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In particle physics, the mathematical combination of a particle's energy and its momentum to give a value for the mass of the particle at rest. The invariant mass is the same for all frames of reference (see Special Relativity).

The invariant mass of a system of decay particles is related to the rest mass of the original particle by the following equation:

${\displaystyle {\mbox{W}}^{2}{\mbox{c}}^{4}=(\Sigma {\mbox{E}})^{2}-(\Sigma {\mbox{pc}})^{2}}$

Where:

${\displaystyle W}$ is the invariant mass of the system of particles

${\displaystyle \Sigma E}$ is the sum of the energies of the particles

${\displaystyle \Sigma pc}$ is the vector sum of the momenta of the particles (includes both magnitude and direction of the momenta) times the speed of light, ${\displaystyle c}$

A simple way of deriving this relation is by using the momentum four-vector (in natural units):

${\displaystyle p_{i}^{\mu }=\left(E_{i},\mathbf {p} _{i}\right)}$

${\displaystyle P^{\mu }=\left(\Sigma E_{i},\Sigma \mathbf {p} _{i}\right)}$

${\displaystyle P^{\mu }P_{\mu }=\eta _{\mu \nu }P^{\mu }P^{\nu }=(\Sigma E_{i})^{2}-(\Sigma \mathbf {p} _{i})^{2}=W^{2}}$, since the norm of any four-vector is invariant.

See also

• Relativistic mass
• Invariant (physics)

• Relativity
• Mass