Conservation of mass in special relativity
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Conservation of mass in the theory of special relativity depends on the definition of mass/matter used. In modern physics, mass and matter are equivalent to energy, and thus the conservation of energy (which always holds) encompasses matter and energy.
"Matter" (not including many types of energy) is generally not conserved in special relativity.
Whether mass is conserved in special relativity depends on which kind of mass is being referred to, and how it is measured and whether or not just one observer measures it for a system.
Relativistic mass is equivalent to relativistic energy, so this type of mass is always conserved in all processes, because total energy is conserved. The caveat is that this type of mass is frame-dependent, so it may not be conserved in a process for an accelerated observer. However, for inertial observers, relativistic mass is conserved.
Invariant mass is observer and frame invariant, which means all observers measure it the same. Invariant mass is sometimes also known as system rest mass even when the masses which make up a system under consideration-- such as vibrating atoms-- are not at rest. It is conserved also, for all single inertial observers, so long as the system is closed.
Sometimes a form of mass is not considered to be conserved in relativistic processes in systems. This can happen for one of two reasons:
1) When invariant mass (as various forms of active energy) has been allowed to escape the system, and this escape has not been kept track of. (See section The mass of systems in mass in special relativity). Complete system closure (including closure to heat and radiation) is needed for system mass to be conserved. When the mass of a system is measured only at standard temperatures, for example, this allows for the escape of mass and energy, as heat.
2) Sometimes a form of mass is not conserved when the mass of a system is found by adding the rest masses of its components. However, for massive particles this amounts to using the measurements of many different observers, and this sort of bookkeeping it is not allowed in special relativity. Even for photons, a single observer and a closed system is required for mass conservation, since photons as considered singly have zero mass, where as pairs or systems of photons moving in different directions will in general exhibit an invariant mass which is associated with the system of photons, but not with any single photon.
Sometimes either of the above processes are equivalently at work. For example, after an energy-releasing transformation, the sum of rest masses of the resulting particles may said to be different from the sum of the rest masses of the particles which began the reaction. But how was this sum measured? If these rest masses were determined with the system closed, this requires that varying frames of references have been used (one for each massive particle, and no mass ascribed to photons). If, however, the rest mass of the system after a reaction has been measured as a whole by a single observer, and found to be changed, this must occur because energy released from the reaction has been allowed to leave the system, as heat or light or other radiation. In the latter case, it will be found that this energy is the missing mass (i.e., it would exhibit the missing mass if captured, confined, and weighed).
Another way of expressing this idea is that if released energy is allowed to remain in a system (for example, as heat, or even trapped radiation), this energy will be measured as, and included in, the ordinary "rest" mass of the system (that is, this energy still contributes to the inertia of the system and its gravitational field). In that case, the mass of the total system will not change in special relativity, for any transformation (including nuclear processes). Only if released energy is allowed to escape the system, will any "defect" in mass appear, as seen by a single observer.
