Invariant mass

"Proper mass" redirects here. For the liturgical mass proper, see Proper (liturgy).

The invariant mass, rest 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 related by Lorentz transformations. When the system as a whole is at rest (or when it equivalently has zero net momentum), the invariant mass is equal to the total energy of the system divided by c². See mass-energy equivalence for a discussion of definitions of mass. Since the mass of systems must be measured with a weight or mass scale in a reference frame in which the system has zero momentum (the center of momentum frame), such a scale always measures the system's invariant mass.

Systems whose four-momentum is a null vector (for example a single photon or many photons moving in exactly the same direction) have zero invariant mass, and are referred to as massless. A physical object or particle moving faster than the speed of light would have space-like four-momenta (such as the theoretical tachyon), and these do not appear to exist.

For an isolated massive system, the four-momentum is a time-like vector and the center of mass moves in a straight line with a steady sub-luminal velocity. Thus, an observer can always be placed to move along with it. In this frame, which is the center of momentum frame, the total momentum is zero, and the system as a whole may be thought of as being "at rest" if it is a bound system (like a bottle of gas). In this frame, which always exists, the invariant mass of the system is equal to the total system energy (in the zero-momentum frame) 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.

Note that for reasons above, such a frame does not exist for single photons, or rays of light moving in one direction. When two or more photons move in different directions, however, a center of mass frame exists. Thus, the mass of a system of several photons moving in different directions is positive, which means that an invariant mass exists for this system.

The example above shows invariant mass of a system may be greater than sum of rest masses of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass).

If a system consists of more than one particle moving in different directions, or the same direction in at different velocities, the particles will be moving relative to each other in the center of momentum frame. They will often also interact through one or more of the fundamental forces, giving them a potential energy of interaction, possibly negative. The kinetic energy of such particles and the potential energy of the force fields increase the total energy above the sum of the particle rest masses, and both terms contribute to the invariant mass of the system. The sum of the particle kinetic energies as calculated by an observer is smallest in the center of momentum frame (again, called the "rest frame" if the system is bound).

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

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. This invariant mass is the same in all frames of reference (see Special Relativity).

$(m_0c^2)^2=E^2-\|\mathbf{p}c\|^2\,$

or in natural units where c = 1,

$m_0^2 = E^2 - \|\mathbf{p}\|^2. \,$

This equation says that the invariant mass is the pseudo-Euclidean length of the four-vector (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. In quantum theory the invariant mass is a parameter in the relativistic Dirac equation for an elementary particle. The Dirac quantum operator corresponds to the particle four-momentum vector.

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:

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

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 momentum of the particles (includes both magnitude and direction of the momenta)

The term invariant mass is also used in inelastic scattering experiments. Given an inelastic reaction with total incoming energy larger than the total detected energy (i.e. not all outgoing particles are detected in the experiment), the invariant mass (also known as the "missing mass") W of the reaction is defined as follows:

$W^2 = \left(\sum E_{in} - \sum E_{out}\right)^2 - \left(\sum \mathbf{p_{in}} - \sum \mathbf{p_{out}}\right)^2$

If there is one dominant particle which was not detected during an experiment, a plot of the invariant mass will show a sharp peak at the mass of the missing particle.

In those cases when the momentum along one direction cannot be measured (i.e. in the case of a neutrino, whose presence is only inferred from the missing energy) the Transverse mass is used.

Example: two-particle collision

In a two-particle collision (or a two-particle decay) the square of the invariant mass (in natural units) is

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

Massless particles

The invariant mass of a system made of two massless particles whose momenta form an angle θ has a convenient expression:

$M^2 \,$	$= (E_1+E_2)^2-\|\textbf{p}_1 + \textbf{p}_2\|^2 \,$
	$= [(p_1,0,0,p_1)+(p_2,0,p_2\sin\theta, p_2\cos\theta)]^2 = (p_1+p_2)^2 -p_2^2\sin^2\theta -(p_1 + p_2\cos\theta)^2 \,$
	$= 2 p_1p_2(1 - \cos\theta). \,$

Collider Experiments

In particle collider experiments, one often defines the angular position of a particle in terms of a azimuthal angle φ and pseudorapidity η. Additionally the transverse momentum, p_T, is usually measured. In this case if the particles are massless, or highly relativistic ( E > > m,) then the invariant mass becomes:

$M^2 \,$

$= 2 p_{T 1} p_{T 2} ( \cosh(\eta_1 - \eta_2) - \cos (\phi_1 - \phi_2) ) .\,$

Rest energy

The rest energy E₀ of a particle is defined as:

$\ E_0=m c^2$,

where c is the speed of light in a vacuum. In general, only differences in energy have physical significance.

The concept of rest energy follows from the Special theory of relativity that leads to Einstein's famous conclusion about equivalence of energy and mass. See background for mass-energy equivalence.

On the other hand, the concept of the equivalent Dirac invariant rest mass may be defined in terms of the self energy corresponding to the product of a geometric matter current and a generalized potential as part of a single definition of mass in a geometric unified theory.

See also

• Mass in special relativity
• Invariant (physics)
• Transverse mass

References

• Landau, L.D., Lifshitz, E.M. (1975). The Classical Theory of Fields: 4-th revised English Edition: Course of Theoretical Physics Vol. 2. Butterworth Heinemann. ISBN 0-7506-2768-9. 
• Okun, L.B. (1989). The Concept of Mass. http://www.worldscibooks.com/etextbook/6833/6833_02.pdf. 
• Halzen, Francis; Martin, Alan (1984). Quarks & Leptons: An Introductory Course in Modern Particle Physics. John Wiley & Sons. ISBN 0-471-88741-2.