Further information: Mass in special relativity and relativistic center of mass for details. Conservation of energy The equations become more complicated in the more fa- miliar three-dimensional vector calculus formalism, due In physics, relativistic mechanics refers to mechanics to the nonlinearity in the Lorentz factor, which accu- compatible with special relativity (SR) and general rel- rately accounts for relativistic velocity dependence and ativity (GR). It provides a non-quantum mechanical de- the speed limit of all particles and ﬁelds. However, scription of a system of particles, or of a ﬂuid, in cases they have a simpler and elegant form in four-dimensional where the velocities of moving objects are comparable spacetime, which includes ﬂat Minkowski space (SR) and to the speed of light c. As a result, classical mechanics curved spacetime (GR), because three-dimensional vec- is extended correctly to particles traveling at high veloc- tors derived from space and scalars derived from time ities and energies, and provides a consistent inclusion of can be collected into four vectors, or four-dimensional electromagnetism with the mechanics of particles. This tensors. However, the six component angular momentum was not possible in Galilean relativity, where it would be tensor is sometimes called a bivector because in the 3D permitted for particles and light to travel at any speed, in- viewpoint it is two vectors (one of these, the conventional cluding faster than light. The foundations of relativistic angular momentum, being an axial vector). mechanics are the postulates of special relativity and gen- eral relativity. The uniﬁcation of SR with quantum me- chanics is relativistic quantum mechanics, while attempts 1 Relativistic kinematics for that of GR is quantum gravity, an unsolved problem in physics. Main article: Four-velocity As with classical mechanics, the subject can be di- vided into "kinematics"; the description of motion by The relativistic four-velocity, that is the four-vector rep- specifying positions, velocities and accelerations, and resenting velocity in relativity, is deﬁned as follows: "dynamics"; a full description by considering energies, momenta, and angular momenta and their conservation laws, and forces acting on particles or exerted by particles. ( ) dX cdt dx There is however a subtlety; what appears to be “mov- U = = , dτ dτ dτ ing” and what is “at rest”—which is termed by "statics" in classical mechanics—depends on the relative motion In the above, τ is the proper time of the path through of observers who measure in frames of reference. spacetime, called the world-line, followed by the object Although some deﬁnitions and concepts from classical velocity the above represents, and mechanics do carry over to SR, such as force as the time derivative of momentum (Newton’s second law), the work done by a particle as the line integral of force ex- X = (ct, x) erted on the particle along a path, and power as the time derivative of work done, there are a number of signiﬁ- is the four-position; the coordinates of an event. Due to cant modiﬁcations to the remaining deﬁnitions and for- time dilation, the proper time is the time between two mulae. SR states that motion is relative and the laws of events in a frame of reference where they take place at the physics are the same for all experimenters irrespective same location. The proper time is related to coordinate of their inertial reference frames. In addition to modi- time t by: fying notions of space and time, SR forces one to recon- sider the concepts of mass, momentum, and energy all of dτ 1 which are important constructs in Newtonian mechanics. = SR shows that these concepts are all diﬀerent aspects of dt γ(v) the same physical quantity in much the same way that it where γ(v) is the Lorentz factor: shows space and time to be interrelated. Consequently, another modiﬁcation is the concept of the center of mass of a system, which is straightforward to deﬁne in classi- √ 1 √ 1 cal mechanics but much less obvious in relativity - see γ(v) = ⇌ γ(v) = . 1 − v · v/c2 1 − (v/c)2
1 2 2 RELATIVISTIC DYNAMICS
(either version may be quoted) so it follows: is often called the relativistic mass of the object in the given frame of reference. This makes the relativistic relation between the spatial ve- U = γ( )(c, ) v v locity and the spatial momentum look identical. How- The ﬁrst three terms, excepting the factor of γ(v), is the ever, this can be misleading, as it is not appropriate in velocity as seen by the observer in their own reference special relativity in all circumstances. For instance, ki- frame. The γ(v) is determined by the velocity v between netic energy and force in special relativity can not be writ- the observer’s reference frame and the object’s frame, ten exactly like their classical analogues by only replac- which is the frame in which its proper time is measured. ing the mass with the relativistic mass. Moreover, un- This quantity is invariant under Lorentz transformation, der Lorentz transformations, this relativistic mass is not so to check to see what an observer in a diﬀerent refer- invariant, while the rest mass is. For this reason many ence frame sees, one simply multiplies the velocity four- people ﬁnd it easier use the rest mass (thereby introduce vector by the Lorentz transformation matrix between the γ through the 4-velocity or coordinate time), and discard two reference frames. the concept of relativistic mass. Lev B. Okun suggested that “this terminology [...] has no rational justiﬁcation today”, and should no longer be 2 Relativistic dynamics taught. Other physicists, including Wolfgang Rindler and T. R. 2.1 Relativistic energy and momentum Sandin, have argued that relativistic mass is a useful con- cept and there is little reason to stop using it. See mass There are a couple of (equivalent) ways to deﬁne mo- in special relativity for more information on this debate. mentum and energy in SR. One method uses conservation Some authors use m for relativistic mass and m for rest laws. If these laws are to remain valid in SR they must be 0 mass, others simply use m for rest mass. This article true in every possible reference frame. However, if one uses the former convention for clarity. does some simple thought experiments using the Newto- nian deﬁnitions of momentum and energy, one sees that The energy and momentum of an object with invariant these quantities are not conserved in SR. One can rescue mass m0 are related by the formulas the idea of conservation by making some small modiﬁ- cations to the deﬁnitions to account for relativistic veloc- ities. It is these new deﬁnitions which are taken as the E2 − (pc)2 = (m c2)2 correct ones for momentum and energy in SR. 0 The four-momentum of an object is straightforward, 2 identical in form to the classical momentum, but replac- pc = Ev . ing 3-vectors with 4-vectors: The ﬁrst is referred to as the relativistic energy–momentum relation. While the energy E and the momentum p de- pend on the frame of reference in which they are mea- P = m0U = (E/c, p) sured, the quantity E2 − (pc)2 is invariant, and arises as The energy and momentum of an object with invariant −c2 times the squared magnitude of the 4-momentum 2 mass m0 (also called rest mass), moving with velocity v vector which is −(m0c) . with respect to a given frame of reference, are respectively It should be noted that the invariant mass of a system given by
√ E = γ(v)m c2 2 − 2 0 Etot (ptotc) m0tot = 2 p = γ(v)m0v c
The factor of γ(v) comes from the deﬁnition of the four- is diﬀerent from the sum of the rest masses of the parti- velocity described above. The appearance of the γ factor cles of which it is composed due to kinetic energy and has an alternative way of being stated, explained next. binding energy. Rest mass is not a conserved quan- tity in special relativity unlike the situation in Newtonian 2.2 Rest mass and relativistic mass physics. However, if an object is not changing internally, then its rest mass will not change and can be calculated The quantity with the same result in any frame of reference. A particle whose rest mass is zero is called massless. Photons and gravitons are thought to be massless; and m = γ(v)m0 neutrinos are nearly so. 2.5 Closed (isolated) systems 3
2.3 Mass–energy equivalence This is the invariant mass of any system which is mea- sured in a frame where it has zero total momentum, such Main article: Mass–energy equivalence as a bottle of hot gas on a scale. In such a system, the mass which the scale weighs is the invariant mass, and The relativistic energy–momentum equation holds for all it depends on the total energy of the system. It is thus more than the sum of the rest masses of the molecules, particles, even for massless particles for which m0 = 0. In this case: but also includes all the totaled energies in the system as well. Like energy and momentum, the invariant mass of isolated systems cannot be changed so long as the system remains totally closed (no mass or energy allowed in or E = pc out), because the total relativistic energy of the system When substituted into Ev = c2p, this gives v = c: massless remains constant so long as nothing can enter or leave it. particles (such as photons) always travel at the speed of An increase in the energy of such a system which is light. caused by translating the system to an inertial frame Notice that the rest mass of a composite system will gen- which is not the center of momentum frame, causes an erally be slightly diﬀerent from the sum of the rest masses increase in energy and momentum without an increase in 2 of its parts since, in its rest frame, their kinetic energy will invariant mass. E = m0c , however, applies only to iso- increase its mass and their (negative) binding energy will lated systems in their center-of-momentum frame where decrease its mass. In particular, a hypothetical “box of momentum sums to zero. light” would have rest mass even though made of parti- Taking this formula at face value, we see that in relativity, cles which do not since their momenta would cancel. mass is simply energy by another name (and measured in Looking at the above formula for invariant mass of a sys- diﬀerent units). In 1927 Einstein remarked about special tem, one sees that, when a single massive object is at rest relativity, “Under this theory mass is not an unalterable (v = 0, p = 0), there is a non-zero mass remaining: m0 magnitude, but a magnitude dependent on (and, indeed, = E/c2. The corresponding energy, which is also the to- identical with) the amount of energy.” tal energy when a single particle is at rest, is referred to as “rest energy”. In systems of particles which are seen from a moving inertial frame, total energy increases and 2.5 Closed (isolated) systems so does momentum. However, for single particles the rest mass remains constant, and for systems of particles the In a “totally-closed” system (i.e., isolated system) the total invariant mass remain constant, because in both cases, energy, the total momentum, and hence the total invari- the energy and momentum increases subtract from each ant mass are conserved. Einstein’s formula for change in 2 other, and cancel. Thus, the invariant mass of systems mass translates to its simplest ΔE = Δmc form, however, of particles is a calculated constant for all observers, as is only in non-closed systems in which energy is allowed to the rest mass of single particles. escape (for example, as heat and light), and thus invariant mass is reduced. Einstein’s equation shows that such sys- tems must lose mass, in accordance with the above for- 2.4 The mass of systems and conservation mula, in proportion to the energy they lose to the sur- of invariant mass roundings. Conversely, if one can measure the diﬀer- ences in mass between a system before it undergoes a re- See also: Center of mass (relativistic) action which releases heat and light, and the system after the reaction when heat and light have escaped, one can estimate the amount of energy which escapes the system. For systems of particles, the energy–momentum equation requires summing the momentum vectors of the particles: 2.5.1 Chemical and nuclear reactions
2 − · 2 2 4 E p pc = m0c In both nuclear and chemical reactions, such energy rep- resents the diﬀerence in binding energies of electrons in The inertial frame in which the momenta of all particles atoms (for chemistry) or between nucleons in nuclei (in sums to zero is called the center of momentum frame. atomic reactions). In both cases, the mass diﬀerence be- In this special frame, the relativistic energy–momentum tween reactants and (cooled) products measures the mass equation has p = 0, and thus gives the invariant mass of of heat and light which will escape the reaction, and thus the system as merely the total energy of all parts of the (using the equation) give the equivalent energy of heat 2 system, divided by c and light which may be emitted if the reaction proceeds. In chemistry, the mass diﬀerences associated with the ∑ −9  2 emitted energy are around 10 of the molecular mass. m0, system = En/c n However, in nuclear reactions the energies are so large 4 2 RELATIVISTIC DYNAMICS
that they are associated with mass diﬀerences, which this gram of mass in the objects that absorb them. can be estimated in advance, if the products and reac- tants have been weighed (atoms can be weighed indi- rectly by using atomic masses, which are always the same 2.6 Angular momentum for each nuclide). Thus, Einstein’s formula becomes im- portant when one has measured the masses of diﬀerent Main article: Relativistic angular momentum atomic nuclei. By looking at the diﬀerence in masses, one can predict which nuclei have stored energy that can In relativistic mechanics, the time-varying mass moment be released by certain nuclear reactions, providing im- portant information which was useful in the development of nuclear energy and, consequently, the nuclear bomb. N = m (x − tv) Historically, for example, Lise Meitner was able to use the mass diﬀerences in nuclei to estimate that there was and orbital 3-angular momentum enough energy available to make nuclear ﬁssion a favor- able process. The implications of this special form of Einstein’s formula have thus made it one of the most fa- L = x × p mous equations in all of science. of a point-like particle are combined into a four- dimensional bivector in terms of the 4-position X and the 2.5.2 Center of momentum frame 4-momentum P of the particle:
2 The equation E = m0c applies only to isolated systems in their center of momentum frame. It has been popularly M = X ∧ P misunderstood to mean that mass may be converted to en- ergy, after which the mass disappears. However, popular where ∧ denotes the exterior product. This tensor is addi- explanations of the equation as applied to systems include tive: the total angular momentum of a system is the sum open (non-isolated) systems for which heat and light are of the angular momentum tensors for each constituent of allowed to escape, when they otherwise would have con- the system. So, for an assembly of discrete particles one tributed to the mass (invariant mass) of the system. sums the angular momentum tensors over the particles, or integrates the density of angular momentum over the Historically, confusion about mass being “converted” to extent of a continuous mass distribution. energy has been aided by confusion between mass and "matter", where matter is deﬁned as fermion particles. In Each of the six components forms a conserved quantity such a deﬁnition, electromagnetic radiation and kinetic when aggregated with the corresponding components for energy (or heat) are not considered “matter”. In some other objects and ﬁelds. situations, matter may indeed be converted to non-matter forms of energy (see above), but in all these situations, the matter and non-matter forms of energy still retain their 2.7 Force original mass. In special relativity, Newton’s second law does not hold For isolated systems (closed to all mass and energy ex- in the form F = ma, but it does if it is expressed as change), mass never disappears in the center of momen- tum frame, because energy cannot disappear. Instead, this equation, in context, means only that when any energy dp F = is added to, or escapes from, a system in the center-of- dt momentum frame, the system will be measured as having where p = γ(v)m v is the momentum as deﬁned above gained or lost mass, in proportion to energy added or re- 0 and m is the invariant mass. Thus, the force is given by moved. Thus, in theory, if an atomic bomb were placed in 0 a box strong enough to hold its blast, and detonated upon
a scale, the mass of this closed system would not change, 3 and the scale would not move. Only when a transparent F = γ(v) m0 a∥ + γ(v)m0 a⊥ “window” was opened in the super-strong plasma-ﬁlled box, and light and heat were allowed to escape in a beam,
and the bomb components to cool, would the system lose 3 the mass associated with the energy of the blast. In a 21 Consequently in some old texts, γ(v) m0 is referred to as kiloton bomb, for example, about a gram of light and heat the longitudinal mass, and γ(v)m0 is referred to as the is created. If this heat and light were allowed to escape, transverse mass, which is numerically the same as the the remains of the bomb would lose a gram of mass, as relativistic mass. See mass in special relativity. it cooled. In this thought-experiment, the light and heat If one inverts this to calculate acceleration from force, one carry away the gram of mass, and would therefore deposit gets 2.10 Classical limit 5
2.10 Classical limit ( ) 1 (v · F)v − The Lorentz factor γ(v) can be expanded into a Taylor a = F 2 . m0γ(v) c series or binomial series for (v/c)2 < 1, obtaining: The force described in this section is the classical 3-D force which is not a four-vector. This 3-D force is the ∞ ( ) n ( ) ( ) ( ) ( ) 1 ∑ v 2n ∏ 2k − 1 1 v 2 3 v 4 5 v 6 appropriate concept of force since it is the force which γ = √ = = 1+ + + +··· − 2 c 2k 2 c 8 c 16 c obeys Newton’s third law of motion. It should not be 1 (v/c) n=0 k=1 confused with the so-called four-force which is merely the 3-D force in the comoving frame of the object trans- and consequently formed as if it were a four-vector. However, the den- sity of 3-D force (linear momentum transferred per unit 1 3 m v4 5 m v6 four-volume) is a four-vector (density of weight +1) when − 2 2 0 0 ··· E m0c = m0v + 2 + 4 + ; combined with the negative of the density of power trans- 2 8 c 16 c ferred. 1 m v2v 3 m v4v 5 m v6v p = m v + 0 + 0 + 0 + ··· . 0 2 c2 8 c4 16 c6 2.8 Torque For velocities much smaller than that of light, one can neglect the terms with c2 and higher in the denominator. The torque acting on a point-like particle is deﬁned as the These formulas then reduce to the standard deﬁnitions of derivative of the angular momentum tensor given above Newtonian kinetic energy and momentum. This is as it with respect to proper time: should be, for special relativity must agree with Newto- nian mechanics at low velocities.
dM Γ = = X ∧ F dτ 3 See also or in tensor components: • Introduction to special relativity • Twin paradox Γ = X F − X F αβ α β β α • Relativistic equations where F is the 4d force acting on the particle at the event • Relativistic heat conduction X. As with angular momentum, torque is additive, so for an extended object one sums or integrates over the distri- • Classical electromagnetism and special relativity bution of mass. • Relativistic system (mathematics)
2.9 Kinetic energy 4 References The work-energy theorem says the change in kinetic energy is equal to the work done on the body. In special 4.1 Notes relativity:  Philip Gibbs, Jim Carr and Don Koks (2008). “What is relativistic mass?". Usenet Physics FAQ. Retrieved 2008- 2 ∆K = W = [γ1(v1) − γ0(v0)]m0c . 09-19. Note that in 2008 the last editor, Don Koks, rewrote a signiﬁcant portion of the page, changing it from a view extremely dismissive of the usefulness of relativis- tic mass to one which hardly questions it. The previous version was: Philip Gibbs and Jim Carr (1998). “Does If in the initial state the body was at rest, so v = 0 and 0 mass change with speed?". Usenet Physics FAQ. Archived γ0(v0) = 1, and in the ﬁnal state it has speed v1 = v, setting from the original on 2007-06-30. γ1(v1) = γ(v), the kinetic energy is then;  Lev B. Okun (July 1989). “The Concept of Mass” (subscription required). Physics Today 42 (6): 31–36. 2 Bibcode:1989PhT....42f..31O. doi:10.1063/1.881171. K = [γ(v) − 1]m0c ,  T. R. Sandin (November 1991). “In defense of relativis- a result that can be directly obtained by subtracting tic mass” (subscription required). American Journal of 2 the rest energy m0c from the total relativistic energy Physics 59 (11): 1032. Bibcode:1991AmJPh..59.1032S. 2 γ(v)m0c . doi:10.1119/1.16642. 6 4 REFERENCES
 See, for example: Feynman, Richard (1998). “The special • Concepts of Modern Physics (4th Edition), A. theory of relativity”. Six Not-So-Easy Pieces. Cambridge, Beiser, Physics, McGraw-Hill (International), 1987, Massachusetts: Perseus Books. ISBN 0-201-32842-9. ISBN 0-07-100144-1  Einstein on Newton • C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. ISBN 0-07-  Randy Harris (2008). Modern Physics: Second Edition. 051400-3. Pearson Addison-Welsey. p. 38. ISBN 0-8053-0308-1. • T. Frankel (2012). The Geometry of Physics (3rd  E. F. Taylor and J. A. Wheeler, Spacetime Physics, W.H. ed.). Cambridge University Press. ISBN 978-1107- Freeman and Co., New York. 1992. ISBN 0-7167-2327- 602601. 1, see pp. 248-9 for discussion of mass remaining constant after detonation of nuclear bombs, until heat is allowed to • L.H. Greenberg (1978). Physics with Modern Appli- escape. cations. Holt-Saunders International W.B. Saunders and Co. ISBN 0-7216-4247-0.  R. Penrose (2005). The Road to Reality. Vintage books. pp. 437–438, 566–569. ISBN 978-00994-40680. Note: • A. Halpern (1988). 3000 Solved Problems in Some authors, including Penrose, use Latin letters in this Physics, Schaum Series. Mc Graw Hill. ISBN 978- deﬁnition, even though it is conventional to use Greek in- 0-07-025734-4. dices for vectors and tensors in spacetime.
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• G. Woan (2010). The Cambridge Handbook of • J.A. Wheeler, C. Misner, K.S. Thorne (1973). Physics Formulas. Cambridge University Press. Gravitation. W.H. Freeman & Co. ISBN 0-7167- ISBN 978-0-521-57507-2. 0344-0. • • P.A. Tipler, G. Mosca (2008). Physics for Scien- J.A. Wheeler, I. Ciufolini (1995). Gravitation and tists and Engineers: With Modern Physics (6th ed.). Inertia. Princeton University Press. ISBN 978-0- W.H. Freeman and Co. ISBN 9-781429-202657. 691-03323-5. • R.J.A. Lambourne (2010). Relativity, Gravitation, • R.G. Lerner, G.L. Trigg (2005). Encyclopaedia and Cosmology. Cambridge University Press. ISBN of Physics (2nd ed.). VHC Publishers, Hans War- 9-780521-131384. limont, Springer. ISBN 978-0-07-025734-4. 7
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