Mass in special relativity

The word "mass" has two meanings in special relativity: invariant mass is an invariant quantity which is the same for all observers in all reference frames, while the relativistic mass is dependent on the velocity of the observer. According to the concept of mass–energy equivalence, invariant mass is equivalent to rest energy, while relativistic mass is equivalent to relativistic energy.
The term "relativistic mass" tends not to be used in particle and nuclear physics and is often avoided by writers on special relativity, in favor of referring to the body's relativistic energy. In contrast, "invariant mass" is usually preferred over rest energy. The measurable inertia of a body in a given frame of reference is determined by its relativistic mass, not merely its invariant mass. For example, photons have zero rest mass but contribute to the inertia of any system containing them.
The concept is generalized in mass in general relativity.

Rest mass

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 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, as when any closed system 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, which is the total energy of the system divided by c².
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.
It is often convenient in calculation that the invariant mass of a system is the total energy of the system in the COM frame. 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 to all influences.

Relativistic mass

The relativistic mass is the sum total quantity of energy in a body or system. Thus, the mass in the formula
is the relativistic mass. For a particle of non-zero rest mass moving at a speed relative to the observer, one finds
In the center of momentum frame, and the relativistic mass equals the rest mass. In other frames, the relativistic mass includes a contribution from the "net" kinetic energy 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 isolated systems, the relativistic mass is also a conserved quantity.
The relativistic mass is also the proportionality factor between velocity and momentum,
Newton's second law remains valid in the form
When a body emits light of frequency and wavelength as a photon of energy, the mass of the body decreases by, which some interpret as the relativistic mass of the emitted photon since it also fulfills. 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. It explains simply and quantitatively why a body subject to a constant acceleration cannot reach the speed of light, and why the mass of a system emitting a photon decreases. In relativistic quantum chemistry, relativistic mass is used to explain electron orbital contraction in heavy elements.
The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.
Relativistic mass is not referenced in nuclear and particle physics, and a survey of introductory textbooks in 2005 showed that only 5 of 24 texts used the concept, although it is still prevalent in popularizations.
If a stationary box contains many particles, its weight increases in its rest frame the faster the particles are moving. Any energy in the box 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, 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, 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 vs. rest 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 this fact remains true only because the system on average must be at rest to be weighed. For example, if an electron in a cyclotron is moving in circles with a relativistic velocity, the mass 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 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. 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. 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. For compound objects and sets of unbound objects, 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 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". However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference.

Invariant mass

The invariant mass is the ratio of four-momentum to four-velocity:
and is also the ratio of four-acceleration to four-force when the rest mass is constant. The four-dimensional form of Newton's second law is:

Relativistic energy–momentum equation

The relativistic expressions for and obey the relativistic energy–momentum relation:
where the m is the rest mass, or the invariant mass for systems, and is the total energy.
The equation is also valid for photons, which have :
and therefore
A photon's momentum is a function of its energy, but it is not proportional to the velocity, which is always.
For an object at rest, the momentum is zero, therefore
Note that the formula is true only for particles or systems with zero momentum.
The rest mass is only proportional to the total energy in the rest frame of the object.
When the object is moving, the total energy is given by
To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to, is the only four-vector associated with the particle's motion, so that if there is a conserved four-momentum, it must be proportional to this vector. This allows expressing the ratio of energy to momentum as
resulting in a relation between and :
This results in
and
these expressions can be written as
where the factor
When working in units where, known as the natural unit system, all the relativistic equations are simplified and the quantities energy, momentum, and mass have the same natural dimension:
The equation is often written this way because the difference is the relativistic length of the energy momentum four-vector, a length which is associated with rest mass or invariant mass in systems. Where and, this equation again expresses the mass–energy equivalence.