Energy–momentum relation

In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating any object's rest (intrinsic) mass, total energy, and momentum: E 2 = ( p c ) 2 + ( m 0 c 2 ) 2 {displaystyle E^{2}=(pc)^{2}+left(m_{0}c^{2} ight)^{2},}     ( 1) ( ∑ n E n ) 2 = ( ∑ n p n c ) 2 + ( M 0 c 2 ) 2 {displaystyle left(sum _{n}E_{n} ight)^{2}=left(sum _{n}mathbf {p} _{n}c ight)^{2}+left(M_{0}c^{2} ight)^{2}}     ( 2) ( ω c ) 2 = k 2 + ( m 0 c ℏ ) 2 . {displaystyle left({frac {omega }{c}} ight)^{2}=k^{2}+left({frac {m_{0}c}{hbar }} ight)^{2},.}     ( 3) In physics, the energy–momentum relation, or relativistic dispersion 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 m0, total energy E, and a momentum of magnitude p, where the constant c is the speed of light, assuming the special relativity case of flat spacetime. The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energy-momentum equation. The energy–momentum relation is consistent with the familiar mass–energy relation in both its interpretations: E = mc2 relates total energy E to the (total) relativistic mass m (alternatively denoted mrel or mtot ), while E0 = m0c2 relates rest energy E0 to (invariant) rest mass m0. Unlike either of those equations, the energy–momentum equation (1) relates the total energy to the rest mass m0. All three equations hold true simultaneously. A more general form of relation (1) 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 E and its relativistic momentum p 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 E and p, while the other frame measures E′ and p′, where E′ ≠ E and p′ ≠ p, 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 E, p, E′, p′ 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 m0 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, E′ and p′ as an observer moving with the particle would conclude to be) and measured in the lab frame (i.e. E and p as determined by particle physicists in a lab, and not moving with the particles).