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.

The invariant mass (also known as the rest mass, intrinsic mass or the proper mass ) is an observer-independent quantity.
The relativistic mass (also known as the apparent mass) depends on one's frame of reference.

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

For a discussion of mass in General relativity, see mass in general relativity. For a general discussion including mass in Newtonian mechanics, see Mass.

1 Summary
2 The relativistic mass concept
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 References
6 External links

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

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).

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, however, the observer-dependent kind of relativistic mass is usually still presented, as it allows certain equations from nonrelativistic mechanics to retain their form (see below). Also, 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. The modifications to this formula needed for general use with invariant mass are discussed later in "The relativistic energy-momentum equation".

As noted above, relativistic mass and invariant mass are equal in some reference frames. These frames include the rest frame of compound objects (such as a solid composed of many particles), as well as 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). The invariant mass of such composite 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).

[edit] 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 will also observe an increased inertia for the object associated with the increase in relativistic mass.

Designating the relativistic mass as $M\!$ we start from $E = Mc^2 \,\!$ to immediately get the general formula (Tolman, 1987, pg 49):

$M = \frac{E}{c^2}\!$

which works for all particles, including those moving at the speed of light. Note that this general formula states that a photon or other hypothetical particle moving at the speed of light has a non-zero relativistic mass as long as it has a non-zero energy. It is correct (though now dated) to say that a photon has relativistic mass. By modern usage, it is incorrect to say that a photon has mass, or to say that a photon has invariant mass. [1] [2]

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

$M = \gamma m \!$

where

m is the invariant mass, and

$\gamma = {1 \over {\sqrt{1 - \frac{u^2}{c^2}}}} \!$ 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, 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

$\mathbf{p}=M\mathbf{u}$

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

However some relations do not work right by doing so. For example, even though Newton's second law remains valid in the form

$\mathbf{f}=\frac{d(M\mathbf{u})}{dt}, \!$

the derived form $\mathbf{f}=M\mathbf{a}$ is invalid as $M\,$ in ${d(M\mathbf{u})}\!$ is generally not a constant [3]. The correct relativistic expression relating force and acceleration for a particle with non-zero rest mass moving in the x direction with velocity u and associated Lorentz factor γ is

$f_x = \gamma^3 m a_x = \gamma^2 M a_x, \,$

$f_y = \gamma m a_y = M a_y, \,$

$f_z = \gamma m a_z = M a_z. \,$

For this reason, the use of the concept of relativistic mass is limited. Because of these equations, $\gamma^3 m\,$ has (rarely) been called longitudinal mass and γ m has (rarely) been called transverse mass.

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, the modern practice is to use invariant mass only. In doing so when one relates four-force to invariant mass and four-acceleration Newton's second law is restored to the form

$F^\mu = mA^\mu.\!$

[edit] The mass of composite systems

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 $\vec{p}$ 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 $\vec{p}$ , where p=|| $\vec{p}$ ||, the invariant mass of a composite system can be computed by the relativistic energy-momentum relationship:

$m = \frac {\sqrt{E^2 - (pc)^2}}{c^2}$

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).

$m = \frac {E}{c^2}$ (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 most famous equation 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.

[edit] Kinetic energy

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

$E = Mc^2 = \gamma m c^2 = {{mc^2} \over {\sqrt{1 - {{v^2} \over {c^2}}}}}$

The corresponding Taylor series is:

$E = mc^2 + \frac{mv^2}{2} + \frac{3mv^4}{8c^2} + \frac{5mv^6}{16c^4} + \dots$

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:

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

Looking at only the kinetic part, this recovers the commonly used formula for kinetic energy in Newton's system: $E_k = \begin{matrix} \frac{1}{2} \end{matrix}mv^2$ .

[edit] 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. 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):

$E^2 - (pc)^2 = 0 \,\!$

$E = pc \,\!$

$p = E/c \,\!$

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

$m^2 = E^2 - p^2 \,\!$

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 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.

[edit] Mass and the momentum 4-vector

In an inertial frame of reference, mass is a constant times the "length" of the momentum 4-vector.

[edit] References

"Does light have mass?" by Philip Gibbs, 1997, retrieved Aug 10 2006
"What is the mass of a photon?" by Matt Austern et al., 1998, retrieved Aug 10 2006
"Does mass change with velocity?" by Philip Gibbs et al., 2002, retrieved Aug 10 2006

Tolman, Richard C (1987). "Relativistic Thermodynamics and Cosmology". ISBN 0-486-65383-8. (paperback reprint of Oxford 1934 original)