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Relativistic Mechanics Relativistic mechanics 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- p 1 p 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.[1] 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.[2] 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.[3] 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,[4] 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 p 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.
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