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Concept
Energy–momentum relation
Summary
In physics, the energy–momentum relation, or relativistic dispersion relation, is the relativistic equation relating total energy (which is also called relativistic energy) to invariant mass (which is also called rest mass) and momentum. It is the extension of mass–energy equivalence for bodies or systems with non-zero momentum. It can be written as the following equation:
This equation holds for a body or system, such as one or more particles, with total energy E, invariant mass m0, and momentum of magnitude p; the constant c is the speed of light. It assumes the special relativity case of flat spacetime and that the particles are free. Total energy is the sum of rest energy and kinetic energy, while invariant mass is mass measured in a center-of-momentum frame.
For bodies or systems with zero momentum, it simplifies to the mass–energy equation , where total energy in this case is equal to rest energy (also written as E0).
The Dirac sea model, which was used to predict the existence of antimatter, is closely related to the energy–momentum relation.
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 () relates the total energy to the rest mass m0. All three equations hold true simultaneously.
If the body is a massless particle (m0 = 0), then () reduces to E = pc. 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 v is much less than c, then () reduces to E = 1/2m0v2 + m0c2; that is, the body's total energy is simply its classical kinetic energy (1/2m0v2) plus its rest energy.
If the body is at rest (v = 0), i.e.
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