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Invariant mass

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The invariant mass or intrinsic mass or proper mass or rest mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. It is known as rest mass for objects, because it is the mass of an object that an observer would measure in the inertial frame of reference in which the object is at rest. For any frame of reference, the invariant mass may be determined from an object's total energy and momentum.

The definition of the invariant mass of a system of particles shows that it is equal to total system energy (divided by c) when the system is viewed from an inertial reference frame which minimizes the total system energy. This reference frame is that in which the velocity of the system's center of mass is zero, and the system's total momentum is zero. This frame is also known as the "center of mass" or "center of momentum" frame.

Particle physics

In particle physics, the invariant mass is often calculated as a 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 which originate from a single originating particle, is related to the rest mass of the original particle by the following equation:

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

Where:

W {\displaystyle W} is the invariant mass of the system of particles, equal to the rest mass of the decay particle.
Σ E {\displaystyle \Sigma E} is the sum of the energies of the particles
Σ p c {\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, c {\displaystyle c}

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

p i μ = ( E i , p i ) {\displaystyle p_{i}^{\mu }=\left(E_{i},\mathbf {p} _{i}\right)}
P μ = ( Σ E i , Σ p i ) {\displaystyle P^{\mu }=\left(\Sigma E_{i},\Sigma \mathbf {p} _{i}\right)}
P μ P μ = η μ ν P μ P ν = ( Σ E i ) 2 − ( Σ p i ) 2 = W 2 {\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

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