Editing Invariant mass (section)

{{Short description|Motion-independent mass, equals total mass when at rest}}
{{redirect|Proper mass|the liturgical mass proper|Proper (liturgy)}}
{{multiple issues|
{{more citations needed|date=March 2011}}
{{Cleanup rewrite|Outdated, incorrect, see talk|date=February 2016}}
}}
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 [[physical body|object]] or [[physical system|system]] of objects that is independent of the overall motion of the system. More precisely, it is a characteristic of the system's total [[energy]] and [[momentum]] that is the same in all [[frames of reference]] related by [[Lorentz transformation]]s.<ref name="LawrenceS">Lawrence S. Lerner. [https://books.google.com/books?id=Nv5GAyAdijoC&pg=PA1073 Physics for Scientists and Engineers, Volume 2, page 1073]. 1997.</ref> If a [[center-of-momentum frame]] exists for the system<!-- note there are massless particles -->, 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 non-zero, the total mass (a.k.a. [[relativistic mass]]) of the system is greater than the invariant mass, but the invariant mass remains unchanged.

Because of [[mass–energy equivalence]], the '''rest energy''' of the system is simply the invariant mass times the [[speed of light]] squared. Similarly, the total energy of the system is its total (relativistic) mass times the speed of light squared.

Systems whose [[four-momentum]] is a [[null vector]], a [[Minkowski space#Causal structure|light-like vector]] within the context of Minkowski space (for example, a single [[photon]] or many photons moving in exactly the same direction) have [[zero]] invariant mass and are referred to as ''[[massless particle|massless]]''. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the hypothesized [[tachyon]]), 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 ''[[speed of light|c]]''<sup>2</sup>. See [[mass–energy equivalence]] 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]] of the system moves in a straight line with a steady subluminal [[velocity]] (with a velocity depending on the [[frame of reference|reference frame]] 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 {{math|''c''<sup>2</sup>}}. 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]]s, or rays of [[light]] 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.

[[Image:Rest mass 0 and 1.svg|thumb|right|160px|Possible 4-momenta of particles. <span style="color:blue;">'''One'''</span> has zero invariant mass, <span style="color:green;">'''the other'''</span> is massive]]