Search

Google Website Translator Gadget

Sunday, January 09, 2011

Releation between mass and momentum

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).

0 comments: