Mass in Special Relativity

Mass in special relativity incorporates the general understandings from the concept of mass-energy equivalence. Added to this concept is an additional complication resulting from the fact that "mass" is defined in two different ways in special relativity: one way defines mass ("rest mass" or "invariant mass") as an invariant quantity which is the same for all observers in all reference frames; in the other definition, the measure of mass ("relativistic mass") is dependent on the velocity of the observer.

The term mass in special relativity usually refers to the rest mass of the object, which is the Newtonian mass as measured by an observer moving along with the object. The invariant mass is another name for the rest mass of single particles. The more general invariant mass (calculated with a more complicated formula) loosely corresponds to the "rest mass" of a "system." Thus, invariant mass is a natural unit of mass used for systems which are being viewed from their center of momentum frame (COM frame), as when any closed system (for example a bottle of hot gas) is weighed, which requires that the measurement be taken in the center of momentum frame where the system has no net momentum. Under such circumstances the invariant mass is equal to the relativistic mass (discussed below), which is the total energy of the system divided by c (the speed of light) squared.

The concept of invariant mass does not require bound systems of particles, however. As such, it may also be applied to systems of unbound particles in high-speed relative motion. Because of this, it is often employed in particle physics for systems which consist of widely separated high-energy particles. If such systems were derived from a single particle, then the calculation of the invariant mass of such systems, which is a never-changing quantity, will provide the rest mass of the parent particle (because it is conserved over time).

It is often convenient in calculation that the invariant mass of a system is the total energy of the system (divided by c2) in the COM frame (where, by definition, the momentum of the system is zero). However, since the invariant mass of any system is also the same quantity in all inertial frames, it is a quantity often calculated from the total energy in the COM frame, then used to calculate system energies and momenta in other frames where the momenta are not zero, and the system total energy will necessarily be a different quantity than in the COM frame. As with energy and momentum, the invariant mass of a system cannot be destroyed or changed, and it is thus conserved, so long as the system is closed. (In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it).

The term relativistic mass is also sometimes used. This is the sum total quantity of energy in a body or system (divided by c2). As seen from the center of momentum frame, the relativistic mass is also the invariant mass, as discussed above (just as the relativistic energy of a single particle is the same as its rest energy, when seen from its rest frame). For other frames, the relativistic mass (of a body or system of bodies) includes a contribution from the "net" kinetic energy of the body (the kinetic energy of the center of mass of the body), and is larger the faster the body moves. Thus, unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for closed systems, the relativistic mass is also a conserved quantity.

Although some authors present relativistic mass as a fundamental concept of the theory, it has been argued that this is wrong as the fundamentals of the theory relate to space-time. There is disagreement over whether the concept is pedagogically useful. The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.

For a discussion of mass in general relativity, see mass in general relativity. For a general discussion including mass in Newtonian mechanics, see the article on mass.

Read more about Mass In Special Relativity:  Terminology, Invariant Mass, The Relativistic Energy-momentum Equation, The Mass of Composite Systems, Conservation Versus Invariance of Mass in Special Relativity

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