Invariant mass

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. 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. 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. 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.



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).


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

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. 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


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