Editing Mass (section)

=== Special relativity ===
{{main|Mass in special relativity}}
In some frameworks of [[special relativity]], physicists have used different definitions of the term. In these frameworks, two kinds of mass are defined: [[rest mass]] (invariant mass),<ref group="note">It is possible to make a slight distinction between "rest mass" and "invariant mass". For a system of two or more particles, none of the particles are required be at rest with respect to the observer for the system as a whole to be at rest with respect to the observer. To avoid this confusion, some sources will use "rest mass" only for individual particles, and "invariant mass" for systems.</ref> and [[relativistic mass]] (which increases with velocity). Rest mass is the Newtonian mass as measured by an observer moving along with the object.  ''Relativistic mass'' is the total quantity of energy in a body or system divided by [[speed of light|''c'']]<sup>2</sup>. The two are related by the following equation:
: <math>m_\mathrm{relative}=\gamma (m_\mathrm{rest})\!</math>
where <math>\gamma</math> is the [[Lorentz factor]]:
: <math qid=Q599404>\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}</math>

The invariant mass of systems is the same for observers in all inertial frames, while the relativistic mass depends on the observer's [[frame of reference]]. In order to formulate the equations of physics such that mass values do not change between observers, it is convenient to use rest mass. The rest mass of a body is also related to its energy ''E'' and the magnitude of its momentum '''p''' by the [[relativistic energy-momentum equation]]:
: <math qid=Q96941619>(m_\mathrm{rest})c^2=\sqrt{E_\mathrm{total}^2-(|\mathbf{p}|c)^2}.\!</math>

So long as the system is [[Closed system|closed]] with respect to mass and energy, both kinds of mass are conserved in any given frame of reference. The conservation of mass holds even as some types of particles are converted to others. Matter particles (such as atoms) may be converted to non-matter particles (such as photons of light), but this does not affect the total amount of mass or energy. Although things like heat may not be matter, all types of energy still continue to exhibit mass.<ref group="note">For example, a nuclear bomb in an idealized super-strong box, sitting on a scale, would in theory show no change in mass when detonated (although the inside of the box would become much hotter). In such a system, the mass of the box would change only if energy were allowed to escape from the box as light or heat. However, in that case, the removed energy would take its associated mass with it. Letting heat or radiation out of such a system is simply a way to remove mass. Thus, mass, like energy, cannot be destroyed, but only moved from one place to another.</ref><ref>{{cite book |last1=Taylor |first1=E.F. |last2=Wheeler |first2=J.A. |date=1992 |title=Spacetime Physics |pages=[https://archive.org/details/spacetimephysics00edwi_0/page/248 248–149] |publisher=W.H. Freeman |isbn=978-0-7167-2327-1 |url=https://archive.org/details/spacetimephysics00edwi_0/page/248 }}</ref> Thus, mass and energy do not change into one another in relativity; rather, both are names for the same thing, and neither mass nor energy ''appear'' without the other.

Both rest and relativistic mass can be expressed as an energy by applying the well-known relationship [[Mass–energy equivalence|''E''&nbsp;= ''mc''<sup>2</sup>]], yielding [[rest energy]] and "relativistic energy" (total system energy) respectively:
: <math qid=Q11663629>E_\mathrm{rest}=(m_\mathrm{rest})c^2\!</math>
: <math qid=Q35875> E_\mathrm{total}=(m_\mathrm{relative})c^2\!</math>

The "relativistic" mass and energy concepts are related to their "rest" counterparts, but they do not have the same value as their rest counterparts in systems where there is a net momentum. Because the relativistic mass is [[Mass–energy equivalence|proportional to the energy]], it has gradually fallen into disuse among physicists.<ref>{{cite arXiv |author=G. Oas |date=2005 |title=On the Abuse and Use of Relativistic Mass |eprint=physics/0504110}}</ref> There is disagreement over whether the concept remains useful [[Pedagogy|pedagogically]].<ref name="okun">{{cite journal |last=Okun |first=L.B. |date=1989 |title=The Concept of Mass |url=http://www.itep.ru/theor/persons/lab180/okun/em_3.pdf |journal=[[Physics Today]] |volume=42 |issue=6 |pages=31–36 |bibcode=1989PhT....42f..31O |doi=10.1063/1.881171 |url-status=dead |archive-url=https://web.archive.org/web/20110722130232/http://www.itep.ru/theor/persons/lab180/okun/em_3.pdf |archive-date=22 July 2011}}</ref><ref>{{cite journal |last1=Rindler |first1=W. |last2=Vandyck |first2=M.A. |last3=Murugesan |first3=P. |last4=Ruschin |first4=S. |last5=Sauter |first5=C. |last6=Okun |first6=L.B. |date=1990 |title=Putting to Rest Mass Misconceptions |url=http://www.itep.ru/theor/persons/lab180/okun/em_4.pdf |journal=[[Physics Today]] |volume=43 |issue=5 |pages=13–14, 115, 117 |bibcode=1990PhT....43e..13R |doi=10.1063/1.2810555 |url-status=dead |archive-url=https://web.archive.org/web/20110722130328/http://www.itep.ru/theor/persons/lab180/okun/em_4.pdf |archive-date=22 July 2011}}</ref><ref>{{cite journal |last1=Sandin |first1=T.R. |date=1991 |title=In Defense of Relativistic Mass |journal=[[American Journal of Physics]] |volume=59 |issue=11 |page=1032 |bibcode=1991AmJPh..59.1032S |doi=10.1119/1.16642}}</ref>

In bound systems, the [[binding energy]] must often be subtracted from the mass of the unbound system, because binding energy commonly leaves the system at the time it is bound. The mass of the system changes in this process merely because the system was not closed during the binding process, so the energy escaped. For example, the binding energy of [[atomic nuclei]] is often lost in the form of gamma rays when the nuclei are formed, leaving [[nuclide]]s which have less mass than the free particles ([[nucleon]]s) of which they are composed.

[[Mass–energy equivalence]] also holds in macroscopic systems.<ref>{{Citation|author=Planck, Max|date=1907|title=Zur Dynamik bewegter Systeme|journal=Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften, Berlin|volume=Erster Halbband|issue=29|pages=542–570|url=https://archive.org/details/sitzungsberichte1907deutsch|bibcode=1908AnP...331....1P|doi=10.1002/andp.19083310602}}
:English Wikisource translation: [[s:Translation:On the Dynamics of Moving Systems|On the Dynamics of Moving Systems]] (''See paragraph 16.'')</ref> For example, if one takes exactly one kilogram of ice, and applies heat, the mass of the resulting melt-water will be more than a kilogram: it will include the mass from the [[thermal energy]] ([[latent heat]]) used to melt the ice; this follows from the [[conservation of energy]].<ref name=hecht>{{cite journal | last1 = Hecht | first1 = Eugene | year = 2006 | title = There Is No Really Good Definition of Mass | url = http://www.physicsland.com/Physics10_files/Mass.pdf| journal = The Physics Teacher | volume = 44 | issue = 1| pages = 40–45 | doi = 10.1119/1.2150758 | bibcode = 2006PhTea..44...40H }}</ref> This number is small but not negligible: about 3.7 nanograms. It is given by the [[latent heat]] of melting ice (334 kJ/kg) divided by the speed of light squared (''c''<sup>2</sup> ≈ {{val|9|e=16|u=m<sup>2</sup>/s<sup>2</sup>}}).