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# Energy–momentum relation: Wikis

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# Encyclopedia

In special relativity, the energy-momentum relation is a relation between the energy, momentum and the mass of a body:

$E^2 = m^2 c^4 + p^2 c^2 , \;$

where c is the speed of light, $E \;$ is total energy, $m \;$ is invariant mass, and $p\;$ is momentum.

For a body in its rest frame, the momentum is zero, so the equation simplifies to

$E = mc^2 \;$

If the object is massless then the energy momentum relation reduces to

$E = pc \;$

as is the case for a photon.

In natural units the energy-momentum relation can be expressed as

$\omega^2 = m^2 + k^2 \;$

where $\omega \;$ is angular frequency, $m \;$ is rest mass and $k \;$ is wave number.

In Minkowski space, energy and momentum (the latter multiplied by a factor of c) can be seen as two components of a Minkowski four-vector. The norm of this vector is equal to the square of the rest mass of the body, which is a Lorentz invariant quantity and hence is independent of the frame of reference.

When working in units where c = 1, known as the natural unit system, the energy-momentum equation reduces to

$m^2 = E^2 - p^2 \,\!$

In particle physics, energy is typically given in units of electron volts (eV), momentum in units of eV/c, and mass in units of eV/c2. In electromagnetism, and because of relativistic invariance, it is useful to have the electric field E and the magnetic field B in the same unit (gauss), using the cgs (gaussian) system of units, where energy is given in units of erg, momentum in g.cm/s and mass in grams.

Energy may also in theory be expressed in units of grams, though in practice it requires a large amount of energy to be equivalent to masses in this range. For example, the first atomic bomb liberated about 1 gram of heat, and the largest thermonuclear bombs have generated a kilogram or more of heat. Energies of thermonuclear bombs are usually given in tens of kilotons and megatons referring to the energy liberated by exploding that amount of trinitrotoluene (TNT).