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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:
:English translations: "Does the Inertia of a Body Depend Upon Its Energy Content?". Translation by George Barker Jeffery and Wilfrid Perrett in The Principle of Relativity, London: Methuen and Company, Ltd. (1923). :Used the newly formulated theory of special relativity to introduce the mass energy formula. One of the Annus Mirabilis papers.
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 ...
The relativistic four-velocity, that is the four-vector representing velocity in relativity, is defined as follows: = = (,) In the above, is the proper time of the path through spacetime, called the world-line, followed by the object velocity the above represents, 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 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.
This theory made many predictions which have been experimentally verified, including the relativity of simultaneity, length contraction, time dilation, the relativistic velocity addition formula, the relativistic Doppler effect, relativistic mass, a universal speed limit, mass–energy equivalence, the speed of causality and the Thomas precession.