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

- Lawrence S. Lerner - Science - 1997 If a^{2}. See

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''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 thefundamental 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

Inparticle 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 ...

:
:$m\_0^2\; c^2\; =\; \backslash left(\; \backslash frac\; \backslash right)\; ^2\; -\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; ^2$
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 ,
:$m\_0^2\; =\; E^2\; -\; \backslash left\backslash ,\; \backslash mathbf\; \backslash right\backslash ,\; ^2\; .$
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 ...

(see also 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:
: $\backslash left(\; W\; c^2\; \backslash right)\; ^2=\; \backslash left(\; \backslash sum\; E\; \backslash right)\; ^2\; -\; \backslash left\backslash ,\; \backslash sum\; \backslash mathbf\; c\; \backslash right\backslash ,\; ^2\; ,$
where
* $W$ is the invariant mass of the system of particles, equal to the mass of the decay particle.
* $\backslash sum\; E$ is the sum of the energies of the particles
* $\backslash sum\; \backslash mathbf$ 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):
: $W^2\; =\; \backslash left(\; \backslash sum\; E\_\backslash text\; -\; \backslash sum\; E\_\backslash text\; \backslash right)\; ^2\; -\; \backslash left\backslash ,\; \backslash sum\; \backslash mathbf\_\backslash text\; -\; \backslash sum\; \backslash mathbf\_\backslash text\; \backslash right\backslash ,\; ^2\; .$
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 (innatural 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
: $\backslash begin\; M^2\; \&=\; (\; E\_1\; +\; E\_2\; )\; ^2\; -\; \backslash left\backslash ,\; \backslash textbf\_1\; +\; \backslash textbf\_2\; \backslash right\backslash ,\; ^2\; \backslash \backslash \; \&=\; m\_1^2\; +\; m\_2^2\; +\; 2\; \backslash left(\; E\_1\; E\_2\; -\; \backslash textbf\_1\; \backslash cdot\; \backslash textbf\_2\; \backslash right)\; .\; \backslash end$
Massless particles

The invariant mass of a system made of two massless particles whose momenta form an angle $\backslash theta$ has a convenient expression: : $\backslash begin\; M^2\; \&=\; (E\_1\; +\; E\_2)\; ^2\; -\; \backslash left\backslash ,\; \backslash textbf\_1\; +\; \backslash textbf\_2\; \backslash right\backslash ,\; ^2\; \backslash \backslash \; \&=;\; href="/html/ALL/s/(\_p\_1\_,\_0\_,\_0\_,\_p\_1\_)\_+\_(\_p\_2\_,\_0\_,\_p\_2\_\backslash sin\_\backslash theta\_,\_p\_2\_\backslash cos\_\backslash theta\_)\_.html"\; ;"title="(\; p\_1\; ,\; 0\; ,\; 0\; ,\; p\_1\; )\; +\; (\; p\_2\; ,\; 0\; ,\; p\_2\; \backslash sin\; \backslash theta\; ,\; p\_2\; \backslash cos\; \backslash theta\; )\; ">(\; p\_1\; ,\; 0\; ,\; 0\; ,\; p\_1\; )\; +\; (\; p\_2\; ,\; 0\; ,\; p\_2\; \backslash sin\; \backslash theta\; ,\; p\_2\; \backslash cos\; \backslash theta\; )$Collider experiments

In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle $\backslash phi$ andpseudorapidity, 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 ...

$\backslash eta$. Additionally the transverse momentum, $p\_$, is usually measured. In this case if the particles are massless, or highly relativistic ($E\; \backslash gg\; m$) then the invariant mass becomes:
$$M^2\; =\; 2\; p\_\; p\_\; (\; \backslash cosh(\backslash eta\_1\; -\; \backslash eta\_2)\; -\; \backslash cos\; (\backslash phi\_1\; -\; \backslash phi\_2)\; )\; .$$
Rest energy

The rest energy $E\_0$ of aparticle
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.
See also

*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
References

* *Citations

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