# Invariant mass

The **invariant mass**, **intrinsic mass**, **proper mass** or just **mass** is a characteristic of the total energy and momentum of an object or a system of objects that is the same in all frames of reference. When the system as a whole is at rest, the invariant mass is equal to the total energy of the system divided by *c*^{2}, which is equal to the mass of the system as measured on a scale. If the system is one particle, the invariant mass may also be called the **rest mass**.

Since the center of mass of an isolated system moves in a straight line with a steady velocity, an observer can always move along with it. In this frame, the center of momentum frame, the total momentum is zero, the system as a whole may be thought of as being "at rest" (though in a disconnected system, parts may be moving away from each other), and the invariant mass of the system is equal to the total system energy divided by *c*^{2}. 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.

## Particle physics

In particle physics, the invariant mass is a mathematical combination of a particle's energy *E* and its momentum **p** which is equal to the mass in the rest frame. This **invariant mass** is the same in all frames of reference (see Special Relativity).

or in natural units where *c* = 1,

This equation says that the invariant mass is the relativistic length of the four-vector (*E*, **p**), calculated using the relativistic version of the pythagorian 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.

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

- is the invariant mass of the system of particles, equal to the mass of the decay particle.
- is the sum of the energies of the particles
- is the vector sum of the momenta of the particles (includes both magnitude and direction of the momenta)

## Example: two particle collision

In a two particle collision (or a two particle decay) the square of the invariant mass (in natural units) is

## See also

## References

- Halzen, Francis; Martin, Alan (1984).
*Quarks & Leptons: An Introductory Course in Modern Particle Physics*. John Wiley & Sons. ISBN 0-471-88741-2.

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