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Energy and Momentum

Note that when we used the term 'energy' we mean γmc2, which is a particle's total energy. The particle's 'kinetic energy', however, is the excess energy due to its motion, over and above the energy it has when at rest: KE = γmc2 - mc2. Thus any particle has an amount of energy mc2 when at rest; this is the famous mass-energy relationship which explains the energy release in many nuclear reactions, and explains, for example, why all stable nuclei have a mass that is less than their constituent particles. Because of this kinetic energy is not always conserved it a collision or decay: it is the total energy γmc2, as we have seen, that is conserved.

There is also an extremely important relationship between energy and momentum:


E2 - |    
= γ2m2c41 -     
= m2c4    

Since m2c4 is a constant, independent of reference frame, the quantity E2 - | must also be frame invariant (the same in every inertial frame). Another important relation is that = .

The above equation suggests that there is a special relationship between energy and momentum. Consider a frame F' moving with speed v with respect to frame F along their mutual x/x'-direction (just like when we derived the Lorentz transformations). There is a particle in F' that has energy E' and momentum p' (and is moving also in the x-direction). What is E and p in the frame F? The answer looks very familiar:


ΔE = γv(ΔE' + vΔp')    
Δp = γv(Δp' + vΔE'/c2)    

γv is the γ factor associated with the relative speed between the frames (v). Not surprisingly these transformations look precisely like the Lorentz transformations between space and time in different frames. These equations also hold if E and p represent total energy and total momentum of a system of particles. Moreover they make it clear that if E and p are conserved in one frame, then they are conserved in any other inertial frame; this is very important to make the conservation laws we derived above meaningful. This arises just because E and p in one frame must be linear functions of E' and p' in another frame. Since the latter quantities are both conserved any linear function of them must be conserved also. Note that, just as with the spacetime transformations the above applies only to the x-direction (there is nothing special about x, except that we have arbitrarily chosen it to be our direction of motion) and py = py' and pz = pz'.

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