen der Physik|language=de|volume=331|issue=6|pages=1–34|doi=10.1002/andp.19083310602|bibcode=1908AnP...331....1P}}</ref> Subsequently, in October 1907, this was rewritten as {{math|1=''M''<sub>0</sub> = {{sfrac|''E''<sub>0</sub>|''c''<sup>2</sup>}}}} and given a quantum interpretation by German physicist [[Johannes Stark]], who assumed its validity and correctness.<ref>{{cite journal|author=Stark, J.|url=https://archive.org/details/physikalischeze00unkngoog|title=Elementarquantum der Energie, Modell der negativen und der positiven Elekrizitat|journal=Physikalische Zeitschrift|page=881|volume=24|issue= 8 |date=1907|language=de}}</ref> In December 1907, Einstein expressed the equivalence in the form {{math|1=''M'' = ''μ'' + {{sfrac|''E''<sub>0</sub>|''c''<sup>2</sup>}}}} and concluded: "A mass {{math|''μ''}} is equivalent, as regards inertia, to a quantity of energy {{math|''μc<sup>2</sup>''}}. […] It appears far more natural to consider every inertial mass as a store of energy."<ref>{{Cite journal|last=Einstein|first=Albert|date=1908|title=Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen|url=http://adsabs.harvard.edu/abs/1908JRE.....4..411E|trans-title=[https://einsteinpapers.press.princeton.edu/vol2-trans/266?ajax On the Relativity Principle and the Conclusions Drawn From it]|journal=Jahrbuch der Radioaktivität und Elektronik|language=de|volume=4|page=411|bibcode=1908JRE.....4..411E}}</ref><ref>{{Cite journal|last=Schwartz|first=H. M.|date=September 1977|title=Einstein's comprehensive 1907 essay on relativity, part II|url=http://aapt.scitation.org/doi/10.1119/1.11053|journal=American Journal of Physics|language=en|volume=45|issue=9|pages=811–817|doi=10.1119/1.11053|bibcode=1977AmJPh..45..811S|issn=0002-9505|access-date=2020-10-14|archive-date=2019-05-28|archive-url=https://web.archive.org/web/20190528171924/https://aapt.scitation.org/doi/10.1119/1.11053|url-status=live}}</ref> American [[physical chemist]]s [[Gilbert N. Lewis]] and [[Richard C. Tolman]] used two variations of the formula in 1909: {{math|1=''m'' = {{sfrac|''E''|''c''<sup>2</sup>}}}} and {{math|1=''m''<sub>0</sub> = {{sfrac|''E''<sub>0</sub>|''c''<sup>2</sup>}}}}, with {{mvar|E}} being the relativistic energy (the energy of an object when the object is moving), {{math|''E''<sub>0</sub>}} is the rest energy (the energy when not moving), {{mvar|m}} is the relativistic mass (the rest mass and the extra mass gained when moving), and {{math|''m''<sub>0</sub>}} is the rest mass.<ref>{{Cite journal|last1=Lewis|first1=Gilbert N.|last2=Tolman|first2=Richard C.|date=1909|title=The Principle of Relativity, and Non-Newtonian Mechanics|url=https://www.jstor.org/stable/20022495|journal=Proceedings of the American Academy of Arts and Sciences|language=en|volume=44|issue=25|page=711|doi=10.2307/20022495|jstor=20022495}}</ref> The same relations in different notation were used by Lorentz in 1913 and 1914, though he placed the energy on the left-hand side: {{math|1=''ε'' = ''Mc''<sup>2</sup>}} and {{math|1=''ε''<sub>0</sub> = ''mc''<sup>2</sup>}}, with {{mvar|ε}} being the total energy (rest energy plus kinetic energy) of a moving material point, {{math|''ε''<sub>0</sub>}} its rest energy, {{mvar|M}} the relativistic mass, and {{mvar|m}} the invariant mass.<ref>{{Cite book|last=Lorentz|first=Hendrik Antoon|url=https://books.google.com/books?id=89PPAAAAMAAJ|title=Das Relativitätsprinzip: drei Vorlesungen gehalten in Teylers Stiftung zu Haarlem|date=1914|publisher=B.G. Teubner|language=de|trans-title=The principle of relativity: three lectures given in Teyler's foundation in Haarlem|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080259/https://books.google.com/books?id=89PPAAAAMAAJ|url-status=live}}</ref>

In 1911, German physicist [[Max von Laue]] gave a more comprehensive proof of {{math|1=''M''<sub>0</sub> = {{sfrac|''E''<sub>0</sub>|''c''<sup>2</sup>}}}} from the [[stress–energy tensor]],<ref>{{Cite journal|last=Laue|first=M.|date=1911|title=Zur Dynamik der Relativitätstheorie|trans-title=[[s:Translation:On the Dynamics of the Theory of Relativity|On the Dynamics of the Theory of Relativity]]|url=http://doi.wiley.com/10.1002/andp.19113400808|journal=Annalen der Physik|language=de|volume=340|issue=8|pages=524–542|doi=10.1002/andp.19113400808|bibcode=1911AnP...340..524L}}</ref> which was later generalized by German mathematician [[Felix Klein]] in 1918.<ref>{{Citation|author=Klein, Felix|title=Über die Integralform der Erhaltungssätze und die Theorie der räumlich-geschlossenen Welt|trans-title=On the integral form of the conservation laws and the theory of the spatially closed world|journal=Göttinger Nachrichten|date=1918|pages=394–423|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN243240503&DMDID=DMDLOG_0051}}</ref>

Einstein returned to the topic once again after [[World War II]] and this time he wrote {{math|1=''E'' = ''mc''<sup>2</sup>}} in the title of his article<ref>{{cite magazine|last=Einstein|first=A.|date=April 1946|title={{math|1=E = mc<sup>2</sup>}}: the most urgent problem of our time|url=http://alberteinstein.info/vufind1/Record/EAR000034164|magazine=[[Science Illustrated]]|publisher=[[Bonnier Publications International]]|volume=1|issue=1|at=item 417 in the ''Bibliography''. pp. 16-17|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080301/http://alberteinstein.info/vufind1/Record/EAR000034164|url-status=live}}</ref> intended as an explanation for a general reader by analogy.<ref>{{Cite book|last=Schilpp, Paul Arthur|url=https://www.worldcat.org/oclc/134995|title=Albert Einstein: philosopher-scientist.|date=1970|publisher=Open Court|isbn=978-0-87548-286-6|edition=3d |location=La Salle, Ill.|at=M.C. Shields ''Bibliography of the Writings of Albert Einstein to May 1951''|oclc=134995}}</ref>

====Alternative version====
An alternative version of Einstein's [[thought experiment]] was proposed by American theoretical physicist [[Fritz Rohrlich]] in 1990, who based his reasoning on the [[Doppler effect]].<ref>{{Cite journal|last=Rohrlich|first=Fritz|date=April 1990|title=An elementary derivation of {{math|1=''E'' = ''mc''<sup>2</sup>}}|url=http://aapt.scitation.org/doi/10.1119/1.16168|journal=American Journal of Physics|language=en|volume=58|issue=4|pages=348–349|doi=10.1119/1.16168|issn=0002-9505|access-date=2020-10-14|archive-date=2021-02-21|archive-url=https://web.archive.org/web/20210221080302/https://aapt.scitation.org/doi/10.1119/1.16168|url-status=live}}</ref> Like Einstein, he considered a body at rest with mass {{mvar|M}}. If the body is examined in a frame moving with nonrelativistic velocity {{mvar|v}}, it is no longer at rest and in the moving frame it has momentum {{math|1=''P'' = ''Mv''}}. Then he supposed the body emits two pulses of light to the left and to the right, each carrying an equal amount of energy {{math|{{sfrac|''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. However, if the same process is considered in a frame that moves with velocity {{math|''v''}} to the left, the pulse moving to the left is [[redshift]]ed, while the pulse moving to the right is [[blue shift]]ed. 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 has not 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. The velocity is small, so the right-moving light is blueshifted by an amount equal to the nonrelativistic [[Doppler shift]] factor {{math|1 − {{sfrac|''v''|''c''}}}}. The momentum of the light is its energy divided by {{mvar|c}}, and it is increased by a factor of {{math|{{sfrac|''v''|''c''}}}}. So the right-moving light is carrying an extra momentum {{math|Δ''P''}} given by:
:<math> \Delta P = {v \over c}{E \over 2c} .</math>

The left-moving light carries a little less momentum, by the same amount {{math|Δ''P''}}. So the total right-momentum in both light pulses is twice {{math|Δ''P''}}. This is the right-momentum that the object lost.
:<math> 2\Delta P = v {E\over c^2} .</math>

The momentum of the object in the moving frame after the emission is reduced to this amount:
:<math> P' = Mv - 2\Delta P = \left(M - {E\over c^2}\right)v .</math>

So the change in the object's mass is equal to the total energy lost divided by {{math|''c''<sup>2</sup>}}. 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.

===Radioactivity and nuclear energy===
[[File:Einstein - Time Magazine - July 1, 1946.jpg|right|thumb|The popular connection between Einstein, the equation {{math|1=''E'' = ''mc''<sup>2</sup>}}, and the [[nuclear weapon|atomic bomb]] was prominently indicated on the cover of ''[[Time (magazine)|Time]]'' magazine in July 1946.]]

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, raising the question of where the energy comes 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 New Zealand physicist [[Ernest Rutherford]] and British radiochemist [[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: "If it were ever found possible to control at will the rate of disintegration of the radio-elements, an enormous amount of energy could be obtained from a small quantity of matter."<ref>{{Cite book|last=Rutherford, Ernest|url=https://www.worldcat.org/oclc/850842708|title=Radio-activity|date=2007|pages=336–338|publisher=Juniper Grove|isbn=978-1-60355-058-1|edition=2nd|location=New York|oclc=850842708}}</ref><ref>{{Cite book|last=Heisenberg|first=Werner|title=Physics And Philosophy The Revolution In Modern Science|url=https://archive.org/details/physicsandphilos010613mbp|date=1958|pages=118–119|publisher=HarperCollins|isbn=978-0-06-120919-2|language=en}}</ref>

Einstein's equation does not explain the large energies released in radioactive decay, but can be used to quantify them. The theoretical explanation for radioactive decay is given by the [[nuclear force]]s responsible for holding atoms together, though these forces were still unknown in 1905. The enormous energy released from radioactive decay had previously been measured by Rutherford and was much more easily measured than the small change in the gross mass of materials as a result. Einstein's equation, by theory, can give these energies by measuring mass differences before and after reactions, but in practice, these mass differences in 1905 were still too small to be measured in bulk. Prior to this, the ease of measuring radioactive decay energies with a [[calorimeter]] was thought possibly likely to allow measurement of changes in mass difference, as a check on Einstein's equation itself. Einstein mentions in his 1905 paper that mass–energy equivalence might perhaps be tested with radioactive decay, which was known by then to release enough energy to possibly be "weighed," when missing from the system. However, radioactivity seemed to proceed at its own unalterable pace, and even when simple nuclear reactions became possible using proton bombardment, the idea that these great amounts of usable energy could be liberated at will with any practicality, proved difficult to substantiate. Rutherford was reported in 1933 to have declared that this energy could not be exploited efficiently: "Anyone who expects a source of power from the transformation of the atom is talking [[moonshine]]."<ref>{{Cite book|last=Reed|first=Bruce Cameron|url=https://books.google.com/books?id=RN5xCgAAQBAJ|title=Atomic Bomb: The Story of the Manhattan Project: How nuclear physics became a global geopolitical game-changer|date=2015-06-01|publisher=Morgan & Claypool Publishers|isbn=978-1-62705-992-3|language=en|section=The neutrino, artificial radioactivity and new elements|quote=We might in these processes obtain very much more energy than the proton supplied, but on the average we could not expect to obtain energy in this way. It was a very poor and inefficient way of producing energy, and anyone who looked for a source of power in the transformation of the atoms was talking moonshine. But the subject was scientifically interesting because it gave insight into the atoms.|at=Second page of section 2.2}}</ref>

This outlook changed dramatically in 1932 with the discovery of the neutron and its mass, allowing mass differences for single [[nuclide]]s and their reactions to be calculated directly, and compared with the sum of masses for the particles that made up their composition. In 1933, the energy released from the reaction of [[lithium-7]] plus protons giving rise to two [[alpha particle]]s, allowed Einstein's equation to be tested to an error of ±0.5%.<ref>{{cite journal | last1 = Oliphant |first1=M. L. E. | authorlink1 = Mark Oliphant | last2 = Kinsey | first2 = B. B. | last3 = Lord Rutherford | authorlink3 = Ernest Rutherford | title = The Transformation of Lithium by Protons and by Ions of the Heavy Isotope of Hydrogen | year = 1933 | journal = Proceedings of the Royal Society | volume = 141 | issue = 845 | pages = 722–733 |doi=10.1098/rspa.1933.0150 |s2cid=93342501 | url = https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1933.0150}}</ref> However, scientists still did not see such reactions as a practical source of power, due to the energy cost of accelerating reaction particles.
After the very public demonstration of huge energies released from nuclear fission after the [[atomic bombings of Hiroshima and Nagasaki]] in 1945, the equation {{math|1=''E'' = ''mc''<sup>2</sup>}} became directly linked in the public eye with the power and peril of nuclear weapons. The equation was featured on 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)|Time]]'' magazine prominently featured a picture of Einstein next to an image of a [[mushroom cloud]] emblazoned with the equation.<ref>{{Cite web|title=TIME Magazine -- U.S. Edition -- July 1, 1946 Vol. XLVIII No. 1|url=http://content.time.com/time/magazine/0,9263,7601460701,00.html|access-date=2020-10-14|website=content.time.com|language=en-us|archive-date=2020-10-15|archive-url=https://web.archive.org/web/20201015042338/http://content.time.com/time/magazine/0,9263,7601460701,00.html|url-status=live}}</ref> Einstein himself had only a minor role in the [[Manhattan Project]]: he had [[Einstein–Szilárd letter|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 in theoretical terms. It was inconsequential, on account of Einstein not being given sufficient information to fully work on the problem.<ref>{{Cite book|last=Isaacson, Walter|url=https://www.worldcat.org/oclc/76961150|title=Einstein: his life and universe|chapter=The bomb|date=10 April 2007|isbn=978-0-7432-6473-0|location=New York|oclc=76961150|access-date=14 October 2020|archive-date=22 August 2020|archive-url=https://web.archive.org/web/20200822170147/http://www.worldcat.org/oclc/76961150|url-status=live}}</ref>

While {{math|1=''E'' = ''mc''<sup>2</sup>}} is useful for understanding the amount of energy potentially released in a fission reaction, it was not strictly necessary to develop the weapon, once the fission process was known, and its energy measured at 200&nbsp;[[MeV]] (which was directly possible, using a quantitative [[Geiger counter]], at that time). The physicist and Manhattan Project participant [[Robert Serber]] noted that somehow "the popular notion took hold long ago that Einstein's theory of relativity, in particular his famous equation {{math|1=''E'' = ''mc''<sup>2</sup>}}, plays some essential role in the theory of fission. 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."{{refn|group=note|{{Cite book|last=Serber|first=Robert|url=http://dx.doi.org/10.2307/j.ctvw1d5pf|title=The Los Alamos Primer|date=2020-04-07|publisher=University of California Press|doi=10.2307/j.ctvw1d5pf|isbn=978-0-520-37433-1|page=7|s2cid=91948043}}. The quotation is taken from Serber's 1992 version, and is not in the original 1943 [[Los Alamos Primer]] of the same name.}} There are other views on the equation's importance to nuclear reactions. In late 1938, the Austrian-Swedish and British physicists [[Lise Meitner]] and [[Otto Robert Frisch]]—while on a winter walk during which they solved the meaning of Hahn's experimental results and introduced the idea that would be called atomic fission—directly used Einstein's equation to help them understand the quantitative energetics of the reaction that overcame the "surface tension-like" forces that hold 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 used ''packing fraction'', or nuclear [[binding energy]] values for elements. These, together with use of {{math|1=''E'' = ''mc''<sup>2</sup>}} allowed them to realize on the spot that the basic fission process was energetically possible.{{refn|group=note|{{block quote|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 a very wobbly, 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 {{math|1=''E'' = ''mc''<sup>2</sup>}}, and… the mass was just equivalent to 200 MeV; it all fitted!|author=Lise Meitner<ref>{{Cite book|last=Sime, Ruth Lewin|url=https://www.worldcat.org/oclc/42855101|title=Lise Meitner: a life in physics|date=1996|publisher=University of California Press|isbn=978-0-520-91899-3|location=Berkeley|pages = 236–237 | oclc=42855101}}</ref>}}}}

===Einstein's equation written===
According to the Einstein Papers Project at the [[California Institute of Technology]] and [[Hebrew University of Jerusalem]], there remain only four known copies of this equation as written by Einstein. One of these is a letter written in [[German language|German]] to [[Ludwik Silberstein]], which was in Silberstein's archives, and sold at [[auction]] for $1.2 million, [[RR Auction]] of [[Boston, Massachusetts]] said on May 21, 2021.<ref>{{cite news
| author =<!--not stated-->
| title =Handwritten example of Einstein equation fetches $1.2M
| url =https://apnews.com/article/lifestyle-science-4ed59f3bd9d9969354dd40ec363fe0e5
| work =Associated Press
| date =May 21, 2021
| access-date =April 11, 2023
}}</ref>

==See also==
{{Portal|Physics}}
{{cmn|colwidth=30em|
* [[Energy density]]
* [[Index of energy articles]]
* [[Index of wave articles]]
* [[Lorentz transform]]
* [[Length contraction]]
* [[Outline of energy]]
* [[Relativity of simultaneity]]
}}

==Notes==
{{Reflist|group=note}}

==References==
{{reflist}}

==External links==
{{Wikisourcepar|Relativity: The Special and General Theory}}
{{Commons category|Einstein formula}}
* [http://www.mathpages.com/home/kmath600/kmath600.htm Einstein on the Inertia of Energy] – MathPages
*[https://history.aip.org/exhibits/einstein/voice1_text.htm Einstein-on film explaining a mass energy equivalence]
* [http://profmattstrassler.com/articles-and-posts/particle-physics-basics/mass-energy-matter-etc/mass-and-energy/ Mass and Energy] – Conversations About Science with Theoretical Physicist Matt Strassler
* [http://plato.stanford.edu/entries/equivME The Equivalence of Mass and Energy] – Entry in the ''Stanford Encyclopedia of Philosophy''
* {{cite web|last=Merrifield|first=Michael|title=E=mc<sup>2</sup> – Mass–Energy Equivalence|url=http://www.sixtysymbols.com/videos/emc2.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]|author2=Copeland, Ed |author3=Bowley, Roger|ref=none }}

{{Einstein}}
{{Relativity}}

{{DEFAULTSORT:Mass-Energy Equivalence}}
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