The invariant mass or intrinsic mass or proper mass or just mass is a measurement or calculation of the mass of an object that is the same for all frames of reference. For any frame of reference, the invariant mass may be determined from a calculation involving an object's total energy and momentum. The term rest mass is applied for the mass of a body that is isolated (free) and at rest relative to the observer.[clarify] By the mass-energy equivalence, the rest mass is equal to the energy content of the isolated (free) body at rest, divided by c2. This result is valid also for a composite body, when the body is viewed from an inertial reference frame in which the body's center of mass is at rest and its total linear momentum is zero, called the center of momentum frame.
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Particle physics
In particle physics, the 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 that is the same for all observers. This invariant mass is the same for all frames of reference (see Special Relativity). A mass for a particle is m in the equation
- <math>\left(mc^2\right)^2=E^2-(\mathbf{p}c)^2</math>
The invariant mass of a system of decay particles which originate from a single originating particle, is related to the mass of the original particle by a similar equation:
- <math>\left(Wc^2\right)^2= \left(\sum E\right)^2-\left(\sum \mathbf{p}c\right)^2</math>
Where:
- <math>W</math> is the invariant mass of the system of particles, equal to the mass of the decay particle.
- <math>\sum E</math> is the sum of the energies of the particles
- <math>\sum \mathbf{p}c</math> is the vector sum of the momenta of the particles (includes both magnitude and direction of the momenta) times the speed of light, <math>c</math>
A simple way of deriving this relation is by using the momentum four-vector (in natural units):
- <math>p_i^\mu=\left(E_i,\mathbf{p}_i\right)</math>
- <math>p^\mu=\left(\Sigma E_i,\Sigma \mathbf{p}_i\right)</math>
- <math>p^\mu p_\mu=\eta_{\mu\nu}p^\mu p^\nu=(\Sigma E_i)^2-(\Sigma \mathbf{p}_i)^2=W^2</math>, since the norm of any four-vector is invariant.
Example two particle collision
In a two particle collision (or a two particle decay) the square of the mass (in natural units) is
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<math>M^2 \,</math> <math>= (p_1 + p_2)^2 \,</math> <math>= p_1^2 + p_2^2 + 2p_1p_2 \,</math> <math>= m_1^2 + m_2^2 + 2\left(E_1 E_2 - \vec{p}_1 \cdot \vec{p}_2 \right) \,</math>
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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.
