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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.
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:
Arnold Sommerfeld derived the relativistic solution of atomic energy levels. [5] We will start this derivation [ 10 ] with the relativistic equation for energy in the electric potential W = m 0 c 2 ( 1 1 − v 2 c 2 − 1 ) − k Z e 2 r {\displaystyle W={m_{\mathrm {0} }c^{2}}\left({\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}-1\right)-k{\frac ...
Any solution of the free Dirac equation is, for each of its four components, a solution of the free Klein–Gordon equation. Despite historically it was invented as a single particle equation the Klein–Gordon equation cannot form the basis of a consistent quantum relativistic one-particle theory, any relativistic theory implies creation and ...
The equation is often written this way because the difference is the relativistic length of the energy momentum four-vector, a length which is associated with rest mass or invariant mass in systems. Where m > 0 and p = 0, this equation again expresses the mass–energy equivalence E = m.
The relativistic energy–momentum equation holds for all particles, even for massless particles for which m 0 = 0. In this case: = When substituted into Ev = c 2 p, this gives v = c: massless particles (such as photons) always travel at the speed of light.
The relativistic Lagrangian can be derived in relativistic mechanics to be of the form: = (˙) (, ˙,). Although, unlike non-relativistic mechanics, the relativistic Lagrangian is not expressed as difference of kinetic energy with potential energy, the relativistic Hamiltonian corresponds to total energy in a similar manner but without including rest energy.
The two-body problem in general relativity (or relativistic two-body problem) is the determination of the motion and gravitational field of two bodies as described by the field equations of general relativity. Solving the Kepler problem is essential to calculate the bending of light by gravity and the motion of a planet orbiting its sun.