Invariant mass

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 transformations.

[1] 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 non-zero, the total mass (a.k.a.

Because of mass–energy equivalence, the rest energy of the system is simply the invariant mass times the speed of light squared.

Systems whose four-momentum is a null vector (for example, a single photon or many photons moving in exactly the same direction) have zero 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), and these do not appear to exist.

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 c2.

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.

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 reference frame used to view 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 c2.

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 photons, or rays of light moving in one direction.

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.

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).

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 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).

This invariant mass is the same in all frames of reference (see also special relativity).

This equation says that the invariant mass is the pseudo-Euclidean length of the four-vector (E, p), calculated using the relativistic version of the Pythagorean theorem which has a different sign for the space and time dimensions.

In quantum theory the invariant mass is a parameter in the relativistic Dirac equation for an elementary particle.

The Dirac quantum operator 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:

where 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") W of the reaction is defined as follows (in natural units):

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.

The invariant mass of a system made of two massless particles whose momenta form an angle

[5] 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 Special relativity § Relativistic dynamics and invariance.