Mass in special relativity

From Wikipedia, the free encyclopedia

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. However, the more general invariant mass (calculated with a more complicated formula) may also be applied to systems of particles in relative motion, and because of this, is usually reserved for systems which consist of widely separated high-energy particles. The invariant mass of systems is the same for all observers and inertial frames, and cannot be destroyed, and is thus conserved, so long as the system is closed. In this case, "closure" implies that an idealized boundary is drawn around the system, and no mass/energy is allowed across it.

The term relativistic mass is also used, and this is the total quantity of energy in a body (divided by c²). The relativistic mass (of a body or system of bodies) includes a contribution from the kinetic energy of the body, and is larger the faster the body moves, so unlike the invariant mass, the relativistic mass depends on the observer's frame of reference.

Because the relativistic mass is proportional to the energy, it has gradually fallen into disuse in among physicists^[1]. There is disagreement over whether the concept remains pedagogically useful.^[2]^[3]

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.

1 Terminology
2 The relativistic mass concept
- 2.1 Early developments: transverse and longitudinal mass
- 2.2 Modern relativistic concepts
3 The relativistic energy-momentum equation
4 The mass of composite systems
5 Conservation versus invariance of mass in special relativity
6 Controversy
7 See also
8 References
9 External links

[edit] Terminology

If a box contains many particles, it weighs more 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 in 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 it.

Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass is just a redundant name for the total energy. The relativistic mass may be said to be the mass of the system as it would be measured on a scale, but a scale cannot be used to measure mass unless the mass is confined or bound (such as a container of gas, or a rotating object) and in this case, relatistic and invariant masses are the same, since the scale will not work unless it is in the center of mass inertial frame of the bound object, and in which frame, relativistic and invariant masses are equal. If a single object is stopped and weighed, or the scale is sent after it, again it is not moving with respect to the scale, and again the relativistic and rest masses are the same.

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 (and where, again, it could in theory be weighed). This is why the invariant mass is also called the rest mass. This special frame 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 (another way of stating this is that it is the frame in which the linear momenta of the system's parts add to zero). For compound objects (made of many smaller objects, some of which may be moving) and sets of unbound objects (some of which may also be moving), 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.

If an particle (such as a photon, or a theoretical graviton) is moving at the speed of light, it is never at rest in any frame. In this case the total energy of the object becomes smaller and smaller in frames which move faster and faster in the same direction. The rest mass (and invariant mass) of such an object is zero, and the only type of "mass" which the object has, is relativistic mass-- a quantity which depends on the observer.

[edit] The relativistic mass concept

See also: Special relativity#Mass-energy equivalence

[edit] Early developments: transverse and longitudinal mass

It was recognized by J. J. Thomson in 1881 ^[4] that a charged body is harder to set in motion than an uncharged body, which was worked out on more detail by Oliver Heaviside (1889) and George Frederick Charles Searle (1896). ^[5] So the electrostatic energy behaves as having some sort of electromagnetic mass, which can increase the normal mechanical mass of the bodies. Later Wilhelm Wien (1900), ^[6] Max Abraham (1902), ^[7] came to the conclusion that the total mass of the bodies is identical to its electromagnetic mass. And because the em-mass depends on the em-energy, the formula for the energy-mass-relation given by Wien (1900) was $m = (4 / 3) E / c 2$ .

It was pointed out by Thomson and Searle, that this electromagnetic mass also increases with velocity. This was also recognized by Hendrik Lorentz (1899, 1904) in the framework of Lorentz's Theory of Electrons. He defined mass as the ratio of force to acceleration not as the ratio of momentum to velocity, so he needed to distinguish between the mass $m L = γ 3 m 0$ parallel to the direction of motion and the mass $m T = γ m 0$ perpendicular to the direction of motion. Only when the force is perpendicular to the velocity is Lorentz's mass equal to what is now called "relativistic mass". (Where $\gamma = 1/\sqrt{1 - v^2/c^2}$ is the Lorentz factor, v is the relative velocity between the aether and the object, and c is the speed of light). Abraham (1902) called $m L$ longitudinal mass and $m T$ transverse mass, (whereby Abraham's own expressions were more complicated than Lorentz's relativistic ones). So, according to this theory no body can reach the speed of light because the mass becomes infinitely large at this velocity. ^[8] ^[9]

The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass $m$ moving in the x direction with velocity v and associated Lorentz factor $γ$ is

$f_x = m \gamma^3 a_x = m_L a_x, \,$

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

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

Einstein calculated the longitudinal and transverse mass (which are equivalent to those of Lorentz, but for a mistake in $m T$ , which was later corrected ) in his 1905 electrodynamics paper and in another paper in 1906. ^[10] ^[11] However, in his first paper on $E = m c 2$ (1905) he treated m as what would now be called the rest mass. ^[12] Some claim that (in later years) he did not like the idea of "relativistic mass": ^[13]

It is not good to introduce the concept of the mass $M = m/\sqrt{1 - v^2/c^2}$ of a moving body for which no clear definition can be given. It is better to introduce no other mass concept than the ’rest mass’ m. Instead of introducing M it is better to mention the expression for the momentum and energy of a body in motion.

– Albert Einstein in letter to L Barnett (quote from L. B. Okun, “The Concept of Mass,” Phys. Today 42, 31, June 1989.)

[edit] Modern relativistic concepts

In special relativity, as in Lorentz's ether theory, an object that has a mass cannot travel at the speed of light. As the object approaches the speed of light, the object's energy and momentum increase without bound.

The velocity dependent mass of Lorentz and Abraham were replaced by the concept of relativistic mass, an expression which was first defined 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.”^[14]

In 1934, Tolman also defined relativistic mass as^[15]

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

which holds for all particles, including those moving at the speed of light. Even a photon, a particle which moves at the speed of light, has relativistic mass.

For a slower than light particle, a particle with a nonzero rest mass, the formula becomes

$M = \gamma m \!$

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."^[15]

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.

In the formula for momentum

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

the mass that occurs is the relativistic mass. In other words, the relativistic mass is the proportionality constant between the velocity and the momentum.

Newton's second law remains valid in the form

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

the derived form $\mathbf{f}=M\mathbf{a}$ is not valid because $M\,$ in ${d(M\mathbf{v})}\!$ is generally not a constant [1] (see the section above on transverse and longitudinal mass).

The rest mass is the ratio of four-momentum to four-velocity:

$p^\mu = m v^\mu\,$

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:

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

[edit] The relativistic energy-momentum equation

Dependency between the rest mass and E, given in 4-momentum (p₀,p₁) coordinates;
p₀c = E

The relativistic expressions for E and p obey the relativistic energy-momentum equation:

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

the m is the rest mass.

The equation is also valid for photons, which have m=0:

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

$E = pc \,\!$

a photon's momentum is a function of its energy, but it is not proportional to the velocity, which is always c.

For an object at rest, the momentum p is zero,

$E = mc^2 \,\!$

And the rest mass is only equal to the total energy in the rest frame of the object.

If the object is moving, the total energy is

$E = \sqrt{ (mc^2)^2 + (pc)^2 } \,\!$

Which has both positive and negative solutions. In classical physics, the negative energy solutions are spurious, and as the momentum increases with the increase of the velocity v, so does the total energy.

To find the form of the momentum and energy as a function of velocity, note that the four-velocity, which is propotional to $(c,\vec v)$ , is the only four-dimensional arrow associated to the particle's motion, so that if there is a conserved four-momentum $(E,\vec pc)$ , it must be proportional to this vector. This gives the ratio of energy and momentum:

$pc=E {v \over c}$

Which makes the energy-momentum equation a relation between E and v.

$E^2 = (mc^2)^2 + E^2 {v^2\over c^2}$

Which gives E

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

and P.

$p = {mv\over \sqrt{1-{v^2\over c^2}}}$

The relativistic mass equation is the formula for E divided by c²

$m_{\mathrm{rel}} = { m \over \sqrt{1-{v^2\over c^2}}}$

When working in units where c = 1, known as the natural unit system, all relativistic equations simplify, in particular all three quantities E,p,m have the same dimensions.

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

The equation is often written in this way because the difference $E 2 - p 2$ is the relativistic length of the energy momentum four-vector. In the rest frame where p = 0, the equation above just states that E=m, again revealing that the rest mass is the energy in the rest frame.

[edit] The mass of composite systems

The rest mass of a composite system is not the sum of the rest masses of the parts, unless all the parts are at rest. The total mass of a composite system includes the kinetic energy and field energy in the 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 length (magnitude) p of the total momentum vector $\vec{p}$ , the invariant mass is given by:

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

In a mathematical system where c = 1, for systems of particles (whether bound or unbound) the total system invariant mass is given equivalently by the following:

$m^2 = \left(\sum E\right)^2 - \left\|\sum \vec{p}\right\|^2$

Where, again, the particle momenta $\vec{p}$ are first summed as vectors, and then the square of their resulting total magnitude (Euclidean norm) is used. This results in a scalar number, which is subtracted from the scalar value of the square of the total energy.

For such a system, in the special center of momentum frame where momenta sum to zero, again the system mass (called the invariant mass) is the same as the total system energy. This invariant mass for a system remains the same quantity in any inertial frame, although the system total energy and total momenta are functions of the particular inertial frame which is chosen, and will vary in such a way between inertial frames as to keep the invariant mass the same for all observers. Invariant mass thus functions for systems of particles in the same capacity as "rest mass" does for single particles.

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. It is widely used in particle physics, because the invariant mass of a particle's decay products is equal to its rest mass. This is used to make measurements of the mass of particles like the Z boson or the top quark.

[edit] Conservation versus invariance of mass in special relativity

Total energy is an additive conserved quantity (for single observers) in systems and in reactions between particles, but rest mass (in the sense of being a sum of particle rest masses) is not conserved. One of the reasons for this is that finding the sum of individual particle rest masses would require multiple observers, one for each particle rest inertial frame. But conservation laws require a single observer and a single inertial frame.

In general, relativistic mass is conserved, but is not invariant. Invariant mass, however, is both conserved and invariant.

The relativistic mass is synonymous with the energy, so conservation of energy automatically means that relativistic mass is conserved for any given observer and inertial frame. However, this quantity, like the total energy of a particle, is not invariant. This means that, even though it is conserved for any observer during a reaction, its absolute value will change with the frame of the observer, and for different observers in different frames.

By contrast, the rest mass and invariant masses of systems and particles are both conserved and also invariant ("invariant" meaning that all single observers always measure or calculate the same quantity for the particle or system). For example: A closed container of gas has a system "rest mass" in the sense that it can be weighed on a resting scale, even while it contains moving components. This mass is the invariant mass, which is equal to the total relativistic energy of the container (including the kinetic energy of the gas) only because it is being measured in the center of momentum frame. The "rest mass" of such a container of gas does not change when it is in motion.

The container may even be subjected to a force which gives it an over-all velocity, or else (equivalently) it may be viewed from an inertial frame in which it has an over-all velocity (that is, technically, a frame in which its center of mass has a velocity). However, in such a situation, although the container's total relativistic energy and total momenta increase, these energy and momentum increases subtract out in the invariant mass definition, so that the moving container's invariant mass will be calculated as the same value as if it were measured at rest, on a scale.

All conservation laws in special relativity (for energy, mass, and momentum) require closed systems. If a system is closed, then both total energy and total momentum in the system are conserved for any observer in any single inertial frame, though their absolute values will vary, according to different observers in different inertial frames. The invariant mass of the system is also conserved, but does not change with different observers. This is also the familiar situation with single particles: all observers calculate the same particle rest mass (a special case of the invariant mass) no matter how they move (what inertial frame they choose), but different observers see different total energies and momenta for the same particle.

Conservation of invariant mass requires the system to be enclosed so that no heat and radiation (and thus invariant mass) can escape. As in the example above, a physically enclosed or bound system does not need to be completely isolated from external forces for its mass to remain constant, because for bound systems these merely act to change the inertial frame of the system or the observer. Though such actions may change the total energy or momentum of the system, they do not change its invariant mass.

On the other hand, for systems which are unbound, the "closure" of the system need only be by an idealized surface, inasmuch as no mass-energy can be allowed into or out of the test-volume, if conservation of invariant mass is to hold. If a force is allowed to act on only one part of an unbound system, this is equivalent to allowing energy into or out of the system, and closure is no longer the case. In this case, conservation of invariant mass of the system also will no longer hold.

Again, in special relativity, the rest mass of a system is not required to be equal to the sum of the rest masses of the parts (a situation which analogous to gross mass-conservation in chemistry). For example, a massive particle can decay into photons which individually have no mass, but which (as a system) preserve the invariant mass of the particle which produced them. Also a box of moving particles (or even photons) will have a larger invariant mass than the sum of the rest masses of the particles which compose it, because the total energy of all particles as seen in the center of momentum frame will also contribute to the system's invariant mass.

In special relativity, mass is not "converted" to energy, for all types of energy still retain their associated mass after a transformation in which mass is present at any time. Invariant mass cannot be destroyed in special relativity, and is thus conserved. Thus, a system invariant mass changes only because invariant mass is allowed to escape, perhaps as light or heat. Thus, when reactions (whether chemical or nuclear) release energy in the form of heat and light, if the heat and light is not allowed to escape (the system is closed), the energy will continue to contribute to the system rest mass, and the system mass will not change. Only if the energy is released to the environment will the mass be lost; this is because the associated mass has been allowed out of the system, where it contributes to the mass of the surroundings. ^[16].

[edit] Controversy

According to Lev Okun,^[2] Einstein himself always meant the invariant mass when he wrote "m" in his equations, and never used an unqualified "m" symbol for any other kind of mass. Okun and followers reject the concept of relativistic mass. Arnold B. Arons has argued against teaching the concept of relativistic mass:^[17]

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).]... 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:

$p = {m_0 v \over {\sqrt{1 - \frac{v^2}{c^2}}}} \!$

rather than through relativistic mass....

On the other hand, T. R. Sandin has written:^[18]

The concept of relativistic mass brings a consistency and simplicity to the teaching of special relativity to introductory students. For example, $E = m c 2$ 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's important to notice that a relationship between speed and mass such as

$m = {m_0 \over {\sqrt{1 - \frac{v^2}{c^2}}}} \!$

implies that the velocity is measured relative to a frame of reference.

[edit] See also

Mass-energy equivalence

[edit] References

^ url=http://arxiv.org/abs/physics/0504110>
^ ^a ^b Lev B. Okun (July 1989), "The Concept of Mass", Physics Today 42 (6): 31–36, doi:10.1063/1.881171, http://www.physicstoday.org/vol-42/iss-6/vol42no6p31_36.pdf
^ T. R. Sandin (Nov. 1991), "In defense of relativistic mass", American Journal of Physics 59 (11): 1032, doi:10.1119/1.16642, http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000059000011001032000001&idtype=cvips&gifs=yes
^ Thomson, J.J. (1881), "On the Effects produced by the Motion of Electrified Bodies", Phil. Mag. 11: 229
^ Searle, G.F.C. (1896), "Problems in electric convection", Phil. Trans. Roy. Soc. 187: 675–718, doi:10.1098/rsta.1896.0017, http://gallica.bnf.fr/ark:/12148/bpt6k559925/f724.table
^ Wien, W. (1900/1901), "Über die Möglichkeit einer elektromagnetischen Begründung der Mechanik", Annalen der Physik 5: 501–513, http://gallica.bnf.fr/ark:/12148/bpt6k153157/f560.chemindefer
^ Abraham, M. (1903), "Prinzipien der Dynamik des Elektrons", Annalen der Physik 10: 105–179, http://www.weltderphysik.de/intern/upload/annalen_der_physik/1903/Band_315_105.pdf
^ Lorentz, H.A. (1899), "Simplified Theory of Electrical and Optical Phenomena in Moving Systems", Proc. Roy. Soc. Amst.: 427–442, http://www.historyofscience.nl/search/detail.cfm?pubid=209&view=image&startrow=1
^ Abraham, M. (1902), "Prinzipien der Dynamik des Elektrons]", Physikalische Zeitschrift 4 (1b): 57–62, http://www.soso.ch/wissen/hist/SRT/A-1902.pdf
^ Einstein, A. (1905), "Zur Elektrodynamik bewegter Körper", Annalen der Physik 17: 891–921, doi:10.1002/andp.19053221004, http://www.physik.uni-augsburg.de/annalen/history/papers/1905_17_891-921.pdf English translation
^ Einstein, A. (1906), "Über eine Methode zur Bestimmung des Verhältnisses der transversalen und longitudinalen Masse des Elektrons", Annalen der Physik 21: 583–586, doi:10.1002/andp.19063261310, http://www.physik.uni-augsburg.de/annalen/history/papers/1906_21_583-586.pdf
^ Einstein, A. (1905), "Ist die Trägheit eines Körpers von dessen Energieinhalt abhängig?", Annalen der Physik 18: 639–643, doi:10.1002/andp.19053231314 url=http://www.physik.uni-augsburg.de/annalen/history/papers/1905_18_639-641.pdf See also the English translation
^ usenet physics FAQ
^ R. Tolman, Philosophical Magazine 23, 375 (1912).
^ ^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.
^ 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.
^ Arnold B. Arons, A Guide to Introductory Physics Teaching (1990, page 263); also in Teaching Introductory Physics (2001, page 308)
^ T. R. Sandin (Nov. 1991), "In defense of relativistic mass", American Journal of Physics 59 (11): 1032, doi:10.1119/1.16642, http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&id=AJPIAS000059000011001032000001&idtype=cvips&gifs=yes

[edit] External links

Gary Oas On the Abuse and Use of the Relativistic Mass, 2005, arXiv.org:physics/0504110.
Usenet Physics FAQ
- "Does light have mass?" by Philip Gibbs, 1997, retrieved Aug 10, 2006
- "Does mass change with velocity?" by Philip Gibbs et al., 2002, retrieved Aug 10 2006
- "What is the mass of a photon?" by Matt Austern et al., 1998, retrieved Jun 27, 2007
- Max Jammer (1997). Concepts of Mass in Classical and Modern Physics. Courier Dover Publications, pp. 177-178. ISBN 0486299988.

Retrieved from "http://en.wikipedia.org/wiki/Mass_in_special_relativity"

Categories: Special relativity | Mass