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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).
These tensor fields should obey any relevant physical laws (for example, any electromagnetic field must satisfy Maxwell's equations).Following a standard recipe which is widely used in mathematical physics, these tensor fields should also give rise to specific contributions to the stress–energy tensor. [1]
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:
Metric tensors resulting from cases where the resultant differential equations can be solved exactly for a physically reasonable distribution of energy–momentum are called exact solutions. Examples of important exact solutions include the Schwarzschild solution and the Friedman-Lemaître-Robertson–Walker solution.
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.
Mass–energy equivalence: E = mc 2, energy and mass are equivalent and transmutable. Relativistic mass, idea used by some researchers. [9] The defining feature of special relativity is the replacement of the Galilean transformations of classical mechanics by the Lorentz transformations. (See Maxwell's equations of electromagnetism.)
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}