# Mass

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In physics, mass is a property of a physical body which determines resistance to being accelerated by a force and the strength of its mutual gravitational attraction with other bodies. The SI unit of mass is the kilogram (kg).

Mass is not the same thing as weight, even though we commonly calculate an object's mass by measuring its weight. A woman standing on the Moon would weigh less than she would on Earth because of the lower gravity, but she would have the same mass.

For everyday objects and energies well-described by Newtonian physics, mass describes the amount of matter in an object. However, at very high speeds or for subatomic particles, special relativity shows that energy is an additional source of mass. Thus, any stationary body having mass has an equivalent amount of energy, and all forms of energy resist acceleration by a force and have gravitational attraction.

There are several distinct phenomena which can be used to measure mass. Although some theorists have speculated some of these phenomena could be independent of each other,[1] current experiments have found no difference among any of the ways used to measure mass:

• Inertial mass measures an object's resistance to being accelerated by a force (represented by the relationship F=ma).
• Active gravitational mass measures the gravitational force exerted by an object.
• Passive gravitational mass measures the gravitational force experienced by an object in a known gravitational field.
• Mass-Energy measures the total amount of energy contained within a body, using E=mc²

The mass of an object determines its acceleration in the presence of an applied force. This phenomenon is called inertia. 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 or is affected by a gravitational field. If a first body of mass mA is placed at a distance r (center of mass to center of mass) from a second body of mass mB, each body experiences an attractive force Fg = GmAmB/r2, where G = Template:Val is the "universal gravitational constant". This is sometimes referred to as gravitational mass.[note 1] Repeated experiments since the 17th century have demonstrated that inertial and gravitational mass are identical; since 1915, this observation has been entailed a priori in the equivalence principle of general relativity.)))

## Units of mass

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The kilogram is one of the seven SI base units; one of three which is defined ad hoc, without reference to another base unit.

The standard International System of Units (SI) unit of mass is the kilogram (kg). The kilogram is 1000 grams (g), which were first defined in 1795 as one cubic decimeter of water at the melting point of ice. Then in 1889, the kilogram was redefined as the mass of the international prototype kilogram, and as such is independent of the meter, or the properties of water. As of January 2013, there are several proposals for redefining the kilogram yet again, including a proposal for defining it in terms of the Planck constant.[2]

Other units are accepted for use in SI:

Outside SI system, other units include:

## Definitions of mass

Template:Properties of mass 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:[3] Every experiment to date has shown these seven values to be proportional, and in some cases equal, and this proportionality gives rise to the abstract concept of mass.

• The amount of matter in certain types of samples can be exactly determined through electrodepositionTemplate:Clarify or other precise processes. The mass of an exact sample is determined in part by the number and type of atoms or molecules it contains, and in part by the energy involved in binding it together (which contributes a negative "missing mass," or mass deficit).
• Inertial mass is a measure of an object's resistance to changing its state of motion 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 [note 3] is a measure of the strength of an object's 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 freely fall and measuring its free-fall acceleration. For example, an object in free-fall near the Moon will experience less gravitational field, and hence accelerate slower 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.
• Energy also has mass according to the principle of mass–energy equivalence. This equivalence is exemplified in a large number of physical processes including pair production, nuclear fusion, and the gravitational bending of light. Pair production and nuclear fusion are processes through which measurable amounts of mass and energy are converted into each other. In the gravitational bending of light, photons of pure energy are shown to exhibit a behavior similar to passive gravitational mass.
• Curvature of spacetime is a relativistic manifestation of the existence of mass. 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.

### Weight vs. mass

{{#invoke:main|main}} In everyday usage, mass and "weight" are often used interchangeably. For instance, a person's weight may be stated as 75 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 Earth's gravitational field at different places, the 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 natural 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 weightless, though they still have mass.[4]

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 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 kilograms but weighs only 81.5 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 W = mg, where g = Template:Val is the 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 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).

Macroscopically, mass is associated with matter—although matter, unlike mass, is poorly defined in science. On the sub-atomic scale, not only fermions, the particles often associated with matter, but also some bosons, the particles that act as force carriers, have rest mass. Another problem for easy definition is that much of the rest mass of ordinary matter derives from the invariant mass contributed to matter by particles and kinetic energies which have no rest mass themselves (only 1% of the rest mass of matter is accounted for by the rest mass of its fermionic quarks and electrons). From a fundamental physics perspective, mass is the number describing under which the representation of the little group of the Poincaré group a particle transforms. In the Standard Model of particle physics, this symmetry is described as arising as a consequence of a coupling of particles with rest mass to a postulated additional field, known as the Higgs field.

### Universal gravitational mass

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

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 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,[14]:397 and inversely proportional to the square of the distance to the body's center.[14]:221[note 6]

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.

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. 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.Template:Clarify

Given two objects A and B, of masses MA and MB, separated by a distance RAB, Newton's law of gravitation states that each object exerts a gravitational force on the other, of magnitude

${\displaystyle \mathbf {F} _{AB}=-GM_{A}M_{B}{\frac {{\hat {\mathbf {R} }}_{AB}}{|\mathbf {R} _{AB}|^{2}}}\ }$,

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

${\displaystyle F=Mg\!}$.

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 beneath the weighing pan, as per Hooke's law, and the scales are 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 balance measures relative weight, giving the relative gravitation mass of each object.

### Inertial mass

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. The simple classical mechanics definition of mass is slightly different than 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

${\displaystyle \mathbf {F} =m\mathbf {a} ,}$

where F is the resultant force acting on the body and a is the acceleration of the body's centre of mass.[note 7] 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 m1 and m2. We isolate the two objects from all other physical influences, so that the only forces present are the force exerted on m1 by m2, which we denote F12, and the force exerted on m2 by m1, which we denote F21. Newton's second law states that

{\displaystyle {\begin{aligned}\mathbf {F_{12}} &=m_{1}\mathbf {a_{1}} ,\\\mathbf {F_{21}} &=m_{2}\mathbf {a_{2}} ,\end{aligned}}}

where a1 and a2 are the accelerations of m1 and m2, 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

${\displaystyle \mathbf {F} _{12}=-\mathbf {F} _{21};}$

and thus

${\displaystyle m_{1}=m_{2}{\frac {|\mathbf {a} _{2}|}{|\mathbf {a} _{1}|}}\!.}$

If |a1| is non-zero, the fraction is well-defined, which allows us to measure the inertial mass of m1. In this case, m2 is our "reference" object, and we can define its mass m as (say) 1 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:

${\displaystyle {\mathbf {p} }=m{\mathbf {v} }}$,

and the body's kinetic energy K to its velocity:

${\displaystyle K={\dfrac {1}{2}}m|\mathbf {v} |^{2}}$.

## Atomic mass

{{#invoke:main|main}} Typically, the mass of objects is measured in relation to that of the kilogram, which is defined as the mass of the international prototype kilogram (IPK), a platinum alloy cylinder stored in an environmentally-monitored safe secured in a vault at the International Bureau of Weights and Measures in France. However, the IPK is not convenient for measuring the masses of atoms and particles of similar scale, as it contains trillions of trillions of atoms, and has most certainly lost or gained a little mass over time despite the best efforts to prevent this. It is much easier to precisely compare an atom's mass to that of another atom, thus scientists developed the atomic mass unit. By definition, 1 u is exactly one twelfth of the mass of a carbon-12 atom, and by extension a carbon-12 atom has a mass of exactly 12 u.

## Mass in relativity

### Special relativity

{{#invoke:main|main}} In Special relativity, there are two kinds of mass: rest mass (or invariant mass),[note 8] 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 c2. The two are related by the following equation:

${\displaystyle m_{\mathrm {relative} }=\gamma (m_{\mathrm {rest} })\!}$
${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$

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:

${\displaystyle (m_{\mathrm {rest} })c^{2}={\sqrt {E_{\mathrm {total} }^{2}-(|\mathbf {p} |c)^{2}}}.\!}$

So long as the system is 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 may be converted to types of energy which are not matter,(e.g. light, kinetic energy, and the potential energy in magnetic, electric, and other fields) but this does not affect the amount of mass. Although things like heat may not be matter, all types of energy still continue to exhibit mass.[note 9][16] 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 E = mc2, yielding rest energy and "relativistic energy" (total system energy) respectively.

${\displaystyle E_{\mathrm {rest} }=(m_{\mathrm {rest} })c^{2}\!}$
${\displaystyle E_{\mathrm {total} }=(m_{\mathrm {relative} })c^{2}\!}$

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 proportional to the energy, it has gradually fallen into disuse among physicists.[17] There is disagreement over whether the concept remains pedagogically useful.[18][19][20]

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. Mass is not conserved in this process because the system is not closed during the binding process. For example, the binding energy of atomic nuclei is often lost in the form of gamma rays when the nuclei are formed, leaving nuclides which have less mass than the free particles (nucleons) of which they are composed.

### General relativity

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In general relativity, the equivalence principle is any of several related concepts dealing with the equivalence of gravitational and inertial mass. At the core of this assertion is 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 pseudo-force experienced by an observer in a non-inertial (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 non-linearity of the Einstein field equations, which makes 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.[21]

## Mass in quantum physics

In classical mechanics, the inert mass of a particle appears in the Euler–Lagrange equation as a parameter m,

${\displaystyle {\frac {\mathrm {d} }{{\mathrm {d} }t}}\ \left(\,{\frac {\partial L}{\partial {\dot {x}}_{i}}}\,\right)\ =\ m\,{\ddot {x}}_{i}}$.

After quantization, replacing the position vector x with a wave function, the parameter m appears in the kinetic energy operator,

${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi ({\mathbf {r} },\,t)=\left(-{\frac {\hbar ^{2}}{2m}}\nabla ^{2}+V({\mathbf {r} })\right)\Psi ({\mathbf {r} },\,t)}$.

In the ostensibly covariant (relativistically invariant) Dirac equation, and in natural units, this becomes

${\displaystyle (-i\gamma ^{\mu }\partial _{\mu }+m)\psi =0\,}$

Where the "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, there is the proposal that this term arises from the coupling of the field ψ to an additional field Φ, the so-called Higgs field. In the case of fermions, the Higgs mechanism results in the replacement of the term mψ in the Lagrangian with ${\displaystyle G_{\psi }{\overline {\psi }}\phi \psi }$. This shifts the explanandum of the value for the mass of each elementary particle to the value of the unknown couplings Gψ. The tentatively confirmed discovery of a massive Higgs boson is regarded as a strong confirmation of this theory. But there is indirect evidence for the reality of the Electroweak symmetry breaking as described by the Higgs mechanism, and the non-existence of Higgs bosons would indicate a "Higgsless" description of this mechanism.

## Notes

1. When a distinction is necessary, M is used to denote the active gravitational mass and m the passive gravitational mass.
2. Since the Avogadro constant NA is defined as the number of atoms in 12 g of carbon-12, it follows that 1 u is exactly 1/(103NA) kg.
3. 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 by laypersons. In most practical applications, Newtonian gravity is used 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".{{#invoke:URL|url|1=http://www2.jpl.nasa.gov/basics/bsf3-2.php%7C2=}} 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 the identical result. The observed property, noted by Galileo, according to which objects of different mass fall with the same rate of acceleration (ignoring air resistance), is an expression of the fact that inertial and gravitational mass are the same.
4. 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:
${\displaystyle \mu =4\pi ^{2}{\frac {{\text{distance}}^{3}}{{\text{time}}^{2}}}\propto {\text{gravitational mass}}}$
5. 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 which predicted that bodies of the same material falling through the same medium would fall at the same speed. See {{#invoke:citation/CS1|citation |CitationClass=book }}
6. These two properties are very useful, as they allow spherical collections of objects to be treated exactly like large individual objects.
7. In its original form, Newton's second law is valid only for bodies of constant mass.
8. 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.
9. 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 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.

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