Mass in Special Relativity - Terminology

Terminology

See also: Special relativity#Mass-energy equivalence

If a stationary box contains many particles, it weighs more in its rest frame, the faster the particles are moving. Any energy in the box (including the kinetic energy of the particles) adds to the mass, so that the relative motion of the particles contributes to the mass of the box. But if the box itself is moving (its center of mass is moving), there remains the question of whether the kinetic energy of the overall motion should be included in the mass of the system. The invariant mass is calculated excluding the kinetic energy of the system as a whole (calculated using the single velocity of the box, which is to say the velocity of the box's center of mass), while the relativistic mass is calculated including invariant mass PLUS the kinetic energy of the system which is calculated from the velocity of the center of mass.

Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass corresponds to the total energy. The relativistic mass is the mass of the system as it would be measured on a scale, but in some cases (such as the box above) this fact remains true only because the system on average must be at rest to be weighed (it must have zero net momentum, which is to say, the measurement is in its center of momentum frame). For example, if an electron in a cyclotron is moving in circles with a relativistic velocity, the weight of the cyclotron+electron system is increased by the relativistic mass of the electron, not by the electron's rest mass. But the same is also true of any closed system, such as an electron-and-box, if the electron bounces at high speed inside the box. It is only the lack of total momentum in the system (the system momenta sum to zero) which allows the kinetic energy of the electron to be "weighed." If the electron is stopped and weighed, or the scale were somehow sent after it, it would not be moving with respect to the scale, and again the relativistic and rest masses would be the same for the single electron (and would be smaller). In general, relativistic and rest masses are equal only in systems which have no net momentum and the system center of mass is at rest; otherwise they may be different.

The invariant mass is proportional to the value of the total energy in one reference frame, the frame where the object as a whole is at rest (as defined below in terms of center of mass). This is why the invariant mass is the same as the rest mass for single particles. However, the invariant mass also represents the measured mass when the center of mass is at rest for systems of many particles. This special frame where this occurs is also called the center of momentum frame, and is defined as the inertial frame in which the center of mass of the object is at rest (another way of stating this is that it is the frame in which the momenta of the system's parts add to zero). For compound objects (made of many smaller objects, some of which may be moving) and sets of unbound objects (some of which may also be moving), only the center of mass of the system is required to be at rest, for the object's relativistic mass to be equal to its rest mass.

A so-called massless particle (such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference. In this case there is no transformation that will bring the particle to rest. The total energy of such particles becomes smaller and smaller in frames which move faster and faster in the same direction. As such, they have no rest mass, because they can never be measured in a frame where they are at rest. This property of having no rest mass is what causes these particles to be termed "massless."