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Invariant mass
 
 
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The invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 and momentum
Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
 of an object or a system of objects that is the same in all frames of reference
Invariant (physics)

In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some Transformation .The gravitational field of the Sun is invariant under a change of time ....
. 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
Mass-energy equivalence

In physics, mass?energy equivalence is the concept that any mass has an associated energy, and that any energy has an associated type of mass. In special relativity this relationship is expressed using the mass?energy equivalence formula...
 of the system as measured on a scale. 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.






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The invariant mass, intrinsic mass, proper mass or just mass is a characteristic of the total energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 and momentum
Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
 of an object or a system of objects that is the same in all frames of reference
Invariant (physics)

In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some Transformation .The gravitational field of the Sun is invariant under a change of time ....
. 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
Mass-energy equivalence

In physics, mass?energy equivalence is the concept that any mass has an associated energy, and that any energy has an associated type of mass. In special relativity this relationship is expressed using the mass?energy equivalence formula...
 of the system as measured on a scale. 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
Center of momentum frame

A center of momentum frame of a system is any inertial frame in which the center of mass is at rest . Note that the center of momentum of a system is not a location, but rather defines a particular inertial frame ....
, the total momentum is zero, the system as a whole may be thought of as being "at rest", and the invariant mass of the system is equal to the total system energy divided by c2. 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.

If the system comprises more than one particle, the particles may be moving relative to each other in the center of momentum frame, and they will generally interact through one or more of the fundamental forces. The kinetic energy of the particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the rest frame.

Particle physics


In particle physics
Particle physics

Particle physics is a branch of physics that studies the elementary particle constituents of matter and radiation, and the interactions between them....
, the invariant mass is a mathematical combination of a particle's energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 E and its momentum
Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
 p which is equal to the mass
Mass

In physical science, mass refers to the degree of acceleration a body acquires when subject to a force: bodies with greater mass are accelerated less by the same force....
 in the rest frame. This invariant mass is the same in all frames of reference (see Special Relativity
Special relativity

Special relativity is the physical theory of measurement in inertial frames of reference proposed in 1905 by Albert Einstein in the paper "Annus Mirabilis Papers#Special relativity"....
).

or in natural units
Natural units

In physics, natural units are physical units of measurement defined in such a way that certain selected universal physical constants are normalized to unity; that is, their numerical value becomes exactly 1 when measured in some system of natural units....
 where c = 1,

This equation says that the invariant mass is the relativistic length of the four-vector
Four-vector

In the theory of relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. It differs from a vector in that it can be transformed by Lorentz transformations....
 (E, p), calculated using the relativistic version of the pythagorean theorem which has a different sign for the space and time dimensions. 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:



where

is the invariant mass of the system of particles, equal to the mass of the decay particle.
is the sum of the energies of the particles
is the vector sum of the momentum
Momentum

In classical mechanics, momentum is the product of the mass and velocity of an object . For more accurate measures of momentum, see the section Momentum#Modern definitions of momentum on this page....
 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
Natural units

In physics, natural units are physical units of measurement defined in such a way that certain selected universal physical constants are normalized to unity; that is, their numerical value becomes exactly 1 when measured in some system of natural units....
) is
  
  


Rest energy

The rest energy or rest energy of a particle
Elementary particle

In particle physics, an elementary particle or fundamental particle is a wiktionary:particle not known to have substructure; that is, it is not known to be made up of smaller particles....
 is defined as:

,

where is the speed of light in a vacuum. In general, only differences in energy
Energy

In physics, energy is a scalar physical quantity that describes the amount of Work_ that can be performed by a force. Energy is an attribute of objects and systems that is subject to a conservation law....
 have physical significance. Defining rest energy puts energy on an absolute scale.

The motivation for defining rest energy is in the Special Theory of Relativity. According to that theory, the mass of a body changes in proportion to its kinetic energy, via:

,

This leads to Einstein's famous conclusion that energy and mass are manifestations of the same phenomena. Defining rest energy as above makes the mathematical expression of mass-energy equivalence more elegant, but is still arbitrary in the way it places energy on an absolute scale. See Mass-energy equivalence#Background
Mass-energy equivalence

In physics, mass?energy equivalence is the concept that any mass has an associated energy, and that any energy has an associated type of mass. In special relativity this relationship is expressed using the mass?energy equivalence formula...


See also


  • Mass in special relativity
    Mass in special relativity

    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 the object....
  • Invariant (physics)
    Invariant (physics)

    In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some Transformation .The gravitational field of the Sun is invariant under a change of time ....


Citations