Mass In Special Relativity

The rest mass or ''' Invariant Mass ''' is an observer-independent quantity.

The Relativistic Mass or '''apparent mass''' depends on one's Frame Of Reference .

In particular, the Relativistic Mass increases with observed Velocity while the rest mass is an intrinsic property of an object, a so-called Invariant .

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, it should be noted that 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 realized that the Invariant Mass was the more fundamental quantity in the theory of relativity. For example, Invariant Mass is the only type of mass which is connected with a gravitational field (thus, no matter how fast an object goes, it cannot become a black hole because of relativistic mass).

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 accepted present usage in the scientific community today (at least in the context of special relativity) considers the Invariant Mass to be the only "mass", while the concept of ''energy'' has replaced the relativistic mass. This usage has been confused by the fact that many kinds of "immaterial" energy (such as light and heat) may present themselves as Invariant Mass in objects or systems (when they observed from the Rest Frame or center-of-momentum frame), and thus some Invariant Mass in objects and systems derives from relativistic effects, 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). 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

\gamma = {1 \over {\sqrt{1 - u^2/c^2}}} \!

is the Lorentz Factor ,

u

c

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=rac{dp}{dt} \!

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\,

\gamma m \,

(e.g., Newton's Second Law in the form

\mathbf{F}=m\mathbf{a}

). This is because other parts of the equation have transformation factors as well (e.g. in ''Newton's second law'', "rest acceleration" 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{F}=m\mathbf{a}

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 Frame s will measure different values, which can be complicated. A more crucial flaw is that

\gamma

is undefined for ''v'' = ''c''; in other words, these equations are not valid for Photon s.

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_{n=1}^{\infty}\left(rac{(2n-1)!}{n!(n-1)!2^{2n-1}} rac{mv^{2n}}{c^{2n-2}}
ight) = mc^2 + rac{mv^2}{2} + rac{3mv^4}{8c^2} + rac{5mv^6}{16c^4} + \dots

The first term (''mc''²) is the rest energy. The other part,

\left(++\dots
ight)

, 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{matrix} rac{1}{2} \end{matrix}mv^2

.

Then it follows that for low velocities,

E \simeq mc^2 + \begin{matrix} rac{1}{2} \end{matrix}mv^2

, which is the relativistic energy and not the Newtonian energy which consists uniquely of the kinetic 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 ''c''². 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 never 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 E&² = m&² 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 Volt s (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 be 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 SYSTEMS

If the mass of a bound system (such as an ordinary object made of bound atoms, or a sealed container of gas or radiation) is measured at rest, then the system as a whole has no net momentum. In such circumstances, the measurement takes place in the "center of momentum" (or "COM") Inertial Frame in which the various momenta of the parts of the system always add up to zero, even if individual parts of the system (such as gas molecules or photons) may be in motion.

In such a system, the individual total relativistic energies of the components must be added to obtain the total energy of the system:

:

E(system) = \Sigma {E(components)} \,\!

From the ''relativistic energy-momentum equation'' above in natural units, the mass of an object is given by:
:

m = } \,\!

Similarly, for a system of objects, the Invariant Mass of the system is given by:

:

\mbox{m}^2=(\Sigma \mbox{E})^2-(\Sigma \mbox{p})^2

Where:

:

m

is the invariant mass of the system of particles
:

\Sigma E

is the sum of the energies of the particles
:

\Sigma p

is the vector sum of the Momenta of the particles (includes both magnitude and direction of the momenta)

Since the total vector sum momentum of such a system is zero, then the mass of the system is simply the total relativistic energy, which is obtained by summing the energies of the components:

:

m = E = E/c^2 \,\!

Note that this invariant mass is the ''rest mass'' of a bound system, but the ''energy'' of each component is its full relativistic energy. This relativistic energy for particles includes not only the rest mass of each particle, but also its kinetic energy. For photons, the energies of the photons are included here. For systems of particles, potential energies must also be included in this expression, so that any work done on a system adds to its energy and mass, and work done BY the system (such as when particles are in the process of being bound) subtracts from the system energy and mass.

In bound systems, then, we see that as a consequence of the cancelling of momenta, that total relativistic energy of the system (even a system with moving parts) equates to the Invariant Mass or what we would ordinarily think of as the Rest Mass of the system. This may cause some confusion, since for single particles in motion, this is not in general the case (as seen in the previous sections of this article). However, the distinction is possible because total relativistic energy for systems with no net momentum is an invariant quantity, while for single particles it is not, unless measured also in the reference frame where the particle has no momentum (is at rest).

To see how the kinetic energy of particles in systems may contribute to the invariant or rest mass of the system, consider a box of ideal monatomic gas on a scale. We neglect the mass of the box itself. When the gas is heated, the energy of the heat appears entirely as increased kinetic energy of the molecules of the gas. An observer in the rest frame of the box sees that for each molecule in the box, an increase in kinetic energy adds to Relativistic Mass but not the molecule's Rest Mass . However, it is not the total rest mass of the molecules which appears as the total mass of gas in the box. In special relativity theory, rest masses cannot be simply summed to find a total mass. When molecules are trapped in the box, their momenta cancel, and thus it is the sum of their Relativistic Mass es in the COM frame which is measured as the mass of the gas in the box, and which is the quantity which is weighed on the scale. This total mass also determines the inertia of the box, and its gravitational field.

In this way, the total kinetic energy of the molecules (in this case, their "heat") adds to the mass of the box, ''without'' adding to the rest mass of any given molecule. This kinetic energy may thus be seen to be ''a system property''. To see that kinetic energy of pairs of particles is an invariant property of a system itself (but not any one particle), consider a model system of two particles, A and B, moving away from each other, each with the same kinetic energy. In this system, the kinetic energy is equally shared between particles A and B. However, in an inertial frame centered on particle A, particle A is at rest and all of the kinetic energy in the system is found in particle B. From the viewpoint of particle B, however, all of the kinetic energy in the system is present in A. It is apparent that kinetic energy in this system is invariant for all observers, but different observers will disagree as to how it is distributed or "located" (the location is relative). Thus, the kinetic energy is a property of the system of two particles, but not of any particle in particular, because apparent location is observer-dependent.

In a similar way, the kinetic energy of single particles in a box of gas does not add to their gravitational fields or inertias, because it is observer-dependent, and may be made to vanish by choice of inertial frame. However, for systems of particles (such as the box of gas), a certain amount of total kinetic energy may reside in the system in the inertial frame where the system has no net momentum, and ''this'' system kinetic energy cannot be removed by choice of inertial frame. This extra energy DOES have weight, invariant mass, inertia, and a gravitational field. However, as noted above, this extra mass (associated with the energy of heat) is not located in any particular place.

Similar considerations apply to photons, and again may cause confusion to the beginning student of special relativity. Single photons have no mass, and their relativistic mass does not give them inertia, or a gravity field. However, systems of photons in which the photons are moving in different directions will generally have a residual invariant mass which cannot be made to vanish by choice of inertial frame. In a box in which the photons are trapped and have zero total vector momentum, this invariant mass is

E/c^2

where

E

is the total photon energy. Thus, while photons do not individually have mass, collections of photons may have an invariant mass associated with them, as a system property. Again, such mass does not reside in any particular location.

Finally, it should be emphasized that the above considerations still hold even for unbound systems, so long as an inertial frame is chosen in which the sum of the momenta of all system components is zero. In this intertial frame, the sum of the relativistic energies of the components determines the invariant mass of the system.

A familiar example is the annihilation of an electron and positron (starting at rest) to form two gamma rays. Before the anihiliation, the rest mass of the system is the sum of the rest masses of the electron and positron. After the annihilation event, each of the gamma rays produced has no Rest Mass , and yet the ''system'' of two gamma rays moving in opposite directions continues to have the same invariant mass as the sum of the electron and positron masses. This invariant mass is not present in either gamma ray (which are individually massless), but remains a property of the system. Its presence insures that the gravitational field associated with the electron and positron does not change, when they are transformed into two photons.

A second familiar example involves the neutron-induced fission of a nucleus of U-235, as in a nuclear reactor. In such circumstances, the two product fission nuclei fly apart with high kinetic energies, under their mutual electrostatic repulsion. In this system, the invariant mass does not change while the product nuclei are in flight, since energy liberated from the electromagnetic field in the nucleus has become kinetic energy, which still contributes to the invariant mass of the system in the same way as for the examples above. Moreover, once the fission product nuclei have been stopped in the material of the fuel rod and their kinetic energy transformed to heat, the mass of the system (including the fuel rod) still is constant, because the various kinds of energy associated with the ''heat'', still contribute to the system's mass. (However, the combined masses of the stopped fission products, if weighed, would now show the appropriate mass defect). Only when the heat of the nuclear reaction is removed through the cooling system, would the mass of the fuel rod decrease. However, the mass of whatever system absorbed the heat (such as the coolant fluid) would now increase by an equal amount.

EXTERNAL LINKS

Usenet Physics FAQ

--- Does mass change with velocity?

--- Does light have mass?

arXiv.org:physics/0504110 : ''On the Abuse and Use of the Relativistic Mass'', Gary Oas, 2005.

-Tolman Experimemt, relativistic mass

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Mass In Special Relativity