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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.
Mass–energy equivalence states that all objects having mass, or massive objects, have a corresponding intrinsic energy, even when they are stationary.In the rest frame of an object, where by definition it is motionless and so has no momentum, the mass and energy are equal or they differ only by a constant factor, the speed of light squared (c 2).
This is the formula for the relativistic doppler shift where the difference in velocity between the emitter and observer is not on the x-axis. There are two special cases of this equation. The first is the case where the velocity between the emitter and observer is along the x-axis. In that case θ = 0, and cos θ = 1, which gives:
The relativistic expressions for E and p obey the relativistic energy–momentum relation: [12] = where the m is the rest mass, or the invariant mass for systems, and E is the total energy. The equation is also valid for photons, which have m = 0 : E 2 − ( p c ) 2 = 0 {\displaystyle E^{2}-(pc)^{2}=0} and therefore E = p c {\displaystyle E=pc}
Several approaches exist as a means of describing the motion of single and multiple relativistic particles, with a prominent example being postulations through the Dirac equation of single particle motion. [2] Since the energy-momentum relation of an particle can be written as: [3]
The stress–energy tensor is the source of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity. Because this tensor has 2 indices (see next section) the Riemann curvature tensor has to be contracted into the Ricci tensor, also with 2 indices.
Yet, because it fails to take into account the electron's spin, the equation predicts the hydrogen atom's fine structure incorrectly, including overestimating the overall magnitude of the splitting pattern by a factor of 4n / 2n − 1 for the n-th energy level. The Dirac equation relativistic spectrum is, however, easily recovered if ...
The second term in the above equation, plays the role of a gravitational force. If f f α {\displaystyle f_{f}^{\alpha }} is the correct expression for force in a freely falling frame ξ α {\displaystyle \xi ^{\alpha }} , we can use then the equivalence principle to write the four-force in an arbitrary coordinate x μ {\displaystyle x^{\mu }} :