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The invariant mass, rest mass, intrinsic mass, proper mass, or in the case of bound systems simply mass, is the portion of the total mass of an
object Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Entity, something that is tangible and within the grasp of the senses ** Object (abstract), an object which does not exist at any particular time or pl ...
or
system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purp ...
of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

and
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

that is the same in all
frames of reference In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...
related by
Lorentz transformation In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "P ...

s.Physics for Scientists and Engineers, Volume 2, page 1073
- Lawrence S. Lerner - Science - 1997
If a
center-of-momentum frame In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
exists for the system, then the invariant mass of a system is equal to its total mass in that "rest frame". In other reference frames, where the system's momentum is nonzero, the total mass (a.k.a.
relativistic mass The word ''mass'' has two meanings in special relativity: ''invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames, while the ''relativistic mass'' is dependent on the velocity ...
) of the system is greater than the invariant mass, but the invariant mass remains unchanged. Due to
mass–energy equivalence In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
, the rest energy of the system is simply the invariant mass times the
speed of light The speed of light in vacuum A vacuum is a space Space is the boundless three-dimensional Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called paramet ...
squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared. Systems whose
four-momentum In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions Three-dimensional space (also: 3-space or, rarely, tri-dimensiona ...

is a
null vector In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
(for example a single photon or many photons moving in exactly the same direction) have
zero 0 (zero) is a number A number is a mathematical object A mathematical object is an abstract concept arising in mathematics. In the usual language of mathematics, an ''object'' is anything that has been (or could be) formally defined, and ...

invariant mass, and are referred to as '' massless''. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized
tachyon A tachyon () or tachyonic particle is a hypothetical particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed sev ...

), and these do not appear to exist. Any time-like four-momentum possesses a reference frame where the momentum (3-dimensional) is zero, which is a center of momentum frame. In this case, invariant mass is positive and is referred to as the rest mass. If objects within a system are in relative motion, then the invariant mass of the whole system will differ from the sum of the objects' rest masses. This is also equal to the total energy of the system divided by '' c''2. See
mass–energy equivalence In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a center of momentum frame in which the entire system has zero momentum, such a scale always measures the system's invariant mass. For example, a scale would measure the kinetic energy of the molecules in a bottle of gas to be part of invariant mass of the bottle, and thus also its rest mass. The same is true for massless particles in such system, which add invariant mass and also rest mass to systems, according to their energy. For an isolated ''massive'' system, the
center of mass In physics, the center of mass of a distribution of mass Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...
of the system moves in a straight line with a steady sub-luminal
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

(with a velocity depending on the
reference frame In physics, a frame of reference (or reference frame) consists of an abstract coordinate system In geometry Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, wi ...

used to view it). Thus, an observer can always be placed to move along with it. In this frame, which is the center-of-momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which exists under these assumptions, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by . This total energy in the center of momentum frame, is the minimum energy which the system may be observed to have, when seen by various observers from various inertial frames. Note that for reasons above, such a rest frame does not exist for single
photon The photon ( el, φῶς, phōs, light) is a type of elementary particle In , an elementary particle or fundamental particle is a that is not composed of other particles. Particles currently thought to be elementary include the fundamental s ...

s, or rays of
light Light or visible light is electromagnetic radiation within the portion of the electromagnetic spectrum that is visual perception, perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nan ...

moving in one direction. When two or more photons move in different directions, however, a center of mass frame (or "rest frame" if the system is bound) exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system even though it does not exist for each photon.

# Sum of rest masses

The invariant mass of a system includes the mass of any kinetic energy of the system constituents that remains in the center of momentum frame, so the invariant mass of a system may be greater than sum of the invariant masses (rest masses) of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass). Consider the simple case of two-body system, where object A is moving towards another object B which is initially at rest (in any particular frame of reference). The magnitude of invariant mass of this two-body system (see definition below) is different from the sum of rest mass (i.e. their respective mass when stationary). Even if we consider the same system from center-of-momentum frame, where net momentum is zero, the magnitude of the system's invariant mass is not equal to the sum of the rest masses of the particles within it. The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound). They will often also interact through one or more of the
fundamental forces In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
, giving them a potential energy of interaction, possibly negative. For an isolated ''massive'' system, the center of mass moves in a straight line with a steady sub-luminal
velocity The velocity of an object is the Time derivative, rate of change of its Position (vector), position with respect to a frame of reference, and is a function of time. Velocity is equivalent to a specification of an object's speed and direction ...

. Thus, an observer can always be placed to move along with it. In this frame, which is the
center of momentum frame In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Sp ...
, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which always exists, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) divided by .

# As defined in particle physics

In
particle physics Particle physics (also known as high energy physics) is a branch of physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which rel ...
, the invariant mass is equal to the
mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value ...
in the rest frame of the particle, and can be calculated by the particle's
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

and its
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

as measured in ''any'' frame, by the
energy–momentum relation In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through Spa ...
: :$m_0^2 c^2 = \left\left( \frac \right\right) ^2 - \left\, \mathbf \right\, ^2$ or in
natural units In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Phy ...
where , :$m_0^2 = E^2 - \left\, \mathbf \right\, ^2 .$ This invariant mass is the same in all
frames of reference In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural science that studies matter, its Motion (physics), motion and behavior through ...

special relativity In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force ...
). This equation says that the invariant mass is the pseudo-Euclidean length of the
four-vector In special relativity In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in oth ...
, calculated using the relativistic version of the Pythagorean theorem which has a different sign for the space and time dimensions. This length is preserved under any Lorentz boost or rotation in four dimensions, just like the ordinary length of a vector is preserved under rotations. In quantum theory the invariant mass is a parameter in the relativistic
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its Dirac equation#Covariant form and relativistic invariance, free form, or including Dirac equation#Comparison with t ...
for an elementary particle. The Dirac
quantum operator In physics, an operator is a function over a space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. Physical space is often conceived in three linear dimensions, although mo ...
corresponds to the particle four-momentum vector. Since the invariant mass is determined from quantities which are conserved during a decay, the invariant mass calculated using the energy and momentum of the decay products of a single particle is equal to the mass of the particle that decayed. The mass of a system of particles can be calculated from the general formula: : $\left\left( W c^2 \right\right) ^2= \left\left( \sum E \right\right) ^2 - \left\, \sum \mathbf c \right\, ^2 ,$ where * $W$ is the invariant mass of the system of particles, equal to the mass of the decay particle. * $\sum E$ is the sum of the energies of the particles * $\sum \mathbf$ is the vector sum of the
momentum In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass Mass is the quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinui ...

of the particles (includes both magnitude and direction of the momenta) The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") of the reaction is defined as follows (in natural units): : $W^2 = \left\left( \sum E_\text - \sum E_\text \right\right) ^2 - \left\, \sum \mathbf_\text - \sum \mathbf_\text \right\, ^2 .$ If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle. In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the missing energy) the transverse mass is used.

# Example: two-particle collision

In a two-particle collision (or a two-particle decay) the square of the invariant mass (in
natural units In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Phy ...
) is : $\begin M^2 &= \left( E_1 + E_2 \right) ^2 - \left\, \textbf_1 + \textbf_2 \right\, ^2 \\ &= m_1^2 + m_2^2 + 2 \left\left( E_1 E_2 - \textbf_1 \cdot \textbf_2 \right\right) . \end$

## Massless particles

The invariant mass of a system made of two massless particles whose momenta form an angle $\theta$ has a convenient expression: :

## Collider experiments

In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle $\phi$ and
pseudorapidity, an angle of zero is usually along the beam axis, and thus particles with high pseudorapidity values are generally lost, escaping through the space in the detector along with the beam. In experimental particle physics Particle physics (also kno ...

$\eta$. Additionally the transverse momentum, $p_$, is usually measured. In this case if the particles are massless, or highly relativistic ($E \gg m$) then the invariant mass becomes: $M^2 = 2 p_ p_ ( \cosh(\eta_1 - \eta_2) - \cos (\phi_1 - \phi_2) ) .$

# Rest energy

The rest energy $E_0$ of a
particle In the Outline of physical science, physical sciences, a particle (or corpuscule in older texts) is a small wikt:local, localized physical body, object to which can be ascribed several physical property, physical or chemical property, chemical p ...
is defined as: :$E_0 = m_0 c^2,$ where $c$ is the
speed of light in vacuum The speed of light in vacuum A vacuum is space devoid of matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ult ...
. In general, only differences in
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

have physical significance. The concept of rest energy follows from the special theory of relativity that leads to Einstein's famous conclusion about equivalence of energy and mass. See . On the other hand, the concept of the equivalent Dirac invariant rest mass may be defined in terms of the self energy corresponding to the product of a geometric matter current and a generalized potential as part of a single definition of mass in a geometric unified theory.

*
Mass in special relativity The word ''mass'' has two meanings in special relativity: ''invariant mass'' (also called rest mass) is an invariant quantity which is the same for all observers in all reference frames; while the ''relativistic mass'' is dependent on the velocity ...
*
Invariant (physics)In theoretical physics, an invariant is an observable of a physical system In physics Physics (from grc, φυσική (ἐπιστήμη), physikḗ (epistḗmē), knowledge of nature, from ''phýsis'' 'nature'), , is the natural scie ...
* Transverse mass

* *

# Citations

{{reflist Theory of relativity Mass Energy (physics) Physical quantities