View source for Mass

{{Short description|Amount of matter present in an object}}
{{about|the scientific concept|the main liturgical service in some Christian churches|Mass (liturgy)|other uses|Mass (disambiguation)}}
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{{Use dmy dates|date=September 2020}}
{{Infobox physical quantity
| name = Mass
| width =
| background =
| image = Poids fonte 2 kg 03 (cropped).jpg
| caption = A {{cvt|2|kg}} cast iron weight used for [[balance (device for weighing)|balances]] 
| unit = [[kilogram]]
| otherunits =
| symbols = m
| baseunits =
| dimension =
| extensive = yes
| conserved = yes
| transformsas =
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}}
{{Classical mechanics|expanded=fundamentals}}

'''Mass''' is an [[Intrinsic and extrinsic properties|intrinsic property]] of a [[physical body|body]]. It was traditionally believed to be related to the [[physical quantity|quantity]] of [[matter]] in a body, until the discovery of the [[atom]] and [[particle physics]]. It was found that different atoms and different [[elementary particle|elementary particles]], theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple [[Mass in special relativity|definitions]] which are conceptually distinct, but physically equivalent. Mass can be experimentally defined as a [[measure (mathematics)|measure]] of the body's [[inertia]], meaning the resistance to [[acceleration]] (change of [[velocity]]) when a [[net force]] is applied.<ref>{{cite web |last1=Bray |first1=Nancy |title=Science |url=https://www.nasa.gov/centers/kennedy/about/information/science_faq.html |website=NASA |access-date=20 March 2023 |date=28 April 2015 |quote=Mass can be understood as a measurement of inertia, the resistance of an object to be set in motion or stopped from motion. |archive-date=30 May 2023 |archive-url=https://web.archive.org/web/20230530095526/https://www.nasa.gov/centers/kennedy/about/information/science_faq.html |url-status=dead }}</ref> The object's mass also determines the [[Force|strength]] of its [[gravitation]]al attraction to other bodies.

The [[SI base unit]] of mass is the [[kilogram]] (kg). In [[physics]], mass is [[Mass versus weight|not the same]] as [[weight]], even though mass is often determined by measuring the object's weight using a [[spring scale]], rather than [[Weighing scale#Balance scales|balance scale]] comparing it directly with known masses. An object on the Moon would weigh less than it does on Earth because of the lower gravity, but it would still have the same mass. This is because weight is a force, while mass is the property that (along with gravity) determines the strength of this force.

In the [[Standard Model]] of physics, the mass of [[elementary particle]]s is believed to be a result of their coupling with the [[Higgs boson]] in what is known as the [[Brout–Englert–Higgs mechanism]].<ref>{{Cite web |date=2024-04-03 |title=The Higgs boson |url=https://home.cern/science/physics/higgs-boson |access-date=2024-04-09 |website=CERN |language=en}}</ref>

== Phenomena ==
There are several distinct phenomena that can be used to measure mass. Although some theorists have speculated that some of these phenomena could be independent of each other,<ref>{{cite magazine |url=https://www.technologyreview.com/2010/06/14/91230/new-quantum-theory-separates-gravitational-and-inertial-mass |title=New Quantum Theory Separates Gravitational and Inertial Mass |magazine=MIT Technology Review |date=14 June 2010|access-date=25 September 2020}}</ref> current experiments have found no difference in results regardless of how it is measured:
* ''Inertial mass'' measures an object's resistance to being accelerated by a force (represented by the relationship [[Newton's laws of motion|{{nowrap|1=''F'' = ''ma''}}]]).
* ''Active gravitational mass'' determines the strength of the gravitational field generated by an object.
* ''Passive gravitational mass'' measures the gravitational force exerted on an object in a known gravitational field.

The mass of an object determines its acceleration in the presence of an applied force. The inertia and the inertial mass describe this property of physical bodies at the qualitative and quantitative level respectively.  According to [[Newton's second law of motion]], if a body of fixed mass ''m'' is subjected to a single force ''F'', its acceleration ''a'' is given by ''F''/''m''. A body's mass also determines the degree to which it generates and is affected by a [[gravitational field]]. If a first body of mass ''m''<sub>A</sub> is placed at a distance ''r'' (center of mass to center of mass) from a second body of mass ''m''<sub>B</sub>, each body is subject to an attractive force {{nowrap|1=''F''<sub>g</sub> = ''Gm''<sub>A</sub>''m''<sub>B</sub>/''r''<sup>2</sup>}}, where {{nowrap|1=''G'' = {{val|6.67|e=-11|u=N⋅kg<sup>−2</sup>⋅m<sup>2</sup>}}}} is the "universal [[gravitational constant]]". This is sometimes referred to as gravitational mass.<ref group="note">When a distinction is necessary, the active and passive gravitational masses may be distinguished.</ref> Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been incorporated ''[[A priori and a posteriori|a priori]]'' in the [[equivalence principle]] of [[general relativity]].

== Units of mass ==

{{further|Orders of magnitude (mass)}}

[[File:SI base unit.svg|thumb|The kilogram is one of the seven [[SI base units]].]]

The [[International System of Units]] (SI) unit of mass is the [[kilogram]] (kg). The kilogram is 1000&nbsp;grams (g), and was first defined in 1795 as the mass of one cubic decimetre of water at the [[melting point]] of ice. However, because precise measurement of a cubic decimetre of water at the specified temperature and pressure was difficult, in 1889 the kilogram was redefined as the mass of a metal object, and thus became independent of the metre and the properties of water, this being a copper prototype of the [[Grave (unit)|grave]] in 1793, the platinum [[Kilogramme des Archives]] in 1799, and the platinum–iridium [[International Prototype of the Kilogram]] (IPK) in 1889.

However, the mass of the IPK and its national copies have been found to drift over time. The [[2019 revision of the SI|re-definition of the kilogram and several other units]] came into effect on 20 May 2019, following a final vote by the [[CGPM]] in November 2018.<ref>{{cite journal
|url = https://www.nature.com/nphys/journal/v13/n2/pdf/nphys4029.pdf
|title = Metrology in 2019 |first=Klaus|last=von Klitzing|journal=Nature Physics|volume=13 |issue=2 |date=February 2017 |page=198|arxiv=1707.06785|bibcode=2017SSPMA..47l9503L|doi=10.1360/SSPMA2017-00044|s2cid = 133817316 }}</ref> The new definition uses only invariant quantities of nature: the [[speed of light]], the [[Caesium standard|caesium hyperfine frequency]], the [[Planck constant]] and the [[elementary charge]].<ref name=Brochure9_2016>
{{cite web
 |title=Draft of the ninth SI Brochure
 |publisher=BIPM
 |url=http://www.bipm.org/utils/common/pdf/si-brochure-draft-2016b.pdf
 |pages=2–9
 |date=10 November 2016
 |access-date=2017-09-10
}}</ref>

Non-SI units accepted for use with SI units include:
* the [[tonne]] (t) (or "metric ton"), equal to 1000&nbsp;kg
* the [[Electronvolt (mass)|electronvolt]] (eV), a unit of [[energy]], used to express mass in units of eV/''c''<sup>2</sup> through [[mass–energy equivalence]]
* the [[Dalton (unit)|dalton]] (Da), equal to 1/12 of the mass of a free [[carbon-12]] atom, approximately {{val|1.66|e=-27|u=kg}}.<ref group="note">The dalton is convenient for expressing the masses of atoms and molecules.</ref>

Outside the SI system, other units of mass include:
* the [[slug (unit)|slug]] (sl), an [[Imperial units|Imperial unit]] of mass (about 14.6&nbsp;kg)
* the [[Pound (mass)|pound]] (lb), a unit of mass (about 0.45&nbsp;kg), which is used alongside the similarly named [[pound (force)]] (about 4.5&nbsp;N), a unit of force<ref group="note">These are used mainly in the United States except in scientific contexts where SI units are usually used instead.</ref>
* the [[Planck mass]] (about {{val|2.18|e=-8|u=kg}}), a quantity derived from fundamental constants
* the [[solar mass]] ({{Solar mass}}), defined as the mass of the [[Sun]], primarily used in astronomy to compare large masses such as stars or galaxies (≈&nbsp;{{val|1.99|e=30|u=kg}})
* the mass of a particle, as identified with its inverse [[Compton wavelength]] ({{nowrap|1&nbsp;cm<sup>−1</sup> ≘ {{val|3.52|e=-41|u=kg}}}})
* the mass of a star or [[black hole]], as identified with its [[Schwarzschild radius]] ({{nowrap|1&nbsp;cm ≘ {{val|6.73|e=24|u=kg}}}}).

=== History of units of mass ===
{{excerpt|History of measurement#Units of mass}}

== Definitions ==

In [[physical science]], one may distinguish conceptually between at least seven different aspects of ''mass'', or seven physical notions that involve the concept of ''mass''.<ref name="Rindler2">{{cite book |author=W. Rindler |date=2006 |title=Relativity: Special, General, And Cosmological |url=https://books.google.com/books?id=MuuaG5HXOGEC&pg=PA16 |pages=16–18 |publisher=Oxford University Press |isbn=978-0-19-856731-8}}</ref> Every experiment to date has shown these seven values to be [[Proportionality (mathematics)|proportional]], and in some cases equal, and this proportionality gives rise to the abstract concept of mass. There are a number of ways mass can be measured or [[operationalization|operationally defined]]:
* Inertial mass is a measure of an object's resistance to acceleration when a [[force]] is applied. It is determined by applying a force to an object and measuring the acceleration that results from that force. An object with small inertial mass will accelerate more than an object with large inertial mass when acted upon by the same force. One says the body of greater mass has greater [[inertia]].
* Active gravitational mass<ref group=note >The distinction between "active" and "passive" gravitational mass does not exist in the Newtonian view of gravity as found in [[classical mechanics]], and can safely be ignored for many purposes. In most practical applications, Newtonian gravity is assumed because it is usually sufficiently accurate, and is simpler than General Relativity; for example, NASA uses primarily Newtonian gravity to design space missions, although "accuracies are routinely enhanced by accounting for tiny relativistic effects".{{URL|http://www2.jpl.nasa.gov/basics/bsf3-2.php}} The distinction between "active" and "passive" is very abstract, and applies to post-graduate level applications of General Relativity to certain problems in cosmology, and is otherwise not used. There is, nevertheless, an important conceptual distinction in Newtonian physics between "inertial mass" and "gravitational mass", although these quantities are identical; the conceptual distinction between these two fundamental definitions of mass is maintained for teaching purposes because they involve two distinct methods of measurement. It was long considered anomalous that the two distinct measurements of mass (inertial and gravitational) gave an identical result. The property, observed by Galileo, that objects of different mass fall with the same rate of acceleration (ignoring air resistance), shows that inertial and gravitational mass are the same.</ref> is a measure of the strength of an object's [[Gauss' law for gravity|gravitational flux]] (gravitational flux is equal to the [[surface integral]] of gravitational field over an enclosing surface). Gravitational field can be measured by allowing a small "test object" to fall freely and measuring its [[free-fall]] acceleration. For example, an object in free-fall near the [[Moon]] is subject to a smaller gravitational field, and hence accelerates more slowly, than the same object would if it were in free-fall near the Earth. The gravitational field near the Moon is weaker because the Moon has less active gravitational mass.
* Passive gravitational mass is a measure of the strength of an object's interaction with a [[gravitational field]]. Passive gravitational mass is determined by dividing an object's weight by its free-fall acceleration. Two objects within the same gravitational field will experience the same acceleration; however, the object with a smaller passive gravitational mass will experience a smaller force (less weight) than the object with a larger passive gravitational mass.
* According to [[theory of relativity|relativity]], mass is nothing else than the [[invariant mass|rest energy]] of a system of particles, meaning the energy of that system in a [[reference frame]] where it has zero [[momentum]]. Mass can be converted into other forms of energy according to the principle of [[mass–energy equivalence]]. This equivalence is exemplified in a large number of physical processes including [[pair production]], [[beta decay]] and [[nuclear fusion]]. Pair production and nuclear fusion are processes in which measurable amounts of mass are converted to [[kinetic energy]] or vice versa.
* Curvature of [[spacetime]] is a relativistic manifestation of the existence of mass. Such [[curvature]] is extremely weak and difficult to measure. For this reason, curvature was not discovered until after it was predicted by Einstein's theory of general relativity. Extremely precise [[atomic clocks]] on the surface of the Earth, for example, are found to measure less time (run slower) when compared to similar clocks in space. This difference in elapsed time is a form of curvature called [[gravitational time dilation]]. Other forms of curvature have been measured using the [[Gravity Probe B]] satellite.
* Quantum mass manifests itself as a difference between an object's quantum [[frequency]] and its [[wave number]]. The quantum mass of a particle is proportional to the inverse [[Compton wavelength]] and can be determined through various forms of [[spectroscopy]].  In relativistic quantum mechanics, mass is one of the irreducible representation labels of the Poincaré group.

=== Weight vs. mass ===
{{main|Mass versus weight}}
[[File:Mass versus weight in earth and mars.svg|300px|right|thumb|Mass and weight of a given object on [[Earth]] and [[Mars]]. Weight varies due to different amount of [[gravitational acceleration]] whereas mass stays the same.]]
In everyday usage, mass and "[[weight]]" are often used interchangeably. For instance, a person's weight may be stated as 75&nbsp;kg. In a constant gravitational field, the weight of an object is proportional to its mass, and it is unproblematic to use the same unit for both concepts. But because of slight differences in the strength of the [[Gravity of Earth|Earth's gravitational field]] at different places, the [[Mass versus weight|distinction]] becomes important for measurements with a precision better than a few percent, and for places far from the surface of the Earth, such as in space or on other planets. Conceptually, "mass" (measured in [[kilograms]]) refers to an intrinsic property of an object, whereas "weight" (measured in [[newtons]]) measures an object's resistance to deviating from its current course of [[free fall]], which can be influenced by the nearby gravitational field. No matter how strong the gravitational field, objects in free fall are [[Weightlessness|weightless]], though they still have mass.<ref>{{cite news |last=Kane |first=Gordon |title=The Mysteries of Mass |newspaper=Scientific American |pages=32–39 |publisher=Nature America, Inc. |date=4 September 2008 |url=http://www.scientificamerican.com/article.cfm?id=the-mysteries-of-mass |access-date=2013-07-05}}</ref>

The force known as "weight" is proportional to mass and [[acceleration]] in all situations where the mass is accelerated away from free fall. For example, when a body is at rest in a gravitational field (rather than in free fall), it must be accelerated by a force from a scale or the surface of a planetary body such as the [[Earth]] or the [[Moon]]. This force keeps the object from going into free fall. Weight is the opposing force in such circumstances and is thus determined by the acceleration of free fall. On the surface of the Earth, for example, an object with a mass of 50&nbsp;kilograms weighs 491 newtons, which means that 491 newtons is being applied to keep the object from going into free fall. By contrast, on the surface of the Moon, the same object still has a mass of 50&nbsp;kilograms but weighs only 81.5&nbsp;newtons, because only 81.5 newtons is required to keep this object from going into a free fall on the moon. Restated in mathematical terms, on the surface of the Earth, the weight ''W'' of an object is related to its mass ''m'' by {{nowrap|1=''W'' = ''mg''}}, where {{nowrap|1=''g'' = {{val|fmt=commas|9.80665|u=m/s<sup>2</sup>}}}} is the acceleration due to [[Earth's gravity|Earth's gravitational field]], (expressed as the acceleration experienced by a free-falling object).

For other situations, such as when objects are subjected to mechanical accelerations from forces other than the resistance of a planetary surface, the weight force is proportional to the mass of an object multiplied by the total acceleration away from free fall, which is called the [[proper acceleration]]. Through such mechanisms, objects in elevators, vehicles, centrifuges, and the like, may experience weight forces many times those caused by resistance to the effects of gravity on objects, resulting from planetary surfaces. In such cases, the generalized equation for weight ''W'' of an object is related to its mass ''m'' by the equation {{nowrap|1=''W'' = –''ma''}}, where ''a'' is the proper acceleration of the object caused by all influences other than gravity. (Again, if gravity is the only influence, such as occurs when an object falls freely, its weight will be zero).

=== Inertial vs. gravitational mass ===
{{See also|Eötvös experiment}}
Although inertial mass, passive gravitational mass and active gravitational mass are conceptually distinct, no experiment has ever unambiguously demonstrated any difference between them. In [[classical mechanics]], Newton's third law implies that active and passive gravitational mass must always be identical (or at least proportional), but the classical theory offers no compelling reason why the gravitational mass has to equal the inertial mass. That it does is merely an empirical fact.

[[Albert Einstein]] developed his [[general theory of relativity]] starting with the assumption that the inertial and passive gravitational masses are the same.  This is known as the [[equivalence principle]].

The particular equivalence often referred to as the "Galilean equivalence principle" or the "[[weak equivalence principle]]" has the most important consequence for freely falling objects. Suppose an object has inertial and gravitational masses ''m'' and ''M'', respectively. If the only force acting on the object comes from a gravitational field ''g'', the force on the object is:
: <math>F = M g.</math>

Given this force, the acceleration of the object can be determined by Newton's second law:
: <math>F = m a.</math>

Putting these together, the gravitational acceleration is given by:
: <math qid=Q30006>a=\frac{M}{m}g.</math>

This says that the ratio of gravitational to inertial mass of any object is equal to some constant ''K'' [[if and only if]] all objects fall at the same rate in a given gravitational field. This phenomenon is referred to as the "universality of free-fall". In addition, the constant ''K'' can be taken as 1 by defining our units appropriately.

The first experiments demonstrating the universality of free-fall were—according to scientific 'folklore'—conducted by [[Galileo Galilei|Galileo]] obtained by dropping objects from the [[Leaning Tower of Pisa]]. This is most likely apocryphal: he is more likely to have performed his experiments with balls rolling down nearly frictionless [[inclined plane]]s to slow the motion and increase the timing accuracy. Increasingly precise experiments have been performed, such as those performed by [[Loránd Eötvös]],<ref>
{{cite journal |last1=Eötvös |first1=R.V. |last2=Pekár |first2=D. |last3=Fekete |first3=E. |date=1922 |title=''Beiträge zum Gesetz der Proportionalität von Trägheit und Gravität'' |journal=[[Annalen der Physik]] |volume=68 |issue=9 |pages=11–66 |bibcode=  1922AnP...373...11E|doi=10.1002/andp.19223730903|url=http://real.mtak.hu/94133/1/300_430_68.pdf }}</ref> using the [[torsion balance]] pendulum, in 1889. {{As of|2008}}, no deviation from universality, and thus from Galilean equivalence, has ever been found, at least to the precision 10<sup>−6</sup>. More precise experimental efforts are still being carried out.<ref>{{cite journal |last1=Voisin |first1=G. |last2=Cognard |first2=I. |last3=Freire |first3=P. C. C. |last4=Wex |first4=N. |last5=Guillemot |first5=L. |last6=Desvignes |first6=G. |last7=Kramer |first7=M. |last8=Theureau |first8=G. |title=An improved test of the strong equivalence principle with the pulsar in a triple star system |journal=Astronomy & Astrophysics |date=June 2020 |volume=638 |pages=A24 |doi=10.1051/0004-6361/202038104 |arxiv=2005.01388 |bibcode=2020A&A...638A..24V |s2cid=218486794 |url=https://www.aanda.org/articles/aa/full_html/2020/06/aa38104-20/aa38104-20.html |access-date=4 May 2022}}</ref>

[[File:Apollo 15 feather and hammer drop.ogv|right|thumb|250px|Astronaut David Scott performs the feather and hammer drop experiment on the Moon.]]

The universality of free-fall only applies to systems in which gravity is the only acting force. All other forces, especially [[friction]] and [[air resistance]], must be absent or at least negligible. For example, if a hammer and a feather are dropped from the same height through the air on Earth, the feather will take much longer to reach the ground; the feather is not really in ''free''-fall because the force of air resistance upwards against the feather is comparable to the downward force of gravity. On the other hand, if the experiment is performed in a [[vacuum]], in which there is no air resistance, the hammer and the feather should hit the ground at exactly the same time (assuming the acceleration of both objects towards each other, and of the ground towards both objects, for its own part, is negligible). This can easily be done in a high school laboratory by dropping the objects in transparent tubes that have the air removed with a vacuum pump. It is even more dramatic when done in an environment that naturally has a vacuum, as [[David Scott]] did on the surface of the [[Moon]] during [[Apollo 15]].

A stronger version of the equivalence principle, known as the ''Einstein equivalence principle'' or the ''strong equivalence principle'', lies at the heart of the [[general relativity|general theory of relativity]]. Einstein's equivalence principle states that within sufficiently small regions of spacetime, it is impossible to distinguish between a uniform acceleration and a uniform gravitational field. Thus, the theory postulates that the force acting on a massive object caused by a gravitational field is a result of the object's tendency to move in a straight line (in other words its inertia) and should therefore be a function of its inertial mass and the strength of the gravitational field.

=== Origin ===

{{main|Mass generation mechanism}}
In [[theoretical physics]], a [[mass generation mechanism]] is a theory which attempts to explain the origin of mass from the most fundamental laws of [[physics]]. To date, a number of different models have been proposed which advocate different views of the origin of mass. The problem is complicated by the fact that the notion of mass is strongly related to the [[gravitational interaction]] but a theory of the latter has not been yet reconciled with the currently popular model of [[particle physics]], known as the [[Standard Model]].

== Pre-Newtonian concepts ==
=== Weight as an amount ===
{{main|Weight}}
[[File:El pesado del corazón en el Papiro de Hunefer.jpg|right|thumb|Depiction of early [[balance scales]] in the [[Papyrus of Hunefer]] (dated to the [[Nineteenth dynasty of Egypt|19th dynasty]], {{circa|1285 BCE}}). The scene shows [[Anubis]] weighing the heart of Hunefer.]]

The concept of [[wikt:amount|amount]] is very old and [[Prehistoric numerals|predates recorded history]]. The concept of "weight" would incorporate "amount" and acquire a double meaning that was not clearly recognized as such.<ref name=":0" />

{{Blockquote|text=What we now know as mass was until the time of Newton called “weight.” ... A goldsmith believed that an ounce of gold was a quantity of gold. ... But the ancients believed that a beam balance also measured “heaviness” which they recognized through their muscular senses. ... Mass and its associated downward force were believed to be the same thing.|author=K. M. Browne|title=The pre-Newtonian meaning of the word “weight”}}

Humans, at some early era, realized that the weight of a collection of similar objects was [[Proportionality (mathematics)|directly proportional]] to the number of objects in the collection:
: <math>W_n \propto n,</math>
where ''W'' is the weight of the collection of similar objects and ''n'' is the number of objects in the collection. Proportionality, by definition, implies that two values have a constant [[ratio]]:
: <math>\frac{W_n}{n} = \frac{W_m}{m}</math>, or equivalently <math>\frac{W_n}{W_m} = \frac{n}{m}.</math>

An early use of this relationship is a [[balance scale]], which balances the force of one object's weight against the force of another object's weight. The two sides of a balance scale are close enough that the objects experience similar gravitational fields.  Hence, if they have similar masses then their weights will also be similar.  This allows the scale, by comparing weights, to also compare masses.

Consequently, historical weight standards were often defined in terms of amounts.  The Romans, for example, used the [[carob]] seed ([[Carat (unit)|carat]] or [[siliqua]]) as a measurement standard.  If an object's weight was equivalent to [http://std.dkuug.dk/JTC1/SC2/WG2/docs/n3138.pdf 1728 carob seeds], then the object was said to weigh one Roman pound.  If, on the other hand, the object's weight was equivalent to [[Ancient Roman units of measurement|144 carob seeds]] then the object was said to weigh one Roman ounce (uncia).  The Roman pound and ounce were both defined in terms of different sized collections of the same common mass standard, the carob seed.  The ratio of a Roman ounce (144 carob seeds) to a Roman pound (1728 carob seeds) was:
: <math>\frac{\mathrm{ounce}}{\mathrm{pound}} = \frac{W_{144}}{W_{1728}} = \frac{144}{1728} = \frac{1}{12}.</math>

=== Planetary motion ===
{{see also|Kepler's laws of planetary motion}}
In 1600 AD, [[Johannes Kepler]] sought employment with [[Tycho Brahe]], who had some of the most precise astronomical data available.  Using Brahe's precise observations of the planet Mars, Kepler spent the next five years developing his own method for characterizing planetary motion. In 1609, Johannes Kepler published his three laws of planetary motion, explaining how the planets orbit the Sun. In Kepler's final planetary model, he described planetary orbits as following elliptical paths with the Sun at a focal point of the [[ellipse]]. Kepler discovered that the [[square (algebra)|square]] of the [[orbital period]] of each planet is directly [[Proportionality (mathematics)|proportional]] to the [[cube (arithmetic)|cube]] of the [[semi-major axis]] of its orbit, or equivalently, that the [[ratio]] of these two values is constant for all planets in the [[Solar System]].<ref group=note>This constant ratio was later shown to be a direct measure of the Sun's active gravitational mass; it has units of distance cubed per time squared, and is known as the [[standard gravitational parameter]]:
: <math qid=Q140028>\mu=4\pi^2\frac{\text{distance}^3}{\text{time}^2}\propto\text{gravitational mass}</math></ref>

On 25 August 1609, [[Galileo Galilei]] demonstrated his first telescope to a group of Venetian merchants, and in early January 1610, Galileo observed four dim objects near Jupiter, which he mistook for stars.  However, after a few days of observation, Galileo realized that these "stars" were in fact orbiting Jupiter.  These four objects (later named the [[Galilean moons]] in honor of their discoverer) were the first celestial bodies observed to orbit something other than the Earth or Sun.  Galileo continued to observe these moons over the next eighteen months, and by the middle of 1611, he had obtained remarkably accurate estimates for their periods.

=== Galilean free fall ===
[[File:Galileo.arp.300pix.jpg|left|upright|thumb|Galileo Galilei (1636)]]
[[File:Falling ball.jpg|thumb|right|upright|Distance traveled by a freely falling ball is proportional to the square of the elapsed time.]]
Sometime prior to 1638, Galileo turned his attention to the phenomenon of objects in free fall, attempting to characterize these motions. Galileo was not the first to investigate Earth's gravitational field, nor was he the first to accurately describe its fundamental characteristics.  However, Galileo's reliance on scientific experimentation to establish physical principles would have a profound effect on future generations of scientists. It is unclear if these were just hypothetical experiments used to illustrate a concept, or if they were real experiments performed by Galileo,<ref>{{cite journal |last=Drake |first=S. |date=1979 |title=Galileo's Discovery of the Law of Free Fall |journal=Scientific American |volume=228 |issue=5 |pages=84–92 |bibcode=1973SciAm.228e..84D |doi=10.1038/scientificamerican0573-84}}</ref> but the results obtained from these experiments were both realistic and compelling.  A biography by Galileo's pupil [[Vincenzo Viviani]] stated that Galileo had dropped [[ball]]s of the same material, but different masses, from the [[Leaning Tower of Pisa]] to demonstrate that their time of descent was independent of their mass.<ref group="note">At the time when Viviani asserts that the experiment took place, Galileo had not yet formulated the final version of his law of free fall. He had, however, formulated an earlier version that predicted that bodies ''of the same material'' falling through the same medium would fall at the same speed. See {{cite book |last=Drake |first=S. |date=1978 |title=Galileo at Work |pages=[https://archive.org/details/galileoatwork00stil/page/19 19–20] |publisher=University of Chicago Press |isbn=978-0-226-16226-3 |url=https://archive.org/details/galileoatwork00stil/page/19 }}</ref> In support of this conclusion, Galileo had advanced the following theoretical argument: He asked if two bodies of different masses and different rates of fall are tied by a string, does the combined system fall faster because it is now more massive, or does the lighter body in its slower fall hold back the heavier body?  The only convincing resolution to this question is that all bodies must fall at the same rate.<ref>{{cite book |last=Galileo |first=G. |date=1632 |title=Dialogue Concerning the Two Chief World Systems|title-link=Dialogue Concerning the Two Chief World Systems }}</ref>

A later experiment was described in Galileo's ''Two New Sciences'' published in 1638.  One of Galileo's fictional characters, Salviati, describes an experiment using a bronze ball and a wooden ramp.  The wooden ramp was "12 cubits long, half a cubit wide and three finger-breadths thick" with a straight, smooth, polished [[Groove (engineering)|groove]].  The groove was lined with "[[parchment]], also smooth and polished as possible".  And into this groove was placed "a hard, smooth and very round bronze ball".  The ramp was inclined at various [[angle]]s to slow the acceleration enough so that the elapsed time could be measured.  The ball was allowed to roll a known distance down the ramp, and the time taken for the ball to move the known distance was measured.  The time was measured using a water clock described as follows:
:a large vessel of water placed in an elevated position; to the bottom of this vessel was soldered a pipe of small diameter giving a thin jet of water, which we collected in a small glass during the time of each descent, whether for the whole length of the channel or for a part of its length; the water thus collected was weighed, after each descent, on a very accurate balance; the differences and ratios of these weights gave us the differences and ratios of the times, and this with such accuracy that although the operation was repeated many, many times, there was no appreciable discrepancy in the results.<ref>{{cite book |last1=Galileo |first1=G. |date=1638 |title=Discorsi e Dimostrazioni Matematiche, Intorno à Due Nuove Scienze |volume=213 |publisher=[[House of Elzevir|Louis Elsevier]]}}, translated in {{cite book |date=1954 |editor1-last=Crew |editor1-first=H. |editor2-last=de Salvio |editor2-first=A. |title=Mathematical Discourses and Demonstrations, Relating to Two New Sciences |url=http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=753&Itemid=99999999 |publisher=[[Dover Publications]] |isbn=978-1-275-10057-2 |access-date=11 April 2012 |archive-date=1 October 2013 |archive-url=https://web.archive.org/web/20131001015122/http://oll.libertyfund.org/index.php?option=com_staticxt&staticfile=show.php%3Ftitle=753&Itemid=99999999 |url-status=dead }} and also available in {{cite book |editor1-last=Hawking |editor1-first=S. |date=2002 |title=On the Shoulders of Giants |pages=[https://archive.org/details/isbn_9780762413485/page/534 534–535] |publisher=[[Running Press]] |isbn=978-0-7624-1348-5 |url=https://archive.org/details/isbn_9780762413485/page/534 }}</ref>

Galileo found that for an object in free fall, the distance that the object has fallen is always proportional to the square of the elapsed time:
: <math>{\text{Distance}} \propto {\text{Time}^2}</math>

Galileo had shown that objects in free fall under the influence of the Earth's gravitational field have a constant acceleration, and Galileo's contemporary, Johannes Kepler, had shown that the planets follow elliptical paths under the influence of the Sun's gravitational mass.  However, Galileo's free fall motions and Kepler's planetary motions remained distinct during Galileo's lifetime.

=== Mass as distinct from weight ===
According to K. M. Browne: "Kepler formed a [distinct] concept of mass ('amount of matter' (''copia materiae'')), but called it 'weight' as did everyone at that time."<ref name=":0">{{Cite journal |last=Browne |first=K. M. |year=2018 |title=The pre-Newtonian meaning of the word "weight"; a comment on "Kepler and the origins of pre-Newtonian mass" [Am. J. Phys. 85, 115–123 (2017)] |journal=American Journal of Physics |volume=86 |issue=6 |pages=471–74|doi=10.1119/1.5027490 |bibcode=2018AmJPh..86..471B |s2cid=125953814 |doi-access=free }}</ref> Finally, in 1686, Newton gave this distinct concept its own name. In the first paragraph of ''Principia'', Newton defined quantity of matter as “density and bulk conjunctly”, and mass as quantity of matter.<ref>{{Cite book |last=Newton |first=I. |url=https://books.google.com/books?id=Tm0FAAAAQAAJ&pg=PP13 |title=The mathematical principles of natural philosophy |publisher=Printed for Benjamin Motte |year=1729 |pages=1–2 |translator-last=Motte |translator-first=A. |orig-date=1686}}</ref>{{Blockquote|text=The quantity of matter is the measure of the same, arising from its density and bulk conjunctly. ... It is this quantity that I mean hereafter everywhere under the name of body or mass. And the same is known by the weight of each body; for it is proportional to the weight.|author=Isaac Newton|title=Mathematical principles of natural philosophy|source=Definition I.}}

== Newtonian mass ==
{| class="wikitable floatright" 
|-
! colspan=2| Earth's Moon !! rowspan=2| Mass of Earth
|-
! [[Semi-major axis]] !! [[Sidereal orbital period]]
|-
|0.002 569 [[Astronomical unit|AU]]||0.074 802 [[sidereal year]]
|rowspan=3|<math qid=Q11376>1.2\pi^2\cdot10^{-5}\frac{\text{AU}^3}{\text{y}^2}=3.986\cdot10^{14}\frac{\text{m}^3}{\text{s}^2}</math>
|-
! Earth's gravity !! Earth's radius
|-
|9.806 65&nbsp;m/s<sup>2</sup>||6 375&nbsp;km
|}

[[File:GodfreyKneller-IsaacNewton-1689.jpg|Isaac Newton, 1689|upright|thumb]]

[[Robert Hooke]] had published his concept of gravitational forces in 1674, stating that all [[celestial bodies]] have an attraction or gravitating power towards their own centers, and also attract all the other celestial bodies that are within the sphere of their activity. He further stated that gravitational attraction increases by how much nearer the body wrought upon is to its own center.<ref>{{cite book |last=Hooke |first=R. |date=1674 |title=An attempt to prove the motion of the earth from observations |url=https://books.google.com/books?id=JgtPAAAAcAAJ&pg=PA1 |publisher=[[Royal Society]]}}</ref> In correspondence with [[Isaac Newton]] from 1679 and 1680, Hooke conjectured that gravitational forces might decrease according to the double of the distance between the two bodies.<ref>{{cite book |editor-last=Turnbull |editor-first=H.W. |date=1960 |title=Correspondence of Isaac Newton, Volume 2 (1676–1687) |page=297 |publisher=Cambridge University Press}}</ref> Hooke urged Newton, who was a pioneer in the development of [[calculus]], to work through the mathematical details of Keplerian orbits to determine if Hooke's hypothesis was correct.  Newton's own investigations verified that Hooke was correct, but due to personal differences between the two men, Newton chose not to reveal this to Hooke.  Isaac Newton kept quiet about his discoveries until 1684, at which time he told a friend, [[Edmond Halley]], that he had solved the problem of gravitational orbits, but had misplaced the solution in his office.<ref>{{cite book |title=Principia |url=https://massless.info/images/Isaac_Newton_Principia_English.pdf |pages=16}}</ref> After being encouraged by Halley, Newton decided to develop his ideas about gravity and publish all of his findings.  In November 1684, Isaac Newton sent a document to Edmund Halley, now lost but presumed to have been titled ''[[De motu corporum in gyrum]]'' (Latin for "On the motion of bodies in an orbit").<ref>
{{cite book |editor-last=Whiteside |editor-first=D.T. |date=2008 |title=The Mathematical Papers of Isaac Newton, Volume VI (1684–1691) |url=https://books.google.com/books?id=lIZ0v23iqRgC|publisher=Cambridge University Press |isbn=978-0-521-04585-8}}</ref> Halley presented Newton's findings to the [[Royal Society]] of London, with a promise that a fuller presentation would follow.  Newton later recorded his ideas in a three-book set, entitled ''[[Philosophiæ Naturalis Principia Mathematica]]'' (English: ''Mathematical Principles of Natural Philosophy'').  The first was received by the Royal Society on 28 April 1685–86; the second on 2 March 1686–87; and the third on 6 April 1686–87.  The Royal Society published Newton's entire collection at their own expense in May 1686–87.<ref name="principia">{{cite book |author1=Sir Isaac Newton |author2=N.W. Chittenden |title=Newton's Principia: The mathematical principles of natural philosophy |url=https://archive.org/details/newtonsprincipi00chitgoog |page=[https://archive.org/details/newtonsprincipi00chitgoog/page/n43 31]|year=1848 |publisher=D. Adee|isbn=9780520009295 }}</ref>{{rp|31}}

Isaac Newton had bridged the gap between Kepler's gravitational mass and Galileo's gravitational acceleration, resulting in the discovery of the following relationship which governed both of these:
: <math>\mathbf{g}=-\mu\frac{\hat{\mathbf{R}}}{|\mathbf{R}|^2}</math>
where '''g''' is the apparent acceleration of a body as it passes through a region of space where gravitational fields exist, ''μ'' is the gravitational mass ([[standard gravitational parameter]]) of the body causing gravitational fields, and '''R''' is the radial coordinate (the distance between the centers of the two bodies).

By finding the exact relationship between a body's gravitational mass and its gravitational field, Newton provided a second method for measuring gravitational mass.  The mass of the Earth can be determined using Kepler's method (from the orbit of Earth's Moon), or it can be determined by measuring the gravitational acceleration on the Earth's surface, and multiplying that by the square of the Earth's radius.  The mass of the Earth is approximately three-millionths of the mass of the Sun.  To date, no other accurate method for measuring gravitational mass has been discovered.<ref>{{cite web |last=Cuk |first=M. |date=January 2003 |title=Curious About Astronomy: How do you measure a planet's mass? |url=http://curious.astro.cornell.edu/question.php?number=452 |work=Ask an Astronomer |access-date=2011-03-12 |url-status=dead |archive-url=https://web.archive.org/web/20030320113723/http://curious.astro.cornell.edu/question.php?number=452 |archive-date=20 March 2003}}</ref>

=== Newton's cannonball ===
[[File:Newton Cannon.svg|thumb|A cannon on top of a very high mountain shoots a cannonball horizontally.  If the speed is low, the cannonball quickly falls back to Earth (A,&nbsp;B). At [[orbital speed|intermediate speeds]], it will revolve around Earth along an elliptical orbit (C,&nbsp;D). Beyond the [[escape velocity]], it will leave the Earth without returning (E).]]
{{main|Newton's cannonball}}

Newton's cannonball was a [[thought experiment]] used to bridge the gap between Galileo's gravitational acceleration and Kepler's elliptical orbits.  It appeared in Newton's 1728 book ''A Treatise of the System of the World''.  According to Galileo's concept of gravitation, a dropped stone falls with constant acceleration down towards the Earth.  However, Newton explains that when a stone is thrown horizontally (meaning sideways or perpendicular to Earth's gravity) it follows a curved path.  "For a stone projected is by the pressure of its own weight forced out of the rectilinear path, which by the projection alone it should have pursued, and made to describe a curve line in the air; and through that crooked way is at last brought down to the ground.  And the greater the velocity is with which it is projected, the farther it goes before it falls to the Earth."<ref name=principia />{{rp|513}} Newton further reasons that if an object were "projected in an horizontal direction from the top of a high mountain" with sufficient velocity, "it would reach at last quite beyond the circumference of the Earth, and return to the mountain from which it was projected."<ref>{{cite book |last1=Newton |first1=Isaac |author-link=Isaac Newton |title=A Treatise of the System of the World |date=1728 |publisher=F. Fayram |location=London |page=6 |url=https://books.google.com/books?id=rEYUAAAAQAAJ&q=ball&pg=PR1 |access-date=4 May 2022}}</ref>

=== Universal gravitational mass ===
[[File:Universal gravitational mass.jpg|An apple experiences gravitational fields directed towards every part of the Earth; however, the sum total of these many fields produces a single gravitational field directed towards the Earth's center.|left|thumb]]
In contrast to earlier theories (e.g. [[celestial spheres]]) which stated that the heavens were made of entirely different material, Newton's theory of mass was groundbreaking partly because it introduced [[Newton's law of universal gravitation|universal gravitational mass]]: every object has gravitational mass, and therefore, every object generates a gravitational field.  Newton further assumed that the strength of each object's gravitational field would decrease according to the square of the distance to that object. If a large collection of small objects were formed into a giant spherical body such as the Earth or Sun, Newton calculated the collection would create a gravitational field proportional to the total mass of the body,<ref name="principia"/>{{rp|397}} and inversely proportional to the square of the distance to the body's center.<ref name="principia"/>{{rp|221}}<ref group="note">These two properties are very useful, as they allow spherical collections of objects to be treated exactly like large individual objects.</ref>

For example, according to Newton's theory of universal gravitation, each carob seed produces a gravitational field.  Therefore, if one were to gather an immense number of carob seeds and form them into an enormous sphere, then the gravitational field of the sphere would be proportional to the number of carob seeds in the sphere.  Hence, it should be theoretically possible to determine the exact number of carob seeds that would be required to produce a gravitational field similar to that of the Earth or Sun.  In fact, by [[unit conversion]] it is a simple matter of abstraction to realize that any traditional mass unit can theoretically be used to measure gravitational mass.

[[File:Cavendish Experiment.png|thumb|right|Vertical section drawing of Cavendish's torsion balance instrument including the building in which it was housed. The large balls were hung from a frame so they could be rotated into position next to the small balls by a pulley from outside. Figure 1 of Cavendish's paper.]]

Measuring gravitational mass in terms of traditional mass units is simple in principle, but extremely difficult in practice.  According to Newton's theory, all objects produce gravitational fields and it is theoretically possible to collect an immense number of small objects and form them into an enormous gravitating sphere.  However, from a practical standpoint, the gravitational fields of small objects are extremely weak and difficult to measure. Newton's books on universal gravitation were published in the 1680s, but the first successful measurement of the Earth's mass in terms of traditional mass units, the [[Cavendish experiment]], did not occur until 1797, over a hundred years later. [[Henry Cavendish]] found that the Earth's density was 5.448 ± 0.033 times that of water.  As of 2009, the Earth's mass in kilograms is only known to around five digits of accuracy, whereas its gravitational mass is known to over nine significant figures.{{clarify|reason="mass in kilograms" does not specify an experimental method. What's the distinction between the methods?|date=January 2014}}

Given two objects A and B, of masses ''M''<sub>A</sub> and ''M''<sub>B</sub>, separated by a [[Displacement (vector)|displacement]] '''R'''<sub>AB</sub>, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude
: <math>\mathbf{F}_{\text{AB}}=-GM_{\text{A}}M_{\text{B}}\frac{\hat{\mathbf{R}}_{\text{AB}}}{|\mathbf{R}_{\text{AB}}|^2}\ </math>,
where ''G'' is the universal [[gravitational constant]]. The above statement may be reformulated in the following way: if ''g'' is the magnitude at a given location in a gravitational field, then the gravitational force on an object with gravitational mass ''M'' is
: <math>F=Mg</math>.
This is the basis by which masses are determined by [[weighing]]. In simple [[spring scales]], for example, the force ''F'' is proportional to the displacement of the [[spring (device)|spring]] beneath the weighing pan, as per [[Hooke's law]], and the scales are [[calibration|calibrated]] to take ''g'' into account, allowing the mass ''M'' to be read off. Assuming the gravitational field is equivalent on both sides of the balance, a [[Beam balance|balance]] measures relative weight, giving the relative gravitation mass of each object.

=== Inertial mass ===
Mass was traditionally believed to be a measure of the quantity of matter in a physical body, equal to the "amount of matter" in an object. For example, [[Adhémar Jean Claude Barré de Saint-Venant|Barre´ de Saint-Venant]] argued in 1851 that every object contains a number of "points" (basically, interchangeable elementary particles), and that mass is proportional to the number of points the object contains.<ref>{{cite journal |last1=Coelho |first1=Ricardo Lopes |title=On the Concept of Force: How Understanding its History can Improve Physics Teaching |journal=Science & Education |date=January 2010 |volume=19 |issue=1 |pages=91–113 |doi=10.1007/s11191-008-9183-1|bibcode=2010Sc&Ed..19...91C |s2cid=195229870 }}</ref> (In practice, this "amount of matter" definition is adequate for most of classical mechanics, and sometimes remains in use in basic education, if the priority is to teach the difference between mass from weight.)<ref>{{cite web |last1=Gibbs |first1=Yvonne |title=Teachers Learn the Difference Between Mass and Weight Even in Space |url=https://www.nasa.gov/centers/armstrong/features/teachers-learn-the-difference-between-mass-and-weight-even-in-space |website=NASA |access-date=20 March 2023 |date=31 March 2017 |archive-date=20 March 2023 |archive-url=https://web.archive.org/web/20230320070434/https://www.nasa.gov/centers/armstrong/features/teachers-learn-the-difference-between-mass-and-weight-even-in-space/ |url-status=dead }}</ref> This traditional "amount of matter" belief was contradicted by the fact that different atoms (and, later, different elementary particles) can have different masses, and was further contradicted by Einstein's theory of relativity (1905), which showed that the measurable mass of an object increases when energy is added to it (for example, by increasing its temperature or forcing it near an object that electrically repels it.) This motivates a search for a different definition of mass that is more accurate than the traditional definition of "the amount of matter in an object".<ref>{{cite journal |last1=Hecht |first1=Eugene |title=There Is No Really Good Definition of Mass |journal=The Physics Teacher |date=January 2006 |volume=44 |issue=1 |pages=40–45 |doi=10.1119/1.2150758|bibcode=2006PhTea..44...40H }}</ref>

[[File:Massmeter.jpg|thumb|Massmeter, a device for measuring the inertial mass of an astronaut in weightlessness. The mass is calculated via the oscillation period for a spring with the astronaut attached ([[Tsiolkovsky State Museum of the History of Cosmonautics]]).]]
''Inertial mass'' is the mass of an object measured by its resistance to acceleration. This definition has been championed by [[Ernst Mach]]<ref>Ernst Mach, "Science of Mechanics" (1919)</ref><ref name=belkind>Ori Belkind, "Physical Systems: Conceptual Pathways between Flat Space-time and Matter" (2012) Springer (''Chapter 5.3'')</ref> and has since been developed into the notion of [[Operationalization|operationalism]] by [[Percy W. Bridgman]].<ref>P.W. Bridgman, ''Einstein's Theories and the Operational Point of View'', in: P.A. Schilpp, ed., ''Albert Einstein: Philosopher-Scientist'', Open Court, La Salle, Ill., Cambridge University Press, 1982, Vol. 2, pp. 335–354.</ref><ref>{{cite journal | last1 = Gillies | first1 = D.A. | year = 1972 | title = PDF | url = https://www.ucl.ac.uk/sts/staff/gillies/documents/1972a_Operationalism.pdf | journal = Synthese | volume = 25 | pages = 1–24 | doi = 10.1007/BF00484997 | s2cid = 239369276 | access-date = 10 April 2016 | archive-date = 26 April 2016 | archive-url = https://web.archive.org/web/20160426201827/https://www.ucl.ac.uk/sts/staff/gillies/documents/1972a_Operationalism.pdf | url-status = dead }}</ref> The simple [[classical mechanics]] definition of mass differs slightly from the definition in the theory of [[special relativity]], but the essential meaning is the same.

In classical mechanics, according to [[Newton's second law]], we say that a body has a mass ''m'' if, at any instant of time, it obeys the equation of motion
: <math>\mathbf{F}=m \mathbf{a},</math>
where '''F''' is the resultant [[force]] acting on the body and '''a''' is the [[acceleration]] of the body's centre of mass.<ref group="note">In its original form, Newton's second law is valid only for bodies of constant mass.</ref> For the moment, we will put aside the question of what "force acting on the body" actually means.

This equation illustrates how mass relates to the [[inertia]] of a body. Consider two objects with different masses. If we apply an identical force to each, the object with a bigger mass will experience a smaller acceleration, and the object with a smaller mass will experience a bigger acceleration. We might say that the larger mass exerts a greater "resistance" to changing its state of motion in response to the force.

However, this notion of applying "identical" forces to different objects brings us back to the fact that we have not really defined what a force is. We can sidestep this difficulty with the help of [[Newton's third law]], which states that if one object exerts a force on a second object, it will experience an equal and opposite force. To be precise, suppose we have two objects of constant inertial masses ''m''<sub>1</sub> and ''m''<sub>2</sub>. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on ''m''<sub>1</sub> by ''m''<sub>2</sub>, which we denote '''F'''<sub>12</sub>, and the force exerted on ''m''<sub>2</sub> by ''m''<sub>1</sub>, which we denote '''F'''<sub>21</sub>. Newton's second law states that
: <math>
\begin{align}
\mathbf{F_{12}} & =m_1\mathbf{a}_1,\\
\mathbf{F_{21}} & =m_2\mathbf{a}_2,
\end{align}</math>
where '''a'''<sub>1</sub> and '''a'''<sub>2</sub> are the accelerations of ''m''<sub>1</sub> and ''m''<sub>2</sub>, respectively. Suppose that these accelerations are non-zero, so that the forces between the two objects are non-zero. This occurs, for example, if the two objects are in the process of colliding with one another. Newton's third law then states that
: <math qid=Q3235565>\mathbf{F}_{12}=-\mathbf{F}_{21};</math>
and thus
: <math>m_1=m_2\frac{|\mathbf{a}_2|}{|\mathbf{a}_1|}\!.</math>

If {{abs|'''a'''<sub>1</sub>}} is non-zero, the fraction is well-defined, which allows us to measure the inertial mass of ''m''<sub>1</sub>. In this case, ''m''<sub>2</sub> is our "reference" object, and we can define its mass ''m'' as (say) 1&nbsp;kilogram. Then we can measure the mass of any other object in the universe by colliding it with the reference object and measuring the accelerations.

Additionally, mass relates a body's [[momentum]] '''p''' to its linear [[velocity]] '''v''':
: <math qid=Q41273>\mathbf{p}=m\mathbf{v}</math>,
and the body's [[kinetic energy]] ''K'' to its velocity:
: <math qid=Q46276>K=\dfrac{1}{2}m|\mathbf{v}|^2</math>.

The primary difficulty with Mach's definition of mass is that it fails to take into account the [[potential energy]] (or [[binding energy]]) needed to bring two masses sufficiently close to one another to perform the measurement of mass.<ref name=belkind/> This is most vividly demonstrated by comparing the mass of the [[proton]] in the nucleus of [[deuterium]], to the mass of the proton in free space (which is greater by about 0.239%—this is due to the binding energy of deuterium). Thus, for example, if the reference weight ''m''<sub>2</sub> is taken to be the mass of the neutron in free space, and the relative accelerations for the proton and neutron in deuterium are computed, then the above formula over-estimates the mass ''m''<sub>1</sub> (by 0.239%) for the proton in deuterium.  At best, Mach's formula can only be used to obtain ratios of masses, that is, as ''m''<sub>1</sub>&nbsp;/&nbsp;''m''<sub>2</sub> = {{abs|'''a'''<sub>2</sub>}}&nbsp;/&nbsp;{{abs|'''a'''<sub>1</sub>}}.  An additional difficulty was pointed out by [[Henri Poincaré]], which is that the measurement of instantaneous acceleration is impossible: unlike the measurement of time or distance, there is no way to measure acceleration with a single measurement; one must make multiple measurements (of position, time, etc.) and perform a computation to obtain the acceleration.  Poincaré termed this to be an "insurmountable flaw" in the Mach definition of mass.<ref>Henri Poincaré. "[https://brocku.ca/MeadProject/Poincare/Poincare_1905_07.html Classical Mechanics]". Chapter 6 in Science and Hypothesis. London: Walter Scott Publishing (1905): 89-110.</ref>

== Atomic masses ==
{{main|Dalton (unit)}}

Typically, the mass of objects is measured in terms of the kilogram, which since 2019 is defined in terms of fundamental constants of nature. The mass of an atom or other particle can be compared more precisely and more conveniently to that of another atom, and thus scientists developed the [[Dalton (unit)|dalton]] (also known as the unified atomic mass unit). By definition, 1&nbsp;Da (one [[Dalton (unit)|dalton]]) is exactly one-twelfth of the mass of a [[carbon-12]] atom, and thus, a carbon-12 atom has a mass of exactly 12&nbsp;Da.

== In relativity ==

=== 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>}}).

=== General relativity ===
{{main|Mass in general relativity}}

In [[general relativity]], the [[equivalence principle]] is the equivalence of [[Gravitational mass|gravitational]] and [[Inertial Mass|inertial mass]]. At the core of this assertion is [[Albert Einstein|Albert Einstein's]] idea that the gravitational force as experienced locally while standing on a massive body (such as the Earth) is the same as the ''[[Fictitious force|pseudo-force]]'' experienced by an observer in a non-[[Inertial frame of reference|inertial]] (i.e. accelerated) frame of reference.

However, it turns out that it is impossible to find an objective general definition for the concept of [[invariant mass]] in general relativity. At the core of the problem is the [[Nonlinear system|non-linearity]] of the [[Einstein field equations]], making it impossible to write the gravitational field energy as part of the [[stress–energy tensor]] in a way that is invariant for all observers. For a given observer, this can be achieved by the [[stress–energy–momentum pseudotensor]].<ref>{{cite book |last1=Misner |first1=C.W. |last2=Thorne |first2=K.S. |last3=Wheeler |first3=J.A. |title=Gravitation |url=https://archive.org/details/gravitation00cwmi |url-access=limited |date=1973 |page=[https://archive.org/details/gravitation00cwmi/page/n498 466] |publisher=W.H. Freeman |isbn=978-0-7167-0344-0}}</ref>

== In quantum physics ==

In [[classical mechanics]], the inert mass of a particle appears in the [[Euler–Lagrange equation]] as a parameter ''m'':
: <math qid=Q875744>\frac{\mathrm{d}}{\mathrm{d}t} \ \left( \, \frac{\partial L}{\partial \dot{x}_i} \, \right) \ = \ m \, \ddot{x}_i .</math>

After quantization, replacing the position vector ''x'' with a [[wave function]], the parameter ''m'' appears in the [[kinetic energy]] operator:
: <math qid=Q165498>i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) =  \left(-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r})\right)\Psi(\mathbf{r},\,t). </math>

In the ostensibly [[Covariance and contravariance of vectors#Informal usage|covariant]] (relativistically invariant) [[Dirac equation]], and in [[natural units]], this becomes:
: <math qid=Q16853908>(-i\gamma^\mu\partial_\mu + m) \psi = 0</math>
where the "[[Invariant mass|mass]]" parameter ''m'' is now simply a constant associated with the [[quantum]] described by the wave function ψ.

In the [[Standard Model]] of [[particle physics]] as developed in the 1960s, this term arises from the coupling of the field ψ to an additional field Φ, the [[Higgs field]]. In the case of fermions, the [[Higgs mechanism]] results in the replacement of the term ''m''ψ in the Lagrangian with <math>G_{\psi} \overline{\psi} \phi \psi</math>. This shifts the [[explanandum]] of the value for the mass of each elementary particle to the value of the unknown [[coupling constant]] ''G''<sub>ψ</sub>.

=== Tachyonic particles and imaginary (complex) mass ===
{{main|Tachyonic field|Tachyon#Mass}}

A [[tachyonic field]], or simply [[tachyon]], is a [[quantum field]] with an [[imaginary number|imaginary]] mass.<ref name=Randall/> Although [[tachyon]]s ([[particle]]s that move [[faster-than-light|faster than light]]) are a purely hypothetical concept not generally believed to exist,<ref name=Randall>Lisa Randall, ''Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions'', p.286: "People initially thought of tachyons as particles travelling faster than the speed of light...But we now know that a tachyon indicates an instability in a theory that contains it.  Regrettably for science fiction fans, tachyons are not real physical particles that appear in nature."</ref><ref name=Tipler>{{cite book| last1 = Tipler| first1 = Paul A.|last2 = Llewellyn| first2 = Ralph A.| title = Modern Physics| url = https://archive.org/details/modernphysicsfif00paul| url-access = limited| publisher = W.H. Freeman & Co.| date = 2008| edition = 5th| location = New York | isbn = 978-0-7167-7550-8| page = [https://archive.org/details/modernphysicsfif00paul/page/n71 54]|quote = ... so existence of particles v > c ... Called tachyons ... would present relativity with serious ... problems of infinite creation energies and causality paradoxes.}}</ref> [[field (physics)|fields]] with imaginary mass have come to play an important [[Higgs boson|role]] in modern physics<ref name=Kutasov>{{cite journal | author = Kutasov, David | author2 = Marino, Marcos | author3 = Moore, Gregory W. | name-list-style = amp | title= Some exact results on tachyon condensation in string field theory | journal=   Journal of High Energy Physics| volume= 2000 | issue = 10 | page= 045 | date= 2000 | doi=10.1088/1126-6708/2000/10/045|arxiv = hep-th/0009148 |bibcode = 2000JHEP...10..045K | s2cid = 15664546 }}</ref><ref name=Sen>{{Cite journal | doi=10.1088/1126-6708/2002/04/048| arxiv=hep-th/0203211| bibcode=2002JHEP...04..048S| title=Rolling Tachyon| year=2002| last1=Sen| first1=Ashoke| journal=Journal of High Energy Physics| volume=2002| issue=4| pages=048| s2cid=12023565}}</ref><ref name=Gibbons>{{cite journal | last1 = Gibbons | first1 = G.W. | year = 2002 | title = Cosmological evolution of the rolling tachyon | journal = Phys. Lett. B | volume = 537 | issue = 1–2| pages = 1–4 | doi=10.1016/s0370-2693(02)01881-6|arxiv = hep-th/0204008 |bibcode = 2002PhLB..537....1G | s2cid = 119487619 }}</ref> and are discussed in popular books on physics.<ref name=Randall/><ref name=Greene>{{cite book |title=The Elegant Universe |title-link=The Elegant Universe |last=Greene |first=Brian |author-link=Brian Greene |publisher=Vintage Books|date=2000}}</ref> Under no circumstances do any excitations ever propagate faster than light in such theories—the presence or absence of a tachyonic mass has no effect whatsoever on the maximum velocity of signals (there is no violation of [[causality]]).<ref name="susskind">{{cite journal |volume = 182 |journal = Phys. Rev. |last1 = Aharonov |first1 = Y. |last2 = Komar |first2 = A. |last3= Susskind |first3 = L. |title = Superluminal Behavior, Causality, and Instability |date = 1969 |doi = 10.1103/PhysRev.182.1400|issue = 5|pages = 1400–1403|bibcode = 1969PhRv..182.1400A }}</ref> While the ''field'' may have imaginary mass, any physical particles do not; the "imaginary mass" shows that the system becomes unstable, and sheds the instability by undergoing a type of [[phase transition]] called [[tachyon condensation]] (closely related to second order phase transitions) that results in [[symmetry breaking]] in [[Standard Model|current models]] of [[particle physics]].

The term "[[tachyon]]" was coined by [[Gerald Feinberg]] in a 1967 paper,<ref name="feinberg67"/> but it was soon realized that Feinberg's model in fact did not allow for [[superluminal]] speeds.<ref name="susskind"/> Instead, the imaginary mass creates an instability in the configuration:- any configuration in which one or more field excitations are tachyonic will spontaneously decay, and the resulting configuration contains no physical tachyons.  This process is known as tachyon condensation.  Well known examples include the [[Higgs mechanism|condensation]] of the [[Higgs boson]] in [[particle physics]], and [[ferromagnetism]] in [[condensed matter physics]].

Although the notion of a tachyonic [[imaginary number|imaginary]] mass might seem troubling because there is no classical interpretation of an imaginary mass, the mass is not quantized.  Rather, the [[scalar field]] is; even for tachyonic [[quantum field theory|quantum fields]], the [[field operator]]s at [[Minkowski space|spacelike]] separated points still [[Canonical commutation relation|commute (or anticommute)]], thus preserving causality. Therefore, information still does not propagate faster than light,<ref name="feinberg67">{{cite journal | first = Gerald | last = Feinberg | author-link = Gerald Feinberg | title = Possibility of Faster-Than-Light Particles | journal = Physical Review | volume = 159 | year = 1967 | pages = 1089–1105 | doi = 10.1103/PhysRev.159.1089|bibcode = 1967PhRv..159.1089F | issue = 5 }}</ref> and solutions grow exponentially, but not superluminally (there is no violation of [[causality]]). [[Tachyon condensation]] drives a physical system that has reached a local limit and might naively be expected to produce physical tachyons, to an alternate stable state where no physical tachyons exist. Once the tachyonic field reaches the minimum of the potential, its quanta are not tachyons any more but rather are ordinary particles with a positive mass-squared.<ref name = "Peskin" />

This is a special case of the general rule, where unstable massive particles are formally described as having a [[complex number|complex]] mass, with the real part being their mass in the usual sense, and the imaginary part being the [[Particle decay#Decay rate|decay rate]] in [[natural units]].<ref name="Peskin">
{{cite book
 |last1=Peskin |first1=M.E.
 |last2=Schroeder |first2=D.V.
 |date=1995
 |title=An Introduction to Quantum Field Theory
 |publisher=Perseus Books
}}</ref> However, in [[quantum field theory]], a particle (a "one-particle state") is roughly defined as a state which is constant over time; i.e., an [[eigenvalue]] of the [[Hamiltonian (quantum mechanics)|Hamiltonian]]. An [[Particle decay|unstable particle]] is a state which is only approximately constant over time; If it exists long enough to be measured, it can be formally described as having a complex mass, with the real part of the mass greater than its imaginary part. If both parts are of the same magnitude, this is interpreted as a [[resonance]] appearing in a scattering process rather than a particle, as it is considered not to exist long enough to be measured independently of the scattering process. In the case of a tachyon, the real part of the mass is zero, and hence no concept of a particle can be attributed to it.

In a [[Lorentz invariant]] theory, the same formulas that apply to ordinary slower-than-light particles (sometimes called "[[bradyon]]s" in discussions of tachyons) must also apply to tachyons. In particular the [[energy–momentum relation]]:
: <math>E^2 = p^2c^2 + m^2c^4 \;</math>
(where '''p''' is the relativistic [[momentum]] of the bradyon and '''m''' is its [[rest mass]]) should still apply, along with the formula for the total energy of a particle:
: <math>E = \frac{mc^2}{\sqrt{1 - \frac{v^2}{c^2}}}.</math>
This equation shows that the total energy of a particle (bradyon or tachyon) contains a contribution from its rest mass (the "rest mass–energy") and a contribution from its motion, the kinetic energy.
When ''v'' is larger than ''c'', the denominator in the equation for the energy is [[imaginary number|"imaginary"]], as the value under the [[square root|radical]] is negative. Because the total [[energy]] must be [[real number|real]], the numerator must ''also'' be imaginary:  i.e. the [[rest mass]] '''m''' must be imaginary, as a pure imaginary number divided by another pure imaginary number is a real number.

== See also ==
{{cols}}
* [[Mass versus weight]]
* [[Effective mass (spring–mass system)]]
* [[Effective mass (solid-state physics)]]
* [[Extension (metaphysics)]]
* [[International System of Quantities]]
* [[2019 revision of the SI base]]
{{colend}}

== Notes ==
{{noteFoot}}

== References ==
{{reflist}}

== External links ==
{{Commons category|Mass (physical property)}}
{{NSRW Poster|Mass}}
* {{cite encyclopedia |url=http://plato.stanford.edu/entries/equivME/ |title=The Equivalence of Mass and Energy|author=Francisco Flores |encyclopedia=Stanford Encyclopedia of Philosophy |date=6 February 2012}}
* {{cite magazine |url=http://www.sciam.com/article.cfm?chanID=sa006&articleID=000005FC-2927-12B3-A92783414B7F0000 |title=The Mysteries of Mass |magazine=Scientific American |author=Gordon Kane |date=27 June 2005|archive-url=https://web.archive.org/web/20071010052029/http://www.sciam.com/article.cfm?chanID=sa006&articleID=000005FC-2927-12B3-A92783414B7F0000 |archive-date=10 October 2007 |url-status=dead }}
* {{cite journal |author=L.B. Okun |title=Photons, Clocks, Gravity and the Concept of Mass |journal=Nuclear Physics B: Proceedings Supplements |volume=110 |pages=151–155 |year=2002 |arxiv=physics/0111134 |doi=10.1016/S0920-5632(02)01472-X |bibcode=2002NuPhS.110..151O |s2cid=16733517 }}
* {{cite web |url=http://video.mit.edu/watch/the-origin-of-mass-and-the-feebleness-of-gravity-9082/ |title=The Origin of Mass and the Feebleness of Gravity |author=Frank Wilczek |date=13 May 2001 |publisher=MIT Video |type=video|author-link=Frank Wilczek }}
* {{cite web |url=http://math.ucr.edu/home/baez/physics/Relativity/SR/mass.html |title=Does mass change with velocity? |author=John Baez |display-authors=etal |year=2012 |author-link=John Baez }}
* {{cite web |url=http://math.ucr.edu/home/baez/physics/ParticleAndNuclear/photon_mass.html |title=What is the mass of a photon? |author=John Baez |display-authors=etal |year=2008 |author-link=John Baez }}
* [[Jim Baggott]] (27 September 2017). [https://www.youtube.com/watch?v=HfHjzomqbZc The Concept of Mass] (video) published by the [[Royal Institution]] on [[YouTube]].

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{{Classical mechanics derived SI units}}

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