Invariant mass - Citizendia

"Proper mass" redirects here. For the liturgical mass proper, see Proper (liturgy). The Proper (Latin proprium) is a part of the Christian liturgy that varies according to the date either representing an observance within the Liturgical Year

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. In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Mathematics and Theoretical physics, an invariant is a property of a system which remains unchanged under some transformation. When the system as a whole is at rest, the invariant mass is equal to the total energy of the system divided by c², which is equal to the mass of the system as measured on a scale. In Physics, mass–energy equivalence is the concept that for particles slower than light any Mass has an associated Energy and vice versa. 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 c². A center of momentum frame (or zero-momentum frame or COM frame of a system is any Inertial frame in which the Center of mass is at rest (has zero velocity 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.

Possible 4-momenta of particles. One has zero invariant mass, the other is massive

1 Particle physics
2 Example: two particle collision
3 See also
4 References

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. Particle physics is a branch of Physics that studies the elementary constituents of Matter and Radiation, and the interactions between them In Physics and other Sciences energy (from the Greek grc ἐνέργεια - Energeia, "activity operation" from grc ἐνεργός In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product Mass is a fundamental concept in Physics, roughly corresponding to the Intuitive idea of how much Matter there is in an object This invariant mass is the same in all frames of reference (see Special Relativity). Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial

$(mc^2)^2=E^2-||\mathbf{p}c||^2\,$

or in natural units where c = 1,

$m^2 = E^2 - ||\mathbf{p}||^2 \,$

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. In Physics, natural units are Physical units of Measurement defined in terms of universal Physical constants, such that some chosen physical In relativity, a four-vector is a vector in a four-dimensional real Vector space, called Minkowski space. 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:

$\left(Wc^2\right)^2= \left(\sum E\right)^2-||\sum \mathbf{p}c||^2$

where

W

is the invariant mass of the system of particles, equal to the mass of the decay particle.

$\sum E$ is the sum of the energies of the particles

$\sum \mathbf{p}$ 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

$M^2 \,$	$= (E_1+E_2)^2-\|\|\textbf{p}_1 + \textbf{p}_2\|\|^2 \,$
	$= m_1^2 + m_2^2 + 2\left(E_1 E_2 - \textbf{p}_1 \cdot \textbf{p}_2 \right) \,$

References

Halzen, Francis; Martin, Alan (1984). In Classical mechanics, momentum ( pl momenta SI unit kg · m/s, or equivalently N · s) is the product In Physics, natural units are Physical units of Measurement defined in terms of universal Physical constants, such that some chosen physical The term Mass in Special relativity usually refers to the Rest mass of the object which is the Newtonian mass as measured by an observer moving along with In Mathematics and Theoretical physics, an invariant is a property of a system which remains unchanged under some transformation. Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.

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Contents

Particle physics

Example: two particle collision

See also

References