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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.
The following notations are used very often in special relativity: Lorentz factor = where = and v is the relative velocity between two inertial frames.. For two frames at rest, γ = 1, and increases with relative velocity between the two inertial frames.
The Schrödinger equation describes the space- and time-dependence of the slow changing (non-relativistic) wave function of a quantum system. The solution of the Schrödinger equation for a bound system is discrete (a set of permitted states, each characterized by an energy level) which results in the concept of quanta.
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac particles", such as electrons and quarks for which parity is a symmetry.
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 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}
These equations, together with the geodesic equation, [8] which dictates how freely falling matter moves through spacetime, form the core of the mathematical formulation of general relativity. The EFE is a tensor equation relating a set of symmetric 4 × 4 tensors. Each tensor has 10 independent components.
Since T 00 is the energy density, T j0 for j = 1, 2, 3 is the jth component of the object's 3d momentum per unit volume, and T ij form components of the stress tensor including shear and normal stresses, the orbital angular momentum density about the position 4-vector X β is given by a 3rd order tensor = (¯) (¯)