Contents 1 Terminology 2 Invariant mass 3 The relativistic energy-momentum equation 4 The mass of composite systems 5 Conservation versus invariance of mass in special relativity 5.1 Closed (meaning totally isolated) systems 5.2 The system invariant mass vs. the individual rest masses of parts of the system 6 The relativistic mass concept 6.1 Transverse and longitudinal mass 6.2 Relativistic mass 6.3 Controversy 7 See also 8 References 9 External links Terminology[edit] 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. The more general invariant mass (calculated with a more complicated formula) loosely corresponds to the "rest mass" of a "system". Thus, invariant mass is a natural unit of mass used for systems which are being viewed from their center of momentum frame (COM frame), as when any closed system (for example a bottle of hot gas) is weighed, which requires that the measurement be taken in the center of momentum frame where the system has no net momentum. Under such circumstances the invariant mass is equal to the relativistic mass (discussed below), which is the total energy of the system divided by c2 (the speed of light squared). The concept of invariant mass does not require bound systems of particles, however. As such, it may also be applied to systems of unbound particles in high-speed relative motion. Because of this, it is often employed in particle physics for systems which consist of widely separated high-energy particles. If such systems were derived from a single particle, then the calculation of the invariant mass of such systems, which is a never-changing quantity, will provide the rest mass of the parent particle (because it is conserved over time). It is often convenient in calculation that the invariant mass of a system is the total energy of the system (divided by c2) in the COM frame (where, by definition, the momentum of the system is zero). However, since the invariant mass of any system is also the same quantity in all inertial frames, it is a quantity often calculated from the total energy in the COM frame, then used to calculate system energies and momenta in other frames where the momenta are not zero, and the system total energy will necessarily be a different quantity than in the COM frame. As with energy and momentum, the invariant mass of a system cannot be destroyed or changed, and it is thus conserved, so long as the system is closed to all influences. (The technical term is isolated system meaning that an idealized boundary is drawn around the system, and no mass/energy is allowed across it.) The term relativistic mass is also sometimes used. This is the sum total quantity of energy in a body or system (divided by c2). As seen from the center of momentum frame, the relativistic mass is also the invariant mass, as discussed above (just as the relativistic energy of a single particle is the same as its rest energy, when seen from its rest frame). For other frames, the relativistic mass (of a body or system of bodies) includes a contribution from the "net" kinetic energy of the body (the kinetic energy of the center of mass of the body), and is larger the faster the body moves. Thus, unlike the invariant mass, the relativistic mass depends on the observer's frame of reference. However, for given single frames of reference and for isolated systems, the relativistic mass is also a conserved quantity. Although some authors present relativistic mass as a fundamental concept of the theory, it has been argued that this is wrong as the fundamentals of the theory relate to space–time. There is disagreement over whether the concept is pedagogically useful.[2][3][4] The notion of mass as a property of an object from Newtonian mechanics does not bear a precise relationship to the concept in relativity.[5] Oxford lecturer John Roche states that relativistic mass is not referenced in nuclear and particle physics, and that about 60% of authors writing about special relativity do not introduce it.[1] If a stationary box contains many particles, it weighs more in its rest frame, the faster the particles are moving. Any energy in the box (including the kinetic energy of the particles) adds to the mass, so that the relative motion of the particles contributes to the mass of the box. But if the box itself is moving (its center of mass 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 (calculated using the single velocity of the box, which is to say the velocity of the box's center of mass), while the relativistic mass is calculated including invariant mass plus the kinetic energy of the system which is calculated from the velocity of the center of mass. Relativistic mass and rest mass are both traditional concepts in physics, but the relativistic mass corresponds to the total energy. The relativistic mass is the mass of the system as it would be measured on a scale, but in some cases (such as the box above) this fact remains true only because the system on average must be at rest to be weighed (it must have zero net momentum, which is to say, the measurement is in its center of momentum frame). For example, if an electron in a cyclotron is moving in circles with a relativistic velocity, the mass of the cyclotron+electron system is increased by the relativistic mass of the electron, not by the electron's rest mass. But the same is also true of any closed system, such as an electron-and-box, if the electron bounces at high speed inside the box. It is only the lack of total momentum in the system (the system momenta sum to zero) which allows the kinetic energy of the electron to be "weighed." If the electron is stopped and weighed, or the scale were somehow sent after it, it would not be moving with respect to the scale, and again the relativistic and rest masses would be the same for the single electron (and would be smaller). In general, relativistic and rest masses are equal only in systems which have no net momentum and the system center of mass is at rest; otherwise they may be different. 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 (as defined below in terms of center of mass). This is why the invariant mass is the same as the rest mass for single particles. However, the invariant mass also represents the measured mass when the center of mass is at rest for systems of many particles. This special frame where this occurs 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 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. A so-called massless particle (such as a photon, or a theoretical graviton) moves at the speed of light in every frame of reference. In this case there is no transformation that will bring the particle to rest. The total energy of such particles becomes smaller and smaller in frames which move faster and faster in the same direction. As such, they have no rest mass, because they can never be measured in a frame where they are at rest. This property of having no rest mass is what causes these particles to be termed "massless." However, even massless particles have a relativistic mass, which varies with their observed energy in various frames of reference, Invariant mass[edit] The invariant mass is the ratio of four-momentum (the four-dimensional generalization of classical momentum) to four-velocity:[6] p μ = m v μ displaystyle p^ mu =mv^ mu , and is also the ratio of four-acceleration to four-force when the rest
mass is constant. The four-dimensional form of
F μ = m A μ . displaystyle F^ mu =mA^ mu . The relativistic energy-momentum equation[edit] This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2016) (Learn how and when to remove this template message) Dependency between the rest mass and E, given in
The relativistic expressions for E and p obey the relativistic energy–momentum relation:[7] E 2 − ( p c ) 2 = ( m c 2 ) 2 displaystyle E^ 2 -(pc)^ 2 =left(mc^ 2 right)^ 2 where the m is the rest mass, or the invariant mass for systems, and E is the total energy. The equation is also valid for photons, which have m = 0: E 2 − ( p c ) 2 = 0 displaystyle E^ 2 -(pc)^ 2 =0 and therefore E = p c displaystyle 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, therefore E 0 = m c 2 displaystyle E_ 0 =mc^ 2 ,! [true only for particles or systems with momentum = 0] The rest mass is only proportional to the total energy in the rest frame of the object. When the object is moving, the total energy is given by E = ( m c 2 ) 2 + ( p c ) 2 displaystyle E= sqrt left(mc^ 2 right)^ 2 +(pc)^ 2 To find the form of the momentum and energy as a function of velocity, it can be noted that the four-velocity, which is proportional to ( c , v → ) displaystyle left(c, vec v right) , is the only four-vector associated with the particle's motion, so that if there is a conserved four-momentum ( E , p → c ) displaystyle left(E, vec p cright) , it must be proportional to this vector. This allows expressing the ratio of energy to momentum as p c = E v c displaystyle pc=E frac v c , resulting in a relation between E and v: E 2 = ( m c 2 ) 2 + E 2 v 2 c 2 , displaystyle E^ 2 =left(mc^ 2 right)^ 2 +E^ 2 frac v^ 2 c^ 2 , This results in E = m c 2 1 − v 2 c 2 displaystyle E= mc^ 2 over sqrt 1-displaystyle v^ 2 over c^ 2 and p = m v 1 − v 2 c 2 . displaystyle p= mv over sqrt 1-displaystyle v^ 2 over c^ 2 . these expressions can be written as E 0 = m c 2 displaystyle E_ 0 =mc^ 2 , , E = γ m c 2 displaystyle E=gamma mc^ 2 , , and p = m v γ . displaystyle p=mvgamma ,. When working in units where c = 1, known as the natural unit system, all the relativistic equations are simplified and the quantities energy, momentum, and mass have the same natural dimension:[8] m 2 = E 2 − p 2 displaystyle m^ 2 =E^ 2 -p^ 2 . The equation is often written this way because the difference E 2 − p 2 displaystyle E^ 2 -p^ 2 is the relativistic length of the energy momentum four-vector, a length which is associated with rest mass or invariant mass in systems. Where m > 0 and p = 0, this equation again expresses the mass-energy equivalence E = m. The mass of composite systems[edit] This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2016) (Learn how and when to remove this template message) 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 p → displaystyle 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 p → displaystyle vec p , the invariant mass is given by: m = E 2 − ( p c ) 2 c 2 displaystyle 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 = ( ∑ E ) 2 − ‖ ∑ p →
‖ 2 displaystyle m^ 2 =left(sum Eright)^ 2 -leftsum vec p right^ 2 Where, again, the particle momenta p → displaystyle 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)
corresponds to the total system energy or, in units where c=1, is
identical to it. 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.
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2016) (Learn how and when to remove this template message) 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) may not be conserved
through an event in which rest masses of particles are converted to
other types of energy, such as kinetic energy. Finding the sum of
individual particle rest masses would require multiple observers, one
for each particle rest inertial frame, and these observers ignore
individual particle kinetic energy. Conservation laws require a single
observer and a single inertial frame.
In general, for isolated systems and single observers, relativistic
mass is conserved (each observer sees it constant over time), but is
not invariant (that is, different observers see different values).
Invariant mass, however, is both conserved and invariant (all single
observers see the same value, which does not change over time).
The relativistic mass corresponds to 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. For example: A closed
container of gas (closed to energy as well) 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 when it is measured in the center of
momentum frame. Just as is the case for single particles, the
calculated "rest mass" of such a container of gas does not change when
it is in motion, although its "relativistic mass" does change.
The container may even be subjected to a force which gives it an
overall velocity, or else (equivalently) it may be viewed from an
inertial frame in which it has an overall velocity (that is,
technically, a frame in which its center of mass has a velocity). In
this case, its total relativistic mass and energy increase. 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.
Closed (meaning totally isolated) systems[edit]
All conservation laws in special relativity (for energy, mass, and
momentum) require isolated systems, meaning systems that are totally
isolated, with no mass-energy allowed in or out, over time. If a
system is isolated, then both total energy and total momentum in the
system are conserved over time 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 also 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 bound system, these two
changes cancel, so that there is no change in the system's invariant
mass. This is just the same result as with single particles: their
calculated rest mass also remains constant no matter how fast they
move, or how fast an observer sees them move.
On the other hand, for systems which are unbound, the "closure" of the
system may be enforced by an idealized surface, inasmuch as no
mass-energy can be allowed into or out of the test-volume over time,
if conservation of system invariant mass is to hold during that time.
If a force is allowed to act on (do work on) only one part of such an
unbound system, this is equivalent to allowing energy into or out of
the system, and the condition of "closure" to mass-energy (total
isolation) is violated. In this case, conservation of invariant mass
of the system also will no longer hold. Such a loss of rest mass in
systems when energy is removed, according to E=mc2 where E is the
energy removed, and m is the change in rest mass, reflect changes of
mass associated with movement of energy, not "conversion" of mass to
energy.
The system invariant mass vs. the individual rest masses of parts of
the system[edit]
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 would be 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 non-interacting particles (e.g., photons, or an ideal gas) will
have a larger invariant mass than the sum of the rest masses of the
particles which compose it. This is because the total energy of all
particles and fields in a system must be summed, and this quantity, as
seen in the center of momentum frame, and divided by c2, is the
system's invariant mass.
In special relativity, mass is not "converted" to energy, for all
types of energy still retain their associated mass. Neither energy nor
invariant mass can be destroyed in special relativity, and each is
separately conserved over time in closed systems. Thus, a system's
invariant mass may change 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 and
isolated), 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.[7]
The relativistic mass concept[edit]
Transverse and longitudinal mass[edit]
Further information: Electromagnetic mass
Concepts that were similar to what nowadays is called "relativistic
mass", were already developed before the advent of special relativity.
For example, it was recognized by
m e m = ( 4 / 3 ) E e m / c 2 displaystyle m_ em =(4/3)E_ em /c^ 2 , which can increase the normal mechanical mass of the bodies.[9] [10]
Then, it was pointed out by Thomson and Searle that this
electromagnetic mass also increases with velocity. This was further
elaborated by
m L = γ 3 m displaystyle m_ L =gamma ^ 3 m parallel to the direction of motion and the mass m T = γ m displaystyle m_ T =gamma m perpendicular to the direction of motion (where γ = 1 / 1 − v 2 / c 2 displaystyle 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). Only when the force is
perpendicular to the velocity, Lorentz's mass is equal to what is now
called "relativistic mass".
m L displaystyle m_ L longitudinal mass and m T displaystyle m_ T transverse mass (although Abraham used more complicated expressions
than Lorentz's relativistic ones). So, according to Lorentz's theory
no body can reach the speed of light because the mass becomes
infinitely large at this velocity.[11] [12] [13]
Also
m T displaystyle m_ T by an unfortunate force definition, which was later corrected), and in another paper in 1906.[14][15] However, he later abandoned velocity dependent mass concepts (see quote at the end of next section). The precise relativistic expression (which is equivalent to Lorentz's) relating force and acceleration for a particle with non-zero rest mass m displaystyle m moving in the x direction with velocity v and associated Lorentz factor γ displaystyle gamma is f x = m γ 3 a x = m L a x , f y = m γ a y = m T a y , f z = m γ a z = m T a z . displaystyle begin aligned f_ x &=mgamma ^ 3 a_ x &=m_ L a_ x ,\f_ y &=mgamma a_ y &=m_ T a_ y ,\f_ z &=mgamma a_ z &=m_ T a_ z .end aligned Relativistic mass[edit]
In special relativity, an object that has nonzero rest 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.
In the first years after 1905, following Lorentz and Einstein, the
terms longitudinal and transverse mass were still in use. However,
those expressions were replaced by the concept of relativistic mass,
an expression which was first defined by
m rel = E c 2 displaystyle m_ text rel = frac E c^ 2 ! , and of a body at rest m 0 = E 0 c 2 displaystyle m_ 0 = frac E_ 0 c^ 2 ! , with the ratio m rel m 0 = γ displaystyle frac m_ text rel m_ 0 =gamma ! . Tolman in 1912 further elaborated on this concept, and stated: “the expression m0(1 - v2/c2)−1/2 is best suited for THE mass of a moving body.”[17][18][19] In 1934, Tolman argued that the relativistic mass formula m rel = E / c 2 displaystyle m_ text rel =E/c^ 2 ! holds for all particles, including those moving at the speed of light, while the formula m rel = γ m 0 displaystyle m_ text rel =gamma m_ 0 ! only applies to a slower than light particle (a particle with a nonzero rest mass). 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."[20] When the relative velocity is zero, γ displaystyle gamma 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 displaystyle gamma approaches infinity. In the formula for momentum p = m rel v displaystyle mathbf p =m_ text rel 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.
While
f = d ( m rel v ) d t , displaystyle mathbf f = frac d(m_ text rel mathbf v ) dt ,! the derived form f = m rel a displaystyle mathbf f =m_ text rel mathbf a is not valid because m rel displaystyle m_ text rel , in d ( m rel v ) displaystyle d(m_ text rel mathbf v ) ! is generally not a constant[21] (see the section above on transverse and longitudinal mass). Even though Einstein initially used the expressions "longitudinal" and "transverse" mass in two papers (see previous section), in his first paper on E = m c 2 displaystyle E=mc^ 2 (1905) he treated m as what would now be called the rest mass.[22] Einstein never derived an equation for "relativistic mass", and in later years he expressed his dislike of the idea:[23] It is not good to introduce the concept of the mass M = m / 1 − v 2 / c 2 displaystyle 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.
—
Controversy[edit] Okun and followers reject the concept of relativistic mass.[2] Also Arnold B. Arons has argued against teaching the concept of relativistic mass:[24] 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 1 − v 2 c 2 displaystyle p= m_ 0 v over sqrt 1- frac v^ 2 c^ 2 ! rather than through relativistic mass. C. Alder takes a similarly dismissive stance on mass in relativity. Writing on said subject matter, he says that "its introduction into the theory of special relativity was much in the way of a historical accident", noting towards the widespread knowledge of E=mc2 and how the public's interpretation of the equation has largely informed how it is taught in higher education.[25] He instead supposes that the difference between rest and relativistic mass should be explicitly taught, so that students know why mass should be thought of as invariant "in most discussions of inertia." Many contemporary authors such as Taylor and Wheeler avoid using the concept of relativistic mass altogether: "The concept of "relativistic mass" is subject to misunderstanding. That's why we don't use it. First, it applies the name mass - belonging to the magnitude of a 4-vector - to a very different concept, the time component of a 4-vector. Second, it makes increase of energy of an object with velocity or momentum appear to be connected with some change in internal structure of the object. In reality, the increase of energy with velocity originates not in the object but in the geometric properties of spacetime itself."[7] While space-time has the unbounded geometry of Minkowski-space, the velocity-space is bounded by c and has the geometry of hyperbolic geometry where relativistic-mass plays an analogous role to that of Newtonian-mass in the barycentric-coordinates of Euclidean geometry.[26] The connection of velocity to hyperbolic-geometry enables the 3-velocity-dependent relativistic-mass to be related to the 4-velocity Minkowski-formalism.[27] See also[edit] Physics portal Mass
References[edit] ^ a b Roche, J (2005). "What is mass?" (PDF). European Journal of
Physics. 26 (2): 225. Bibcode:2005EJPh...26..225R.
doi:10.1088/0143-0807/26/2/002.
^ a b c L. B. Okun (1989), "The Concept of Mass" (PDF), Physics Today,
42 (6): 31–36, Bibcode:1989PhT....42f..31O,
doi:10.1063/1.881171
^ T. R. Sandin (1991), "In defense of relativistic mass", American
Journal of Physics, 59 (11): 1032, Bibcode:1991AmJPh..59.1032S,
doi:10.1119/1.16642
^ L. B. Okun (2009), "
Also in Teaching Introductory Physics, 2001, p. 308 ^ Adler, Carl (September 30, 1986). "Does mass really depend on
velocity, dad?" (PDF). American Journal of Physics. American
Association of Physics Teachers. 55 (8): 739–743.
Bibcode:1987AmJPh..55..739A. doi:10.1119/1.15314 – via HUIT Sites
Hosting.
^ Hyperbolic Triangle Centers: The
External links[edit] Silagadze, Z. K. (2008), "Relativity without tears", Acta Physica Polonica B, 39: 811–885, arXiv:0708.0929 , Bibcode:2008AcPPB..39..811S Oas, Gary (2005), "On the Abuse and Use of Relativistic Mass", arXiv:physics/0504110 Usenet Physics FAQ "Does mass change with velocity?" by Philip Gibbs et al., 2002, retrieved August 10, 2006 "What is the mass of a photon?" by Matt Austern et al., 1998, retrieved June 27, 2007 Max Jammer (1997), Concepts of
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