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The '''invariant mass''' or '''proper mass''' or '''rest mass''' is a measurement or calculation of the ] of an object that is the ]. Also known as ''rest mass'' because it is the mass that an observer in the same frame of reference as the object would measure. It can be determined from an object's ] and ]. The '''invariant mass''' or '''proper mass''' or '''rest mass''' is a measurement or calculation of the ] of an object that is the ]. 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 ] and ].


The definition of the invariant mass of a system of particles shows that it is equal to total system energy (divided by c^2) when the system is viewed from an inertial reference frame which ''minimizes'' the total system energy. This reference frame is that in which the system total momentum is zero, also known as the "center of mass" or "center of momentum" frame. The definition of the invariant mass of a system of particles shows that it is equal to total system energy (divided by c^2) 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 the system's center of mass is zero, and the system's total momentum is zero. This frame is also known as the "]" or "center of momentum" frame.


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


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


:<math>\mbox{W}^2\mbox{c}^4=(\Sigma \mbox{E})^2-(\Sigma \mbox{pc})^2 \,</math> :<math>\mbox{W}^2\mbox{c}^4=(\Sigma \mbox{E})^2-(\Sigma \mbox{pc})^2 \,</math>

Revision as of 01:10, 29 August 2006

The invariant 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^2) 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 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 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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