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The term mass in special relativity is commonly used by physicists to mean a quantity that does not depend on the observer or the inertial frame used to observe it. However, because the term relativistic mass is also used, this occasionally leads to confusion.

The invariant mass of an object (also known as the rest mass, intrinsic mass or proper mass) is an observer-independent quantity that is synonymous with mass. On the other hand, the relativistic mass of an object (also known as apparent mass) increases with its speed and therefore depends on one's frame of reference. The concept of relativistic mass has gradually fallen into disuse in physics since 1950^{[citation needed]}, when particle physics showed the relevance of invariant mass, to the point that relativistic mass is rarely used in 2007 scientific research literature. However, relativity text books of the early 1920s, written by well-respected physicists, made the term "relativistic mass" common in popular discussions and even in textbooks currently in use.

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.

Contents 1 Terminology usage 2 The relativistic mass concept 2.1 Early developments 2.2 Modern relativistic concepts 3 The mass of composite systems 3.1 Kinetic energy 4 The relativistic energy-momentum equation 4.1 Mass and the momentum 4-vector 5 Conservation of mass in special relativity 6 References

Relativistic mass and rest mass (invariant mass) are both traditionally used concepts in physics. However, with the development of Minkowski four-vector notation and general relativity, physicists gradually concluded that the invariant mass is the more fundamental quantity in the theory of relativity, and that the relativistic mass is just a redundant expression for total energy.

According to Lev Okun,^[1] Einstein himself always meant invariant mass when he wrote "m" in his equations, and never used a single "m" symbol for any other kind of mass. Einstein first deduced in 1905 that the mass (inertia) of bodies increases with their internal energy (energy content), and that this mass does not depend on the inertial frame used to observe it (see section below on mass in systems).

Today, many physicists reserve the word "mass" exclusively for invariant mass. When considering a body in one motion, they use the expression for the momentum and energy. The relativistic mass is thus a redundant quantity as it is proportional to the total energy of the body. Nevertheless, many popular books and textbooks, and some other physicists, still teach the relativistic mass concept.

While different kinds of "immaterial" energy (when it is allowed to enter or escape the system as heat or radiation) may present themselves as mass in objects or systems (when they are observed from the rest frame or center-of-momentum frame) and may be created by changes in mass (as in nuclear fission), the invariant mass of a body does not change when its kinetic energy changes, as Einstein first pointed out in 1905.

The value of relativistic mass is equal to invariant mass in one reference frame (except for massless particles like photons that cannot exist at rest). This frame is the rest frame of simple or compound objects (such as a solid composed of many particles), or more generally the center-of-mass inertial frame for systems of particles or objects, whether bound (such as a container of gas) or unbound (such as a system of interacting particles at high speed). Reactions in this special inertial frame (so long as the system remains closed) do not produce changes in invariant mass, relativistic mass, or energy.

Arnold B. Arons has argued against teaching the concept of relativistic mass:^[2]

For many years it was conventional to enter the discussion of dynamics through derivation of the relativistic mass, that is the mass–velocity relation, and this is probably still the dominant mode in textbooks. More recently, however, it has been increasingly recognized that relativistic mass is a troublesome and dubious concept. [See, for example, Okun (1989).] Not only does it get one into the infelicities associated with longitudinal and transverse masses, but it also tempts one to associate relativistic mass (rather than just rest mass) with gravitational effects. The latter association is basically incorrect. The sound and rigorous approach to relativistic dynamics is through direct development of that expression for momentum that ensures conservation of momentum in all frames:

rather than through relativistic mass. Unfortunately, it is more difficult to derive the momentum expression in a simple way than it is to obtain the mass–velocity relation from the collision gedanken experiments prevalent in the literature. [See Peters (1986) for a recent effort to simplify this derivation.]

On the other hand, T. R. Sandin has written:^[3]

The concept of relativistic mass brings a consistency and simplicity to the teaching of special relativity to introductory students. For example, E = mc² then expresses the beautifully simplifying equivalence of mass and energy. Those who claim not to use relativistic mass actually do so—if not by name—when considering systems of particles or photons. Relativistic mass does not depend on the angle between force and velocity—this supposed dependence results from incorrect use of Newton's second law of motion.

It was recognized by J. J. Thomson in 1881 that a charged body is harder to set in motion than an uncharged body. ^[4] So electrostatic charged bodies behave as having a velocity dependent electromagnetic mass besides their normal mechanical mass. George Frederick Charles Searle (1896), Wilhelm Wien (1900) and Max Abraham (1902) concluded that the velocity dependant electromagnetic mass of a body at rest is m=(4/3)E/c². ^{[5] [6]}

In 1899 it was also recognized by Hendrik Lorentz in the framework of Lorentz's Theory of Electrons, that the mass of the electrons depends on their relative speed to the luminiferous aether, and he calculated that mass perpendicular to motion m₀ at rest in the aether grows by the factor m = γm₀, where

is the Lorentz factor,

v is the relative velocity between the aether and the object, and

c is the speed of light.

So, according to this theory no body can reach the speed of light because the mass becomes infinitely large at this velocity. More precisely, Lorentz distinguished between the mass parallel to the direction of motion and the mass perpendicular to the direction of motion. The above equation has therefore not exactly the same meaning as the identical equation for "relativistic mass" which is discussed below.

The correct relativistic expression relating force and acceleration for a particle with non-zero rest mass moving in the x direction with velocity v and associated Lorentz factor γ is

Thus was called longitudinal mass and was called transverse mass. Abraham (1902) was the first to use the terms longitudinal and transvere mass for Lorentz's two masses, but his expressions were more complicated than those of Lorentz. ^{[7] [8]}

Also Einstein used longitudinal and transverse mass concepts in his 1905 electrodynamics paper and in another paper in 1906. ^{[9] [10]}

Like in Lorentz's electron theory, in special relativity an object with mass cannot travel at the speed of light. As such an object approaches the speed of light, a stationary observer will observe that the object's energy and momentum are increasing toward infinity.

The velocity dependent mass of Lorentz and Abraham evolved into the concept of relativistic mass, an expression which was coined by Richard C. Tolman in 1912, who stated: “the expression m₀(1 - v²/c²)^-1/2 is best suited for THE mass of a moving body.”^[11]

In 1934, Tolman also defined relativistic mass as^[12]

which should work for all particles, including those moving at the speed of light. For example, this formula states that a photon (which moves at the speed of light) has relativistic mass.

For a slower than light particle (i.e. non-zero rest mass) the formula becomes

Tolman remarked on this relation that "We have, moreover, of course the experimental verification of the expression in the case of moving electrons to which we shall call attention in §29. We shall hence have no hesitation in accepting the expression as correct in general for the mass of a moving particle."^[12]

When the relative velocity is zero, γ is simply equal to 1, and the relativistic mass is reduced to the rest mass as one can see in the next two equations below. As the velocity increases toward the speed of light c, the denominator of the right side approaches zero, and consequently γ approaches infinity.

The main benefit of using the relativistic mass is that the formula for momentum

from nonrelativistic mechanics retains its module by simply replacing m by M.

Additionally, some relations do not work right (even when taken in module) by doing so. For example, even though Newton's second law remains valid in the form

the derived form is invalid as in is generally not a constant [1] (see the section above on Lorentz and Abraham).

Another downside of this approach is that since γ depends on velocity, observers in different inertial reference frames will measure different values, which can be complicated.

An upside of this approach is that the calculation of the mass of composite systems is straightforward (simple addition), while this is complicated with invariant mass. Nevertheless, it has become popular to use invariant mass. In doing so when one relates four-force to invariant mass and four-acceleration Newton's second law is restored to the form

When discussing the "mass" (meaning invariant mass) of composite systems such as a pair of interacting particles, a little care must be taken. The invariant mass of a composite system can not in general be computed by adding the rest masses of its components, for invariant mass must also account for kinetic and potential energies present in a multi-particle system.

The total energy E of a composite system can be determined by adding together the sum of the energies of its components. The total momentum of the system, a vector quantity, can also be computed by adding together the momenta of all its components. Given the total energy E and the scalar norm p of the total momentum , where p=||||, the invariant mass of a composite system can be computed by the relativistic energy-momentum relationship:

Note that the invariant mass of a closed system is also independent of observer or inertial frame, and is a constant, conserved quantity for closed systems and single observers, even during chemical and nuclear reactions. The invariant mass of a system differs by a constant factor of c² from the rest energy of the composite system (the energy in its rest frame or center of mass frame (COM frame). As such, the measure of invariant mass of systems includes all energy (heat, light, kinetic energy) present in the system, so long as, in measuring the total system energy, an inertial frame is chosen in which total system momentum is zero (COM frame).

(Center of Momentum frame for systems)

Since the COM frame (also called center-of-momentum frame) is chosen as the frame to measure the mass of most compound objects, Einstein's mass-energy equivalence formula E = mc² continues to apply in these circumstances. For example, if the mass of a nuclear bomb were measured by weighing it, this system mass would be a conserved invariant mass and would not change, even after the bomb exploded. However, after the explosion, this total system mass would also include the heat and light from the explosion. Only after the heat and light was removed (resulting in a non-closed system) would the mass of the constituents of the bomb show a decreased mass (in this case, a mass decrease equal to the mass of the heat and light removed).

Invariant mass is a concept widely used in particle physics, as the invariant mass of a particle's decay products is equal to its rest mass. It is this that is used to make measurements of the mass of particles such as the z particle or top quark.

If M is the relativistic mass and m is the rest mass, with E being the total energy, we have:

The corresponding Taylor series is:

The first term (mc²) is energy which does not depend on velocity, and is commonly known as rest energy. The other terms represent kinetic energy.

For low velocities (speeds not sizable fraction of c), the terms with c in the denominator are negligible, so the relativistic energy is approximated by the first two terms:

Looking at only the kinetic part, this recovers the commonly used formula for kinetic energy in Newton's system: .

The relativistic expressions for E and p above can be manipulated into the fundamental relativistic energy-momentum equation:

Note that there is no relativistic mass in this equation; the m stands for the rest mass. This equation is a more general version of Einstein's famous equation "E=mc²", and can be regarded as the defining equation for invariant mass.

The equation is also valid for photons, which are massless (have no rest mass):

Therefore a photon's momentum is a function of its energy; it is not analogous to the momentum in Newtonian mechanics.

Considering an object at rest, the momentum p, in the first equation above, is zero, and we obtain

which reduces to

suggesting that this last well-known relation is only valid when the object is at rest, giving what is known as the rest energy. If the object is in motion, we have

From this we see that the total energy of the object E depends on its rest energy and momentum; as the momentum increases with the increase of the velocity v, so does the total energy.

This E is in fact equivalent to that of the relativistic energy equation in the previous section, and that energy equation differs from the relativistic mass equation by a factor of c². Therefore the relativistic mass is essentially the same as the total energy — but scaled and with different units. Since the energy-momentum equation is more convenient to use (especially with four-vector notation), the relativistic mass is hardly ever used in practice.

When working in units where c = 1, known as the natural unit system, the energy-momentum equation reduces to

The equation is often written in this form to show the invariance of mass (rest mass), as the energy and momenta of single particles changes, when seen from different inertial frames.

The equation above reduces to m² = E² or m = E, when v = 0, showing that proper choice of inertial frame gives the rest mass of a particle as the rest energy. The same reduction happens for systems of particles (where E and p are sums), when the inertial frame is chosen as the center-of-mass frame (COM frame, sometimes called the system rest frame) where total p = 0. Such a frame can always be identified for any system. In this case, again m = E, showing the useful property that in the COM frame of a system, the system mass (invariant mass) is given by the system total energy. Unlike the case of single particles, the system total energy, as a sum, may include kinetic and photon energies. These energies by themselves have no "rest mass," for individual particles, but in the case of systems where paricles are moving, they still contribute to the system mass (the "rest" mass of the system if it were to be enclosed and weighed, or otherwise have its mass measured). For example, a container full of gas molecules has an invariant mass which includes all the kinetic energies of the moving molecules (as well as the mass of all other kinds of energy present as heat). A hollow container also is more massive by the total energy of the black body radiation it contains, which is entirely composed of individually massless photons.

Energy is typically in units of electron volts (eV), momentum in units of eV/c, and mass in units of eV/c². This is the primary unit system in particle physics.
Energy may also in theory be expressed in units of grams, though in practice it requires a great deal of energy to be equivalent to masses in this range, and these energies are expressed in other units. For example, the first atomic bomb liberated about 1 gram of heat, and the largest thermonuclear bombs have generated a kilogram or more of heat. However, such energies are instead always given in tens of kilotons and megatons referring to the energy liberated by exploding that amount of trinitrotoluene (TNT); or terajoules and petajoules.

Invariant mass is a constant times the "length" of the momentum 4-vector. This is true in any inertial frame of reference (it is invariant) for any closed system of particles undergoing any sort of motion (where mass is constant), or interaction (across which mass is conserved, which means it remains the same value).

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 inertial 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 during interactions also, for all single inertial observers, so long as the system is closed.

Sometimes a given form of mass is not conserved in relativistic processes or interactions, 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. 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, whereas 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. ^[13].

1. ^ Lev B. Okun (July 1989). "The Concept of Mass". Physics Today 42 (6): 31–36. 
2. ^ Arnold B. Arons, A Guide to Introductory Physics Teaching (1990, page 263); also in Teaching Introductory Physics (2001, page 308)
3. ^ T. R. Sandin (Nov. 1991). "In defense of relativistic mass". American Journal of Physics 59 (11). 
4. ^ Thomson, J.J. (1881), "On the Effects produced by the Motion of Electrified Bodies", Phil. Mag. 11: 229
5. ^ Searle, G.F.C. (1896), "Problems in electric convection", Phil. Trans. Roy. Soc. 187: 675-718
6. ^ Wien, W. (1900/1901), "Über die Möglichkeit einer elektromagnetischen Begründung der Mechanik", Annalen der Physik 5: 501-513
7. ^ Lorentz, H.A. (1899), "Simplified Theory of Electrical and Optical Phenomena in Moving Systems", Proc. Roy. Soc. Amst.: 427-442
8. ^ Abraham, M. (1902), "Prinzipien der Dynamik des Elektrons", Physikalische Zeitschrift 4 (1b): 57-62
9. ^ Einstein, A. (1905), "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17: 891-921 English translation
10. ^ Einstein, A. (1906), "Über eine Methode zur Bestimmung des Verhältnisses der transversalen und longitudinalen Masse des Elektrons", Annalen der Physik 21: 583-586
11. ^ R. Tolman, Philosophical Magazine 23, 375 (1912).
12. ^ ^{a b} Tolman, R. C. (1934). Relativity, Thermodynamics, and Cosmology. Oxford: Clarendon Press. LCCN 340-32023.  Reissued (1987) New York: Dover ISBN 0-486-65383-8.
13. ^ E. F. Taylor and J. A. Wheeler, Spacetime Physics, W.H. Freeman and Co., NY. 1992. ISBN 0-7167-2327-1, see pp. 248-9 for discussion of mass remaining constant after detonation of nuclear bombs, until heat is allowed to escape.