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In physics, the energy–momentum relation is the relativistic equation relating any object's rest (intrinsic) mass, total energy, and momentum: holds for a system, such as a particle or macroscopic body, having intrinsic rest mass , total energy , and a momentum of magnitude , where the constant ''c'' is the speed of light, assuming the special relativity case of flat spacetime. The energy-momentum relation () is consistent with the familiar mass-energy relation in both its interpretations: relates total energy to the (total) relativistic mass (alternatively denoted or ), while relates rest energy to rest (invariant) mass which we denote . Unlike either of those equations, the energy-momentum equation () relates the ''total'' energy to the ''rest'' mass . All three equations hold true simultaneously. Special cases of the relation () include: #If the body is a massless particle (), then () reduces to . For photons, this is the relation, discovered in 19th century classical electromagnetism, between radiant momentum (causing radiation pressure) and radiant energy. #If the body's speed is much less than , then () reduces to ; that is, the body's total energy is simply its classical kinetic energy () plus its rest energy. #If the body is at rest (), i.e. in its center-of-momentum frame (), we have and ; thus the energy-momentum relation and both forms of the mass-energy relation (mentioned above) all become the same. A more general form of relation () holds for general relativity. The invariant mass (or rest mass) is an invariant for all frames of reference (hence the name), not just in inertial frames in flat spacetime, but also accelerated frames traveling through curved spacetime (see below). However the total energy of the particle and its relativistic momentum are frame-dependent; relative motion between two frames causes the observers in those frames to measure different values of the particle's energy and momentum; one frame measures and , while the other frame measures and , where and , unless there is no relative motion between observers, in which case each observer measures the same energy and momenta. Although we still have, in flat spacetime; : The quantities , , , are all related by a Lorentz transformation. The relation allows one to sidestep Lorentz transformations when determining only the magnitudes of the energy and momenta by equating the relations in the different frames. Again in flat spacetime, this translates to; : Since does not change from frame to frame, the energy–momentum relation is used in relativistic mechanics and particle physics calculations, as energy and momentum are given in a particle's rest frame (that is, and as an observer moving with the particle would conclude to be) and measured in the lab frame (i.e. and as determined by particle physicists in a lab, and not moving with the particles). In relativistic quantum mechanics, it is the basis for constructing relativistic wave equations, since if the relativistic wave equation describing the particle is consistent with this equation – it is consistent with relativistic mechanics, and is Lorentz invariant. In relativistic quantum field theory, it is applicable to all particles and fields.〔 〕 This article will use the conventional notation for the "square of a vector" as the dot product of a vector with itself: . ==Origins of the equation== The equation can be derived in a number of ways, two of the simplest include: #considering the relativistic dynamics of a massive particle, #evaluating the norm of the four-momentum of the system. This is completely general for all particles, and is easy to extend to multi-particle systems (see below). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Energy–momentum relation」の詳細全文を読む スポンサード リンク
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