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" if it is a bound system (like a bottle of gas), 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 center of momentum frame (or rest frame if the system is bound).
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).
or in natural units where c = 1,
This equation says that the invariant mass is the relativistic 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.
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:
In a two particle collision (or a two particle decay) the square of the invariant mass (in natural units) is
The rest energy E or rest energy of a particle is defined as:
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 background for mass-energy equivalence.