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Mass in special relativity

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The term mass in special relativity can be used in different ways, occasionally leading to confusion. Historically, mass can refer to either the invariant mass or the relativistic mass.

In particular, the relativistic mass increases with observed speed while the rest mass is an invariant property of an object: it does not change with a change of reference system.

Contents

Summary

In the earlier years of relativity, relativistic mass was sometimes taken to be the "correct" notion of mass, and the invariant mass was referred to as the rest mass. However, 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), but this mass too, is a kind of invariant mass (see section below on mass in systems).

Gradually, with the development of Minkowski four-vector notation and general relativity, it was concluded that the invariant mass was the more fundamental quantity in the theory of relativity.

Scales and balances always operate in the rest frame of objects being measured. Because in this special frame invariant mass and relativistic mass are equal, scales and balances measure both types of mass.

The common present usage in the scientific community today (at least in the context of particle physics) considers the invariant mass to be the only "mass", while the concept of energy has replaced the relativistic mass. This usage may be confusing because many kinds of "immaterial" energy (such as light and heat) may present themselves as invariant mass in objects or systems (when they are observed from the rest frame or center-of-momentum frame), and thus some invariant mass in objects and systems is subject to variation (when it is allowed to enter or escape the system as heat or radiation), just as Einstein first pointed out in 1905.

In popular science and basic relativity courses, however, the observer-dependent kind of relativistic mass is usually still presented, due to its conceptual simplicity and the fact that certain equations from nonrelativistic mechanics retain their form (namely, Newton's second law). Einstein's famous equation [E = mc^2 \,\!] remains generally true for all observers only if the [m\,\!] in the equation is considered to be relativistic mass. It is true for invariant mass, only in specific circumstances to be discussed.

As noted above, relativistic mass and invariant mass are equal in some reference frames. These frames includes the rest frame of compound objects (such as a solid composed of many particles), and also 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). Peculiarly, the invariant mass of such systems includes the relativistic mass of the components. Reactions in this special inertial frame therefore do not produce changes in either mass or energy by any definition of these terms (so long as the system remains closed).

For other reference frames, and other single observers, mass and energy are separately conserved in reactions, but as noted the value of relativistic mass and total energy in systems varies, as measured by different observers, even though the value is conserved in reactions, as seen by each observer. Invariant mass, however, is both conserved and invariant between observers.

Statements that mass is not conserved in special relativity (which are seen in some presentations of the subject) require one or both of the following conditions to be true:

1) The system is not closed, which means that mass or energy has been allowed to enter or escape. For example, mass is not conserved in a chemical or nuclear reaction if heat or radiation is allowed to escape from the system between measurements, but otherwise mass continues to be conserved (according to single observers, or an unchanged inertial frame).

2) The system mass is measured by multiple obsevers. This happens in effect when the various rest masses of moving parts of a system are simply added to obtain a total "mass." This procedure is valid in Newtonian mechanics, but it misses counting the system mass associated with kinetic energies and radiation, and is not in general valid in special relativity. Statements that mass is not conserved in special relativity sometimes mean merely that the sum of rest masses of products is not the sum of the mass of the initial system. However, this always amounts to some violation of the first or second conditions for measuring nonconservation of mass. In general, if a single observer measures the mass of a system, and no mass or energy of any kind is allowed to escape, the mass of the system will not change during any energy-transforming reaction.

The relativistic mass concept

According to the theory of 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 kinetic energy and momentum is increasing toward infinity. Certain experiments (but not all) will also exhibit an increased inertia for the object associated with the increase in relativistic mass.

The relativistic mass M is then formulated as:

[M = \gamma m \!]
where
m is the rest mass, and
[\gamma = }} \!] is the Lorentz factor,
u is the relative velocity between the observer and the object, and
c is the speed of light.
When the relative velocity is zero, [\gamma] is simply equal to 1, and the relativistic mass is reduced to the rest mass as you 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 [\gamma] approaches infinity.

The main benefit of using the relativistic mass is that the formulas

[F=\frac \!]     and     [p=mv \,\!]
from nonrelativistic mechanics retain their form, and are valid for relativistic situations when used with M in place of m. The first equation is Newton's second law, the second is simply the definition of momentum.

Note, however, that many relations do not work right if one simply replaces [m\,] by [\gamma m \,] (e.g., Newton's second law in the form [\mathbf=m\mathbf]). This is because other parts of the equation have transformation factors as well (e.g. in Newton's second law, "rest acceleration" [in the direction of relative velocities of the observer and object] can be converted to "relativistic acceleration" by a factor of [\gamma^2]). The correct relativistic form of [\mathbf=m\mathbf] is actually

[F_x = \gamma^3 m a_x \,]
[F_y = \gamma m a_y \,]
[F_z = \gamma m a_z \,]
(assuming that the velocity is along the [x] direction). For this reason, the use of the concept of relativistic mass is limited.

Another downside of this approach is that since [\gamma] depends on velocity, observers in different inertial reference frames will measure different values, which can be complicated. It should also be noted that these equations apply to matter and they are unsuited for photons: with v=c and m=0, [\gamma m ] of a photon is undefined.

Kinetic energy

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

[E = Mc^2 = \gamma m c^2 = }}}]
The corresponding Taylor series is:
[ E = mc^2 + \sum_^\left(\frac} \frac}}\right) = mc^2 + \frac + \frac + \frac + \dots]
The first term (mc2) is the rest energy. The other part, [\left(++\dots\right)], is known as the kinetic energy. Except for speeds a sizable fraction of c, the terms with c in the denominator are negligible, therefore we obtain the commonly used formula for kinetic energy in Newton's system: [E_k = \begin \frac \endmv^2].
Then it follows that for low velocities, [E \simeq mc^2 + \begin \frac \endmv^2], which is the relativistic energy.

The relativistic energy-momentum equation

The relativistic expressions for E and p above can be manipulated into the fundamental relativistic energy-momentum equation:
[E^2 - (pc)^2 = (mc^2)^2 \,\!]
Note that there is no relativistic mass in this equation; the m stands for the rest mass. The equation is also valid for photons, which are massless (have no rest mass):
[E^2 - (pc)^2 = 0 \,\!]
[E^2 = (pc)^2 \,\!]
[E = pc \,\!]
[p = E/c \,\!]
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

[E^2 = (mc^2)^2 \,\!]
which reduces to
[E = mc^2 \,\!]
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
[E^2 = (mc^2)^2 + (pc)^2 \,\!]
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

[E^2 - p^2 = m^2 \,\!]
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 E² = m² or E = m when v = 0, showing that proper choice of inertial frame gives the rest energy of a particle as the total energy.

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; or terajoules and petajoules.

The mass of composite systems

When discussing the mass of composite systems such as a pair of interacting particles, a little care must be taken. One definition of the mass of the system is the sum of the rest masses of the separated constituents. With this definition, one finds that the mass of the system will generally not be conserved. Over the course of a reaction, the mass of the system can increase or decrease, as the rest energy of the particles is converted to other forms of energy or vice versa. This is a surprising departure from nonrelativistic mechanics, where the law of conservation of mass dictates that the total mass is always the same. Thus, in terms of the rest masses, special relativity predicts a violation of the law of conservation of mass, though the effect is usually too small to be observed except in nuclear reactions.

The advantage of this definition is that one can quickly calculate the energy exchanged in any nuclear reaction. The rest masses of nuclear particles are generally well-known, so the mass defect, the change in mass between the reactants and products of the reaction, can be easily calculated. The energy released (or absorbed if the mass increases) is then given by the equivalent energy of the mass defect, according to the equation E=mc2. So the energy required to drive an endothermic reaction is exactly given by the increase in mass of the products, while the energy released in an exothermic reaction is given by the decrease in mass.

Another notion that one can speak of is the invariant mass of the system, which 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). It is given by

[m=\sqrt,]
where ET and pT are the total energy and momentum of all the constituents of the system, respectively.

The advantage of this definition is that, as a consequence of the law of conservation of energy, the invariant mass of a system is conserved. Moreover, invariant mass is a Lorentz invariant quantity, which means calculations hold in any reference frame. Also the invariant mass of a system of decay products gives the mass of the unstable parent particle, which may be otherwise unobservable.

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