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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 ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ...
or
system A system is a group of 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 purpose and express ...
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, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
and
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
that is the same in all frames of reference related by Lorentz transformations.Lawrence S. Lerner
Physics for Scientists and Engineers, Volume 2, page 1073
1997.
If a center-of-momentum frame 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. Because of
mass–energy equivalence In physics, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
, the rest energy of the system is simply the invariant mass times the
speed of light The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit fo ...
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 (also called momentum-energy or momenergy ) is the generalization of the classical three-dimensional momentum to four-dimensional spacetime. Momentum is a vector in three dimensions; similarly four-momentum is ...
is a null vector (for example, a single
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
or many photons moving in exactly the same direction) have
zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by Multiplication, multiplying digits to the left of 0 by th ...
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 that always travels faster than light. Physicists believe that faster-than-light particles cannot exist because they are not consistent with the known laws of physics. If such partic ...
), 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, mass–energy equivalence is the relationship between mass and energy in a system's rest frame, where the two quantities differ only by a multiplicative constant and the units of measurement. The principle is described by the physicis ...
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 in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
of the system moves in a straight line with a steady sub-luminal
velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
(with a velocity depending on the
reference frame In physics and astronomy, a frame of reference (or reference frame) is an abstract coordinate system whose origin, orientation, and scale are specified by a set of reference points― geometric points whose position is identified both math ...
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 A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless, so they alwa ...
s, or rays of
light Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 t ...
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, 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 Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity i ...
. Thus, an observer can always be placed to move along with it. In this frame, which is the
center of momentum frame In physics, the center-of-momentum frame (also zero-momentum frame or COM frame) of a system is the unique (up to velocity but not origin) inertial frame in which the total momentum of the system vanishes. The ''center of momentum'' of a system is ...
, 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 or high energy physics is the study of fundamental particles and forces that constitute matter and radiation. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) an ...
, the invariant mass is equal to the
mass Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different ele ...
in the rest frame of the particle, and can be calculated by the particle's
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
  and its
momentum In Newtonian mechanics, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
  as measured in ''any'' frame, by the energy–momentum relation: m_0^2 c^2 = \left( \frac \right) ^2 - \left\, \mathbf \right\, ^2 or in natural units where , m_0^2 = E^2 - \left\, \mathbf \right\, ^2 . This invariant mass is the same in all frames of reference (see also
special relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
). This equation says that the invariant mass is the pseudo-Euclidean length of the four-vector , 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 for an elementary particle. The Dirac
quantum operator In physics, an operator is a function over a space of physical states onto another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Beca ...
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( W c^2 \right) ^2 = \left( \sum E \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, momentum (more specifically linear momentum or translational momentum) is the product of the mass and velocity of an object. It is a vector quantity, possessing a magnitude and a direction. If is an object's mass ...
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( \sum E_\text - \sum E_\text \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) is \begin M^2 &= ( E_1 + E_2 ) ^2 - \left\, \mathbf_1 + \mathbf_2 \right\, ^2 \\ &= m_1^2 + m_2^2 + 2 \left( E_1 E_2 - \mathbf_1 \cdot \mathbf_2 \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: \begin M^2 &= (E_1 + E_2) ^2 - \left\, \textbf_1 + \textbf_2 \right\, ^2 \\ &= ( p_1 , 0 , 0 , p_1 ) + ( p_2 , 0 , p_2 \sin \theta , p_2 \cos \theta ) ^2 \\ &= (p_1 + p_2) ^2 - p_2 ^2 \sin^2 \theta - ( p_1 + p_2 \cos \theta ) ^2 \\ &= 2 p_1 p_2 ( 1 - \cos \theta ) . \end


Collider experiments

In particle collider experiments, one often defines the angular position of a particle in terms of an azimuthal angle  \phi and
pseudorapidity In experimental particle physics, pseudorapidity, \eta, is a commonly used spatial coordinate describing the angle of a particle relative to the beam axis. It is defined as :\eta \equiv -\ln\left tan\left(\frac\right)\right where \theta is the ...
\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 physical sciences, a particle (or corpuscule in older texts) is a small localized object which can be described by several physical or chemical properties, such as volume, density, or mass. They vary greatly in size or quantity, from ...
is defined as E_0 = m_0 c^2, where c is the
speed of light in vacuum The speed of light in vacuum, commonly denoted , is a universal physical constant that is important in many areas of physics. The speed of light is exactly equal to ). According to the special theory of relativity, is the upper limit ...
. In general, only differences in
energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of ...
have physical significance. The concept of rest energy follows from the
special theory of relativity In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates: # The law ...
that leads to Einstein's famous conclusion about equivalence of energy and mass. See background for mass–energy equivalence.


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 which remains unchanged under some transformation. Invariance, as a broader term, also applies to the no change of form of physical laws under a transformation, and is clos ...
* Transverse mass


References

* *


Citations

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